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Mathematics Course One
Student Textbook California Standards-Driven Program
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CGP
Education
California
Mathematics Course One
Student Textbook California Standards-Driven Program
California
Mathematics Course One
Student Textbook California Standards-Driven Program
Contents This Textbook provides comprehensive coverage of all the California grade 6 Standards. The Textbook is divided into seven Chapters. Each of the Chapters is broken down into small, manageable Lessons and each Lesson covers a specific Standard or part of a Standard.
California Standard
Chapter 1 — Ordering and Manipulating Numbers
NS 1.1 NS 2.3
Section 1.1 — Integers and the Number Line Exploration Zero Pairs ............................................................................................................................. 2 Lesson 1.1.1 Comparing Integers .............................................................................................................. 3 Lesson 1.1.2 Adding and Subtracting Integers .......................................................................................... 6
NS 2.3
Section 1.2 — Multiplication and Division by Integers Exploration Multiplication with Integer Tiles ............................................................................................. Lesson 1.2.1 Multiplying with Integers ....................................................................................................... Lesson 1.2.2 Dividing with Integers ............................................................................................................ Lesson 1.2.3 Integers in Real Life ..............................................................................................................
NS 1.1
Section 1.3 — Decimals Exploration A Decimal Strip ..................................................................................................................... 22 Lesson 1.3.1 Decimals ............................................................................................................................... 23 Lesson 1.3.2 Ordering Decimals ................................................................................................................ 26
MR 2.1 MR 2.3 MR 2.6
Section 1.4 — Estimation Lesson 1.4.1 Rounding Numbers ............................................................................................................... Lesson 1.4.2 Using Rounded Numbers ..................................................................................................... Lesson 1.4.3 Estimation ............................................................................................................................. Lesson 1.4.4 Using Estimation ...................................................................................................................
9 10 14 18
29 32 35 38
Chapter 1 — Investigation Populations .............................................................................................................................................. 41
Chapter 2 — Expressions and Equations
ii
AF 1.2 AF 1.3 AF 1.4
Section 2.1 — Expressions Exploration Algebra Tile Expressions ...................................................................................................... Lesson 2.1.1 Variables ............................................................................................................................... Lesson 2.1.2 Expressions .......................................................................................................................... Lesson 2.1.3 Multi-Variable Expressions ................................................................................................... Lesson 2.1.4 Order of Operations ..............................................................................................................
43 44 48 51 55
AF 1.1
Section 2.2 — Equations Exploration Equations with Algebra Tiles ................................................................................................ Lesson 2.2.1 Equations .............................................................................................................................. Lesson 2.2.2 Manipulating Equations ........................................................................................................ Lesson 2.2.3 Solving + and – Equations .................................................................................................... Lesson 2.2.4 Solving × and ÷ Equations .................................................................................................... Lesson 2.2.5 Graphing Equations ..............................................................................................................
59 60 63 66 69 72
California Standard
Chapter 2 — Continued
AF 1.2 AF 1.3 AF 3.1 AF 3.2 MR 1.3
Section 2.3 — Geometrical Expressions Lesson 2.3.1 Expressions About Length .................................................................................................... Lesson 2.3.2 Expressions About Area ....................................................................................................... Lesson 2.3.3 Finding Complex Areas ........................................................................................................ Lesson 2.3.4 The Distributive Property ...................................................................................................... Lesson 2.3.5 Using the Distributive Property ............................................................................................. Lesson 2.3.6 Squares and Cubes .............................................................................................................. Lesson 2.3.7 Expressions and Angles .......................................................................................................
MR 1.1 MR 1.3 MR 2.2 MR 2.4 MR 2.5 MR 2.7
Section 2.4 — Problem Solving Lesson 2.4.1 Analyzing Problems .............................................................................................................. 97 Lesson 2.4.2 Important Information ........................................................................................................... 100 Lesson 2.4.3 Breaking Up a Problem ......................................................................................................... 104
75 79 82 85 88 90 94
Chapter 2 — Investigation Design a House ............................................................................................................................................ 108
Chapter 3 — Fractions and Percentages
NS 1.1
Section 3.1 — Fractions Lesson 3.1.1 Understanding Fractions ....................................................................................................... Lesson 3.1.2 Improper Fractions ................................................................................................................ Lesson 3.1.3 More on Fractions ................................................................................................................. Lesson 3.1.4 Fractions and Decimals ........................................................................................................
110 114 118 121
NS 2.1 NS 2.2
Section 3.2 — Multiplying Fractions Exploration Multiplying Fractions: an Area Model .................................................................................... Lesson 3.2.1 Multiplying Fractions by Integers ........................................................................................... Lesson 3.2.2 More on Multiplying Fractions by Integers ............................................................................ Lesson 3.2.3 Multiplying Fractions by Fractions .........................................................................................
124 125 129 133
NS 2.1 NS 2.2
Section 3.3 — Dividing Fractions Lesson 3.3.1 Dividing by Fractions ............................................................................................................. 136 Lesson 3.3.2 Solving Problems by Dividing Fractions ................................................................................ 140
NS 1.1 NS 2.1 NS 2.4
Section 3.4 — Adding and Subtracting Fractions Exploration Adding Fractions: an Area Model .......................................................................................... Lesson 3.4.1 Making Equivalent Fractions ................................................................................................. Lesson 3.4.2 Finding the Simplest Form .................................................................................................... Lesson 3.4.3 Fraction Sums ....................................................................................................................... Lesson 3.4.4 Fractions with Different Denominators .................................................................................. Lesson 3.4.5 Least Common Multiples ...................................................................................................... Lesson 3.4.6 Mixed Numbers and Word Questions ...................................................................................
144 145 149 152 155 160 163
NS 1.4
Section 3.5 — Percents Exploration Percents with a Double Number Line ................................................................................... Lesson 3.5.1 Fractions and Percents ......................................................................................................... Lesson 3.5.2 Percents and Decimals ......................................................................................................... Lesson 3.5.3 Percents of Numbers ............................................................................................................ Lesson 3.5.4 Circle Graphs and Percents .................................................................................................. Lesson 3.5.5 Percent Increase ................................................................................................................... Lesson 3.5.6 Percent Decrease ................................................................................................................. Lesson 3.5.7 Simple Interest ......................................................................................................................
167 168 171 174 177 181 185 189
iii
Contents California Standard
Chapter 3 — Continued Chapter 3 — Investigation Wildlife Trails .............................................................................................................................................. 193
Chapter 4 — Ratio, Proportion and Rate
NS 1.2 NS 1.3
Section 4.1 — Ratio and Proportion Exploration Billboard Ratios ..................................................................................................................... Lesson 4.1.1 Ratios .................................................................................................................................... Lesson 4.1.2 Equivalent Ratios .................................................................................................................. Lesson 4.1.3 Proportions ........................................................................................................................... Lesson 4.1.4 Proportions and Cross-Multiplication ....................................................................................
NS 1.2 NS 1.3
Section 4.2 — Proportion in Geometry Lesson 4.2.1 Similarity ............................................................................................................................... 209 Lesson 4.2.2 Proportions and Similarity ..................................................................................................... 213 Lesson 4.2.3 Scale Drawings ..................................................................................................................... 217
AF 2.1 MR 3.1
Section 4.3 — Converting Units Exploration Plotting Conversions ............................................................................................................. Lesson 4.3.1 Customary and Metric Units ................................................................................................. Lesson 4.3.2 Conversions and Proportions ............................................................................................... Lesson 4.3.3 Converting Between Unit Systems ....................................................................................... Lesson 4.3.4 Other Conversions ................................................................................................................
221 222 226 229 233
AF 2.2 AF 2.3
Section 4.4 — Rates Exploration Running Rates ...................................................................................................................... Lesson 4.4.1 Rates .................................................................................................................................... Lesson 4.4.2 Using Rates .......................................................................................................................... Lesson 4.4.3 Finding Speed ....................................................................................................................... Lesson 4.4.4 Finding Time and Distance ................................................................................................... Lesson 4.4.5 Average Rates ......................................................................................................................
236 237 240 244 247 250
195 196 199 202 205
Chapter 4 — Investigation Sunshine and Shadows ................................................................................................................................ 253
Chapter 5 — Data Sets
SDP 1.1 SDP 1.3 SDP 1.4
iv
Section 5.1 — Statistical Measures Exploration Estimation Line-Up ............................................................................................................... Lesson 5.1.1 Median and Mode ................................................................................................................. Lesson 5.1.2 Mean and Range .................................................................................................................. Lesson 5.1.3 Extreme Values ..................................................................................................................... Lesson 5.1.4 Comparing Data Sets ...........................................................................................................
255 256 259 262 265
California Standard
Chapter 5 — Continued
SDP 1.1 SDP 1.2
Section 5.2 — Adding Extra Data Lesson 5.2.1 Including Additional Data: Mode, Median, and Range .......................................................... 268 Lesson 5.2.2 Including Additional Data: The Mean .................................................................................... 272
SDP 1.1 SDP 2.3
Section 5.3 — Data Displays Lesson 5.3.1 Analyzing Graphs ................................................................................................................. 275 Lesson 5.3.2 Finding the Mean and Median from Graphs ......................................................................... 279 Lesson 5.3.3 Other Types of Graph ........................................................................................................... 282
SDP 2.1 SDP 2.2 SDP 2.4
SDP 2.3 SDP 2.5
Section 5.4 — Sampling Exploration Sampling Survey ................................................................................................................... Lesson 5.4.1 Using Samples ...................................................................................................................... Lesson 5.4.2 Convenience, Random, and Systematic Sampling ............................................................... Lesson 5.4.3 Samples and Accuracy ......................................................................................................... Lesson 5.4.4 Questionnaire Surveys .........................................................................................................
286 287 290 293 296
Section 5.5 — Statistical Claims Lesson 5.5.1 Evaluating Claims ................................................................................................................. 300 Lesson 5.5.2 Evaluating Displays .............................................................................................................. 304 Chapter 5 — Investigation Selling Cookies ............................................................................................................................................. 309
Chapter 6 — Probability
SDP 3.1
Section 6.1 — Outcomes and Diagrams Lesson 6.1.1 Listing Possible Outcomes .................................................................................................... 311 Lesson 6.1.2 Tree Diagrams ...................................................................................................................... 314 Lesson 6.1.3 Tables and Grids ................................................................................................................... 317
SDP 3.1 SDP 3.3 SDP 3.4
Section 6.2 — Theoretical Probability Exploration Heads or Tails ....................................................................................................................... Lesson 6.2.1 Probability ............................................................................................................................. Lesson 6.2.2 Expressing Probability .......................................................................................................... Lesson 6.2.3 Counting Outcomes .............................................................................................................. Lesson 6.2.4 Probability of an Event Not Happening ................................................................................ Lesson 6.2.5 Venn Diagrams ..................................................................................................................... Lesson 6.2.6 Combining Events .................................................................................................................
320 321 324 327 330 333 336
SDP 3.4 SDP 3.5
Section 6.3 — Dependent and Independent Events Exploration Pick a Card ........................................................................................................................... Lesson 6.3.1 Independent and Dependent Events .................................................................................... Lesson 6.3.2 Events and Probabilities ....................................................................................................... Lesson 6.3.3 Calculating Probabilities of Independent Events ..................................................................
339 340 343 346
SDP 3.2
Section 6.4 — Experimental Probability Exploration Shooting Baskets .................................................................................................................. 349 Lesson 6.4.1 Relative Frequency ............................................................................................................... 350 Lesson 6.4.2 Making Predictions ............................................................................................................... 353 Chapter 6 — Investigation A Game of Chance ....................................................................................................................................... 356
v
Contents California Standard
Chapter 7 — Geometry
AF 3.1 AF 3.2 MG 1.1 MG 1.2
Section 7.1 — Circles Exploration Circle Ratios ......................................................................................................................... 358 Lesson 7.1.1 Parts of a Circle .................................................................................................................... 359 Lesson 7.1.2 Circumference and p ......................................................................................................................... 362 Lesson 7.1.3 Area of a Circle ..................................................................................................................... 366
MG 2.1 MG 2.2
Section 7.2 — Angles Exploration Folding Angles ...................................................................................................................... Lesson 7.2.1 Describing Angles ................................................................................................................. Lesson 7.2.2 Pairs of Angles ..................................................................................................................... Lesson 7.2.3 Supplementary Angles ......................................................................................................... Lesson 7.2.4 The Triangle Sum .................................................................................................................. Lesson 7.2.5 Complementary Angles ........................................................................................................
369 370 374 377 381 384
MG 2.3
Section 7.3 — Two-Dimensional Figures Exploration Sorting Shapes ..................................................................................................................... Lesson 7.3.1 Classifying Triangles by Angles ............................................................................................ Lesson 7.3.2 Classifying Triangles by Side Lengths .................................................................................. Lesson 7.3.3 Types of Quadrilaterals ........................................................................................................ Lesson 7.3.4 Drawing Quadrilaterals .........................................................................................................
388 389 393 398 402
MG 1.3
Section 7.4 — Three-Dimensional Figures Exploration Building Cylinders ................................................................................................................. Lesson 7.4.1 Three Dimensional Figures ................................................................................................... Lesson 7.4.2 Volume of Rectangular Prisms ............................................................................................. Lesson 7.4.3 Volume of Triangular Prisms and Cylinders .......................................................................... Lesson 7.4.4 Volume of Compound Solids ................................................................................................
405 406 410 413 416
MR 1.2 MR 3.1 MR 3.2 MR 3.3
Section 7.5 — Generalizing Results Lesson 7.5.1 Generalizing Results ............................................................................................................. 419 Lesson 7.5.2 Proving Generalizations ........................................................................................................ 424 Chapter 7 — Investigation Reclining Chairs ............................................................................................................................................ 427
Additional Questions Additional Questions for Chapter 1 ............................................................................................................... Additional Questions for Chapter 2 ............................................................................................................... Additional Questions for Chapter 3 ............................................................................................................... Additional Questions for Chapter 4 ............................................................................................................... Additional Questions for Chapter 5 ............................................................................................................... Additional Questions for Chapter 6 ............................................................................................................... Additional Questions for Chapter 7 ...............................................................................................................
428 431 438 445 451 457 462
Appendixes Glossary .............................................................................................................................................. 471 Formula Sheet .............................................................................................................................................. 473 Index .............................................................................................................................................. 475
vi
California Grade Six Mathematics Standards The following table lists all the California Mathematics Content Standards for grade 6 with cross references to where each Standard is covered in this Textbook. Each Lesson begins by quoting the relevant Standard in full, together with a clear and understandable objective. This will enable you to measure your progression against the California grade 6 Standards as you work your way through the Program. California Standard
Number Sense
1.0
Students compare and order positive and negative fractions, decimals, and mixed numbers. Students solve problems involving fractions, ratios, proportions, and percentages:
Chapters 1, 3, 4
1.1
Compare and order positive and negative fractions, decimals, and mixed numbers and place them on a number line.
Chapters 1, 3
1.2
Interpret and use ratios in different contexts (e.g., batting averages, miles per hour) to show the relative sizes of two quantities, using appropriate notations (a/b, a to b, a:b).
Chapter 4
1.3
Use proportions to solve problems (e.g., determine the value of N if = , find the length of a side of 7 21 a polygon similar to a known polygon). Use cross-multiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse.
Chapter 4
1.4
Calculate given percentages of quantities and solve problems involving discounts at sales, interest earned, and tips.
Chapter 3
2.0
Students calculate and solve problems involving addition, subtraction, multiplication, and division:
Chapters 1, 3
2.1
Solve problems involving addition, subtraction, multiplication, and division of positive fractions and explain why a particular operation was used for a given situation.
Chapter 3
2.2
Explain the meaning of multiplication and division of positive fractions and perform the calculations
Chapter 3
4
(e.g.,
N
5 15 5 16 2 ÷ = × = ). 8 16 8 15 3
2.3
Solve addition, subtraction, multiplication, and division problems, including those arising in concrete situations, that use positive and negative integers and combinations of these operations.
Chapter 1
2.4
Determine the least common multiple and the greatest common divisor of whole numbers; use them to solve problems with fractions (e.g., to find a common denominator to add two fractions or to find the reduced form for a fraction).
Chapter 3
California Standard
Algebra and Functions
1.0
Students write verbal expressions and sentences as algebraic expressions and equations; they evaluate algebraic expressions, solve simple linear equations, and graph and interpret their results:
Chapter 2
1.1
Write and solve one-step linear equations in one variable.
Chapter 2
1.2
Write and evaluate an algebraic expression for a given situation, using up to three variables.
Chapter 2
1.3
Apply algebraic order of operations and the commutative, associative, and distributive properties to evaluate expressions; and justify each step in the process.
Chapter 2
1.4
Solve problems manually by using the correct order of operations or by using a scientific calculator.
Chapter 2
2.0
Students analyze and use tables, graphs, and rules to solve problems involving rates and proportions:
Chapter 4
2.1
Convert one unit of measurement to another (e.g., from feet to miles, from centimeters to inches).
Chapter 4
2.2
Demonstrate an understanding that rate is a measure of one quantity per unit value of another quantity.
Chapter 4
2.3
Solve problems involving rates, average speed, distance, and time.
Chapter 4
vii
California Grade Six Mathematics Standards 3.0
Students investigate geometric patterns and describe them algebraically:
3.1
Use variables in expressions describing geometric quantities (e.g., P = 2w + 2l, A = bh, C = pd — the 2 formulas for the perimeter of a rectangle, the area of a triangle, and the circumference of a circle, respectively).
Chapter 2, 7
3.2
Express in symbolic form simple relationships arising from geometry.
Chapter 2, 7
California Standard
1
Measurement and Geometry
1.0
Students deepen their understanding of the measurement of plane and solid shapes and use this understanding to solve problems:
Chapter 7
1.1
Understand the concept of a constant such as p; know the formulas for the circumference and area of a circle.
Chapter 7
1.2
Know common estimates of p (3.14; ) and use these values to estimate and calculate the 7 circumference and the area of circles; compare with actual measurements.
Chapter 7
1.3
Know and use the formulas for the volume of triangular prisms and cylinders (area of base × height); compare these formulas and explain the similarity between them and the formula for the volume of a rectangular solid.
Chapter 7
2.0
Students identify and describe the properties of two-dimensional figures:
Chapter 7
2.1
Identify angles as vertical, adjacent, complementary, or supplementary and provide descriptions of these terms.
Chapter 7
2.2
Use the properties of complementary and supplementary angles and the sum of the angles of a triangle to solve problems involving an unknown angle.
Chapter 7
2.3
Draw quadrilaterals and triangles from given information about them (e.g., a quadrilateral having equal sides but no right angles, a right isosceles triangle).
Chapter 7
California Standard
viii
Chapter 2, 7
22
Statistics, Data Analysis and Probability
1.0
Students compute and analyze statistical measurements for data sets:
Chapter 5
1.1
Compute the range, mean, median, and mode of data sets.
Chapter 5
1.2
Understand how additional data added to data sets may affect these computations.
Chapter 5
1.3
Understand how the inclusion or exclusion of outliers affects these computations.
Chapter 5
1.4
Know why a specific measure of central tendency (mean, median) provides the most useful information in a given context.
Chapter 5
2.0
Students use data samples of a population and describe the characteristics and limitations of the samples:
Chapter 5
2.1
Compare different samples of a population with the data from the entire population and identify a situation in which it makes sense to use a sample.
Chapter 5
2.2
Identify different ways of selecting a sample (e.g., convenience sampling, responses to a survey, random sampling) and which method makes a sample more representative for a population.
Chapter 5
2.3
Analyze data displays and explain why the way in which the question was asked might have influenced the results obtained and why the way in which the results were displayed might have influenced the conclusions reached.
Chapter 5
2.4
Identify data that represent sampling errors and explain why the sample (and the display) might be biased.
Chapter 5
2.5
Identify claims based on statistical data and, in simple cases, evaluate the validity of the claims.
Chapter 5
3.0
Students determine theoretical and experimental probabilities and use these to make predictions about events:
Chapter 6
3.1
Represent all possible outcomes for compound events in an organized way (e.g., tables, grids, tree diagrams) and express the theoretical probability of each outcome.
Chapter 6
3.2
Use data to estimate the probability of future events (e.g., batting averages or number of accidents per mile driven).
Chapter 6
3.3
Represent probabilities as ratios, proportions, decimals between 0 and 1, and percentages between 0 and 100 and verify that the probabilities computed are reasonable; know that if P is the probability of an event, 1 – P is the probability of an event not occurring.
Chapter 6
3.4
Understand that the probability of either of two disjoint events occurring is the sum of the two individual probabilities and that the probability of one event following another, in independent trials, is the product of the two probabilities.
Chapter 6
3.5
Understand the difference between independent and dependent events.
Chapter 6
California Standard
Mathematical Reasoning
1.0
Students make decisions about how to approach problems:
1.1
Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, identifying missing information, sequencing and prioritizing information, and observing patterns.
1.2
Formulate and justify mathematical conjectures based on a general description of the mathematical question or problem posed.
1.3
Determine when and how to break a problem into simpler parts.
2.0
Students use strategies, skills, and concepts in finding solutions:
2.1
Use estimation to verify the reasonableness of calculated results.
2.2
Apply strategies and results from simpler problems to more complex problems.
2.3
Estimate unknown quantities graphically and solve for them by using logical reasoning and arithmetic and algebraic techniques.
2.4
Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning.
2.5
Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language; support solutions with evidence in both verbal and symbolic work.
2.6
Indicate the relative advantages of exact and approximate solutions to problems and give answers to a specified degree of accuracy.
2.7
Make precise calculations and check the validity of the results from the context of the problem.
3.0
Students move beyond a particular problem by generalizing to other situations:
3.1
Evaluate the reasonableness of the solution in the context of the original situation.
3.2
Note the method of deriving the solution and demonstrate a conceptual understanding of the derivation by solving similar problems.
3.3
Develop generalizations of the results obtained and the strategies used and apply them in new problem situations.
Illustrated in Chapters 1, 2, and 7, and throughout Program
Shows key standards
ix
Published by CGP Education Editor: Andy Park Core material written by: Harold I. Lawrance and Bonnie Spence Harold is a freelance mathematics consultant after working as a High School Math teacher in California for fourteen years. Bonnie worked as a writer on the STEM Project through the University of Montana and previously worked as a Middle School Math teacher in Oklahoma and Colorado. California General Advisor: Marcia Lomneth Marcia has worked in several mathematics education positions, including staff support and curriculum development. She now works as a California Mathematics Consultant. Section Exploration Writer: Alex May Alex is a science teacher in Colorado. He previously worked as Co-Director of the Irvine Mathematics Project at the University of California. California English Language Learner Advisor: Judith A. McGinty Judith is a Middle School Math teacher in Oakland Unified School District, California. She has vast experience of teaching English Language Leaners and holds a CLAD certificate (Cross-cultural Language and Academic Development). California Advanced Learner Advisor: William D. Nolan William is a Middle School Math teacher in New York State and is experienced in delivering math courses to diverse student groups. Additional Writers: Cristina M. Dube and Heather M. Heise Cristina is a freelance math consultant and test item writer, and has previously worked as an Elementary Math and General Education teacher in Texas. Heather is a freelance math consultant and test item writer, and previously worked as an Elementary Math teacher in Minnesota and Texas. Additional Contributors: Paul R. Allrich and Darrell Dean Ballard Supporting Editors: Tim Burne Mary Falkner Tom Harte Sarah Hilton Kate Houghton Paul Jordin
Sharon Keeley John Kitching Simon Little Tim Major Ali Palin
Glenn Rogers Emma Stevens Ami Snelling Claire Thompson Julie Wakeling
Proofreading: Joiclyn Austin, Heather M. Heise, and Jeff Probst Graphic Design: Caroline Batten, Russell Holden, Jane Ross and Ash Tyson
Mathematics Content Standards for California Public Schools reproduced by permission, California Department of Education, CDE Press, 1430 N Street, Suite 3207, Sacramento, CA 95814. ISBN 13: 978 1 60017 034 8 website: www.cgpeducation.com Printed by Elanders Hindson Ltd, UK and Johnson Printing, Boulder, CO Clipart sources: CorelDRAW and VECTOR. Text, design, layout, and illustrations © CGP, Inc. 2007 All rights reserved.
x
Chapter 1 Ordering and Manipulating Numbers Section 1.1
Exploration — Zero Pairs .............................................. 2 Integers and the Number Line ....................................... 3
Section 1.2
Exploration — Multiplication with Integer Tiles.............. 9 Multiplication and Division with Integers ..................... 10
Section 1.3
Exploration — A Decimal Strip .................................... 22 Decimals ...................................................................... 23
Section 1.4
Estimation.................................................................... 29
Chapter Investigation — Populations .................................................. 41
1
Section 1.1 introduction — an exploration into:
Zer oP air s Zero Pair airs Lots of math involves adding and subtracting with positive and negative numbers. Positive and negative numbers “cancel each other out” — for example, if you add the numbers 1 and –1, you get zero. You can use this fact to add and subtract other numbers too. 5
You can represent positive and negative integers using blue and red tiles. Each blue tile represents 1. So the number 5 would be shown using 5 blue tiles. Each red tile represents –1. So the number –3 would be shown using 3 red tiles.
–3
A red and a blue tile together make zero. This is called a zero pair.
1 + (–1) = 0
Because zero pairs have a value of zero, they do not affect the value of other groups of tiles. Example 5
Add 5 + (–3) with integer tiles. Solution
Take 5 blue tiles, and add 3 red tiles. You can remove the zero pairs. This leaves 2 blue tiles. So 5 + (–3) = 2.
+ –3
Removing the zero pairs leaves 2 blue tiles.
Example The total value of the tiles is still 4.
Subtract 4 – (–2) with integer tiles. Solution
Take 4 blue tiles. To subtract –2, you need to remove 2 red tiles. But there are no red tiles. However, you can add two zero pairs — zero pairs do not change anything.
=
Now you can remove 2 red tiles. This leaves 6 blue tiles. So 4 – (–2) = 6.
Exercises 1. Use integer tiles to model and solve each of the following: a. –6 + 5 b. –2 + (–7) c. 8 + (–3) d. 5 – 7
e. –3 – 2
f. 2 – (–3)
2. How can 4 – (–6) be solved using red and blue tiles? Show how removing 6 red tiles has the same effect as adding 4 blue tiles.
Round Up You can use tiles to represent lots of addition and subtraction problems. To do an addition, you need to add tiles. To do a subtraction, you need to remove tiles. But remember, you can always add or remove a zero pair without affecting anything. 2
Section 1.1 Explor a tion — Zero Pairs Explora
Lesson
Section 1.1
1.1.1
Comparing Integers
California Standard:
This Lesson is about some of the different kinds of numbers you need to use, and how you can use the number line to put them in order. The number line shows whether numbers are greater than or less than each other.
Number Sense 1.1 Compare and order positive and negative fractions, decimals, and mixed numbers and place them on a number line.
The Number Line Shows the Order of Numbers
What it means for you: You’ll be able to arrange positive and negative numbers on the number line, and you’ll learn about different types of numbers.
Key words: • • • • • • •
natural number whole number integer origin number line greater than less than
The number line is a useful diagram that shows how numbers are ordered. Numbers to the right of 0 are positive and numbers to the left of 0 are negative. Example
Positive numbers are to the right of 0
–6 –5 –4 –3 –2 –1 0
1
2
3 4
5
6
0 is called the origin
1
Arrange the numbers 2, 8, and 12 on the number line. Solution
Don’t forget: 0 isn’t positive or negative.
Negative numbers are to the left of 0
8
2
0
12
Just count to the right of 0 by 2, 8, and 12 steps.
You always count steps from 0, but if it’s a negative number then remember to count to the left.
Don’t forget: Negative and positive versions of the same number are the same distance from the origin. For example, 3 and –3 are both 3 steps away from 0 on the number line, but in different directions. The negative side of the number line is like a mirror image of the positive side.
Example
2
Arrange the numbers –3, –5, 3, and 7 on the number line. Solution
–5
–3
0
3
7
Remember that –3 and –5 are to the left of 0.
Guided Practice 1. Arrange the numbers 3, 4, –1, and 8 on the number line. Say which side of the origin the numbers in Exercises 2–5 would go on. 2. –23 4. 6
3. 9 5. –1
Section 1.1 — Integers and the Number Line
3
The Number Line Lets You Compare Numbers Check it out: The further to the right on a number line, the greater a number is. It doesn’t matter which side of the origin the numbers are.
If two numbers are plotted on the number line, then the number to the left is less than the number to the right. You could also say that the number to the right is greater than the number to the left. There are two important symbols used to order and compare two numbers. The symbol > means “greater than” and the symbol < means “less than.”
Example Check it out: –4 is less than –2, even though 4 is greater than 2. That’s because –4 goes on the left of –2 on the number line.
3
By placing them on the number line, show that 1 is greater than –4. Solution
1 is to the right of –4, so 1 > –4. –4
0 1
Guided Practice Write these numbers with the correct < or > symbol, then use the number line to say which number in each Exercise is greater. 6. 7, 3 8. –1, 10 10. –9, –11
7. 2, 3 9. –8, 0 11. –1, 1
Whole and Natural Numbers Are Types of Integers You’ve already been dealing with a few different types of numbers on the number line — now it’s time to learn their names. Check it out: If you’re just counting objects, you use natural numbers to say how many you have.
Check it out: For example, –3, –2, –1, 0, 1, 2, and 3 are all integers but –0.5 is not an integer.
4
Natural numbers are also called counting numbers because they are the numbers used to count with — 1, 2, 3, and so on.
–6 –5 –4 –3 –2 –1 0
Whole numbers are like the natural numbers except that they include the number zero.
–6 –5 –4 –3 –2 –1 0
Integers include all the natural numbers, the negative versions of them, and zero.
Section 1.1 — Integers and the Number Line
–6 –5 –4 –3 –2 –1 0
1
2
3 4
5
6
Natural numbers
1
2
3 4
5
6
Whole numbers
1
Integers
2
3 4
5
6
Example Check it out: Natural numbers and whole numbers have something in common — numbers less than zero are not included.
4
Describe the following numbers as natural, whole, or integer. There may be more than one answer for each. 1. 1
2. 0
3. –3
Solution
1. 1 is greater than zero so it is a natural number. All natural numbers are also whole numbers and integers. Don’t forget: 0 is a whole number but not a natural number.
2. 0 is not a natural number, but it is a whole number. As it is a whole number it is also an integer. 3. –3 is less than zero so it is not a natural number and not a whole number. But it is an integer.
Guided Practice Describe each number as a natural number, a whole number, or an integer. There may be more than one answer for each number. 12. 7
13. –9
14. –400
15. 1
16. 55
17. 0
18. 26
19. –16
20. –32
21. 3.2
Independent Practice 1. Place 3, 1, 5, and –2 on a number line. Now try these: Lesson 1.1.1 additional questions — p428
Answer each of Exercises 2–9 as true or false. 2. 4 < 9 3. 23 > –40 4. 0 < –1 5. –1 > –2 6. 4 > –5 7. 0 is a positive number. 8. Whole numbers can be negative. 9. All integers are whole numbers. 10. Hector is thinking of a number that is a whole number but is not a natural number. What number is Hector thinking of? Exercises 11–13 make use of the number line shown below. 11. How many natural numbers are between –3 and 5? 12. How many whole numbers are between –3 and 5? 13. How many integers are between –3 and 5? –3
0
5
Round Up Thinking in terms of the number line is a good way to compare integers so you can say which is greater than the other. It also helps you to understand what happens when you add positive and negative numbers, which is all part of the next Lesson. Section 1.1 — Integers and the Number Line
5
Lesson
1.1.2 California Standard: Number Sense 2.3 Solve addition, subtraction, multiplication, and division problems, including those arising in concrete situations, that use positive and negative integers and combinations of these operations.
What it means for you: You’ll learn to add positive and negative numbers to integers.
Key words: • positive • negative • integer
Adding and Subtracting Integers You’ve met positive and negative numbers, and you know how to arrange them on the number line. This Lesson covers addition and subtraction of integers, including negative numbers.
Adding a Positive Moves Right on the Number Line In the last Lesson, you saw that numbers on the number line are greater the farther to the right they are. When you add a positive integer to any number, you’re making it greater, so you move to the right. To add a positive integer, move that number of positions to the right on the number line. It doesn’t make any difference whether the original number is positive or negative — you always move to the right. Example
1
Using the number line, add 9 to –4. Don’t forget:
Solution +1
Integers include all the natural numbers, the negative versions of them, and zero. So –2, –1, 0, 1, and 2 are all integers.
+2
+3
+4
+5
–4 –3 –2 –1 0
+6
1
+7
2
+8
+9
4
3
5
Counting 9 spaces to the right of –4 on the number line shows that the answer is 5.
Adding a Negative Moves Left on the Number Line When you add a negative integer, move that number of positions to the left on the number line. Example
2
Using the number line, add –6 to 5. Solution 6
–1 0
5
4
1
3
2
2
3
1
4
5
Counting 6 spaces to the left of 5 on the number line shows that the answer is –1.
6
Section 1.1 — Integers and the Number Line
When you’re adding a negative number, it doesn’t matter whether you’re adding it to a positive or negative number — you always move to the left. Example
3
Using the number line, add –5 to –1. Solution 5
4
3
2
1
–6 –5 –4 –3 –2 –1 0
Counting 5 spaces to the left of –1 on the number line shows that the answer is –6.
Guided Practice Answer the following Exercises without using a calculator. Use a number line to help. 1. 7 + 3 2. –2 + 2 3. –8 + 4 4. 1 + 3 5. –81 + 8 6. –9 + 11 7. 9 + (–2) 8. 35 + (–10) 9. –5 + (–1) 10. 0 + (–5) 11. 36 + (–10) 12. –15 + (–7)
Subtracting a Positive is Like Adding a Negative You already know that to add a negative integer, you move that number of positions to the left on the number line. The method for subtracting a positive integer is exactly the same. Example
4
Using the number line, calculate 8 – 2. Solution
The 2 tells you that you need to move 2 steps, and the “–” sign tells you that you need to move left. So the answer is 6.
2
6
1
8
Guided Practice Answer the following Exercises without using a calculator. Use a number line to help. 13. 7 – 3 14. –1 – 10 15. 8 – 8 16. 3 – 8 17. –2 – 9 18. 1 – 10 Section 1.1 — Integers and the Number Line
7
Subtracting a Negative Is Like Adding a Positive You’ve already seen that adding a negative means you move left on the number line. So subtracting a negative must mean you move right on the number line. Another way of saying this is that subtracting a negative is the same as adding a positive. Example
5
Calculate 8 – (–3). Solution
8 – (–3) The – sign means "go left on the number line"...
...but this – sign reverses the direction again
You end up moving 3 steps right on the number line. So you can just write 8 – (–3) as 8 + 3 = 11.
1
8
2
3
9 10 11
Guided Practice Write Exercises 19–24 as additions, then find the solutions. 19. 14 – (–9) 21. –9 – (–15) 23. –4 – (–8)
20. 23 – (–10) 22. 4 – (–4) 24. –26 – (–17)
Independent Practice
Now try these: Lesson 1.1.2 additional questions — p428
1. The water level of a lake in early April was 3 inches above normal. The water level had dropped to –4 inches below normal by early May. What integer can be added to 3 to get the water level, in inches, in early May? Write each of Exercises 2–5 as an addition problem. 2. 4 – 3 3. 7 – 8 4. –1 – 26 5. 4 – 5
5 4 3 2 1 0 -1 -2 -3 -4
Normal water level
Write each of Exercises 6–9 as a subtraction problem. 6. 5 + (–9) 7. 1 + (–3) 8. 2 + (–1) 9. 433 + (–348) Find the answer to each of Exercises 10–13. 10. 4 + (–15) 11. 18 – 9 12. 23 + 6 13. 15 – (–3)
Round Up Adding to and subtracting from integers is a matter of deciding which direction to move, then moving the correct distance. When you’re deciding whether to go left or right you only need to think about whether you’re adding a positive or negative number — it doesn’t matter what you’re adding it to. 8
Section 1.1 — Integers and the Number Line
Section 1.2 introduction — an exploration into:
Multiplica tion with Inte ger Tiles Multiplication Integ You can use colored tiles to show what happens when you multiply numbers together. Multiplication is really just the same as doing lots of addition — for example, 3 × 2 is the same as 2 + 2 + 2, or even 3 + 3. In this Exploration, blue tiles show positive numbers, while red tiles show negative numbers. For example, the number 5 is represented by 5 blue tiles, and –4 is represented by 4 red tiles. A blue and a red tile together have a total value of zero, and are called a zero pair. You can multiply numbers by adding together equal groups of tiles.
1 + (–1) = 0
Example Find 3 × (–4) using integer tiles. 3 groups of –4 (12 red tiles in total)
Solution
This multiplication problem means: “Make 3 groups of –4.” So 3 × (–4) = –12.
Making 3 groups of something is easy to think about — it involves adding things together. Making –4 groups of something is harder to understand, but it involves subtracting. Example 12 zero pairs: total value = 0
Find –4 × 3 using integer tiles. Solution
This multiplication problem means: “Make –4 groups of 3.” Making –4 groups of 3 is the same as subtracting 4 groups of 3 blue tiles. But before you can do this, you need some blue tiles. Use zero pairs. Start with 12 zero pairs. Now remove 4 groups of 3 blue tiles, or 12 blue tiles. That leaves 12 red tiles, or –12. So –4 × 3 = –12.
Remove 12 blue tiles
Exercises 1. Use integer tiles to model and solve each of the following: a. 5 × 3
b. 2 × (–6)
c. 4 × (–3)
d. –4 × 2
e. –3 × (–2)
f. –2 × (–5)
2. Use your results to find a rule for the product of a positive and a negative number. 3. Use your results to find a rule for the product of two negative numbers.
Round Up Multiplying using red and blue tiles can help you work out the answers to some difficult questions. In this Section, you’ll see other ways of thinking about multiplication with negative numbers. But the answers will always be the same as you get with tiles. Section 1.2 Explor a tion — Multiplication with Integer Tiles Explora
9
Lesson
Section 1.2
1.2.1
Multiplying with Integers
California Standard:
You’ve seen how to use a number line to show what happens when you add or subtract positive and negative integers. In this Lesson you’ll see how it can be useful for doing multiplication problems too.
Number Sense 2.3 Solve addition, subtraction, multiplication, and division problems, including those arising in concrete situations, that use positive and negative integers and combinations of these operations.
What it means for you: You’ll see what happens when you multiply positive and negative whole numbers.
Key words: • integer • product • factor
Multiplication Is All About Grouping Things Multiplication is really a way of adding together groups of objects. For instance, 2 × 3 just means “2 groups of 3.”
+
Doing “3 groups of 2” gives the same result.
+
There are 6 blocks in total, so 2 × 3 = 6.
=
+
There are still 6 blocks, so 3 × 2 = 6.
=
You can do the same kind of grouping and counting on the number line. Example
1
Show the answer to 2 × 3 using a number line. Don’t forget: You can work out the product of two numbers in either order — a × b = b × a. This is called the commutative property of multiplication — see Lesson 2.3.2 for more.
Solution
You can show the answer with 2 arrows, each of length 3: 3 3 2 × 3 is twice as far from 0 as 3 is
0
4
2
6
8
You could also show the same answer with 3 arrows of length 2: 2 2 2 0
4
2
6
8
3 × 2 is three times as far from 0 as 2 is
Guided Practice Don’t forget:
What multiplication is shown on each number line?
The answer to a multiplication is called a product. The numbers being multiplied together are called factors.
1. 7
0
14
21
2. 0
2
4
6
8 10 12 14 16 18 20
3. 0
10
4
8
Section 1.2 — Multiplication and Division with Integers
12
16
Multiplying by a Negative Changes the Direction Even if you’re multiplying by a negative, you’re still dealing with groups. So 3 × (–2) still means “3 groups of –2.” Don’t forget: On the number line, positive numbers are on the right, negative numbers are on the left.
Just like in Example 1, there are 3 arrows of length 2 on the number line, but this time the negative sign means they’re pointing left. –2 –2 –2 You can see from this number line that 3 × (–2) = –6. –6 0 –4 –2 Example
2
Calculate 4 × (–1). Solution –1 –4
–1 –3
–1 –2
Example
–1 –1
You can see from this number line that 4 × (–1) = –4.
0
3
The outside temperature at midnight was 0 °F. Every hour after that, the temperature dropped by 3 °F. What was the temperature at 5 a.m.? Solution
The change in temperature is –3 °F each hour for five hours, so you need to solve 5 × (–3).
–3
–15
–3
–12
–3
–9
–3
–6
–3
–3
0
This shows that 5 × (–3) = –15, so at 5 a.m. the temperature was –15 °F.
Guided Practice What multiplication is shown on each number line? 4.
5. –40
–30
–20
–10
0
–40
–32
–24
–16
–8
0
6. What conclusion can you make from Exercises 4 and 5? Check it out: A negative change such as in Exercises 11–13 usually shows something’s getting smaller or lower.
Calculate the following multiplications: 7. 7 × (–4) 8. –7 × 3 9. 30 × (–2)
10. –2 × 30
11. A submarine changes its depth in the water by –25 feet per minute. What is its total change in depth in four minutes? 12. A bird is flying toward the ground. Its height changes by –16 feet per second. What is the bird’s total change in height in 5 seconds? 13. The amount of fuel in a racing car changes by –6 gallons per lap. What is the change in its fuel load over 7 laps? Section 1.2 — Multiplication and Division with Integers
11
A Negative Times a Negative Equals a Positive On the last page you saw that multiplying a positive integer by a negative integer results in a negative solution. But if you multiply one negative number by another, their “–” signs cancel each other out. Example
4
Calculate –3 × (–2). Solution
You know that 3 × (–2) means “3 groups of –2”, and 3 × (–2) = –6. The extra negative sign in –3 × (–2) just changes the sign again. The answer must be positive: –3 × –2 = 6
If you’ve got several negative integers to multiply, you can do it bit by bit. Example
5
Calculate –3 × (–2) × (–5). Solution
[–3 × (–2)] × –5 Work it out in smaller parts = 6 × (–5)
First multiply two of the numbers: –3 × (–2) = 6
= –30
Now, positive × negative = negative
So you can multiply any two integers using these rules:
Rules for multiplying integers positive × positive = positive positive × negative = negative negative × positive = negative negative × negative = positive
You can still use these rules even if you’re multiplying more than two numbers together. Just count the number of “–” signs in the question. If there’s an even number of negative factors, they’ll cancel out in pairs, and the answer will be positive. If there’s an odd number of negative factors, you’ll end up with one that doesn’t cancel out, so the final answer will be negative.
12
Section 1.2 — Multiplication and Division with Integers
Example
6
Solve –2 × 5 × (–4) × (–10). Solution
There are three minus signs. This is an odd number, so the answer will be negative. Work out the “size” of the number by finding: 2 × 5 × 4 × 10 = 400 So the answer must be –400. To prove it, break the question down into smaller parts: –2 × 5 × (–4) × (–10) = –10 × (–4) × –10
Negative × positive = negative
= 40 × (–10)
Negative × negative = positive
= –400
Positive × negative = negative
Guided Practice Say whether the following will give positive or negative answers. (You don’t need to work out the actual solutions.) 14. –8 × (–3) 15. –2 × 9 16. 2 × (–3) × (–5) 17. –27 × (–13) × (–7) × (–17) 18. –6 × 11 × (–19) × (–83) 19. –1 × 2 × (–3) × 4 × (–5) 20. 225 × (–311) × (–277) × (–1008) × 47 × (–119)
Independent Practice
Now try these: Lesson 1.2.1 additional questions — p428
In Exercises 1–6, use a number line to solve the multiplication. 1. 5 × 7 2. –3 × 12 3. 11 × 4 4. 6 × (–6) 5. 21 × (–2) 6. –8 × (–3) 7. Ms. Ross is overdrawn on her bank account. Her balance is –$30. Mr. Banks is overdrawn on his bank account by 5 times the amount Ms. Ross is overdrawn. What is Mr. Banks’s account balance? 8. Sara multiplied two negative integers together. She then multiplied her answer by another negative number. Is her final result positive or negative? 9. Pablo multiplied two integers together. The answer that he got was –28. What integers might he have multiplied together?
Round Up It’s important to know what happens when you multiply by negative integers, because they appear in lots of math topics. You’ll need the rules for multiplying again when you learn about dividing with negative integers in the next Lesson. Section 1.2 — Multiplication and Division with Integers
13
Lesson
1.2.2
Dividing with Integers
California Standard:
You already know how to divide by positive whole numbers. Now you’re going to see how a number line can be helpful in answering division questions with positive and negative integers. This is similar to what you saw in the previous Lesson with multiplication.
Number Sense 2.3 Solve addition, subtraction, multiplication, and division problems, including those arising in concrete situations, that use positive and negative integers and combinations of these operations.
Division Means Breaking a Number into Equal Parts “Divide” is another way of saying “share out equally.”
What it means for you: You’ll learn what to do when you see positive and negative integers in division questions.
Key words: • integer • quotient • positive • negative
Example
1
A large bag contains 12 apples. The apples are shared equally among 4 students. How many apples does each student get? Solution
You can see from the picture that if the apples are shared out equally, each student will get 3 apples.
You can also think of division as the opposite of multiplication. Example 1 is asking you to find 12 ÷ 4. You can rewrite that as 12 ÷ 4 = ? which is the same as saying 4 × ? = 12 Check it out: When you do a division, the result is called the quotient.
Then you can use “guess and check” to work out what the question mark must represent: 4 × 1 = 12 8 4 × 2 = 12 8 4 × 3 = 12 9
Try substituting 1: Try substituting 2: Try substituting 3:
Guided Practice Solve without a calculator: 1. 18 ÷ 3 4. 26 ÷ 2 7. 88 ÷ 11
14
2. 21 ÷ 7 5. 100 ÷ 4 8. 70 ÷ 5
Section 1.2 — Multiplication and Division with Integers
3. 12 ÷ 6 6. 54 ÷ 9 9. 48 ÷ 12
Division Can Also Be Shown on a Number Line Below is a number line showing the multiplication 4 × 6 = 24. Check it out:
6
One-half of a number means dividing it by 2. One-third of a number means dividing it by 3. One-fourth of a number means dividing it by 4. And so on...
0
6
6
6
12
6
24
18
You can think about this line in another way. It shows that the number 24 can be divided into 4 equal parts, each of size 6. It also shows that the number 6 is one-fourth of the way from 0 to 24. So the number line also shows that 24 ÷ 4 = 6
Check it out: There are two ways of using a number line to represent a division like 35 ÷ 5.
Example
Use the number line below to solve 35 ÷ 5.
1) You can do it as in Example 2, by saying “35 ÷ 5 means you have to divide 35 into 5 equal parts.” 2) Or you can say “35 ÷ 5 means working out how many 5s there are in 35.” This gives you a slightly different number line, as shown below.
0
5
10
15
20
25
30
2
0
7
14
21
28
35
Solution
The number line shows the line from 0 to 35 divided into 5 equal parts. You can see that 7 is one-fifth of the way from 0 to 35, so the answer must be 35 ÷ 5 = 7.
35
Both ways are equally good, but in this book, we’ve used approach 1) above.
Guided Practice Use the number line below to answer Exercises 10–12:
0
17
10. 102 ÷ 2
34
51
11. 102 ÷ 3
68
85
102
12. 85 ÷ 5
Use the number line below to answer Exercises 13–18:
0
13. 24 ÷ 4 16. 36 ÷ 2
6 12 18 24 30 36 42 48 54 60
14. 60 ÷ 5 17. 36 ÷ 3
15. 54 ÷ 3 18. 48 ÷ 4
Section 1.2 — Multiplication and Division with Integers
15
Negative Numbers Can Be Used in Division Dividing with negative integers is similar to dividing with positive integers. You still have to work out how many equal parts go into the number. Example
3
Calculate –42 ÷ 7 using the number line. Solution
Check it out: Another way of thinking about Example 3 is to say that –42 can be split into 7 equal groups of –6.
–36
–42
–30
–24
–18
–12
–6
0
The number –6 is one-seventh of the way from 0 to –42 on the number line. So the answer is –6. You can check this by working out 7 × (–6) = –42
Guided Practice Use the number line below to answer Exercises 19–22:
-64
-48
-56
-40
19. –64 ÷ 8 21. –40 ÷ 5
-32
-24
-16
-8
0
20. –64 ÷ 2 22. –48 ÷ 3
The rules for dividing integers are really similar to the multiplication rules you saw in the last Lesson.
Rules for dividing integers positive ÷ positive = positive positive ÷ negative = negative negative ÷ positive = negative negative ÷ negative = positive Example
4
Check it out: Another way of thinking about Example 4 is to say that –42 ÷ (–7) means “how many equal groups of –7 are in –42?”.
16
Calculate –42 ÷ (–7). Solution
You can use a number line to work out that 42 divides into 7 equal 0 6 12 18 24 groups of 6, so 42 ÷ 7 = 6. Then using the above rules, you can see that –42 ÷ (–7) = 6
Section 1.2 — Multiplication and Division with Integers
30
36
42
Example
5
Calculate –120 ÷ (–40). Solution
You can use the rules for dividing integers instead of the number line. You are dividing a negative by a negative, so the answer is positive. The answer will be the same as 120 ÷ 40. –120 ÷ (–40) = 120 ÷ 40 = 3 Check: –40 × 3 = –120
Guided Practice Say whether the following will give positive or negative answers. (You don’t need to work out the actual solutions.) 23. –18 ÷ 3
24. –27 ÷ (–9)
25. –17 ÷ (–1)
26. 625 ÷ (–25)
27. –363 ÷ 11
28. –1008 ÷ (–24)
Independent Practice Now try these: Lesson 1.2.2 additional questions — p429
Solve: 1. 16 ÷ 2 4. 66 ÷ (–11)
2. 45 ÷ 5 5. –27 ÷ (–9)
Use a number line to solve: 7. 63 ÷ 7 8. 371 ÷ 7 10. –126 ÷ 6 11. 188 ÷ (–4)
3. –36 ÷ 12 6. –126 ÷ 9 9. 40 ÷ (–5) 12. –64 ÷ (–4)
13. The Garcias are going to the store. Mrs. Garcia is going to divide $42 equally among her three children so they have pocket money. How much money does each child get? 14. A submarine is –400 m from the surface of the ocean. It rises to half of this depth. At what depth is the submarine now? 15. Emma’s bank account is overdrawn. Her balance is –$240. She pays some money into the account so that her overdraft is reduced to one-third of the old balance. What is the new balance on Emma’s account?
Round Up You’ve now learned rules for adding, subtracting, multiplying, and dividing integers. In the next Lesson, you’ll see these rules applied in real-life situations. Section 1.2 — Multiplication and Division with Integers
17
Lesson
1.2.3
Integers in Real Life
California Standard:
Now you’ve learned about positive and negative integers, and how to add, subtract, multiply, and divide them. In this Lesson you’ll use what you have learned in real-life problems.
Number Sense 2.3 Solve addition, subtraction, multiplication, and division problems, including those arising in concrete situations, that use positive and negative integers and combinations of these operations.
What it means for you: You’ll answer real-life questions that involve addition, subtraction, multiplication, and division.
Real Situations Often Involve Addition and Subtraction You always need to read the problem and identify the operations being used — either add, subtract, multiply, or divide. Then you should always rewrite the question in math language to help you solve it. Example
1
At 10 a.m. the outside temperature was 70 °F. It was 5 °F warmer by midday. What was the midday temperature? Key words: • integer • range • operation
Solution
The question involves adding 5 °F to 70 °F, which is written 70 + 5. So the midday temperature was 70 + 5 = 75 °F.
Check it out: Remember to write the solution as an answer to the original question — for instance, in Example 1 you must remember to add the °F symbol.
Integer subtraction can be used to find out how much a quantity has changed. It can also be used to calculate the range of a set of numbers. Example
2
A student in Ms. Chang’s English class scored 12 points on a quiz. Another scored 27 points. What is the range of those scores? Check it out:
Solution
To find the range of a set of numbers, you subtract the least from the greatest.
The range is found by subtracting 12 from 27. So the answer is 27 – 12 = 15.
Guided Practice 1. Jerry has 23 ballpoint pens. His friend George gives him 16 more pens. How many pens does Jerry have now? 2. One day there are 32 passengers on a bus. The next day there are 67. What is the range of these two numbers? 3. A scuba diver is 16 meters below sea level. How far below sea level is the diver if she goes 15 meters lower?
18
Section 1.2 — Multiplication and Division with Integers
4. Althea has 37 bags of potato chips. She gives 10 to Jackie. How many bags does Althea have now? 5. In one day, Isaac runs the 400 m and 1600 m races. How many meters has he run during these two races? 6. After a drought, the water level of Lake Pinebrook was 19 inches below normal. After a long rainy season, the water level rose to 22 inches above normal. Find the range of the water levels.
Watch Out for Multiplication and Division Problems Look out for words like “times” or “double” or “triple” — these words are clues that the problem will involve multiplication. Example
3
A small fish is swimming 5 feet below the surface of the ocean. A larger fish is swimming at 12 times the depth of the small fish. How far below the surface is the larger fish? Solution
The question tells you how deep the small fish is. You need to find 12 times that figure, so you need to multiply. Written as a multiplication problem, the question is 5 × 12 = 60. The larger fish is 60 feet below the surface of the ocean. Division problems often have the word “divided” in them, or they may have a fraction written out in words. Example
4
A large fish is swimming 145 feet below the surface of the ocean. A smaller fish is swimming at one-fifth the depth of the large fish. How far below the surface is the smaller fish? Solution
This time you need to find one-fifth of 145, so you need to divide. Written as a division problem, the question is 145 ÷ 5 = 29. The smaller fish is 29 feet below the surface of the ocean.
Guided Practice 7. Kia tripled her CD collection this year. She started with 36 CDs. How many does she have now? 8. It takes Marcus 2.5 hours to paint his bathroom, but 5 times longer to paint his hall. How long does he spend painting the hall? 9. A hot-air balloon is 96 feet from the ground. It then sinks to one-quarter of that height. What is the new height of the balloon? Section 1.2 — Multiplication and Division with Integers
19
Some Questions Involve More Than One Operation Sometimes you’ll need to carry out more than one operation to find the answer to a question. If that happens, you will need to break the question up into parts, then solve them in the right order. Check it out: Watch out for clue words that will help you figure out what to do to solve the question. Here are some examples of clue words: Add – sum, total, increase, more than Subtract – take away, fewer, difference, decrease, less than Multiply – times, product, double, triple Divide – share, quotient, split, one-half, one-third
Example
5
A bird is flying 25 feet above the ground. The bird triples its height, and then drops 60 feet to meet another bird. How high above the ground are the birds when they meet? Solution
This question has two parts. “The bird triples its height” means you need to multiply by 3: 25 × 3 = 75 feet. A drop of 60 feet means subtract 60: 75 – 60 = 15 The birds are 15 feet above the ground when they meet.
Guided Practice For Exercises 10–13, decide what operations are needed, then write out the question as a math problem and solve it. 10. Rory is driving his car at a speed of 40 mph. Before getting on the highway, he reduces his speed by one-half, then triples his speed when he gets on the highway. What is Rory's speed on the highway? 11. Kelly has $12. Her friend Elena lends her another $28. Kelly then spends one-fourth of her money on a book. How much does the book cost? 12. After low sales in January, a clothing store was overdrawn and had a bank balance of –$2400. In February, the store was overdrawn by one-third of what it was in January. In March, the overdraft was onefourth of what it was in February. What was the balance in March? 13. A fire truck holds 500 gallons of water. A hose attached to the truck uses 12 gallons of water each minute to put out a fire. The hose was turned on for 8 minutes to put out a fire. How much water is left in the fire truck after the fire?
20
Section 1.2 — Multiplication and Division with Integers
Independent Practice 1. Jermaine is baking. He makes an extra-large pie by multiplying the amounts of ingredients in the recipe by three. If the original recipe says to use 350 grams of apples, how many grams of apples does Jermaine use? 2. A bank clerk notices two accounts that are overdrawn. Account 1 has a balance of –$24, and account 2 is overdrawn 3 times that amount. What is the balance of account 2? Now try these: Lesson 1.2.3 additional questions — p429
3. On Monday, Laura rides 21 miles on her bike. On Tuesday, she rides one-third the distance she did on Monday. On Wednesday, she rides for 10 miles more than on Tuesday. How far did Laura ride on Wednesday? 4. Three boats set out from a harbor. Boat A travels 112 miles. Boat B travels 45 miles less than boat A. Boat C travels 5 times as far as boat B. How far did boat C travel? 5. Karl has 99 red balloons. He gives one-third to Libby, then another 20 each to Billy and Rosario. How many balloons does Karl have left? 6. Manuel plants a 4-foot-tall tree in his backyard. Ten years later, the tree is five times as tall as when he planted it. The tree is then struck by lightning, and the top half breaks off exactly halfway up the trunk. How tall is the tree after it is struck by lightning? 7. Susan, Stephanie, Connor, and Gino open a store together. The business is not successful and makes a loss of $19,500. After Paul buys their stock for $1500, the rest of the loss is split between the partners. How much money has each partner lost? 8. The water level in a lake drops during the summer to 23 inches below its normal level. During a period of heavy rain, the water level rises again by 5 inches per day. After 7 days of rain, how far above the normal level is the water in the lake? 9. Ms. Alano has a box of 100 paper clips. She gives 3 to each of her students, and then has 10 left over. How many students does Ms. Alano have? 10. Doug and Jim are playing golf. They both start off 300 yards away from the hole. Jim’s ball lands 175 yards from the hole. Doug’s ball lands half as far away as Jim’s, but then rolls down a slope which takes it 30 yards farther away from the hole. How far from the hole does Doug’s ball finish up?
Round Up Remember to read the question carefully, think about what you’re being asked, and convert everything into math language to help you solve it. The more you practice, the better you’ll get at spotting the different clue words for addition, subtraction, multiplication, and division. Section 1.2 — Multiplication and Division with Integers
21
Section 1.3 introduction — an exploration into:
A Decimal Strip In this Exploration, you’re going to measure things in meters. You can measure any length in meters. Big things (like a classroom) might be many meters long. But small things (like your pencil) probably measure much less than one meter — this is where decimals are useful. length = 1
Look at your meter strip — its length in meters is 1. To measure the length of the classroom, you could lay several strips end to end. The length of the classroom would be the number of strips you use. But you need a different method for things less than a meter long.
length = 5
Example Show how you could measure the length of your pencil using your meter strip. Solution
You could divide your strip into smaller sections. If you divide the strip into ten equal sections, each section will be one tenth, or 0.1, of a meter. Fold your strip into ten equal parts. Label each of the folds using tenths: 0.1, 0.2, 0.3, ..., 1.0 Now you can use your meter strip to measure things to the nearest tenth of a meter.
0 0. 1
0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1.0 This pencil is 0.2 meters long (to the nearest tenth of a meter).
Exercises 1. Using your strip, measure five objects in the classroom to the nearest tenth of a meter. Example Show how you could make your strip even more accurate for measuring. Solution
This is 0.21 meters long (to the nearest hundredth of a meter).
0
0. 1
0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28
0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19
Divide your strip like this one, using a ruler. Label each mark as shown.
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
You can divide each small section on your strip into ten smaller equal parts. This means the whole strip is in 100 parts. Each part is one hundredth, or 0.01, of a meter.
0. 2
Now you can use your meter strip to measure things to the nearest hundredth of a meter.
Exercises 2. Measure the same five objects as in Exercise 1, but to the nearest hundredth of a meter.
Round Up Each hundredth of your strip was one centimeter long. You could divide each centimeter into ten smaller parts. Then each part would be 0.001 (one thousandth) of a meter long (a millimeter). 22
Section 1.3 Explor a tion — A Decimal Strip Explora
Lesson
Section 1.3
1.3.1
Decimals
California Standard:
You’ve seen decimals in earlier grades — but now you’re going to deal with different place values and negative decimals too.
Number Sense 1.1 Compare and order positive and negative fractions, decimals, and mixed numbers and place them on a number line.
What it means for you: You’ll learn more about the place values of decimals.
Key words: • • • •
decimal point tenth hundredth thousandth
Decimals Are Numbers Between Integers Decimals are used to represent numbers between integers. Digits after the decimal point represent part of an integer. Each one on the number line is divided into 10 tenths. 0.8 means “0 plus 8 tenths,” and 1.4 means “1 plus 4 tenths.” Each tenth is then divided into 10 hundredths. In other words, each one is divided into 100 hundredths.
0.8 0
1.4 1
2
1.4
1.5
1.44
Similarly, each one on the number line is divided into 1000 thousandths. Check it out: To read a decimal number, say the whole-number part in the normal way — and then add the part after the decimal point, reading it as the number of tenths, or the number of hundredths, or similar. Also, say the word “and” to make it clear where the decimal place is. For example, 5.7 is read “five and seven tenths,” while 12.75 is read “twelve and seventy-five hundredths.”
Example
1
Explain the meaning of each of the digits in the number 6.582. Between which two integers does the number 6.582 lie? Solution
• The number 6.582 means: 6 ones plus 5 tenths plus 8 hundredths plus 2 thousandths. • The number 6.582 is 6 ones plus numbers after a decimal point. So the number 6.582 lies between 6 and 7. Example
2
David spent 1 hour and 30 minutes playing basketball. Express this number of hours as a decimal. Solution
• 30 minutes is 0.5 of an hour, so David played basketball for 1 whole hour plus an extra 0.5 hours. • Putting those together, David played basketball for 1.5 hours.
Guided Practice In Exercises 1–4, first say which digit is in the hundredths place, then which digit is in the thousandths place. 1. 34.251 2. 128.734 3. 0.163 4. 3514.902 5. A runner completed a marathon in 4 hours and 15 minutes. Express this number as a decimal. Section 1.3 — Decimals
23
Decimals Can Be Shown on the Number Line Check it out: The numbers 1.230 and 1.23 are the same because the 0 on the end of 1.230 just means “1.23 and then 0 thousandths extra.” For the same reason, any number of 0s can be added or removed if they come at the very end of a number after a decimal point.
To put a decimal on the number line, read it as an instruction. For example, with the number 2.68: • first move 2 ones along the number line, • then move 6 tenths in the same direction, • then move 8 hundredths in the same direction.
0
If the number line is only divided into tenths, you’ll have to estimate the hundredths position by placing the cross between the two lines.
1
2
3
× 2.68
Example Check it out:
2 ones + 6 tenths + 8 hundredths
2.68 equals
3
Put 1.43 on the number line. Solution
1.43 is 1 along the number line, then 4 tenths, and then 3 hundredths. 0
1
2
3
× 1.43
You can plot negative decimals in the same way. But this time you always move in the negative direction — to the left. Example
4
Put –1.27 on the number line. Solution
–1.27 is 1 along the number line in the negative direction, and then 2 tenths in the same direction, and then 7 hundredths in the same direction. –1
0
1
× –1.27
Guided Practice Place the numbers in Exercises 6–9 onto a number line. 6. 1.4 7. –2.8 8. 6.3 9. –0.5
24
Section 1.3 — Decimals
Draw Zoomed-in Number Lines to Be More Accurate It is hard to show thousandths on a normal number line accurately. Instead, you can draw a zoomed in number line. This one has a scale marked in tenths rather than integers, with divisions for hundredths. –0.2
Example
–0.1
0
0.1
0.2
5
Put –0.183 on the number line. Check it out: Example 5 uses a “close up” number line that only shows the origin and a small section of the number line immediately to the left. It’s as though someone has zoomed in on the number line, allowing you to see the really small hundredth marks.
Solution
–0.183 is 1 tenth along the number line in the negative direction, and then 8 hundredths in the same direction, and then 3 thousandths in the same direction. –0.2
–0.1
0
× –0.183
Guided Practice Place the numbers in Exercises 10–13 onto a number line. 10. –1.73 11. –5.78 12. –5.624 13. 0.597
Independent Practice Now try these: Lesson 1.3.1 additional questions — p429
1. The temperature in Celsius of a substance during a science experiment is twelve and twenty-three hundredths below zero. Represent this temperature using a decimal. Place the numbers in Exercises 2–5 onto a number line. 2. 23.5 3. –2.28 4. –19.23 5. 3.246
Round Up Decimals look complicated, but they can be broken up into tenths, hundredths, and thousandths, and put on the number line. In the next Lesson you’ll see how that helps you put decimals in order. Section 1.3 — Decimals
25
Lesson
1.3.2
Ordering Decimals
California Standard:
Now it’s time to learn how to order decimals. If you’re given two decimals, you need to be able to say which is greater.
Number Sense 1.1 Compare and order positive and negative fractions, decimals, and mixed numbers and place them on a number line.
What it means for you: You’ll learn two different methods for ordering decimals.
Key words: • greater than (>) • less than (<)
There are a couple of ways of ordering decimals — either using the trusty number line or without it.
The Number Line Shows the Order of Decimals In Lesson 1.1.1 you saw that if any two integers are plotted on the number line, the number to the left is less than the number to the right. The same thing applies to decimals — and it doesn’t make any difference whether the numbers are positive or negative. The symbol < is used to mean “less than” and the symbol > is used to mean “greater than.”
Don’t forget: You saw the “<“ and “>” symbols in Lesson 1.1.1, when you were comparing integers.
Example
1
Using the number line, explain why the statement –1.7 > 1.2 is not true. Solution
Place –1.7 and 1.2 on a number line: –1.7
1.2
×
×
–2
–1
0
1
2
–1.7 is to the left of 1.2, so –1.7 is less than 1.2. So –1.7 < 1.2, which means it is false to say that –1.7 > 1.2.
Guided Practice In Exercises 1–4, use a number line to say which number is greater. 1. 23.45, 10.2 3. 8.163, 8.162
2. –2.3, 1.9 4. –1.2, –0.8
Use a number line to say whether each statement in Exercises 5–8 is true or false. 5. 3.9 < 3.14 7. 0.5 < –1.5
26
Section 1.3 — Decimals
6. 2.1 > 2.09 8. –1.3 > –1.2
Positive Decimals Can Be Compared Digit by Digit You can also compare decimals without using the number line. Example Don’t forget: Adding 0s to the end of decimals doesn’t change the value at all. It just means “...and no extra tenths, hundredths, or thousandths.”
2
Without using the number line, say whether 23.52 is greater than 23.5. Solution
Before you compare, make sure the decimal points line up. Add zeros to the end of either decimal to make them the same length.
Line up the
23.52 decimal points zeros 23.50 Add at the end
23.52 23.50
Work from left to right and compare each of the digits. same different The first three digits are the same in both numbers. The digits in the hundredths column are different, so use them to find the answer. 2 is greater than 0, so 23.52 is greater than 23.5.
Guided Practice In Exercises 9–12, say which is the lesser number. 9. 0.23, 0.24 11. 2.5, 2.23
10. 3.5, 13.5 12. 0.1, 0.01
13. By comparing each with the others, put these decimals in order, from the least to the greatest: 23.45, 0.356, 3829.34, 7.5, 0.2
Negative Decimals Can Be Compared Too Comparing negative decimals is the same as comparing positive decimals. Example
3
Without using the number line, say which of –0.3 and –0.317 is greater. Solution
Don’t forget:
Add zeros to –0.3 to make it the same number of decimal places as –0.317, then compare each digit starting from the left.
0 isn’t positive or negative.
–0.317 –0.300 same
different
The first two digits are the same in both numbers. The digits in the hundredths column are different, so they can be used to find the answer. But remember that these digits are all negative. 0 is greater than –1, so –0.3 is greater than –0.317. Section 1.3 — Decimals
27
Don’t forget: Negative numbers always go on the left of the origin, and positive numbers always go on the right of the origin. So, positive numbers are always greater than negative numbers, whether they are decimals or not.
Guided Practice In Exercises 14–17, say which is the greater number. 14. –0.003, –0.017 15. –2.1, 13.5 16. –18.9, –25.73 17. –0.01, –0.001 18. By comparing each with the others, put these decimals in order, from the least to the greatest: –2.0, –12.7, –194.644, –0.5, –7.0
Independent Practice 1. Explain why you can say that –3.234 is less than 2.855 without using a number line or comparing any digits.
Now try these: Lesson 1.3.2 additional questions — p430
Say whether each statement in Exercises 2–7 is true or false. 2. 23.3 > 23.25 3. 4.83 > 4.9 4. –34.234 > –2.334 5. –4.234 < 0.001 6. 14.2 < 19.3 7. 15.345 < 393.3 8. By comparing each with the others, put these decimals in order: –3.344, 28.0, –0.7, 0.523, 0.001, 18.534, –3.345, 0.01 9. Adan works for a delivery company. A customer comes in with a package that weighs 1.023 pounds. Adan knows that packages that weigh less than 1.7 pounds cost $2 but packages that weigh 1.7 pounds or more cost $3. How much should he charge the customer? 10. Helen and Morty go scuba diving. Helen swims down 10.9 m below the surface of the water. Morty swims to 10.28 m. Who is lower? 11. Annie is teaching Alvy how to drive. He looks at the fuel gauge to check if they need to stop for gasoline, and sees this: E 1 2 3 4 The gauge shows how much gasoline is left in gallons. How much is left? Exercises 12–14 make use of the table below. Mrs. Bueller is observing a substance Day as part of a biotech experiment at Monday work. The table shows the Tuesday temperatures of the substance on Wednesday Monday, Tuesday, and Wednesday.
Temperature –20.3 ºC 21.5 ºC 19.8 ºC
12. On what day was the temperature the coldest? 13. On what day was the temperature the warmest? 14. The temperature was 2 and 25 hundredths warmer on Thursday compared to Wednesday. What was the temperature of the substance on Thursday?
Round Up Comparing decimals can be tricky. Just make sure that before you compare numbers digit by digit, you make sure the decimal points are correctly lined up. 28
Section 1.3 — Decimals
Lesson
Section 1.4
1.4.1
Rounding Number s Numbers
California Standard:
Rounding involves replacing one number with another number that’s easier to work with. You’ll use rounded numbers in the next couple of Lessons to check and to estimate answers.
Ma thema tical R easoning 2.1 Mathema thematical Reasoning Use estima tion to vverify erify the estimation reasona b leness of easonab calcula ted rresults esults calculated esults..
What it means for you: You’ll learn about rounding exact figures to make them easier to work with.
Suppose you wanted to find 18 × 43, but had lost your calculator. You could find an answer close to 18 × 43 by rounding to the nearest ten. “Rounding to the nearest ten” means replacing a number with the nearest multiple of 10. Replacing a number with a higher number is called rounding up. Replacing a number with a lower number is called rounding down.
Key words: • • • • • •
Rounded Number s Can Be Easier To Use Numbers
rounding place value digit decimal tenth hundredth
Example
1
Round 18 and 43 to the nearest ten. Don’t forget: A multiple of ten is any number that ends in 0: 10, 20, 100, 2500, 58 340, and so on.
Check it out: If you place 18 on the number line, you can see that it’s nearer to 20 than 10. So 20 is the nearest ten to 18.
20
10
×
8 away from 10
Solution
You need to decide whether to round up or down. Look at the digit in the ones place: • If the ones digit is 5 or more, round up. • If the ones digit is 4 or less, round down. Start with 18:
The digit in the ones place is 8. 8 is more than 5, so round up. 18 rounded up to the nearest ten is 20.
Next, 43:
The digit in the ones place is 3. 3 is less than 4, so round down. 43 rounded down to the nearest ten is 40.
18 2 away from 20
43 is nearer to 40 than 50. So 40 is the nearest ten to 43.
50
40
By rounding, you can replace 18 × 43 with 20 × 40. This is much easier to solve: 20 × 40 = 800 800 is fairly close to the real answer: 18 × 43 = 774
×
43 3 away 7 away from 40 from 50
Guided Practice In Exercises 1–8, round the numbers to the nearest ten. 1. 36 2. 84 3. 199 4. 4006 5. 267 6. 7161 7. 2994 8. 2995 9. During a science experiment, a group of students observed that there were 415 ants in a colony. Round this amount to the nearest ten.
Section 1.4 — Estimation
29
You Can R ound to Dif ent Place Values Round Difffer erent You can round numbers to place values other than tens. Write the number. Underline the digit in the position you want to round to. • If the digit to the right of the underlined digit is 5 or more, round up. • If the digit to the right of the underlined digit is 4 or less, round down. Example
2
Round 25,281 to the nearest hundred. Solution
Write the number, and underline the hundreds digit:
25,281
You’re rounding to the nearest hundred, so that’s going to be either 25,200 or 25,300. The digit to the right of the underline is 8. That’s greater than 5, so round up.
25,300
So 25,281 rounds up to 25,300, to the nearest hundred.
Guided Practice In Exercises 10–13, round the numbers to the nearest hundred. 10. 38,383 11. 5756 12. 8128 13. 40,079 In Exercises 14–17, round the numbers to the nearest thousand. 14. 11,905 15. 8117 16. 2,599,582 17. 464,333 18. Clara lives in a city that has a population of 82,458 people. Write this population rounded to the nearest thousand.
Don’t forget: The positions after the decimal point have special names.
You Can R ound Decimals JJust ust Lik e Whole Number s Round Like Numbers You round decimals in the same way as whole numbers. Instead of rounding to the nearest ten, hundred, and so on, you round to the nearest one, tenth, hundredth, or any other number of decimal places. Example
3
Round 0.0815 to the nearest thousandth. See Lesson 1.3.1 for more about decimals.
Solution
Write the number, and underline the thousandths digit: Check it out: If you’re rounding decimals, don’t add zeros to the right of the underlined digit. So 0.83 rounded to the nearest tenth is 0.8, not 0.80.
30
Section 1.4 — Estimation
0.0815
The nearest thousandth will be either 0.081 or 0.082. Look to the right of the underline. The digit to the right is 5, so round up. So 0.0815 rounds up to 0.082, to the nearest thousandth.
0.082
Guided Practice Check it out: Always read the question carefully. Don’t mix up hundreds and hundredths, or thousands and thousandths.
In Exercises 19–22, round the numbers to the nearest tenth. 19. 28.0634 20. 2.247 21. 5.78 22. 6.892 In Exercises 23–26, round the numbers to the nearest hundredth. 23. 0.1066 24. 15.596 25. 409.4902 26. 7.734 In Exercises 27–30, round the numbers to the nearest thousandth. 27. 9.46071 28. 1.7254 29. 5.226822 30. 3.1007 31. A distance of 1 mile is equal to 1.609344 km. Write this to the nearest hundredth of a kilometer. Exercises 32–34 are about Malik, who has $12.57 in his pocket. 32. How much money does Malik have to the nearest dollar? 33. How much money does Malik have to the nearest dime? 34. How much money does Malik have to the nearest quarter?
Independent Practice In Exercises 1–5, round the number 94,521.8375: 1. to the nearest hundred 3. to the nearest thousandth 5. to the nearest one Now try these: Lesson 1.4.1 additional questions — p430
2. to the nearest hundredth 4. to the nearest thousand
6. The number 3478 was rounded to 3480. To what place value was the number rounded? 7. Raul’s thermometer shows that the temperature is 91.5 °F. What is the temperature to the nearest degree? 8. Mount Whitney is 14,505 feet high. Write this figure to the nearest hundred feet. 9. The average distance from the Earth to the Moon is 238,857 miles. What is this distance to the nearest thousand miles? 10. The speed of light is 299,792,458 meters per second. What is this to the nearest million m/s? 11. Jessica has $17.33. What is this amount to the nearest quarter? 12. A square inch is equal to 6.4516 cm². Convert 6 in² to cm², then round your answer to the nearest hundredth.
Round Up Rounding numbers makes them easier to deal with. You can use rounded numbers to quickly find an approximate answer when you don’t need an exact one. They can also help you to check your work, by letting you get an idea of how big your answer should be. Section 1.4 — Estimation
31
Lesson
1.4.2
Using R ounded Number s Rounded Numbers
California Standards:
This Lesson will tell you more about using rounded numbers. You’ll think about how much certain numbers should be rounded. You’ll also see how rounded numbers are useful for checking your work.
Ma thema tical R easoning 2.1 Mathema thematical Reasoning Use estima tion to vverify erify the estimation reasona b leness of easonab calcula ted rresults esults calculated esults.. Ma thema tical R easoning 2. 6 Mathema thematical Reasoning 2.6 Indica te the rrela ela ti ve Indicate elati tiv ad v anta ges of e xact and adv antag exact a ppr oxima te solutions to ppro ximate pr ob lems and gi ve ans wer s prob oblems giv answ ers to a specified de g r ee of deg accur ac y. accurac acy
What it means for you: You’ll learn when it’s a good idea to use rounding, and how to check your answers using rounded numbers.
Key words: • • • • • •
rounding place value accuracy check reasonable estimate
Check it out: The important thing when you choose how much to round is how much information you need to give. If you don’t round enough, the number might give details that aren’t useful. If you round too much, you might give information that is misleading.
Sometimes You Need to Choose Ho w Muc h to R ound How Much Round People round numbers to different place values depending on what the numbers are being used for. Example
1
Below are three situations connected with the distance between Town A and Town B. Match each situation to the most suitable level of rounding. Situation A road sign in Town A shows the distance to Town B. Jada lives in Town B. She tells a friend from another state how far away she lives from Town A.
Distance 203.56 miles 204 miles 200 miles
A mapping company is making an accurate map of the area.
Solution
A road sign wouldn’t give distances to the nearest hundredth of a mile. The road sign would give the distance as 204 miles. This is accurate enough, and can be read easily by a passing driver. Jada’s friend would only want a rough idea of how far away Town A is. Jada could tell her friend that she lives 200 miles from Town A. The exact distance doesn’t matter to someone who lives far away. The mapping company needs to know exact distances to draw an accurate map. They would use the figure of 203.56 miles.
Guided Practice Exercises 1–4 give three figures: one exact and two rounded numbers. Choose which you think is most suitable, and explain your choice. (In each case, the first figure given is the exact answer.) 1. Ana Lucia writes in a history essay: Thomas Jefferson lived to the age of 83 / 80 / 100. 2. On a form, Gavin gives his height as 160.67 cm/161 cm/200 cm. 3. A school records how many students are in school each day. Today there are 3914/3900/4000 students attending. 4. Mr. Anderson returns from a vacation in England with £10 left over from his extra cash. £1 is worth $1.85965. He changes the money at the bank and receives $18.5965/$18.60/$19.
32
Section 1.4 — Estimation
The Amount of R ounding Af ac y Rounding Afffects the Accur Accurac acy If you use rounding to estimate a sum, be careful how much you round. Rounding to higher place values usually gives an estimate farther from the actual answer than rounding to lower place values. Example
2
Check it out: Using rounded numbers to check your answer won’t ever tell you that your answer is definitely right, only whether it is reasonable. Your answer might be close to the real answer but could still be wrong.
Lucas wants to add 3439 and 5482. He doesn’t need an exact answer, so he decides to use rounding. Look at Lucas's work below. How could he have found a more accurate answer? 3000 + 5000 8000
Solution
Lucas rounded to the nearest thousand, so he got an estimate of 8000. If he had rounded to the nearest hundred, he would have gotten 3400 + 5500 = 8900, which is much closer to the actual answer of 8921.
Guided Practice 5. Estimate 962 – 246 by rounding to the nearest hundred. 6. Estimate 962 – 246 by rounding to the nearest ten. 7. Calculate 962 – 246 exactly. Which of your two estimates was closer to the actual result?
Rounded Number s Can Be Used to Chec k Wor k Numbers Check ork Many times you’ll want to check your work without doing the calculation all over again. Rounding is a way to see if your answer is reasonable. Example
3
Calculate 2343 + 5077. Then check your work by rounding to the nearest hundred. Solution
Actual sum: 2343 + 5077 7420
Rounded sum: 2300 + 5100 7400
These are rounded to the nearest hundred
The answer to the rounded sum is close to the answer to the actual sum, so the answer to the actual sum is reasonable.
Section 1.4 — Estimation
33
Example
4
Martin is trying to solve 29.6 × 9.8. He gets the answer 192.08. Check Martin’s answer by rounding to the nearest ten. Solution
Rounded sum:
30 × 10 = 300
Martin’s answer is a long way from the rounded answer, so it looks like his answer of 192.08 might be wrong. In fact
29.6 × 9.8 = 290.08
This is much closer to the rounded estimate.
Guided Practice In Exercises 8–15, check the answers given by rounding to the place value shown in parentheses. Say whether the answers are reasonable. 8. 1818 + 700 = 3918 (hundred) 10. 490 + 770 = 1260 (hundred) 12. 2.85 × 52.1 = 96.385 (one) 14. 32.815 + 84.565 = 117.38 (ten) 15. 10.48 × 67.02 = 902.3696 (one)
9. 22 × 79 = 738 (ten) 11. 642 – 369 = 273 (ten) 13. 68 × 47 = 5032 (ten)
Independent Practice
Now try these: Lesson 1.4.2 additional questions — p430
In Exercises 1–6, use rounding to check the answers given, and say whether or not you think they are reasonable: 1. 6898 + 517 = 7415 2. 97 × 411 = 13,677 3. 547 × 695 = 527,855 4. 74,861 – 2940 = 65,621 5. 96 × 7973 = 526,218 6. 4362 – 1855 = 2507 7. Darnell is trying to work out the answer to 52 + 871. Darnell thinks the answer is 923. He asks Zoe and Enrique if his answer looks about right. Zoe thinks the answer should be near to 1000. Enrique thinks the answer should be about 920. Why might Zoe and Enrique have gotten different answers? 8. In the Olympic 100 meters final, the first three athletes finish the race in times of 9.89 seconds, 9.94 seconds, and 9.99 seconds, to the nearest hundredth of a second. Why would it not be a good idea to round these times any further? 9. Town C is putting up a new sign showing its population. The population is 45,691 people, but this figure is changing all the time so the town decides to use a rounded figure. What figure do you think they should use: 45,690, 45,700, 46,000, or 50,000? Explain your answer.
Round Up Remember that a rounded number is usually not the same as the real figure. It only gives you a guide to how big the real number is. 34
Section 1.4 — Estimation
Lesson
1.4.3
Estimation
California Standard:
Estimation means “making a good guess.” You can use it if you don’t need to know an exact answer, or if a question has no exact right answer.
Mathematical Reasoning 2.3 Estimate unknown quantities graphically and solve for them by using logical reasoning and arithmetic and algebraic techniques.
What it means for you: You’ll practice working out an approximate answer when you can’t find, or don’t need, an exact one.
Key words: • estimate • compare • exact answer
You Can Estimate When There’s No Exact Answer Sometimes in math there is no exact right answer. You can use the information you do have to make an estimate. Example
1
Carla has a tall bookshelf and a short bookshelf. When full, the tall bookshelf can hold about 60 books. Estimate from the picture how many books the small bookshelf will hold. Solution
There is no exact number of books you can fit on a bookshelf, because not all books are the same size. To estimate the answer, compare the bookshelves. The tall one has 3 shelves, and the small one only 2. All the shelves are the same size, so the small bookshelf will hold around two-thirds the number of books. So you can estimate that the small bookshelf will hold about 40 books. Check it out: Use rounding to work out rough amounts — see Lessons 1.4.1 and 1.4.2 for more on rounding.
Guided Practice 1. This crate is partially filled with 18 apples. Estimate the total number of apples the crate can hold. 2. Mr. Lawrence is loading rolls of carpet of varying thickness into his van. He has put 5 in so far. Estimate how many rolls he will get in the van in total. 3. Jermaine collects seashells, and gets two new cases to display some of them. He has filled the smaller one with 19 shells. Estimate how many will fit in the larger case. Section 1.4 — Estimation
35
You Can Estima te If You Don wer Estimate Don’’ t Need an Exact Ans Answ You don’t always need to use an exact figure. Sometimes an estimate is enough. 2
This graph shows how many people visited a theme park last week. The manager only wants a rough idea of how many visitors there were each day. Estimate the total number of people who visited the park over Friday and Saturday. Solution
25,000
Number of visitors
Example
20,000 15,000 10,000 5000 0
M
T
W F Th Day of week
S
Sun
The value for Friday is about halfway between 15,000 and 20,000 on the vertical axis, so a good estimate for Friday would be about 17,500 people. A good estimate for Saturday might be 21,000, because the value for Saturday is just above 20,000. This gives an estimate of 21,000 + 17,500 = 38,500 for the total number of people visiting the park over Friday and Saturday.
Guided Practice 4. Use the graph from Example 2 to estimate the number of theme park visitors on Monday, Tuesday, Wednesday, Thursday, and Sunday. 5. This car is 20 feet long. Estimate the length of the bicycle.
6. The house in this picture is 30 feet tall. Estimate the height of the tree.
7. Carlos spent a total of 50 minutes doing his English and math homework. The circle graph shows how much of the time he spent on math and how much on English. Estimate how many minutes Carlos spent doing his math homework.
36
Section 1.4 — Estimation
Math
8. This bar graph shows the number of points scored by four students in a math test. Alex scored 40 points. Estimate the number of points scored by Lupe, Aisha, and Joe.
40
Alex
English
Lupe
Aisha
Joe
Independent Practice 1. George has two sunflowers in his garden, shown in the picture. The taller flower is 62 inches tall. Estimate the height of the shorter sunflower. 2. Casey is baking cookies. She has put 9 cookies on the cookie sheet as shown. Estimate the total number of cookies Casey can fit on this tray. 3. Ms. Marquez is putting up pictures the students in her art class have painted. Her display boards are shown below. She has put up 7 pictures on the smaller board. Estimate how many pictures she can fit on the larger board.
The bar graph shows the number of 6th grade boys and girls who went on a trip to the museum. 4. Estimate how many girls went to the museum. 5. Estimate how many boys went to the museum.
30 15
Girls
Now try these: Lesson 1.4.3 additional questions — p431
Boys
6. The circle graph shows how many soccer balls, footballs, and basketballs a sports store has in stock. The total number of balls in stock is 60. Estimate how many of each type of ball are in stock.
Number of sales
The graph below shows the sales of sunglasses from one store for six months of the year. 500 Use it to answer Exercises 7–9. 400 7. Estimate the number of pairs of 300 sunglasses sold in April. 200 8. Estimate the total number of pairs of sunglasses sold in June, July, and August. 100 9. Estimate the average monthly sales for 0 Apr May Jun Jul Aug Sep the period. Month of year
Round Up Some estimates are better than others, but remember there’s often no exact answer when you’re giving an estimate. Make sure you always check your answer to make sure the estimate is reasonable. Section 1.4 — Estimation
37
Lesson
1.4.4
Using Estima tion Estimation
California Standards:
Estimation is really useful in a lot of real-life situations, where you might not be able, or don’t need, to do an exact calculation. There are other times when it’s better to figure out the exact answer.
Ma thema tical R easoning 2. 1 Mathema thematical Reasoning 2.1 Use estima tion to vverify erify the estimation reasona b leness of easonab calcula ted rresults esults calculated esults.. Ma thema tical R easoning 2. 6 Mathema thematical Reasoning 2.6 Indica te the rrela ela ti ve Indicate elati tiv ad vanta ges of e xact and adv antag exact a ppr oxima te solutions to ppro ximate pr ob lems and gi ve ans wer s prob oblems giv answ ers to a specified de g ree of deg accur ac y. accurac acy
What it means for you: You’ll learn when it’s a good idea to use estimation in calculations, and when it’s not. You’ll also learn about another type of estimation, called front-end estimation.
Estima tes Ar en ways a Good Idea Estimates Aren en’’ t Al Alw There are some situations where you definitely shouldn’t use an estimate. Example
1
Dr. Brown is giving medication to a patient. He needs to give 1.5 milligrams of the medicine for every pound that the patient weighs. Is it appropriate for Dr. Brown to estimate the amount of medicine for a particular patient? Solution
Dr. Brown needs to make precise measurements of the amount of medicine — it’s very important that he gives the correct dose.
Key words: • estimate • precise • front-end estimation
Guided Practice In Exercises 1–7, say whether each situation needs a precise figure, or if an estimate would be more suitable, and give a reason for your answer: 1. Mr. Bishop is deciding how much gas to put in his car at the start of a long journey. 2. Mrs. Suarez is figuring out how much wallpaper she needs for her bedroom. 3. Sasha’s pumpkin is weighed for the annual “heaviest pumpkin” competition. 4. Professor Elliott is finding the heights of a class of children for a scientific study. 5. Peter is calculating how many wins and losses his baseball team had this season. 6. Susie is deciding how many sandwiches to make for a party. 7. Ms. Ryan is figuring out the grade point averages of the students in her class.
38
Section 1.4 — Estimation
Front-End Estima tion Is Another Kind of R ounding Estimation Rounding Another type of estimate is front-end estimation. To do this, you add the first digit in each number, and then make a rough estimate of the value of the remaining digits. Example
2
Manuel is standing in line in the cafeteria. He wants to buy a sandwich, some soup, and a salad, but he only has a $10 bill. Check it out: Front-end estimation is most often used to figure out prices. It’s a type of estimation that people often do in their heads in stores and restaurants.
Use front-end estimation to estimate how much Manuel’s lunch will cost. Solution
MENU (All prices inc lude tax.) include
Sandwich Salad Soup French fries Soda Water
$4.49 $1.79 $3.69 $1.10 $1.29 $1.15
Manuel wants to estimate $4.49 + $3.69 + $1.79. Add the frontend digits:
Then estimate 4.49 the rest: 3.69 + 1.79 + 8 about
4.49 3.69 1.79 2.00
And add the two parts:
8.00 + 2.00 10.00
Front-end estimation gives an estimated cost of $10 for Manuel’s meal. He should calculate the exact total to see if he can buy the 3 items. $4.49 + $3.69 + $1.79 = $9.97, so he can buy all 3 items — but the estimate was too close to be sure.
Guided Practice In Exercises 8–12, use the menu shown in Example 2. Estimate each total using front-end estimation. 8. A sandwich, a bowl of soup, and a water. 9. A salad and two bags of French fries. 10. A bag of French fries, a soda, and two salads. 11. Two bowls of soup, two bags of French fries, and two sodas. 12. A salad, a bowl of soup, two sandwiches, three bags of French fries, and a water. 13. Which of the sums from Exercises 8–12 would you calculate exactly to see if you could afford them if you only had $10?
Section 1.4 — Estimation
39
Independent Practice In Exercises 1–6, say whether each situation needs a precise figure, or if an estimate would be more suitable, and give a reason for your answer: 1. Dr. Burke is taking the temperature of a patient with a fever. 2. Ms. Bennett is figuring out how long it will take her to grade the homework from her math class. 3. Clifton is weighing his suitcase before he goes on vacation to make sure he doesn’t have to pay for excess baggage. 4. The police tell news reporters the height of a suspect based on descriptions from witnesses. 5. A biologist is figuring out the size of the population of wild rabbits in California. 6. A butcher is figuring out how much to charge for 4 pounds of steak. 7. Max wants to buy 3 books, priced $7.95, $6.45, and $8.88. Estimate the total cost of the books using front-end estimation. 8. Julieta wants to buy a scarf for $9.29, a pair of gloves for $8.59, and 2 hats for $5.85 each. Use front-end estimation to estimate how much money she will need.
Now try these: Lesson 1.4.4 additional questions — p431
9. Mr. Benitez owns a pet store. He aims to make $1000 every 3 days. The receipts for the last 3 days came to $370, $365, and $397. Estimate the total receipts for these 3 days by front-end estimation. 10. Shantice is eating in a restaurant. Her appetizer costs $6.95, the main course is $15.20, and the dessert costs $9.65. Use front-end estimation to find the approximate cost of Shantice’s meal. To the right is a price list of items from a sporting goods store. Use front-end estimation to estimate the cost of the following combinations of items:
Baseball bat Baseball Basketball Football Football helmet Hockey stick
$46.50 $11.49 $32.69 $52.99 $89.99 $63.49
11. A football and a football helmet. 12. A hockey stick and a baseball bat. 13. A baseball, a basketball, and a football. 14. 2 baseball bats and 2 baseballs. 15. One of everything on the list.
Round Up Estimating can be a really useful method, but you shouldn’t use it all the time. You always need to decide whether estimating will give you an answer that’s useful for solving the problem — and sometimes you need to figure out the precise solution even after you’ve tried doing an estimation. 40
Section 1.4 — Estimation
Chapter 1 Investigation
Popula tions opulations You’ve been asked to write a report about the population of nine different counties in California. The table below contains some data you will find useful — it shows the population of the nine counties on a particular day in two different years. Part 1: You have been asked to include some specific information in your report:
County Alpine
• Lists of the counties in order of population: (i) in 2004 (ii) in 2005. • A list showing the counties’ population change between 2004 and 2005. Include comments on which counties’ populations increased and decreased most.
Population (2004) 1,274
Population (2005) 1,242
Number of births 11
Amador
37,552
38,221
271
Lassen
35,325
35,696
312
Modoc
9,828
9,813
92
Colusa
20,663
21,315
359
Plumas
21,378
21,557
181
Mono
13,529
13,512
179
Sierra
3,537
3,514
19
Trinity
13,826
14,025
117
• A bar graph showing the populations in each year. Before plotting the bar graph, each figure must be rounded to the nearest hundred. The vertical axis of your graph should be labeled from 0 to 40, and must show the populations in “numbers of thousands.” Part 2: If the number of births in each county remained the same for the next 10 years, and nothing else happened to cause a change in the population, what would the population of each county be in: (i) 2008 (ii) 2015? Do you think these predictions are realistic? Explain your answer. Extension • Order all nine counties in order of their numbers of births, from least to greatest. How does this list compare to the lists that you obtained in Part 1? Describe some similarities and differences. • A county's birthrate is the number of births per thousand people. To find it, divide the number of births by the population (in 2005), and then multiply the result by 1000. How does the list of birthrates compare to the list of numbers of births?
Open-ended Extension Write the report on the populations in the nine counties. You may need to do some further research. • You should include all the information you’ve found out so far. • Include any other information or graphs you think are useful. • You may also extend the report to include information about other counties.
Round Up The number skills you used in this Investigation are ones that you may use every day for the rest of your life. For example, when you add up prices in a store, these are exactly the skills you use. Cha pter 1 In vestig a tion — Populations Chapter Inv estiga
41
Chapter 2 Expressions and Equations Section 2.1
Exploration — Algebra Tile Expressions ..................... 43 Expressions ................................................................. 44
Section 2.2
Exploration — Equations with Algebra Tiles ............... 59 Equations .................................................................... 60
Section 2.3
Geometrical Expressions ............................................ 75
Section 2.4
Problem Solving .......................................................... 97
Chapter Investigation — Design a House ........................................ 108
42
Section 2.1 introduction — an exploration into:
Alg ebr a Tile Expr essions Alge bra Expressions You can represent values using blue and red tiles — blue tiles represent positive values, while red tiles represent negative values. But it’s not just numbers like 3, 4, or –5 that you can show. Tiles can also be used to show more complicated expressions involving variables — such as x + 5. In this Exploration, blue square tiles have a value of +1, and red square tiles have a value of –1. You can use these tiles to represent numbers, like 3, 4, or –5. Variables are things whose value you don’t know. They are usually given names like x or y. These funny-shaped tiles show the variables x and –x.
=1
= –1
=x
= –x
Example Show the expression x + 5 with algebra tiles. Solution
The blue shape represents the variable x, and the 5 blue tiles represent +5. Together the group of tiles shows the expression x + 5.
x
+
5
Example This also represents x + (–2). x – 2 and x + (–2) mean the same thing.
Show the expression x – 2 with algebra tiles. Solution
The blue shape represents the variable x, and the 2 red tiles represent –2. Together the group of tiles shows the expression x – 2.
x
–
2
Example Show the expression –3x with algebra tiles. Solution
The expression –3x means the same as “3 groups of –x,” or (–x) + (–x) + (–x). So the tiles on the right show the expression –3x.
–3x
Exercises 1. Use algebra tiles to model each of these expressions: a. x + 7 d. 4x
b. x – 4 e. –2x
c. –x + 2 f. 2x – 1
2. Write the expression represented by the diagram.
Round Up In math, expressions with variables are really useful because you can use them to represent just about anything. The next few Lessons show you how to write down many different amounts as mathematical expressions. Section 2.1 Explor a tion — Algebra Tile Expressions Explora
43
Lesson
Section 2.1
2.1.1
Varia bles ariab
California Standard:
One difficulty with math problems is that you can’t get started if you’re not given all the numbers for a problem. For instance, you can’t work out the area of a rectangle unless you’re given the width and height. That’s where variables come in.
Alg ebr a and Functions 1.2 Alge bra Write and evaluate an alg ebr aic e xpr ession ffor or a alge braic expr xpression gi ven situa tion, using giv situation, up to thr ee vvaria aria bles three ariab les..
What it means for you: You’ll learn a way of using letters or symbols in place of numbers that you don’t yet know.
Key words: • variable • unknown
A Varia ble Is an Unkno wn Amount ariab Unknown Sometimes you need to write out a math problem but you don’t know all the numbers. All you need to do is write a letter or symbol to represent each unknown number. The letters or symbols are called variables. You can use any letter or symbol as a variable — but the most common variable is x. Example
1
You are trying to write out a math problem that involves the length of a line. However, you don’t know the length of the line. What can you do? Solution
You can replace the number you don’t know with a variable. You can use whatever letter or symbol you like to represent the unknown length. For example, you might choose x. You can write the problem out with an x where the length of the line should be.
Example Check it out: Try to choose sensible letters for variables — letters that will make it easy to remember what they represent. For example, w is a good letter to use for the amount of money owned by Mr. Wilson. But r would be better if the person was Mrs. Richards. This is especially important if you need to use more than one variable — see Lesson 2.1.3.
44
Section 2.1 — Expressions
2
Mr. Harris has $7.50 more than Mr. Wilson. What extra information do you need to find out how much money Mr. Harris has? How could you represent this unknown variable? Solution
You need to know how much money Mr. Wilson has. In the problem, you’re not told how much Mr. Wilson has, so it’s an unknown amount. That means you need a variable to represent “the amount of money Mr. Wilson has.” Pick a variable that will be easy to remember, and then write a sentence like: Let w represent the amount of money Mr. Wilson has.
Guided Practice Identify an unknown in Exercises 1–4 that could then be used to find the required quantity. Choose a letter to represent this quantity. 1. Emilio and James started doing their homework at the same time. Emilio finished 10 minutes after James. You want to know how long Emilio spent on his homework. 2. Julianne finished a test at 2:45 p.m. You want to know the time when she started the test. 3. Mr. Mason’s truck weighs 4 times as much as his car. You want to know the weight of Mr. Mason’s truck. 4. Adam finished the race 2 seconds before Juan. You want to know how long it took Adam to finish the race.
Varia bles Wor k JJust ust Lik e Nor mal Number s ariab ork Like Normal Numbers You can add and subtract with variables just like you would with normal numbers. Example
3
4 is added to the variable, x. Show this on a number line. Check it out: This is exactly the same kind of picture that you saw in Section 1.1 — only with variables instead of numbers.
Solution
To add 4 to x, start at x and then move 4 to the right. It doesn’t matter that you don’t know exactly where x is on the number line. You just need to show what you would do if someone told you what x represents. Move 4 to the right
x
Example
x+4
4
10 is subtracted from the variable, y. Show this on a number line. Solution
To subtract 10 from y, start at y and then move 10 to the left. Move 10 to the left
y – 10
y
Section 2.1 — Expressions
45
Guided Practice Show the sums in Exercises 5–16 on number lines. 5. x + 2 6. x – 3 7. x – (–3) 8. y + (–5) 9. y – 9 10. y + 14 11. z + (–15) 12. z + 23 13. z – (–2) 14. a + 3 – 1 15. a + 6 + 5 16. a – 9 – (–7)
Don’t forget: Look at the diagrams in Section 1.1 if you’re not sure what these should look like.
You Can Multipl y and Di vide with Varia bles Too Multiply Divide ariab When you multiply or divide with variables, you could be multiplying or dividing by either positive or negative numbers. This might seem to make things complicated, but you just have to remember that the variable will still behave like any other number. Example Check it out: In this example, the variable y is a positive number. You need to remember that if the variable is negative, the arrows on the number lines would all point in the opposite direction. For example, if y is negative, 3 × y looks like this:
5
Use a number line to show the following operations with the positive variable y: y multiplied by 3 y multiplied by –3 y divided by 4 y divided by –4 Solution
3×y
3y
3 × y looks like this:
3×y 3y y
0
0
y ÷ (–4) looks like this:
y For 3 × y, you move a distance of y three times.
y ÷ (–4)
y
–3 × y looks like this: 0
–3 × y
–y
–3y
–3 × y is just like 3 × y, except the negative sign means the result is on the opposite side of 0.
y
0
Check it out: Notice how –3 × y is the same as 3 × (–y). And y ÷ (–4) is the same as –y ÷ 4.
y÷4
y ÷ 4 looks like this:
y
0
For y ÷ 4, you move a fourth of the distance to y.
y ÷ (–4) looks like this: –y
y ÷ (–4)
y
0
y ÷ (–4) is just like y ÷ 4, except the negative sign means the result is on the opposite side of 0.
46
Section 2.1 — Expressions
Guided Practice Don’t forget: Look at the diagrams in Example 5 if you’re not sure what these should look like.
Represent these calculations on number lines. Do each problem twice — once for a positive variable, and once for a negative. 17. q × 6 18. q × 5 19. q × (–2) 20. r ÷ 3 21. r ÷ (–3) 22. r ÷ 5 23. s × 2 24. s × (–5) 25. s ÷ 1
Independent Practice Identify an unknown in Exercises 1–9 that could then be used to find the required quantity. Choose a letter to represent this unknown. 1. Karen eats 5 more cookies than Lupe. You want to know how many cookies Karen has eaten. 2. Laura and Kenny set up lemonade stands. On the first day, Laura sells five more cups of lemonade than Kenny. You want to know how many cups of lemonade Laura sold. Now try these: Lesson 2.1.1 additional questions — p431
3. Deandre gets 36% more than Ana on a history test. You want to know Deandre’s score. 4. Madison measures her arm with a piece of string. It is exactly 5 times as long as the string. You want to know the length of Madison’s arm. 5. Gregory has 25 cakes. He shares them equally between his friends. You want to know how many cakes each friend gets. 6. Carolyn takes one-fourth as long as Nina to do her homework. You want to know how long Nina takes. 7. Tanya is selling some old toys that she no longer plays with and Imelda buys half of them. You want to know how many toys Imelda bought. 8. A rectangle has one side of length 5 inches. You want to know the area of the rectangle. 9. Tim has one-fourth of the amount of money that he had last week. You want to know how much money Tim has. Show the sums in Exercises 10–15 on a number line. 10. h + 7 11. h – 1 12. h – 2 13. m + (–5) 14. m – (–3) 15. m + 3 Draw the calculations in Exercises 16–21 on the number line. Show the cases both where the variable is positive, and where it is negative. 16. v × 5 17. v × 2 18. v × (–3) 19. w ÷ 9 20. w ÷ 3 21. w ÷ (–4)
Round Up At first it can look kind of confusing if you see a sum with letters in it — but just remember that the letters are just standing in for unknown numbers. Section 2.1 — Expressions
47
Lesson
2.1.2
Expr essions Expressions
California Standard:
Variables are all well and good, but they’re only useful when you use them to solve math problems. You can use variables and numbers to describe a problem in math terms — it’s called an expression.
Alg ebr a and Functions 1.2 Alge bra valua te an ev aluate Write and e alg ebr aic e xpr ession ffor or a alge braic expr xpression gi ven situa tion giv situation tion, using up to three variables.
What it means for you: You’ll practice how to make your own expressions, and how to work out the value of an expression once you’re told the values of the variables.
An Expr ession Is a Pr ob lem Written in Ma th Ter ms Expression Prob oblem Math erms Expressions are used to write word problems in math terms. Expressions are like instructions that tell you what you have to do to a number or variable.
Key words: • variable • expression • evaluate
Check it out: Mathematical expressions can be numerical or algebraic. Numerical expressions only contain numbers — for example, 2 + 4, 3 × 4 – 7, and so on. Algebraic expressions contain at least one variable — for example, x + 4, 3x – 7.
In Words
Expression
A number, x, is increased by 7
x+7
A number, y, is decreased by 7
y–7
A number, a, is multiplied by 7
a×7
A number, k, is divided by 7
k÷7
Sometimes you might have to describe a real-life situation using a mathematical expression. You need to imagine what would happen to a quantity, and write that down using variables, and +, –, ×, and ÷. Example
1
Peter has $165 in his savings account. Lottie has x more dollars than Peter. Write an expression for the amount of money that Lottie has. Solution
Lottie has more money than Peter. To find how much she has, you need to add $x to the amount Peter has. So the expression for the number of dollars Lottie has is 165 + x.
Guided Practice In Exercises 1–6, write an expression for each word problem. 1. A number, x, is decreased by 10. 2. A number, q, is increased by 27. 3. The variable w is multiplied by 20.
48
Section 2.1 — Expressions
Don’t forget: You have to calculate things in parentheses before any other parts of the expression.
4. A computer repairman charges $25 per hour to fix computers. How many dollars should he charge for h hours of work? 5. Aisha’s jump rope is 2 feet shorter than Lisa’s. If y represents the length of Lisa’s rope in feet, then write an expression to represent the length of Aisha’s rope (also in feet). 6. Describe the expression 100 – (4 × z) in your own words.
You Can Ev alua te if You Kno w the Value of the Varia ble Evalua aluate Know ariab If you’re given a value to use for a variable, you can work out the value of the expression. It’s called evaluating the expression. Example
2
Evaluate the expression 7 × h, if the value of h is 6. Solution
Start with 7 × h. But h = 6, so replace h with the number 6: 7 × 6 This means evaluating the expression gives 7 × 6 = 42 Example
3
The expression 120 – s is used to find out how many seats are available in a movie theater after s people are already seated. How many seats are available if 73 people are already seated? Solution
Start with 120 – s. You’re told that s = 73, so replace s with the number 73 This means that 120 – 73 = 47 seats are available.
Guided Practice Evaluate the expressions given in Exercises 7–12. 7. t + 28, given that t = 19 8. y – 7, given that y = 5 9. d × 9, given that d = 3 10. g ÷ 5, given that g = 25 11. x + (–15), given that x = 13 12. c – (–3), given that c = 10 13. The expression 15 + t gives Earl’s height in centimeters, where t is his friend Tony’s height in cm. If Tony is 135 cm tall, how tall is Earl? 14. The amount of money made by a hot dog stand is $2.5h, where h is the number of hot dogs sold. How much money does the stand make if it sells 271 hot dogs in one day? Section 2.1 — Expressions
49
Ther e Ar e Thr ee w ays of Writing Multiplica tion here Are hree wa Multiplication Check it out: Watch out — the • can be a bit confusing because it looks like a decimal point. The • sign just means “multiplied by.”
In an expression, there are three ways of writing a multiplication. The first is to use the × sign.
4×p
Sometimes a raised period, •, is used instead of ×.
4•p
If you’re just multiplying a variable by a number, then you can leave out the symbol completely.
4p
They all mean the same thing
Guided Practice Rewrite Exercises 15–18 using • as a multiplication symbol. 15. 5 × i 16. 12 × t 17. 8r 18. –9k Rewrite Exercises 19–22 using × as a multiplication symbol. 19. 11•d 20. 4•q 21. 4q 22. 53z Rewrite Exercises 23–26 without a multiplication symbol. 23. 7•w 24. 13 × t 25. –3 × y 26. 4•c
Independent Practice In Exercises 1–6, evaluate the expression 19 – y for each value of y. 1. y = 11 2. y = –11 3. y = 26 4. y = 19 5. y = 0 6. y = –8 7. A variable, d, is multiplied by 15. Write this as an expression. Evaluate the expression for d = 4. 8. One palm tree is sixteen feet taller than another. Write an expression for the height of the shorter palm tree. 9. Jessica shares q potato chips between 12 friends. Write an expression for the number of potato chips received by each friend. Evaluate the expression if q = 96.
Now try these: Lesson 2.1.2 additional questions — p432
Exercises 10 and 11 are about Manuel, the manager at a restaurant. 10. Manuel collected t dollars in tips and wants to share it equally among his n employees. Write an expression that can be used to calculate how much each employee will get in tips. You can assume that Manuel does not get any tips. 11. If there are 5 employees and Manuel collected $90 in tips, use your expression from Exercise 10 to calculate how much each employee will get. Rewrite Exercises 12–14 to show all three ways of writing multiplication. 12. 3 × w 13. 7•t 14. –9r
Round Up To evaluate an expression, you just need to substitute numbers in place of variables. They take some getting used to, but expressions are a really good way to write down and then solve math problems. 50
Section 2.1 — Expressions
Lesson
2.1.3
Multi-V aria ble Expr essions Multi-Varia ariab Expressions
California Standard:
Some expressions contain more than one variable — you’ve met some of those before but there are plenty more in this Lesson. You’ll also get some more practice substituting values for variables.
Alg ebr a and Functions 1.2 Alge bra valua te an ev aluate Write and e alg ebr aic e xpr ession ffor or a alge braic expr xpression gi ven situa tion, using up to giv situation, thr ee vvaria aria bles three ariab les..
What it means for you: You’ll see expressions containing two or three variables.
Some Expr essions Use Two Varia bles Expressions ariab If there are two numbers you don’t know, that’s not a problem. You can use two different variables, one for each unknown number.
Key words: • • • •
variable expression evaluate term
Remember these key words: sum — this means you have to add two or more numbers dif ence — this means you difffer erence have to subtract one number from another pr oduct — this means you product have to multiply two or more numbers quotient / rra a tio — both these words mean you have to divide one number by another
If part of an expression is written in parentheses, you should always work out that part first.
The sum of a and b
a+b
The product of v and w
v × w, v•w, or vw
p is subtracted from 9q
9q – p
1
DeAndre, Kia, and Anthony each thought of a number. DeAndre’s number was twice as big as the sum of Kia and Anthony’s numbers. Write an expression for DeAndre’s number. Solution
Represent Kia’s number by k, and Anthony’s number by a. Then DeAndre’s number is 2 × (k + a). Evaluating expressions with two or more variables is the same as for one variable. Just substitute the numbers you’re given and find the result. Example
Don’t forget:
Expression
You can use expressions with two (or more) variables to represent situations with more than one unknown quantity. Example
Check it out:
In Words
2
Evaluate the expression x – y for x = 20 and y = 8. Solution
Replace x with 20 and y with 8. That means that x – y becomes 20 – 8, so the answer is 12. Example
3
Evaluate the expression (15 – f ) × (7 + z) for f = 2 and z = 3. Solution
Replace f with 2 and z with 3. The expression becomes (15 – 2) × (7 + 3) = 13 × 10 = 130.
Section 2.1 — Expressions
51
Guided Practice In Exercises 1–4, evaluate the expressions for the values given. 1. e + f when e = 5 and f = 7 2. q ÷ s when q = 18 and s = 3 3. y – (5t) when y = 10 and t = 1 4. (k × 9) + z when k = 11 and z = 6 Write an expression for each of Exercises 5–7. 5. w is divided by v. 6. r is multiplied by t. 7. k is multiplied by 3, t is divided by 4, then the results are added together.
Some Expr essions Use Thr ee Varia bles Expressions hree ariab If a calculation contains three unknowns then you’ll need to include three variables in your expression. Example
4
Wesley has w cookies and Maribel has m cookies. They decide to eat c cookies between them. Write an expression for the total number of cookies left. Solution
The number of cookies Wesley and Maribel have before they eat any is w+m So once they have taken away c by eating them, there will be w+m–c
Guided Practice In Exercises 8–12, evaluate the expression z + y – x for the values given. 8. x = 5, y = 5, z = 2 9. x = 1, y = 1, z = 1 10. x = 2, y = 3, z = –2 11. x = 9, y = –3, z = 1 12. x = 2, y = 0.5, z = 3
Check it out: In an expression without parentheses such as ab – c, always carry out the multiplication before the addition. So here, multiply a by b, then subtract c from the result.
52
Section 2.1 — Expressions
13. An elevator leaves the first floor and goes up x floors. The elevator then goes down y floors, then goes up another z floors. Write an expression for the floor that the elevator is on now. 14. The amount that Mr. Jackson is paid each week is calculated by first multiplying his hours worked: 24 hourly wage, a, by the number of hours he worked that week, b. Then the amount, c, that hourly wage: he owes in taxes is subtracted from this amount. $9.10 This is represented by the expression ab – c. taxes: Use the chart on the right to calculate the $39.80 amount of Mr. Jackson’s paycheck this week.
Expr essions Ar e Made Up of Ter ms Expressions Are erms Check it out: In an expression, any quantity that isn’t a variable is called a constant.
Parts of an expression that are separated only by + or – are called terms. Each “+” or “–” belongs to the term that follows. Example
5
What are the terms in the expression 3 × l – 2 × m + n – 7? Solution
Check it out: “+ n” is the same as “n.”
Terms are separated by + or – signs, so the terms are: (i) 3 × l, (ii) –2 × m, (iii) n, and (iv) –7.
When you evaluate multi-variable expressions with more than one term, it makes sense to evaluate each term individually first. Check it out: Evaluating terms individually (and writing down the results) also makes it easier to find any mistakes you make along the way.
Example
6
Evaluate 5a + 3b + 2c, when a = 2, b = 3, c = 5 Solution
First, you need to put the values of a, b, and c into the expression: (5 × 2) + (3 × 3) + (2 × 5) You can now evaluate each term: 10 + 9 + 10 Finally, you can evaluate the simplified sum: 10 + 9 + 10 = 29
Guided Practice Identify the terms in Exercises 15–20. 15. v – 16 16. d + 22 18. x – (–2) + y 19. m – 14 + 2n
17. f + (–g) 20. 13b + 5c – d
In Exercises 21–26, evaluate the given expressions by first evaluating the terms. 21. r + 4s + 3t, when r = 1, s = –4, t = 7 22. 3r – 3s + 5t, when r = 5, s = 8, t = 2 23. 5r + 10s – 2t, when r = 6, s = –3, t = 15 24. r + s + st, when r = 2, s = 9, t = 5 25. 6 + rs + 7t, when r = 10, s = 11, t = 0 26. 200 – 4rs – 3st – 2rt – rst, when r = 5, s = 1, t = 7
Section 2.1 — Expressions
53
Independent Practice Evaluate the expression xy using the values of x and y given in Exercises 1–6. 1. x = 0.5 and y = 20 2. x = –6 and y = –3 3. x = 0 and y = 5 4. x = –1 and y = 1 5. x = –10 and y = 10 6. x = 0.75 and y = 4 Now try these: Lesson 2.1.3 additional questions — p432
Exercises 7 and 8 are both about the same Math Club. 7. The Math Club had $160 in its treasury. The club raised another j dollars from selling hot dogs after school and k dollars from selling snacks during school. Write an expression that represents how much money the Math Club has now. 8. Use your expression from Exercise 7 to work out how much money the Math Club have if they raised $25 selling hot dogs and $45 selling snacks. Exercises 9 and 10 are both about Mrs. Roca. 9. Mrs. Roca took $50 with her to the movies. She spent x dollars on tickets and y dollars on food. Write an expression that represents how much money Mrs. Roca had after the movie. 10. She spent $18 on tickets and $10 on food. Use your expression from Exercise 9 to calculate how much money she had left over after the movie. Write an expression for each of Exercises 11–15. 11. d is divided by g. 12. The sum of r, s, and t. 13. k is subtracted from the sum of w and p. 14. The product of x, y, and z. 15. 2a is added to the product of b and c.
Don’t forget: The perimeter is the distance all the way around a shape.
Exercises 16–17 use the diagram to the right, showing the field that will be used to keep a horse. 16. Write an expression for the perimeter of the p field. 17. Use your expression to calculate the 30 ft. q perimeter of the field when p = 42 feet, r q = 38 feet, and r = 32 feet. 18. Jeff is moving his desk, TV, and couch to his new apartment. His desk weighs x pounds, his TV weighs y pounds, and his couch weighs z pounds. He loads all three items into the back of his truck. His truck can carry up to 1000 pounds. Write an expression that represents how much more weight the truck can carry after the three items are loaded in the truck.
Round Up No need to panic — when you’re dealing with expressions that contain more than one variable, the rules are just the same as for single-variable expressions. 54
Section 2.1 — Expressions
Lesson
2.1.4
Or der of Oper a tions Order Opera
California Standards:
Sometimes you’ll meet expressions containing different combinations of operations (such as ×, ÷, +, and –). When you do, you need to be sure to do all of them in the right order.
Alg ebr a and Functions 1.3 Alge bra der of pply alge braic order A ppl y alg ebr aic or oper a tions and the opera commutative, associative, and distributive properties to evalua te e xpr essions; and aluate expr xpressions; justify eac h ste p in the each step pr ocess process ocess.. Alg ebr a and Functions 1.4 Alge bra Solv e pr ob lems man uall y Solve prob oblems manuall ually by using the cor der corrrect or order of oper a tions or by using a opera scientific calculator.
What it means for you: You’ll see that it matters in which order you evaluate an expression, and you’ll learn the correct order.
The Or der In Whic h You Ev alua te Mak es a Dif ence Order hich Evalua aluate Makes Difffer erence The expression 2 + 3 × 7 looks like it could give two different answers, depending on how you work it out. For example, working out “2 + 3” first gives: 2 + 3 × 7 = 5 × 7 = 35, while working out “3 × 7” first gives: 2 + 3 × 7 = 2 + 21 = 23. To avoid this situation, the following rule is used for all math expressions: Always do multiplication before addition, unless parentheses tell you to do otherwise. Example
1
Evaluate 2 + 3 × 7, and (2 + 3) × 7. Solution
Key words: • • • • • •
parentheses exponent multiply and divide add and subtract expression evaluate
2 + 3 × 7 = 2 + 21 = 23
Wor k out the m ultiplica tion fir st ork multiplica ultiplication first
(2 + 3) × 7 = 5 × 7 = 35
The par entheses tell yyou ou to do the ad dition fir st parentheses addition first
In fact, you do all multiplications and divisions before additions and subtractions (unless parentheses tell you otherwise). Example
2
Evaluate: (i) 32 – 3 × 8,
(ii) 56 – 8 ÷ 2.
Solution
(i) 32 – 3 × 8 = 32 – 24 =8
Wor k out the m ultiplica tion fir st ork multiplica ultiplication first
(ii) 56 – 8 ÷ 2 = 56 – 4 = 52
Wor k out the di vision fir st ork division first
You do multiplications and divisions working from left to right. You also do additions and subtractions from left to right — after you’ve finished all the multiplications and divisions. Example
3
Evaluate: 45 + 34 ÷ 17 – 2 × 3. Solution
45 + 34 ÷ 17 – 2 × 3 = 45 + 2 – 2 × 3 = 45 + 2 – 6 = 47 – 6 = 41
× and ÷ fir st, fr om left to right first, from Then + and –, fr om left to right from Section 2.1 — Expressions
55
Guided Practice Evaluate the expressions shown in Exercises 1–8. 1. 4 + 2 × 5 2. 3 + 7 × 4 3. 12 ÷ 2 + 4 4. 8 × 7 + 1 5. 11 – 1 × 4 6. 88 + 20 ÷ 2 7. (88 – 3) ÷ 5 8. 8 + 9 ÷ 3 × 5 – 3 Don’t forget: Here, exponents means expressions like 45, y2, x3, and so on — they show repeated multiplication. The exponent is actually the little number written above and to the right of a larger number (or variable) called the base. The exponent shows the number of bases you need to multiply together. So y3 = y × y × y.
Remember... multiplications and divisions before additions and subtractions — unless parentheses tell you otherwise. More generally, the order you should perform operations is as follows:
Check it out: An easier way to remember the order of operations is to remember the word PEMD AS PEMDAS AS, which stands for: 1. P arentheses 2. E xponents Division 3. M ultiplication / D 4. A ddition / S ubtraction
Order of Operations: ()[]
1. Evaluate anything inside parentheses (or brackets)
y2
2. Evaluate exponents
×÷
3. Multiply and divide from left to right
+–
4. Add and subtract from left to right
Example
4
Evaluate: 32 × 5 – (20 – 2) ÷ 32. Solution
32 × 5 – (20 – 2) ÷ 32 = 32 × 5 – 18 ÷ 32 = 9 × 5 – 18 ÷ 9 = 45 – 18 ÷ 9 = 45 – 2 = 43
Example
Par entheses fir st arentheses first Then e xponents exponents × and ÷, from left to right + and –, from left to right
5
Alan and Jessica are each trying to calculate the expression 24 ÷ 3 – 22. Their work is shown below. Who has the correct answer? Alan: 24 ÷ 3 – 22 = 8 – 22 = 62 = 36
Jessica: 24 ÷ 3 – 22 = 24 ÷ 3 – 4 =8–4 =4
Solution
Alan performed the division, then the subtraction, then evaluated the exponent. That is incorrect — he should have found the exponent first. Jessica evaluated the exponent first, then performed the division, then the subtraction. This is the right order, so Jessica has the right answer. 56
Section 2.1 — Expressions
Guided Practice Check it out: Use the same order of operations inside parentheses as well. For example: 10 – (42 + 2 × 3) You need to evaluate the parentheses first, and you follow the normal PEMDAS rules to do that. So evaluate 42 + 2 × 3 in this order: exponent, multiplication, addition. Therefore 10 – (42 + 2 × 3) 16 + 2 × 3) = 10 – (16 = 10 – (16 + 6 ) = 10 – 22 = –12
Evaluate the expressions shown in Exercises 9–18. 9. 8 + (10 – 2) ÷ 4 10. 4 – 2 × 16 11. 5 × 7 + 7 × 5 12. 7 × 3 – 8 ÷ 2 + 6 14. 23 – 9 ÷ 3 13. 32 + 4 × 9 16. 3 × (42 – 5) + 11 15. (4 + 1)2 – 12 ÷ 4 – 1 17. (62 – 32) ÷ (6 – 3)2 18. (15 ÷ 3 – 2) × (33 – 5 × 4)
Or der R ules Also A ppl y to Expr essions with Varia bles Order Rules pply Expressions ariab The order rules apply to all types of expressions, including those with variables. Example
6
Evaluate x + yz when x = –3, y = 9, and z = 10. Solution
Substitute in your values for x, y, and z
–3 + 9 × 10
There are no parentheses or exponents
–3 + 9 × 10
Carry out the multiplication
–3 + 90
Carry out the addition, giving the answer
87
Example
7
Evaluate 19 – f 2 × (w + q) when f = 3, w = 5, and q = 2. Solution
Substitute in your values for f, w, and q
19 – 32 × (5 + 2)
Evaluate the parentheses
19 – 32 × 7
Evaluate the exponents
19 – 9 × 7
Carry out the multiplication
19 – 63
Carry out the subtraction, giving the answer
–44
Guided Practice Evaluate the expressions shown in Exercises 19–21, given that f = 2, j = 3, and g = 19. 19. g – j × 8 20. (f + g) × j + 4 21. (j 2 – f 2) ÷ 5 Evaluate the expressions shown in Exercises 22–24, given that s = 10, k = 3, and q = 1. 22. s × k + qs + 23 23. (k 2 – 1) ÷ 4q 24. q – sk 3 Section 2.1 — Expressions
57
Independent Practice Evaluate Exercises 1–4, given that w = 2, b = 8, c = –0.5. 1. b × (w – c) 2. w × b + w × c 2 3. (6 – w) 4. (b + w + 2 × c) ÷ (b – 2 × c) 5. Felipe and Sylvia are trying to evaluate the expression 2 × 18 – 6.
Don’t forget: If there are no parentheses, evaluate multiplications and divisions from left to right.
Felipe: 2 × 18 – 6 Sylvia: 2 × 18 – 6 = 36 – 6 = 2 × 12 = 30 = 24 Who is correct? What has the other person done wrong? Which of the expressions in Exercises 6–11 are true, and which are false? For those that are false, explain what is wrong. 6. (2 + 3) × 6 = 2 + 3 × 6 7. (7 + 5) – 3 = 7 + 5 – 3 8. (100 ÷ 2) ÷ 25 = 100 ÷ 2 ÷ 25 9. 10 ÷ 2 × 5 = 10 ÷ (2 × 5) 10. 3 + (6 × 9) = 3 + 6 × 9 11. 6 × 52 = (6 × 5)2 12. Carly wants to exercise for a total of 50 hours this month. She exercised 1.5 hours for each of the first eight days. Then, she exercised 2 hours for each of the next four days. Write and evaluate an expression for the remaining number of hours she has to exercise.
Now try these: Lesson 2.1.4 additional questions — p432
13. Mr. Chang earns $12 per hour baking bread at the bakery. He worked 7 hours on Thursday and c hours on Friday. Write an expression of the form a × (b + c) that can be used to calculate how much Mr. Chang earned on Thursday and Friday. 14. Given that Mr. Chang worked for 8.5 hours on Friday, find the value of the expression you wrote for Exercise 13. Which of the expressions in Exercises 15–19 are true, and which are false? For those that are false, explain what is wrong. 15. h – r + q = h – (r + q) 16. a × (5 + g) = a × 5 + g 17. 16 × f ÷ 3 + 2 = 16 × f ÷ (3 + 2) 18. 5 + j – w = (5 + j) – w 19. 18y – 2j = (18 – 2) × (y – j)
Round Up Hopefully you can now see how important the order of operations is when evaluating expressions. You’ll need to be able to get the order right from now on — so make sure you memorize it. 58
Section 2.1 — Expressions
Section 2.2 introduction — an exploration into:
Equa tions with Alg ebr a Tiles Equations Alge bra You can use red and blue tiles to represent math equations. Equations show that two things are “balanced,” and you’re not allowed to do anything that might make them unbalanced. This means you can add or remove tiles, as long as you do the same to both sides. You can use tiles to represent equations. These tiles show that x + 2 = 5.
=
When you have an equation, you can add or remove tiles — as long as you add or remove exactly the same tiles on both sides. Remove 2 squares from If you can get x by itself, you can find its value. Here, x = 3.
=
both sides sides.
Example Show the equation x + 3 = 5 with algebra tiles, and then find the value of x. Solution
The equation x + 3 = 5 looks like the picture on the right.
=
Remove 3 squares from both sides to get x by itself. This tells you that x = 2. Remove 3 tiles from both sides sides. For some equations, you need to use zero pairs before you can get x by itself. Example Show x – 4 = 7 with tiles, and find the value of x. =
Solution
Add 4 to both sides and remove zero pairs.
The equation x – 4 = 7 looks like this.
Add 4 tiles to both sides, and then remove zero pairs. This leaves x on one side of the equation, and 11 blue tiles on the other side. So x = 11. Example Show 3x = –15 with tiles, and find the value of x. Solution
=
3x = –15 looks like the picture on the right. Put the x’s in a line, and then put the same number of tiles next to every x. For every x, there are 5 red tiles. This tells you that x = –5.
Exercises 1. Use algebra tiles to model each of these equations, and then find the value of x. a. x + 6 = 9
b. x – 3 = 8
c. x + 2 = –5
d. 2x = 10
2. Write the equation that is represented by these tiles. Then solve the equation to find the value of x.
e. 3x = –12 =
Round Up Keeping equations balanced by doing the same to both sides is one of the most important things you’ll ever learn in math. There will be a lot more about this in the next Section. Section 2.2 Explor a tion — Equations with Algebra Tiles Explora
59
Lesson
2.2.1
Equations
California Standard:
You’ve done plenty of work with expressions — now it’s time to tackle equations. They’re both quite similar — both involve variables, numbers, and combinations of the usual +, –, ×, and ÷ operations.
Algebra and Functions 1.1 Write and solve one-step linear equations in one variable.
What it means for you: You’ll learn about equations and how they’re related to expressions.
Key words: • equation • guess and check • solve
An Equation Tells You That Two Things are Equal An equation tells you that something is equal to something else. These are all examples of equations: 14 + 6 = 20 9–3=4+2 4y + x = x – 7 Writing an equation is similar to writing an expression. It may involve a real-life problem, and you may have to use variables to represent unknown values. Example
Check it out: An equation involving variables can be true for all values of the variable — for example, y + y = 2y (this kind of equation is usually called an identity). Or it can be true for only particular values of the variable — for example, 2y + 3 = 11, which is true only if y = 4. Finding the values that make an equation true is called solving the equation, and there’s more about this later. (The equations in Examples 1 and 2 are only true for particular values.)
1
Richard spent $3.50 on fruit and $7.26 on vegetables. He then spent $x on meat, making a total of exactly $20. Write an equation to represent this. Solution
Richard spent 3.50 + 7.26 + x dollars, which is equal to $20. Writing this as an equation, 3.50 + 7.26 + x = 20. This can be simplified to 10.76 + x = 20. Example
2
Edward and Bianca each wrote a story in English class. Bianca’s story has 31 words more than Edward’s. Write an equation to represent this. Solution
Create variables — call the number of words in Bianca’s story b and the number of words in Edward’s story e. Although both b and e are unknown amounts, the question says that b is 31 greater than e. So, a possible equation is e + 31 = b. You could also have written b – 31 = e. Both equations show that b is 31 greater than e.
Guided Practice Write equations to represent the situations described in Exercises 1–6. 1. Jose scored 10 percent more than William on a class test. 2. Theresa owns two pairs of shoes more than Kimberly. 60
Section 2.2 — Equations
3. Richard and Tim each bake cakes. Richard notices that Tim’s cake is three times as heavy as his. 4. Isaac goes to the county fair. Because he has been saving up, he has seven times as much money as his sister to spend on rides. 5. Laura’s dad runs a newsstand that sells bottles of water. Laura knows that five small bottles of water contain the same amount as two large bottles. 6. Monica has a candy bar that is a third the size of Julio’s.
Equations Can Be Used to Find the Value of Variables Equations like y + 6 = 16 tell you enough information to find out y. One way to find y is to guess at it again and again until you find the right answer. Sometimes this is called the “guess and check” method. Check it out: Using an equation to find the value of an unknown variable is sometimes called solving the equation.
Example
3
Find the value of y if y + 6 = 16. Solution
First you need to guess a value for y. Then you can use it in the equation and check if it makes the equation true. Try y = 5:
5 + 6 = 16 is not true. In fact, 5 + 6 = 11, which is too small.
Try y = 12: 12 + 6 = 16 is also not true. In fact, 12 + 6 = 18, which is too big. Try y = 10: 10 + 6 = 16, which is true. So y does equal 10. This means the solution to the equation is y = 10. Check it out: As shown in Example 4, one way of doing “guess and check” is thinking of the equation as a question. Once you can answer “yes” to the question “7h = 56?”, you’ve got the right value of h.
Example
4
Julia uses the expression 7h to find out how many dollars she earned doing yard work. h represents the number of hours she worked. If Julia earned $56, then how many hours did she work? Solution
An equation for the situation is 7h = 56. Try guessing different values of h, then see if they’re right. h
7h
7h = 56?
7
49
No
10
70
No
8
56
Yes
If Julia worked for 7 hours, she would have earned only $49, which is too little. If Julia worked for 10 hours, she would have earned $70, which is too much.
Julia must have worked for 8 hours — because when h = 8, the equation 7h = 56 is true. Section 2.2 — Equations
61
Guided Practice Use “guess and check” to solve the equations in Exercises 7–10. 7. x + 14 = 30 8. 2z + 16 = 38 9. 9a – 12 = 69 10. 3z ÷ 7 = 6 11. Luis uses the expression $3 × m to work out how much money he spends calling his relatives in Europe for m minutes. His last phone call cost $60. Write an equation to represent this situation and then find out how long he spoke for. 12. Pedro works out how much he must pay toward a food bill by using the formula d – $3.24, where d is the cost of the entire bill. He ends up paying $2.97. Write an equation for this situation and then find out how much the food bill must have been. 13. What value of y satisfies the equation 111 = 19y + 16?
Independent Practice Now try these: Lesson 2.2.1 additional questions — p433
Use “guess and check” to solve the equations in Exercises 1–4. 1. a + 20 = 57 2. 3b + 9 = 15 3. 7c – 22 = 20 4. 12d ÷ 8 = 6 5. Brandon has $t in his savings jar. He puts in an extra $12. If the total amount in the jar is n, write an equation linking n and t. 6. Solve the equation you wrote for Exercise 5 when n = 48. 7. Cody is playing an old video game. In it you get 400 points every time you collect a ring and 800 points every time you collect an egg. Write an equation about p, the total number of points you get for collecting r rings and e eggs. 8. Guadalupe has started going to dance class. Each class costs $3 and she spent $23.50 on dance shoes. Including the shoes, she has spent $71.50 in total. Write an equation for this and solve it to find out how many lessons Guadalupe has been to. 9. It takes 2 eggs to make 20 cookies, and 4 eggs to make 40 cookies. Which equation best describes the number of eggs, x, that it takes to make y cookies? y = 10x x = 10y y + x = 10 y – x = 10
Round Up Equations look like expressions, but they contain a “=” — which tells you that something is equal to something else. Don’t worry if you’re not 100% comfortable with equations yet — all of the Lessons in this Section are about equations, so you’ll have many chances to practice.
62
Section 2.2 — Equations
Lesson
2.2.2
Manipulating Equations
California Standard:
Now that you’ve met equations, it’s time to look at the things you can do to change an equation without making it untrue.
Algebra and Functions 1.1 Write and solve one-step linear equations in one variable.
What it means for you: You’ll learn about how to manipulate linear equations, which you’ll use in Lessons 2.2.3 and 2.2.4 when you’re solving them.
Key words: • equation • manipulate
You Have to Keep Equations Balanced An equation is like a scale. The bit before the equals sign has the same value as the bit after the equals sign, so the scale is balanced.
5+4
When manipulating equations, you have to keep the scale balanced. You can’t take 4 from one side and not from the other because then the two sides aren’t equal.
5
The only way to keep the scale balanced is to always do the same thing to both sides.
=
3+6
3+6
Here’s another way to show the same thing:
5+4
=
3+6
= If you take any 3 boxes from the left-hand side, you also need to take 3 boxes from the right-hand side to keep the equation balanced.
There are a total of 9 boxes on each side of the equals sign — so the equation is balanced.
= Both sides still have the same total number of colored boxes.
You still have to keep both sides of an equation balanced when there are variables involved. Example
1
The variables x and y are linked by the equation x + y = 10. Show how to keep the equation balanced if: (i) 3 is added to the left-hand side, (ii) y is subtracted from the right-hand side. Solution
(i) If 3 is added to the left-hand side, then you also have to add 3 to the right-hand side. This gives: x + y + 3 = 10 + 3, which can be simplified to x + y + 3 = 13. Check it out: Notice how x + y – y can be simplified to just x. The “+ y – y” part cancels out.
(ii) If y is subtracted from the right-hand side, then you also have to subtract y from the left-hand side. This gives: x + y – y = 10 – y, which can be simplified to x = 10 – y.
Section 2.2 — Equations
63
Example Check it out: Notice how the “+ 7 – 7” part cancels out.
2
Subtract 7 from both sides of the equation x + 7 = 23. What does the result tell you about the value of x? Solution
Check it out: Notice how you’ve found the value of x by subtracting something from both sides of the equation.
So So
x + 7 = 23 x + 7 – 7 = 23 – 7 x = 16.
Example
3
Felipe has the equation j + 26 = 32. He subtracts 26 from the left-hand side to get j = 32. Is that the right value for j? Solution
Felipe has changed one side of the equation without changing the other in the same way. It is no longer balanced. This value for j won’t make the equation true. He should have done the same to both sides. j + 26 = 32 j + 26 – 26 = 32 – 26 Subtract 26 from both sides j=6 Then simplify to get j on its own
Guided Practice Fill in the missing parts of Exercises 1–6. w = 23 y=3 1. 2. w×6=? y+5=? 4.
b=7 b+9 = ?
5.
x = 15 x ÷5 = ?
3.
a = 27 a÷9=?
6.
h = 12 h× 4 = ?
Equations can involve more than one variable. As always, keep both sides of the equation balanced, by doing the same to both sides of the equation. Example
4
Adriana buys 3 wrenches and 2 screwdrivers. Each wrench costs $w and each screwdriver $s. She spends $15.66, so she writes 3w + 2s = 15.66. Check it out: Notice that $46.98 is the cost of 3 lots of “3 wrenches and 2 screwdrivers” — which is the same as 9 wrenches and 6 screwdrivers.
64
Section 2.2 — Equations
She decides to work out the cost of three times as many wrenches and screwdrivers by multiplying the left-hand side of the equation by 3. What must she do to work out the right-hand side? Solution
She needs to multiply the right-hand side by 3. 3 × (3w + 2s) = 3 × 15.66, or 3 × (3w + 2s) = 46.98
Check it out: When you’re manipulating equations, write the different stages down the page, lining them up by their = signs. That makes it easier to see what operations have been carried out on each side.
Guided Practice Fill in the missing parts of Exercises 7–12. 7.
u=f+2 u–5=?
10.
6=q 6 −19 = ?
8.
9. r + j = 9 (r + j )×15 = ?
j = z + 26 j – 26 = ?
11. f = b fz = ?
12. ? = m + 3 2=m
Explain what step of the work is incorrect in Exercises 13–18. 13. f = z + 19 f=z
14.
t = 3k t–3=k
15. 6 + g = 7 5+g=8
16. p = 7 2p = 27
17.
h = 11 + j h – 11 = j – 11
18. 15q = 9 q = 9 × 15
Independent Practice Now try these: Lesson 2.2.2 additional questions — p433
1. Joshua and Ashley both try manipulating the same equation. Their work is shown below. Have either of them made a mistake? If so, what have they done wrong?
Joshua
Ashley
9w = 27 9w + 16 = 27 ÷ 16
9w = 27 9w + 16 = 43
2. Vanessa notices that the cost of a big bag of oranges, b, is exactly $0.45 less than the cost of a sandwich, s. Write an equation involving subtraction to represent this. 3. Add 0.45 to both sides of your equation from Exercise 2. 4. Using your answer for Exercise 3, if a big bag of oranges costs $2.25, then what does a sandwich cost? Fill in the missing parts of Exercises 5–10. 5.
w + p = 22 w+p+9=?
6. 16 – h = 5 ?=5+h
7.
8.
h÷3=4 3×h÷3=?
9. w + j = 4 ?=4+w+j
10. m × 2 = 4 m=?
9h = 9 9h ÷ 9 = ?
Round Up The concepts here are really important — in the next Lesson, you’ll manipulate equations to solve them. Remember — you must do exactly the same thing to both sides of the equation. Always. Section 2.2 — Equations
65
Lesson
2.2.3
Solving + and – Equations
California Standard:
As you might have guessed, all that manipulating equations was leading up to something, and this is it. This Lesson is about a fast way of solving equations by manipulating sides.
Algebra and Functions 1.1 Write and solve one-step linear equations in one variable.
What it means for you: You’ll manipulate + and – equations to solve them without using “guess and check.”
Key words: • equation • solve
Manipulating Is Faster Than Guess and Check You can use an equation to find out the value of a variable. You’ve done this already using “guess and check,” but a quicker way is to manipulate the equation. An equation like y + 9 = 16 is balanced just like one with only numbers. To find the value of y, you need to get the variable alone on one side of the equals sign. If the variable has something added to it, use subtraction to get it on its own. In y + 9 = 16, subtract 9 from both sides to get y on its own.
y+9
=
16
y
=
16 – 9
You can do exactly the same thing without drawing the scales. y + 9 = 16 y + 9 – 9 = 16 – 9 y=7
The “+ 9” and “– 9” cancel each other out.
You can check that y = 7 is the correct solution by substituting it back into the original equation: 7 + 9 = 16 — this is true, so y = 7 is correct. Example
1
Alicia and Paula are each solving the equation x + 19 = 37. Their work is shown below. Who solved the problem correctly? Alicia
Check it out: The “+ 19 – 19” is crossed out because the values cancel. It’s okay to cross things out — but do it lightly, so that all your work can still be seen.
x + 19 = 37 x + 19 – 19 = 37 x = 37
Paula
x + 19 = 37 x + 19 – 19 = 37 – 19 x = 18
Solution
Paula is correct. She has remembered to subtract 19 from both sides of the equation. Alicia’s solution doesn’t make sense — she’s subtracted 19 from one side of the equation, but not the other.
66
Section 2.2 — Equations
Guided Practice
Check it out: Don’t let decimals or negative numbers put you off. Just do exactly the same as you have been doing and you’ll still get the right answer.
Solve the equations given in Exercises 1–14. 1. x + 11 = 17 2. x + 10 = 20 3. k + 75 = 200 4. 14 + x = 14 5. y + 191 = 87 6. w + 8 = –1 7. 3.8 + y = 19.1 8. 33.4 + z = 47.2 9. 1.3 = y + 0.5 10. q + 18 = 20.2 11. g + (–8) = 13 12. –9 = f + 9 13. v + 23 = –10 14. i + (–9) = 10 15. Yesenia is given two new video games for her birthday. She now has 18. Write an equation for this, using v for the number of video games she had before her birthday. Solve it to find v. 16. Alejandro puts $14.50 into his bank account. He now has a total of $34.15. Write an equation for this, using m for the amount he had in the bank before the deposit. Then solve it to find m.
Subtraction Equations Can Be Solved with Addition If the variable has something subtracted from it, then you need to add to both sides. This way, you can get the variable on its own. Taking something away and then giving it back is the same as not doing anything at all. q – 5 = –3 Add 5 to both sides q – 5 + 5 = –3 + 5 Then simplify q=2 Example
2
George and Kayla are each solving the equation u – 23 = 19. Their work is shown below. Who solved the problem correctly? George Kayla u – 23 = 19 u – 23 = 19 u – 23 + 23 = 19 + 23 u – 23 + 23 = 19 u = 42 u = 19 Solution
George is correct. He has remembered to add 23 to both sides of the equation. Kayla’s solution doesn’t make sense — she has added 23 to one side of the equation, but not the other. Example
3
Solve the equation x – (–9) = –14.3. Solution
Don’t forget: There’s more about solving this kind of equation in Lesson 2.2.5.
x – (–9) = –14.3 x – (–9) + ( –9) = –14.3 + ( –9) x = –14.3 + ( –9) x = –23.3 Section 2.2 — Equations
67
Guided Practice Solve the equations given in Exercises 17–30. 17. q – 5 = 12 18. q – 6 = –3 19. x – 3 = 0 20. w – 3 = 6 21. n – 2 = 13 22. x – 12 = 14 23. 12 = h – 6 24. j – 6 = 5 25. 19 = r – 15 26. b – 11 = 2.7 27. g – 0.1 = 0.9 28. y – (–2) = 8 29. e – 0.3 = 1.9 30. 9 = v – 0 31. Ted is playing a game of snap. So far he has put down 5 cards, leaving him holding 7. Write an equation for this, using n for the number of cards Ted had at the start of the game. Solve it to get n.
Independent Practice
Now try these: Lesson 2.2.3 additional questions — p433
Solve the equations given in Exercises 1–8. 1. 14 + k = 15 2. 11 = t + 3 3. j + 1 = 3.9 4. 15 = p + 16 5. c – 3 = 0 6. 1.5 = k – 1.5 7. f – 0.6 = 10.2 8. 1.3 = a – 2.8 In Exercises 9–16, write an equation for the situation described and then solve it. Use x to represent the variable. 9. When eight is subtracted from a number, the result is 25. 10. Three plus a number is 30. 11. Nine more than a number is twenty-three. 12. When a number is decreased by 8.9, the result is 41.4. 13. A number increased by 9.8 is equal to 17. 14. A number decreased by forty-two is equal to ten. 15. Twenty is the sum of a number and three. 16. When 5.6 is subtracted from a number, the result is 17.2. 17. Mr. Jenkins lost 14 pounds on a diet. He now weighs 210 pounds. Write a subtraction equation for this situation, using w to represent his weight before the diet. Then solve the equation to find w. 18. Monique’s hair has grown by an inch and a half since it was last cut and it is now 11 inches long. Write an addition equation to represent this situation. Use h to represent the length of hair right after the last haircut. Then solve the equation to find h. 19. Karen is 61 inches tall, which is 8 inches shorter than her mother. Which equation can be used to represent her mother’s height, h? h + 61 = 8 h = 61 – 8 h – 8 = 61 h = 8 × 61
Round Up In the world of equation manipulation, “+” gets rid of “–”, and “–” gets rid of “+.” There’s really not much more to it than that, as long as you play by the rules — you must always do exactly the same thing to both sides. 68
Section 2.2 — Equations
Lesson
2.2.4
Solving × and ÷ Equations
California Standard:
More equations, but this time with multiplication and division in them. The same rules as last Lesson apply — you need to get the variable alone on one side of the equals sign.
Algebra and Functions 1.1 Write and solve one-step linear equations in one variable.
Divide to Cancel Out a Multiplication
What it means for you: You’ll do more equations, this time dealing with equations containing multiplications and divisions.
To solve equations like 5t = –20, you still need to get the variable, t, on its own. The variable has been multiplied by a number, 5 — so you can get the variable on its own by dividing. In this case, you need to divide by 5. 5t = –20 5t ÷ 5 = –20 ÷ 5 t = –20 ÷ 5 t = –4
Key words: • equation • solve
Don’t forget: 5t ÷ 5 = t
Example
divide both sides by the same thing the variable is multiplied by
1
Solve the equation 7r = 42. Solution
7r = 42 7r ÷ 7 = 42 ÷ 7 r = 42 ÷ 7 r=6
Divide both sides by 7, then simplify to work out r.
Exactly the same rule works if the variable has been multiplied by a negative number. Example
2
Solve the equation –6q = 54. Solution
–6q means –6 × q, so you need to divide by –6. –6q = 54 Divide both sides by –6, –6q ÷ (–6) = 54 ÷ (–6) then simplify to work out q. q = 54 ÷ (–6) q = –9
Don’t forget: “•” just means “multiply.” The phrase “5 multiplied by k” can be written as “5•k” or “5 × k” or just “5k.”
Guided Practice Solve the equations given in Exercises 1–8. 1. 3t = 27 2. 16w = 48 3. 7x = 28 4. r × 6 = 7.2 5. 12y = 13.2 6. –9p = –72 7. –5•k = 30 8. h × 8 = –56 Section 2.2 — Equations
69
Multiply to Cancel Out a Division Division and multiplication cancel each other out, so you can use multiplication to solve division equations. m ÷ 11 = 6 m ÷ 11 × 11 = 6 × 11 m = 6 × 11 m = 66
Don’t forget: Division and multiplication are “opposites” — see Lesson 1.2.2 to remind yourself why.
Example
multiply both sides by the same thing the variable is divided by
3
Solve the equation q ÷ 5 = 7. Solution
q÷5=7 q÷5×5=7×5 q=7×5 q = 35
Example
Multiply both sides by 5 then simplify to work out q.
4
Sienna and Raoul are each solving the equation p ÷ 6 = 12. Their work is shown below. Who solved the problem correctly? Raoul Sienna p ÷ 6 = 12 p ÷ 6 = 12 p ÷ 6 × 6 = 12 + 6 p ÷ 6 × 6 = 12 × 6 p = 18 p = 72 Solution
Sienna is correct. Raoul has not done the same to both sides of the equation. He has multiplied the left-hand side by 6, but added 6 to the right-hand side.
Guided Practice Solve the equations given in Exercises 9–20. 9. t ÷ 2 = 3 10. w ÷ 5 = 11 11. q ÷ 4 = 13 12. u ÷ 8 = 5 13. x ÷ 1.5 = 3 14. p ÷ 5 = 2.7 15. g ÷ –3 = 9 16. s ÷ –7 = 6 17. d ÷ –6 = –12 18. j ÷ 13 = –15 19. y ÷ 0.9 = 9.1 20. e ÷ 1.2 = –2.3 21. A builder wants to buy a pipe that is long enough to cut into four 30-inch pieces, as shown below. Write and solve a division equation that can be used to find the length of the pipe, p, that the builder should buy.
70
Section 2.2 — Equations
Independent Practice Now try these: Lesson 2.2.4 additional questions — p434
Solve the equations given in Exercises 1–8. 1. 7x = 28 2. 0.5s = 10 3. w ÷ 8 = 1 4. 34p = 0 5. 5•r = –10 6. t ÷ 0.5 = 10 7. 7 = z ÷ –8 8. –7.2 = –6y 9. Juan likes to swim laps at the local pool. The pool has a length of 25 meters. Juan wants to swim 350 meters. Write and solve a multiplication equation to find out how many lengths Juan needs to swim. 10. Taylor earned $94.50 last week at her job, where she earns $6.75 an hour. Solve the equation 94.50 = 6.75h to find out how many hours, h, she worked.
Don’t forget: To check that an answer is correct, put it back into your original equation. If you put 9 back into the equation 3x = 12, you can see right away that it doesn’t make sense.
11. When Ross solved the equation 12 = 3x, he made a mistake. His work is shown below. What mistake did Ross make? 3x = 12 3x – 3 = 12 – 3 x=9
12. A poster has an area of 800 square inches, and a length of 40 inches. Write and solve a multiplication equation that can be used to find the width. 13. A farmer has a rectangular plot of land that he divides into four smaller plots, each with an area of 5000 square yards. Write and solve a division equation to find the area of the original plot. Solve the equations given in Exercises 14–21. These will give you practice on both “+” and “–” questions and “×” and “÷” questions. 14. f + 9 = 3 15. 12k = 288 16. –7 = n ÷ 4 17. b – 23 = 2.1 18. q – 0.23 = 1.66 19. –2d = 8 20. 4.2 = m ÷ 3 21. y ÷ 5 = –80 In Exercises 22–27, write an equation for the situation described and then solve it. Use x to represent the variable. 22. A number divided by seven is equal to fifteen. 23. The product of a number and three is equal to negative nine. 24. A number multiplied by 2 is equal to 8.4. 25. A number added to 9 is 37. 26. A number divided by negative three is equal to five. 27. Negative fifteen taken away from a number is 6.2.
Round Up By now you should be used to always doing the same thing to both sides of an equation. The main thing from this Lesson is that you can undo multiplication with division and you can undo division with multiplication. Just keep practicing... Section 2.2 — Equations
71
Lesson
2.2.5
Graphing Equations
California Standard:
Solving equations is one of the most important things you do in math. That means you need plenty of practice at it. And you need to understand exactly why this method works. This Lesson starts with a summary of things you’ve seen in the previous Lessons in this Section.
Algebra and Functions 1.1 Write and solve one-step linear equations in one variable.
What it means for you: You’ll see how what you’ve learned about equations matches up to the number line, and you’ll practice deciding what sort of equation you’re dealing with.
Key words: • equation • graph • number line
There Are Different Types of One-Step Equations A one-step equation is one that can be solved in one step by either adding, subtracting, multiplying by, or dividing by one thing. There are four main types. For example: (i) a + 3 = 4.2 — solve by subtracting 3 from both sides to get a = 1.2 (ii) s – 7 = 12 — solve by adding 7 to both sides to get s = 19 (iii) 9m = 27 — solve by dividing both sides by 9 to get m = 3 (iv) d ÷ 8 = 2 — solve by multiplying both sides by 8 to get d = 16 Before you can solve an equation, you must be able to spot what kind of equation you have. Example
Don’t forget: –7y = 14 means –7 × y, so it’s a multiplication equation that you solve by dividing. (It is not the same as y – 7, or 7 – y.)
Don’t forget: Adding a negative number is the same as subtracting a positive number. And subtracting a negative number is the same as adding a positive number.
Don’t forget: Notice how closely addition and subtraction are linked. Addition problems can be turned into subtraction problems. Similarly, subtraction problems can be turned into addition problems.
1
Explain how you would solve each of the following equations. (i) x + 12 = 4.2, (ii) –7y = 14, (iii) a – 87 = 1 Solution
(i) x + 12 = 4.2 (ii) –7y = 14
Solve by subtracting 12 from both sides: x = –7.8
Solve by dividing both sides by –7: y = –2
(iii) a – 87 = 1 Solve by adding 87 to both sides: a = 88 Sometimes you can look at an equation in more than one way. Example
2
Solve n + (–2) = 6.3. Solution
There are two ways to think about n + (–2) = 6.3. (i) You can say that “n has –2 added to it, so solve the equation by subtracting –2 from both sides.” This gives n + (–2) – (–2) = 6.3 – (–2). But 6.3 – (–2) is the same as 6.3 + 2, so this gives n = 8.3 (ii) Or you can rewrite the equation before solving it, using the fact that adding a negative is the same as subtracting a positive: n – 2 = 6.3. You can solve this by adding 2 to both sides. This gives n – 2 + 2 = 6.3 + 2, which gives n = 8.3 Whichever method you use, it all comes down to the same thing.
72
Section 2.2 — Equations
Equations with division come in two forms. You may see the ÷ written in or you may see the equation written as though it is a fraction. Example 3 h =7. Solve −9 Solution
This is the same as h ÷ (–9) = 7. Solve by multiplying by –9. h ÷ (–9) × (–9) = 7 × (–9), which gives h = 7 × (–9) = –63.
Guided Practice Identify the type, and solve the equations in Exercises 1–12. 1. 3t = 27
2. –4 + q = 11
3. g × 2.4 = 4.8
4. –8 + z = 8 u =3 7. 2.9 10. n ÷ 7 = 4
5. d × 19 = 95
6. –4j = 20
8. b – (–3) = –9
9. 12 + j = 9 p = 8.2 12. 12
11. k + (–3) = 14
Show Your Solutions on the Number Line Sometimes in math, you need to draw your solutions on a graph. For the equations in this Lesson, that means drawing a number line. Example
4
Solve the equation n + 5 = 42 and graph your result. Solution
Don’t forget: There’s a lot more information about the number line in Chapter 1.
Solve the equation in the normal way. Here, you need to subtract 5 from both sides to get n = 37. n Now graph this solution 35 36 37 38 39 40 41 42 on a number line.
×
But notice also that a number line can help you solve an equation. Example
5
Check it out:
Solve the equation n + 5 = 42.
This isn’t a new way to solve an equation — it’s just a different way of thinking about the method you’ve already seen. Solving n + 5 = 42 by subtracting 5 from both sides is equivalent to moving 5 places to the left on a number line.
Solution
The equation n + 5 = 42 tells you that there is a position on the number line, n, and that 42 is 5 places to the right of it. That means that n is 5 places to the left of 42. That can be written as n = 42 – 5. So n = 37.
n+5 n 37
42 38
39
40
41
42 – 5 = 37
Section 2.2 — Equations
73
Example
6
Solve the equation 6x = 18 and graph your result. Solution
You can divide both sides by 6 to find the solution x = 3. x
Graphing the result means you have to draw a number line.
× 3
0
If you like, you can even use the number line to solve the equation. The equation 6x = 18 tells you that 18 is 6 times further away from 0 on the number line than x.
6
9
12 15 18
6x x 0
×3 18 ÷ 6 = 3
18
Guided Practice Solve the equations in Exercises 13–18, and graph your results. 13. m + 10 = 9 14. t – 3 = 2 15. 4x = 16 16. r – 3 = –3 17. n + (–1) = 36.5 18. y ÷ 9 = 12 Write an equation represented by each of the number lines in Exercises 19–20. Find the solution to each equation. 19.
w ÷ (–5)
w + 25
20. –9
–2
0
Independent Practice Now try these: Lesson 2.2.5 additional questions — p434
Solve the equations in Exercises 1–12. Show your results on a number line. 1. d – 1 = 2
3. b + 62 = –189
7. 8g = 64
2. h•2.5 = 10 g =6 5. −8 8. v – 0.25 = 1.5
10. r ÷ –12 = –3
11. j + 23 = 26
12. z – (–2) = 6
4. 11 = 12 + k
6. 6 × p = 174 9. 15 = 5k
13. During the week, Ramon drives t miles to school. On the weekend, he drives three times as far to a tennis club. The tennis club is 18 miles away. Write an equation for this and solve it to find out how far Ramon lives from school.
Round Up Equations become much easier to deal with if you practice — solving the equation just means working out how to get the variable alone. In the next Section, you’ll be using expressions and equations to describe real-life situations. 74
Section 2.2 — Equations
Lesson
Section 2.3
2.3.1
Expressions About Length
California Standards:
You’ve seen lots of mathematical expressions — things like x + 4, or y – 7. You’ve also seen equations, and learned how to solve them — things like x + 2 = 6, for example.
Algebra and Functions 1.2 Write and evaluate an algebraic expression for a given situation, using up to three variables. Algebra and Functions 1.3 Apply algebraic order of operations and the commutative, associative, and distributive properties to evaluate expressions; and justify each step in the process. Algebra and Functions 3.1 Use variables in expressions describing geometric quantities (e.g., P = 2w + 2l, A = ½bh, C = pd — the formulas for the perimeter of a rectangle, the area of a triangle, and the circumference of a circle, respectively). Algebra and Functions 3.2 Express in symbolic form simple relationships arising from geometry.
What it means for you: You’ll practice using expressions to represent the length and perimeter of objects.
Now you need to apply what you know to other situations.
Expressions Can Describe Length If you don’t know the length of something, use a variable to represent it. Example
1
Write an expression for the total height of the window on the right.
? in.
35 in. Solution
Choose a letter to represent the unknown height — w, say. Then the total height of the window is (35 + w) inches. Pictures can help you understand questions. So if there’s no picture in a question, you should draw one yourself. Example
2
Imelda’s brother is 56 inches tall. The top of his head comes to Imelda’s shoulders. Write an expression for the height of Imelda. Solution
x
Draw a picture — a simple one will do fine. Key words: • • • • •
length perimeter expression commutative associative
Don’t forget: Always check whether you need to include units in your answer — for example, inches, yards, miles...
The unknown length is the height of Imelda’s head. So choose a letter to represent it — x, say. Then Imelda’s height is (56 + x) inches.
56 in. Imelda’s brother
Imelda
Guided Practice 1. Write an expression for the length of these boxes. 5 yds 5 yds ? yds 2. Susan’s backyard has a swimming pool at the end. She measures the distance from the house to the swimming pool as 50 meters. By choosing a letter to represent the length of the swimming pool, write an expression for the length of the entire yard. Section 2.3 — Geometrical Expressions
75
Perimeter Means the Distance Around a Shape The perimeter is the distance around the outside of a shape. You find a shape’s perimeter by adding together the lengths of its sides. If you don’t know the length of a side, use a variable. Example
3
Write an expression for the perimeter of the shape below. The two unknown sides have the same length, y inches. 14 in. y in.
11 in. y in.
Don’t forget:
Solution
You can simplify expressions by combining terms that contain the same variables. So 7 + x + x + x can be simplified to 7 + 3x.
The perimeter is found by adding the lengths of all the sides together. So, the perimeter of the shape is 11 + 14 + y + y. This can be simplified to (25 + 2y) inches.
When you write an expression for the perimeter of a rectangle, there could be two unknown lengths — the length and the width. So you might need to use two variables. Example
4 l
Write an expression for the perimeter of the rectangle on the right.
w
w l
Solution
The perimeter of the shape is w + l + w + l = 2w + 2l. Or you could write this expression as 2l + 2w. It means exactly the same. Check it out: You use the associative and commutative properties all the time in math. Most times, you probably do it almost without thinking — for example, when you work out 10 + 3 instead of 3 + 10 (an example of the commutative property of addition being used). But now you need to be able to recognize when you do it. There’ll be many examples pointed out throughout this book.
76
Add Numbers Together in Any Order When you add the lengths together to find the perimeter of a shape, you can add numbers together in any order. This is because addition is commutative and associative. Commutative Property of Addition: Associative Property of Addition:
Section 2.3 — Geometrical Expressions
x+y=y+x
(x + y) + z = x + (y + z)
Guided Practice 3. Find the perimeter of a rectangle with length 8 inches and width 5 inches. 4. Write and simplify an expression for the perimeter of the shape shown below. 18 cm 13 cm 11 cm ? cm
5. A triangle has one side of length 3 inches, one side of length 4 inches, and a third side of length q inches. Write and simplify an expression for the perimeter of this triangle.
Solve Geometry Equations by Manipulating Them If you’re given enough information, you can set up an equation to find a missing length or a perimeter. Example
5
Write an expression for the perimeter of the shape below. If the full perimeter is 100 inches, find the length of the unknown side. 19 in. 26 in.
15 in.
? in.
16 in.
Solution
The perimeter is the distance all the way around the outside. Using the picture, this is (15 + 19 + 26 + 16 + x) inches, where x is the length of the unknown side. This can be simplified to 76 + x inches. Don’t forget: You could use the “guess and check” method to solve this equation. But “guess and check” only works well for finding integers.
Now you can write an equation using the other information in the question — the fact that the full perimeter is 100 inches. So 76 + x = 100. You can then solve this equation by subtracting 76 from both sides. x + 76 = 100 76 + x is the same as x + 76 x = 100 – 76 Subtract 76 from both sides, and simplify. x = 24 So the length of the unknown side is 24 inches.
Section 2.3 — Geometrical Expressions
77
Guided Practice 6. By first writing an expression, find the unknown length on the shape shown below, which has a perimeter of 60 m. 11 m
9 m 13 m
? m
5 m
7. Liz knows that the perimeter of her kitchen is 50 yards. She knows from measuring that three of its four walls have a combined length of 32 yards. What is the length of the remaining wall?
Independent Practice Now try these: Lesson 2.3.1 additional questions — p434
1. Write an expression for the length of the piece of wood shown below.
20 in.
? in.
2. Use “guess and check” to find the unknown length in Exercise 1 if the total length of the wood is 32 inches. Write an expression for the perimeter of each shape given in Exercises 3–4. 3.
23 cm
4.
13 cm ? cm
25 cm
? yds 15 yds
8 yds In Exercises 5–6, form and solve an equation to find the unknown length, given that the perimeter of each shape is 100 inches. 31 in. 6. 5. 18 in. 31 in. ? 35 in.
29 in.
?
Round Up Dealing with expressions about lengths isn’t harder than dealing with expressions in any other situation. The main thing is to remember how to make an expression in the first place. Once you’ve mastered that, you should be ready to tackle area... 78
Section 2.3 — Geometrical Expressions
Lesson
2.3.2
Expressions About Area
California Standards:
You’ve used expressions to represent lengths and perimeters, but they can be used to represent just about anything else as well.
Algebra and Functions 1.2 Write and evaluate an algebraic expression for a given situation, using up to three variables. Algebra and Functions 1.3 Apply algebraic order of operations and the commutative, associative, and distributive properties to evaluate expressions; and justify each step in the process. Algebra and Functions 3.1 Use variables in expressions describing geometric quantities (e.g., P = 2w + 2l, A = ½bh, C = pd — the formulas for the perimeter of a rectangle, the area of a triangle, and the circumference of a circle, respectively). Algebra and Functions 3.2 Express in symbolic form simple relationships arising from geometry.
What it means for you:
In this Lesson, they’re used to describe areas. The good thing is that all the ideas about variables and solving equations work exactly the same as before.
Expressions Can Describe Area You’ve already seen the formula for the area of a rectangle — area = base × height. But if you don’t know one or both of these measurements, you have to use variables. Example
1
Write an expression for the area of the rectangle on the right. Then use it to find the area when j = 4. Solution
6 cm
Area = base × height = 6 × j = 6j cm2. Evaluating this for j = 4 gives an area of 6 × 4 = 24 cm2. If there are two unknowns, you need two variables. Example
2
You’ll practice using expressions to represent the area of objects.
Write an expression for the area of the rectangle on the right.
Key words:
Solution
• • • • •
Call the height of the rectangle h, and the base b.
length area expression commutative associative
j cm
height = ?
base = ? Both the base and the height are unknown, so you need to use variables for both of them.
Then the area of the rectangle is b × h, or bh. Check it out: The formula for the area of a rectangle: “Area = base × height,” is often shortened to just “A = bh.”
Guided Practice 1. Write an expression that represents the area of a rectangle with base b cm and height 4 cm. 2. Write an expression that represents the area of a rectangle with base 12 inches and height f inches. 3. Evaluate your expression from Exercise 2 to find the area of the rectangle when the height is 3 inches. Section 2.3 — Geometrical Expressions
79
The Area of a Triangle is Half the Area of a Rectangle Check it out: Notice that the area of a triangle is half the area of the rectangle with the same base and height measurements. 2
h
1 2
1
You’ve seen the formula for the area of a triangle before: 1 area = ×base× height. Just like with rectangles, if you don’t know the 2 length of the base or the height, you’ll need to use a variable. Example
3
Write an equation describing the area of the triangle shown below. Then use your equation to find the triangle’s area if its height is 3 inches and its base is 8 inches.
b In this diagram, the areas marked 1 are equal, and the areas marked 2 are equal. The white triangle has half the area of the rectangle.
height = h base = b
Solution
There are two unknowns, so your expression will need to include two variables — the height h and the base b. 1 2
1 2
Use these variables in the triangle area formula: Area = × b × h = bh Check it out: The formula for the area of a triangle is usually shortened 1 to just “A = 2 bh.”
Use this equation to find the triangle’s area when b = 8 and h = 3. 1 2
1 2
Area = bh = ×8×3 = 12 in 2
Multiply Numbers Together in Any Order Check it out: As with the equivalent properties for addition in the previous lesson, you probably use the associative and commutative properties for multiplication all the time without thinking — for example, when you work out 2 × 9 instead of 9 × 2 (this uses the commutative property of multiplication). Examples of these properties being used will be shown throughout this book.
Like addition, multiplication is also commutative and associative. Commutative Property of Multiplication:
xy = yx
So the area of the rectangle in Example 2 could have been written hb. Associative Property of Multiplication:
(xy)z = x(yz)
This means you don’t need parentheses in the formula for the area of a triangle to show the order of the multiplications.
( 12 ×b)×h is the same as 12 ×(b×h). Guided Practice 4. Write an expression that represents the area of a triangle of height 2 inches and base y inches. 5. Evaluate your expression from Exercise 4 to find the area of the triangle when the base is 7 inches. 6. Write an expression that represents the area of a triangle with base x cm and height y cm.
80
Section 2.3 — Geometrical Expressions
Use Equations to Find Missing Lengths Equations can be used to find missing lengths. First you have to write an equation to describe the situation. Then you can solve it to find the missing value. Example
4
The area of the rectangle shown is 36 in2. Find the length of the unknown side.
3 in. ? in.
Solution
First write an equation linking the different quantities in the question. These are: • the side you know the length of (3 in.), • the side whose length you need to find out — call this b, • the area of the triangle (36 in2). The equation linking these three quantities is: area = base × height, or A = bh. Substituting in the values you know, this gives 36 = 3 × b, or 3b = 36. Don’t forget: The method for solving equations like these was explained in Lesson 2.2.4.
Now solve this equation: 3b = 36 Divide both sides by 12. b = 12 This means the length of the unknown side is 12 inches.
Guided Practice 7. A rectangle has area 12 in2 and its base is 3 inches. Form and solve an equation to find its height. 8. A triangle has area 25 in2 and its height is 5 inches. Form and solve an equation to find the length of its base.
Independent Practice Now try these:
1. Write an expression for the area of a square with side length b cm.
Lesson 2.3.2 additional questions — p435
2. Write an expression for the area of a triangle with base 6 inches and height h inches. 3. Evaluate your expression from Exercise 2 to find the area of the triangle when the height is 3 inches. 4. By breaking it into simple shapes, write an expression for the area of the shape shown to the right.
h in.
8 in.
b in.
Round Up Expressions for areas of triangles have that tricky ½ — so they can be awkward. Focus on working out each little bit of the expression in turn — that’s the best way to avoid mistakes. Section 2.3 — Geometrical Expressions
81
Lesson
2.3.3
Finding Complex Areas
California Standards:
Finding the area of a rectangle or a triangle is one thing. But once you can do that, you can start to find out the areas of some really complicated shapes using those very same techniques.
Algebra and Functions 3.1 Use variables in expressions describing geometric quantities (e.g., P = 2w + 2l, A = ½bh, C = pd — the formulas for the perimeter of a rectangle, the area of a triangle, and the circumference of a circle, respectively). Mathematical Reasoning 1.3 Determine when and how to break a problem into simpler parts.
What it means for you: You’ll see how you can use the formulas for the areas of rectangles and triangles to find areas of much more complex shapes too.
This is an important idea in math — using what you know about simple situations to find out about more complex ones.
Find Complex Areas by Breaking the Shape Up There’s no easy formula for working out the area of a shape like this one.
10 in. 4 in. 8 in.
But you can find the area by breaking the shape up into two smaller rectangles. Example
6 in.
1 10 in.
Find the area of the shape above. 4 in.
Solution
Key words: • complex shape
8 in.
Divide the shape into two rectangles, as shown.
6 in.
h
Area of large rectangle = 10 × 4 = 40 in2. b
Now you need the dimensions of the small rectangle, b and h. b = 10 – 6 = 4 in. And h = 8 – 4 = 4 in. So the area of the small rectangle = bh = 4 × 4 = 16 in2. Check it out: There’s often more than one way you can break a complicated shape up. Try to spot the way that will make your calculations as easy as possible.
So the total area of the shape is 40 + 16 = 56 in2. You don’t always have to break a complicated shape down into rectangles. You just have to break it down into simple shapes that you know how to find the area of. Example
2 9 in.
Find the area of the shape on the right. 4 in. Solution
4 in.
Divide the shape into a rectangle and a triangle. 9
Area of rectangle = 9 × 4 = 36 in2.
3 in.
4
1 2
Area of triangle = × 4 ×3 = 6 in 2 . So the total area of the shape is 36 + 6 = 42 in2.
82
Section 2.3 — Geometrical Expressions
4 3
Guided Practice Check it out: These problems can look tough, so take it step by step. Work out the area of each rectangle or triangle in the shape in turn. Don’t try to do the whole calculation at once.
Find the areas of the shapes below. 3 in. 3 in.
1.
6 cm
2. 2 cm 2 in.
4 in.
4 cm 3 cm
3.
12 cm
4.
8 in. 8 in.
6 cm
6 in.
8 in.
6 in.
6 cm
6 in.
4 cm
4 cm
Complex Areas Can Involve Variables Sometimes you have to use variables for the unknown lengths. But you can write an expression in just the same way. Example
3 x
Find the area of the shape on the right. y Solution
Divide the shape into two rectangles. Area of the large rectangle = xy. Area of the small rectangle = ab.
b
x a
y b
So the total area of the shape is xy + ab.
a
Guided Practice
Check it out: If you’re finding questions 6–8 tough, then look back at Exercises 2–4 above. The shapes are exactly the same, only this time the lengths are variables instead of numbers. Work out the areas in just the same way.
Find the areas of the shapes below. x 5. 6. x
x y b
x
a
x 7.
b b
a
8.
b
b
a a
b
a c
c
Section 2.3 — Geometrical Expressions
83
Check it out: You can divide the shape up differently and get the same answer. For example, you could divide it into two rectangles: q q
You Can Subtract Areas As Well Sometimes it’s easier to find the area of a shape that’s too big, and subtract a smaller area from it. Example
4
q
Calculate the area of the shape on the right.
q
q
q
p
Then the total area is pq + q(p – q). Using the distributive property (covered in the next lesson), you’ll see that this gives exactly the same answer as in Example 4. pq + q(p – q) = pq + qp – q2 = 2pq – q2 Look for another way to divide up the shape, and check that it gives the same expression for the area.
q
p
Solution
q
This time it’s easier to work out the area of the rectangle with the red outline, and subtract the area of the gray square.
q p
Area of red rectangle = p × 2q = 2pq. Area of gray square = q × q = q2. So area of original shape = 2pq – q2.
Guided Practice Use subtraction to find the areas of the shapes below. a 16 in. 9. 10.
6 in.
c
b
5 in.
c
5 in.
Independent Practice Find the areas of the shapes below. Now try these:
1.
5 cm
2.
Lesson 2.3.3 additional questions — p435
4 cm
5 cm 5 cm
4 cm 9 cm
2a
7 cm
2 in.
3.
4. 1 in.
b 3b
4 in.
2a 8 in.
4a
Round Up Remember that it doesn’t matter whether your lengths are numbers or variables — you treat the problems in exactly the same way. That’s one of the most important things to learn in algebra. 84
Section 2.3 — Geometrical Expressions
Lesson
2.3.4
The Distributive Property
California Standard:
The distributive property is the name given to a way of breaking up certain types of multiplication. It can help a lot with math once you understand it properly.
Algebra and Functions 1.3 Apply algebraic order of operations and the commutative, associative, and distributive properties to evaluate expressions; and justify each step in the process.
Use the Distributive Property to Remove Parentheses Look at the shape on the right. a
What it means for you: You’ll learn about the distributive property, which allows certain types of multiplication to be rewritten in other ways and helps simplify mental math.
Key words: • distributive property • multiplication • parentheses
b You can work out its total area in two different ways.
c
• You can find the areas of the two smaller rectangles and add them. Total area = area of 1st rectangle + area of 2nd rectangle = ab + ac. • Or you can find the total width of the whole rectangle by adding b and c, and then multiply by the height. Total area = a(b + c). But these two expressions must be the same, since they both represent the same area. This is an example of the distributive property. Distributive Property:
Example
a(b + c) = ab + ac
1
Check it out:
Verify the distributive property in the case where a = 7, b = 5, and c = 3.
“Verify” means “Check that something is true.” So here, you just put the numbers in the distributive property equation and check that it “works.”
Solution
• First work out ab + ac. This is (7 × 5) + (7 × 3) = 35 + 21 = 56 • Next work out a(b + c). This is 7 × (5 + 3) = 7 × 8 = 56 Both sides are equal, and so the distributive property holds.
The distributive property means you can remove parentheses from an expression. Example Check it out: For an alternative way to do this Example, see Exercise 1 in the next Guided Practice.
2
Remove the parentheses from the expression p(q + r). Solution
Using the distributive property: p(q + r) = pq + pr
Section 2.3 — Geometrical Expressions
85
Guided Practice 1. Verify the distributive property by calculating the area of this rectangle in two different ways.
Don’t forget: a × (b + c) is exactly the same as a(b + c).
q
r
p
Draw a picture as in Exercise 1 to represent the expressions in Exercises 2–5. In each case, verify the distributive property by finding the area in two ways. 2. 2 × (7 + 9) 3. 20 × (9 + 4) 4. h × (3 + h), for h = 10 5. g × (y + d), for g = 3, y = 9, d = 5.
Don’t forget:
Remove the parentheses from the expressions in Exercises 6–8. 6. p × (q + r) 7. r × (s + t) 8. b × (c + d)
Using the commutative property of addition, you can write (q + r) × m as m × (q + r).
Remove the parentheses from the expressions in Exercises 9–11. 9. (q + r) × m 10. (a + b) × c 11. (e + f) × g
The Distributive Property Works with Subtraction Too If the parentheses contain a subtraction, then when you remove the parentheses, the expression will still contain a subtraction.
a(b – c) = ab – ac Example
3
Verify the distributive property for 2 × (6.5 – 0.5). Solution
(2 × 6.5) – (2 × 0.5) = 13 – 1 = 12 2 × (6.5 – 0.5) = 2 × 6 = 12 This rule works for any numbers — even negative ones. Remember to be extra careful if negative numbers are involved. Example
4
Verify the distributive property for –2 × (6.5 – 0.5). Solution
(–2 × 6.5) – (–2 × 0.5) = –13 – (–1) = –13 + 1 = –12 –2 × (6.5 – 0.5) = –2 × 6 = –12
86
Section 2.3 — Geometrical Expressions
Guided Practice 12. Verify the distributive property equation a(b – c) = ab – ac by calculating the area of the shaded rectangle in two different ways.
b a c
Use the distributive property to evaluate the expressions in Exercises 13–20. Show your work. 13. 4 × (8 – 2) 14. 9 × (15 – 8) 15. –5 × (–7 – 5) 16. 9 × (8 – 15) 17. –6(23 – 18) 18. –2(–2 – 6) 19. 3 × (17 – p), given that p = 7 20. m × (n – p), given that m = 8, n = 7, p = 5 Remove the parentheses from the expressions in Exercises 21–23. 21. r × (s – t) 22. a × (b – c) 23. m × (n – p) Say whether each of Exercises 24–27 is true or false. 24. y × (2 – 3) = y × 2 – y × 3 25. f × (g – u) + y = f × g + f × u + f × y 26. k × (l – 12) = k – 12 27. t × 9 = t × 10 – t
Independent Practice Use the distributive property to evaluate the expressions in Exercises 1–8. Show your work. 1. 2 × (6 – 3) 2. 4 × (22 – 14) 3. –4 × (–2 – 8) 4. 2 × (6 – 23) 5. –6 × (–23 + 8) 6. –3 × (–8 – 1) 7. q × (17 – q), given that q = 3 8. b × (c – d), given that b = 10, c = 6, d = 8 Remove the parentheses from the expressions in Exercises 9–11. 9. t × (d – r) 10. (y – h) × z 11. (9 – s) × 3
Now try these: Lesson 2.3.4 additional questions — p436
Say whether each of Exercises 12–15 is true or false. 12. v × (5 – e) = 5v – 5e 13. p × (q – 3) + 7 = pq – 3p + 7p 14. (2 – m) × n = 2n – mn 15. –d × (–9 – r) = 9d + dr
Round Up The distributive property is very useful in math — it comes up again and again. So learn it now... Practice writing it down a few times until you can do it without thinking. It’ll definitely be useful in the next Lesson. Section 2.3 — Geometrical Expressions
87
Lesson
2.3.5
Using the Distributive Property
California Standard: Algebra and Functions 1.3 Apply algebraic order of operations and the commutative, associative, and distributive properties to evaluate expressions; and justify each step in the process.
The distributive property is incredibly useful — not only when you’re doing algebra, but also when you’re trying to work out complicated sums in your head.
What it means for you:
The distributive property can help with mental math.
You’ll learn how to use the distributive property for mental math problems and problems involving patterns.
The idea is to split complicated numbers into numbers that are easy to multiply in your head.
The Distributive Property Makes Mental Math Easier
Example Key words: • distributive property • multiplication • expression
1
Calculate 7 × 14 without using a calculator. Solution
7 × 14 = 7 × (10 + 4) = (7 × 10) + (7 × 4) = 70 + 28 = 98 Example
Write 14 as 10 + 4 Use the distributive property
2
Calculate 5 × 107 without using a calculator. Solution Write 107 as 100 + 7 5 × 107 = 5 × (100 + 7) = (5 × 100) + (5 × 7) Use the distributive property = 500 + 35 = 535
Sometimes it’s easier to use a subtraction. Example
3
Calculate 6 × 29 without using a calculator. Solution
6 × 29 = 6 × (30 – 1) = (6 × 30) – (6 × 1) = 180 – 6 = 174
Write 29 as 30 – 1 Use the distributive property
Guided Practice Don’t forget: Use the commutative property of multiplication to rewrite these Exercises in the easiest order. For example, 21 × 8 is the same as 8 × 21.
88
Using the distributive property, do Exercises 1–8 as mental math. 1. 6 × 13 2. 7 × 21 3. 2 × 17 4. 21 × 8 5. 11 × 13 6. 16 × 5 7. 3 × 33 8. 15 × 9
Section 2.3 — Geometrical Expressions
The Distributive Property Is Used All the Time The distributive property can be used to rewrite expressions from all areas of math. Example
4
This triangle is made out of 2 red sticks and 1 blue stick. The total length of the sticks in one triangle is given by 2r + b, where r is the length of a red stick and b is the length of a blue stick.
Don’t forget: It’s always a good idea to check your answers. Here, you could count the number of red and blue sticks in the final diagram, and see if it matches your expression.
Write an expression for the total length of sticks in the pattern on the right. Solution
The pattern is made up of 6 of the small triangles. So the total length of red and blue sticks is 6 × (2r + b). Using the distributive property, this can be rewritten: 6 × (2r + b) = (6 × 2r) + (6 × b) = 12r + 6b
Guided Practice Exercises 9–11 are about the diagram on the right. 9. Write an expression for the total length of the sides, if each purple side has length p and each green side has length g. 10. Write an expression for the total length of the sides, if 3 of these shapes are arranged in a pattern as shown. Write your expression in two different ways, using the distributive property. 11. Write two expressions for the length of the sides if 100 of the shapes are arranged in a similar pattern.
Independent Practice
Now try these: Lesson 2.3.5 additional questions — p436
1. Carl is at the store to buy school supplies. Pens cost $1.15 each and a package of computer paper costs $3.75. He wants to buy two pens and two packages of paper. Write an expression of the form a × (b + c) that can be used to represent this situation. 2. Use the distributive property to calculate the money Carl spent. Use the distributive property to do Exercises 3–8 as mental math. 3. 9 × 29 4. 14 × 12 5. 19 × 7 6. 17 × 7 7. 15 × 11 8. 9 × 31 Rewrite Exercises 9–12 using the distributive property. 9. a × (11 – b) 10. j × (g + k) 11. 7 × (3 + s) 12. 15 × (q – w)
Round Up This mental math technique might seem hard to begin with, but can be really useful once you’ve had some practice. And if you can manage it, it’s a trick that’ll impress all your friends. Section 2.3 — Geometrical Expressions
89
Lesson
2.3.6
Squares and Cubes
California Standards:
This is mostly a Lesson on things you’ve seen before — expressions, equations, order of operations... But everything comes about from looking at (and thinking about) squares and cubes.
Algebra and Functions 1.2 Write and evaluate an algebraic expression for a given situation, using up to three variables. Algebra and Functions 3.1 Use variables in expressions describing geometric quantities (e.g., P = 2w + 2l, A = ½bh, C = pd — the formulas for the perimeter of a rectangle, the area of a triangle, and the circumference of a circle, respectively). Algebra and Functions 3.2 Express in symbolic form simple relationships arising from geometry.
What it means for you: You’ll practice using expressions to represent areas and volumes of objects.
Key words: • • • • • • • •
square cube edge face expression power area volume
Squares Are a Special Sort of Rectangle You’ve already seen that you can write the formula for the area of a rectangle as A = bh. But a square is just a rectangle whose sides all s have the same length — call it s. So the formula for the area becomes: s A = s × s or A = s2 Remember — s2 (“s squared”) means exactly the same as s × s. Example
Write an expression for the total area of 4 identical squares with sides of length d. Evaluate your expression in the case where d = 3 in. Solution
The area of one square is given by d2. So the area of 4 identical squares is given by 4d2. You now have to evaluate this expression for d = 3. The total area will be 4 × 32 — but remember to evaluate exponents before multiplication. So the total area is 4 × 32 = 4 × 9 = 36 in2.
You can use “guess and check” to find the length of a side if you know the area. Example
Don’t forget: Evaluate exponents before doing multiplication — see PEMDAS in Lesson 2.1.4.
Don’t forget: 32 can be said out loud as either “3 squared” or “3 to the second power.” Both mean exactly the same thing.
90
1
2
A square has area 36 in2. What is the length of its sides? Solution
You need to solve the equation 36 = s2. That means you need to find a value for s that gives 36 when you square it. Use “guess and check”: • Try s = 10. Here s2 comes to 10 × 10 = 100, which is much too large. • Try s = 5. Now s2 comes to 5 × 5 = 25, which is too small. • Try s = 6. Now s2 comes to 6 × 6 = 36. So the square must have sides of length 6 inches.
Section 2.3 — Geometrical Expressions
Guided Practice 1. Write an expression for the area of a square with side length z. Don’t forget: Units such as in2 should be read as “square inches.”
2. A square has area 81 cm2. What is the length of its sides? 3. A square has area 49 in2. What is the length of its sides? 4. A square has area 16 yd2. What is the length of its sides? 5. A square has area 64 m2. What is the length of its sides? 6. A square has area 121 in2. What is the length of its sides?
Check it out: The straight lines that run along the side of faces are known as edges. They are shown red on the diagram on the right — 9 are visible, but there are 12 altogether. See Section 7.4 for more information.
Cubes Are “Three-Dimensional Squares” A cube is a three-dimensional solid with six square faces. Its surface area is the total area of all 6 of its faces. As each face is a square, the formula for surface area of a cube is: A = 6s2
Edge Edge s s
Example
s
3
Without using a calculator, find the surface area of a cube whose edges have length 9 cm. Solution
The surface area of the cube is given by 6s2. If s = 9, this is 6 × 92 = 6 × 81 Calculate the exponent first. Rewrite 81 as 80 + 1. = 6 × (80 + 1) = 6 × 80 + 6 × 1 Use the distributive property. Evaluate × before +. = 480 + 6 2 = 486 cm . Or you might have to solve equations to do with cubes. Example
4
The surface area of a cube is 54 in2. Use “guess and check” to find the length of the cube’s edges. Solution
The surface area of the cube is given by the expression 6s2. You need to find s such that 6s2 = 54. Use “guess and check.” • Try s = 4. Here 6s2 comes to 6 × 42 = 6 × 16 = 96, which is too large. • Try s = 2. Now 6s2 comes to 6 × 22 = 6 × 4 = 24, which is too small. • Try s = 3. Now 6s2 comes to 6 × 32 = 6 × 9 = 54, which is perfect. So the lengths of the cube’s edges must be 3 inches.
Section 2.3 — Geometrical Expressions
91
Guided Practice 7. Write an expression for the surface area of a cube with edge length k. 8. A cube has surface area 96 in2. What is the length of its edges? 9. A cube has surface area 216 cm2. What is the length of its edges? 10. A cube has surface area 150 yd2. What is the length of its edges?
Volume Is a Measurement of 3-D Space Check it out: The volume of a unit cube is always 1 cubic unit (1 unit3). This is because, for a unit cube, s = 1, and so s3 = 1 × 1 × 1 = 1. But notice that if you use different units, your unit cube changes too. For example, if you’re using yards, then your unit cube has sides of length 1 yard, and so its volume is 1 yd3. However, if you’re measuring things in centimeters, then a unit cube will have sides of length 1 cm, and a volume of 1 cm3.
Volume is a measurement of the amount of space inside a three-dimensional object. It’s measured in cubic units and equals the 1 number of unit cubes (cubes whose edges have length 1) that fit inside the object. 1 In the diagram on the right, each side has a s length of 2 units, so two unit cubes fit along 1 each side. (One unit cube is shaded blue.) You can calculate the volume of a cube using the formula:
V=s×s×s Example
or
s
s (= 2 here)
V = s3
5
Draw a picture to show the unit cubes in a cube with edge length 3 cm. Find the volume of the cube by counting unit cubes. Then verify your answer using the equation V = s3. Solution
1 cm
The edges have length 3 cm, so 3 unit cubes (using units of cm) will fit along each edge.
1 cm
3 cm 1 cm Count the unit cubes in layers, starting at the front. 3 cm There are 9 unit cubes in the front layer. 3 cm And there are 3 layers altogether, which means there are 9 × 3 = 27 unit cubes altogether. And since each unit cube has a volume of 1 cm3, the volume must be 27 cm3. Now use the equation V = s3 with s = 3 to check the result. V = 33 = 3 × 3 × 3 = 27 cm3.
92
Section 2.3 — Geometrical Expressions
Guided Practice 11. Using the formula, find the volume of a cube with edges 6 meters long. 12. By counting the number of unit cubes inside, work out the volume of a cube with edges 3 inches long. 13. Use “guess and check” to find the edge length of a cube with a volume of 64 cubic yards.
Independent Practice 1. Find the area of a square whose sides have length 5 in. 2. Find the volume of a cube whose sides have length 7 in. In Exercises 3–7, find the value of p. 3.
4.
p in.
p in.
Check it out: Another way to write 24 in2 is 24 sq. in. They both mean “24 square inches.”
p in.
p in. Area = 144 in2
Area = 36 in2 5.
6.
4 ft. 4 ft. 4 ft.
Now try these: Lesson 2.3.6 additional questions — p436
Volume = p ft3
Edge Length = p cm 2 Surface Area = 150 cm
7.
Edge Length = p in. Volume = 729 in3
Round Up There were a few formulas in this Lesson, but in the end it always comes down to two skills — being able to substitute values into a formula, and being able to work backward to find out what values need to be substituted. Keep practicing those skills. Section 2.3 — Geometrical Expressions
93
Lesson
2.3.7
Expressions and Angles
California Standards:
More expressions, but this time about angles. Some of this you’ll have seen last year, but now you can apply your knowledge of expressions.
Algebra and Functions 1.2 Write and evaluate an algebraic expression for a given situation, using up to three variables. Algebra and Functions 3.1 Use variables in expressions describing geometric quantities (e.g., P = 2w + 2l, A = ½bh, C = pd — the formulas for the perimeter of a rectangle, the area of a triangle, and the circumference of a circle, respectively).
As with the earlier Lessons, expressions are used as a shorthand way of explaining how to work something out. In the world of angles, that usually means many pictures of triangles...
Angles Are Measured Between Two Lines Whenever two lines meet, you get an angle. The first thing you need to know about angles is how to label them. The diagrams below show two different ways of identifying angles: A
Algebra and Functions 3.2 Express in symbolic form simple relationships arising from geometry.
What it means for you:
Here, the marked angle could be called “the angle ABC.” If you follow the path “A then B then C,” you trace out the angle. Or it could simply be called “the angle at B.”
B
r°
More expressions practice — this time with angles.
C
On this diagram, the size (or measure) of the angle has been labeled with the variable r.
Key words: • angle • triangle • expression
Example
1
C
Write an expression for the measure of angle CAT.
80° x°
A Solution
The angle CAT is made up of two parts — an 80° angle and then a further angle of x°.
T
So, the angle CAT must be (80 + x)°.
Guided Practice Don’t forget: The little square symbol in the corner of the angle MED ( ) means that corner is a right angle. A right angle has a measure of 90°.
Exercises 1–2 make use of the diagram on the right.
M L
1. Write an expression for the measure of angle LED if MED = 90° and MEL = r°.
r°
2. Evaluate your expression to find the measure of angle LED when r = 30. E
94
Section 2.3 — Geometrical Expressions
D
The Angles in a Triangle Always Add Up to 180° Check it out: There’s a lot more about angles in triangles in Section 7.2. But for now, just remember that the measures of the angles in a triangle add up to 180°.
The measures of the angles in a triangle always add up to 180°. You can use this fact to make expressions for unknown angles. Example
2
Use the diagram below to write an equation involving x. Solve it, given that j = 70.
j° 80° x° Check it out:
Solution
There are other correct equations you could have, but they would all be equivalent to j + x = 100. For example, you could have: 80 + j + x = 180 x = 100 – j
Use the fact that the angle measures in a triangle add up to 180°. In the form of an equation, this is: 80 + j + x = 180 Or, subtracting 80 from both sides, you get: j + x = 100 To solve this, put j = 70. Then your equation becomes: 70 + x = 100 Now you can subtract 70 from both sides to get x = 30º.
Example
3
Find the values of x and y in the diagrams below. 1
x° x°
2
2y° y°
Solution
There’s a right angle (90°) in the first triangle, so x + x + 90 = 180. This means: x + x = 90, or 2x = 90. Divide both sides of this equation by 2 to get x = 45. This means the two equal angles in the first triangle are 45°. There’s also a right angle (90°) in the second triangle, so y + 2y + 90 = 180. This tells you: 2y + y = 90, or 3y = 90 Divide both sides of this equation by 3 to get y = 30. So the unknown angles in the second triangle measure 30° and 60°.
Section 2.3 — Geometrical Expressions
95
Guided Practice 3. Write an equation for c in the triangle on the right. Then use your equation to find c.
c°
4. Mark draws a triangle and starts measuring the angles. The first angle he measures is 70°. 40º He labels the other two angles’ measures q and w. Write an equation involving q and w. 5. Mark measures w and finds out that it is 50. Use your expression from Exercise 4 to find the value of q.
Independent Practice Write an equation for v in each of Exercises 1–2. Now try these: Lesson 2.3.7 additional questions — p437
1.
2. L The measure of angle PQR = 120° R eº vº aº vº P A Q Exercises 3–4 are about this regular hexagon.
G
3. By considering the shaded triangle, write an equation containing a. Use your equation to find a.
bº
aº 4. Write an equation linking a and b. aº aº Use this equation and your answer to Exercise 3 to find the value of b. Exercises 5–6 are about the triangle on the right. 5. Omar measures the angle LQP and finds that it is 100°. He next measures the angle QLP, and calls it w. Write an equation for m, the measure of angle LPQ.
L w° m° 100°
6. Omar finds that w = 20°. Use your answer to Exercise 5 to find the size of angle LPQ.
Q
Exercises 7–8 are about the ladder shown below. 7. Jasmine leans a ladder against a wall, and sees a right triangle. She measures the angle the ladder makes with the ground, g°. Write an equation for k in terms of g. 8. The angle with measure g° turned out to be 75º. Use your equation from Exercise 7 to find k.
k°
g°
Round Up The facts about angles inside a triangle often provide an excuse for some expression work. The technique is the same as for every other expression question — make an expression then you’ll probably have to work something out using it. 96
Section 2.3 — Geometrical Expressions
P
Lesson
Section 2.4
2.4.1
Analyzing Problems
California Standards:
Part of math is being able to apply it to the real world. That means spotting what math to use and when to use it. Sometimes problems can be reduced to equations, other times you just need to think logically.
Mathematical Reasoning 1.1 Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, identifying missing information, sequencing and prioritizing information, and observing patterns. Mathematical Reasoning 2.4 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning.
Some Problems Involve Patterns When analyzing a problem, you need to try to figure out what is happening. Sometimes this means spotting a pattern in the results. Example
1
Find the next pattern in this sequence: ?
What it means for you: You’ll practice two of the fundamental math skills — being able to identify what is happening in different situations and how to use the information you’re given.
Solution
The shape is rotated clockwise by 90° every time. So the next one is
Example
Key words: • pattern • sequence • relationship
.
2
Find the next pattern in this sequence: ?
Check it out: This is an example of two patterns being combined. The first pattern is that the square rotates, so that the dots are on the next side in the clockwise direction. The second pattern is that the number of dots alternates — 1, then 2, then 1, and so on.
Solution
The pattern alternates between squares with two dots and squares with one dot. So the next square should have two dots. Every two squares, there is a rotation clockwise by 90°, so the next square should have its dots along the left edge of the square. So the next one is
.
Guided Practice In Exercises 1–2, find the next shape in the pattern. 1. 2.
? ? Section 2.4 — Problem Solving
97
Patterns Can Be Number Based Patterns don’t just crop up in questions with pictures — they can occur in questions with numbers too. Example
3
Find the next number in this sequence: 1, 3, 5, 7, 9, 11, ... Solution
Each number is 2 more than the one before it. So the next number must be 13.
Example
4
Find the next number in this sequence: Check it out: It’s always worth looking at the differences between consecutive numbers in a sequence like this. Write the differences down — it can make a pattern easier to spot. So in Example 4, you’d write down 2, 3, 4, 5...
1, 3, 6, 10, 15, ... Solution
Between the first number and the second, 2 is added. Between the second number and the third, 3 is added. Between the third number and the fourth, 4 is added. Between the fourth number and the fifth, 5 is added. So, you should add 6 to get from the fifth number to the sixth. That means that the sixth number is 15 + 6 = 21.
Guided Practice
Check it out: Sometimes it’s not the value of the numbers that’s important, but the pattern they make. Look at Exercise 9, for example.
98
In Exercises 3–10, find the next number in the pattern. 3. 1, 2, 3, 4, 5, 6, ... 4. 10, 9, 8, 7, 6, ... 5. 4, 6, 8, 10, 12, ... 6. 1, 6, 11, 16, 21, ... 7. 1, 11, 111, 1111, 11,111, ... 8. 40, 35, 30, 25, 20, 15, ... 9. 1, 16, 161, 1616, 16,161, ... 10. 1, 2, 4, 7, 11, 16, ...
Section 2.4 — Problem Solving
Some Problems Ask Things in a Strange Order Some problems might look complicated, even though the math behind them is actually quite simple. Example
5
On his birthday, Scott wonders how old the public library was the year he was born. Scott is now 12 and he knows that the public library was 50 years old, 9 years ago. How old was the library when Scott was born? Solution
Check it out: The only math this question asks you to do is adding and subtracting integers. But the way it asks the question is very complicated — and you need to think about it very logically and carefully.
The question gives you two pieces of information about two different times: • Now: Scott is 12. • 9 years ago: The library was 50 years old. You need to work out how old the library was when Scott was born. From your first piece of information, you can see this was 12 years ago. You now need to work out how old the library was 12 years ago. From the second piece of information, you can see it was 50 years old 9 years ago. So 12 years ago it must have been 47 years old.
Guided Practice 11. Eduardo needs another $17 to be able to afford a new video game that costs $40. Yesterday he spent $10 on comic books. How much did he have before he bought the comic books?
Now try these: Lesson 2.4.1 additional questions — p437
Independent Practice In Exercises 1–2, find the next shape in the pattern. 1. 2.
? ?
In Exercises 3–5, find the next number in the pattern. 3. 1, 2, 3, 11, 12, 13, 21, 22, 23, ... 4. 1, 10, 100, 1000, ... 5. 25, 27, 23, 25, 21, 23, 19, 21, 17, ... 6. Mayra has 16 oranges. Her father eats 4 of them, then she shares the rest equally between herself and her two brothers. How many do they get each?
Round Up Analyzing problems is an important part of math and an important thing in real life. Ask yourself what is going on and what information you have, then try to combine the two. It can be difficult to put all the pieces together, but just take it slowly and don’t panic. Section 2.4 — Problem Solving
99
Lesson
2.4.2
Important Information
California Standard:
Sometimes solving a problem might be hard because you have a lot of information, including some that you don’t need. Other times, solving a problem might be hard (or even impossible) because you don’t have enough information. This Lesson is a chance to practice dealing with both kinds of problems.
Mathematical Reasoning 1.1 Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, identifying missing information, sequencing and prioritizing information, and observing patterns.
What it means for you: This Lesson is all about questions that don’t contain the right amount of information. You’ll practice identifying information that doesn’t matter and information that is missing.
Sometimes You Have Too Much Information Sometimes a math problem may try to confuse you by giving information that you don’t need. You need to be able to figure out what information is important and what isn’t. Example
1 Menu Sandwich Salad Drink French fries
The price list for items sold at the school Snack Bar is shown on the right. Maria has $10 and buys a sandwich and a drink. How much change should she get?
$3.55 $2.25 $1.35 $0.90
Solution
Key words: • important • missing • information
Maria has spent $3.55 + $1.35 = $4.90. As she started with $10, she should have $10 – $4.90 = $5.10 change. Although the question shows the price of salad and French fries, Maria doesn’t buy either of those, so you have to ignore that information. Example
2
What is the measure of the angle ABC? A 15°
C 23°
Solution
The measure of angle ABC is 23° + 15° = 38°.
B
D
The question also tells you that angle CBD is a right angle. But you don’t need that information, so you can ignore it.
Guided Practice
Don’t forget: Read the question carefully... Here, you just have to spot the unimportant pieces of information — you don’t have to solve the problems themselves.
100
In Exercises 1–3, identify any pieces of information that aren’t needed. 1. Find the area of a rectangle which is sized 6 inches by 3 inches and has a perimeter of 18 inches. 2. Deidra earns $5 an hour as a babysitter and $6.50 as a lifeguard. Last weekend, she babysat for 4 hours. Mr. Smith’s Bills How much did she earn? $700.00 Rent 3. Mr. Smith’s monthly bills are shown on the right. How much did he spend on rent and groceries?
Section 2.4 — Problem Solving
$58.00 Phone Utilities $75.00 Groceries $186.00
Some Problems Don’t Give Enough Information Sometimes you’ll have a problem that you can’t answer because you haven’t been told enough. When that happens you need to be able to figure out what information is missing. Example
3
Raul is at the store to buy cheese, walnuts, and grapes. He wants to buy one pound of cheese and 1.5 pounds each of grapes and walnuts. If cheese costs $1.75 per pound, how much will Raul spend altogether? Solution
You would need to know the cost of walnuts and the cost of grapes to answer this question. Example
4
The local park has a triangular garden area, as shown on the right. What is the value of x?
x°
55°
Solution
There is not enough information to solve this problem. You need to know the measures of two angles in a triangle to find the measure of the third angle. But the measure of only one angle is given.
Guided Practice In Exercises 4–6, identify the piece of important information not given. 4. Mr. Mendoza is building a rectangular play yard for his children. The length of the play yard is 8 meters. What is the area? 5. Ronald earned $175.00 working at the grocery store last week. How much does he earn per hour working at the grocery store? 6. Bill weighs 10 pounds more than his sister. What does Bill weigh?
It Can Be Hard to Tell If You Have Enough Information Sometimes it’s not obvious if you have enough information or not. You have to start a problem and see where you can get to. Sums to 15
These next examples are about “magic squares.” A magic square is a 3 × 3 grid containing the numbers 1 to 9, where each row, column, and diagonal adds up to the same number (in these examples, 15).
Sums to 15 Sums to 15 Sums to 15 All columns sum to 15
Sums to 15
Section 2.4 — Problem Solving
101
Example
5
9 2 5
Complete the magic square shown. All the rows, columns, and diagonals must sum to 15. Don’t forget:
Solution
Rows go across (they’re horizontal).
The numbers 2, 5, and 9 have already been used. That means you have to put the numbers 1, 3, 4, 6, 7, and 8 in the grid, making sure that all rows, columns, and diagonals sum to 15.
row row row
Columns go down (they’re vertical).
There are a lot of gaps. At this stage, you can’t be 100% sure you have enough information. But you can make a start... First of all, you can fill in the numbers shown in red. But then you have enough information to fill in the numbers in green.
columns
And finally, you can fill in the number in the center-right box. It must be 7.
This must be 15 – (9 + 2), using the top row. This must be 15 – (2 + 5), using one of the diagonals.
4 9 2 3 5 7 8 1 6 This must be 15 – (9 + 5), using the middle column.
So here, you did have enough information.
Example
6
Do you have enough information to complete this magic square? All the rows, columns, and diagonals must sum to 15.
9 5 3
Solution
You can work out the numbers in red straightaway. But then there’s no easy way to carry on. It looks like you don’t have enough information. But... if you try to put the next number in, you discover something. You can’t put “8” in any of the colored squares. • If you put it in the green square, then the top row sums to more than 15. • If you put it in the blue square, then the top row also sums to more than 15. • If you put it in the red square, then the left-hand column will sum to more than 15 (you can’t use 0). So “8” has to go in the bottom-right square. And then you can fill in all the others.
9 7 5 3 1 9 7 5 3 1
2 9 4 7 5 3 6 1 8
So you did have enough information, but you had to think hard about it. 102
Section 2.4 — Problem Solving
Guided Practice 7. Do you have enough information to complete the magic squares on the right? All the rows, columns, and diagonals must sum to 15. What do you discover when you try to complete the second magic square?
2 5 9
5 9 4
Independent Practice In Exercises 1–6, say whether you have too much or too little information to answer the question. If you have too much information, then say which information you don’t need. If you have too little, then say what information you would need. Now try these: Lesson 2.4.2 additional questions — p437
1. At 4:00 p.m., the temperature was 45° Fahrenheit. Each hour, the temperature dropped by the same amount. What was the temperature at 7:00 p.m.? 2. Damian is ten minutes late for school. He set off at 8:30 a.m. and it took him 40 minutes to get to school. During that time he had a 5-minute stop to buy a drink. What time does school start? 3. Marissa is a doctor. On Tuesday, she sees 12 patients in the morning, and in the afternoon she sees a further 10. To help out another doctor, Jorge, she agrees to see 3 of his patients on Wednesday. How many patients did she see in total on Tuesday? 4. The Coleman family goes to the movies. Adult tickets sell for $4.50 and children’s tickets sell for $2.00. How much will the Coleman family pay for tickets? 5. Mario needs to buy 120 meters of pipe, which he intends to divide into 8 equal parts. Pipe costs $10/meter. How much money does he need to buy the pipe? 6. Joshua is playing a game of tic-tac-toe against Luis. Luis says that for every game he loses, he will buy Joshua a soda. In the end, Luis loses four games. How much money must he spend on soda? 7. Do you have enough information to complete the magic squares on the right? All the rows, columns, and diagonals must sum to 15.
5 9 4
4 9 5
Round Up Being able to tell the difference between information that matters and information that doesn’t matter is important in math and in real life. The main thing is not to panic. Just go through each fact and decide whether it is important or not. When you get to the end, see if you have enough information to answer the question. Section 2.4 — Problem Solving
103
Lesson
2.4.3
Breaking Up a Problem
California Standards:
This Lesson is about some very general ideas that you can use in all the other math lessons you do this year, next year, the year after that, and so on. The idea is to make the most of every little bit of math knowledge you have — a very useful thing to be able to do.
Mathematical Reasoning 1.3 Determine when and how to break a problem into simpler parts. Mathematical Reasoning 2.2 Apply strategies and results from simpler problems to more complex problems. Mathematical Reasoning 2.4 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. Mathematical Reasoning 2.5 Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language; support solutions with evidence in both verbal and symbolic work. Mathematical Reasoning 2.7 Make precise calculations and check the validity of the results from the context of the problem.
What it means for you: This is about being smart.
Key words • question technique • units • reasonable
Check it out: Write the numbers in rows of 4 until you’ve got 20 of them. 3, 9, 7, 1 3, 9, 7, 1 3, 9, 7, 1 3, 9, 7, 1 3, 9, 7, 1
Don’t forget: 32 means the same thing as 3 × 3. 33 means the same thing as 3 × 3 × 3, and so on. The small raised number is called the exponent.
104
Don’t Do More Work Than You Need to Sometimes solving a problem directly is very difficult. In those cases, look for any clever shortcuts you can to make things easier. Example
1
Linda is working on an extra-credit assignment. Her teacher wants her to find the last digit of 320 without using a calculator. Linda is, however, allowed to use the table below. How can Linda find the answer most efficiently? n
3n
Last Digit of 3n
n
3n
Last Digit of 3n
1
3
3
6
729
9
2
9
9
7
2187
7
3
27
7
8
6561
1
4
81
1
9
19,683
3
5
243
3
10
59,049
9
Solution
One way would be to work out more and more powers of 3. So she could work out 311 by calculating 59,049 × 3. Then she could find 312 by multiplying this answer by 3, and so on until she got to 320. But it would take a long time, and it would be very easy to make a mistake. However, Linda spots a pattern in the final digits. They go 3, 9, 7, 1, and then repeat. So by writing down 3, 9, 7, 1 again and again until she reaches the 20th number, Linda works out that the last digit of 320 must be 1.
Guided Practice 1. By making a table for small exponents, find the last digit of 430. 2. Marissa is asked to add up every number from 1 to 100. To try to find a pattern, she paired up the numbers in the sum. She added the first number (1) to the last (100), the second number (2) to the second to last (99), and so on. She found that each pair added up to 101. She then realized that she could find the answer by doing just one multiplication. What multiplication should she do?
Section 2.4 — Problem Solving
The next example has been broken up into several stages. It’s not the problem itself that’s the important thing here, but the approach to completing the task.
Break Problems Up into Smaller Parts Sometimes a big, complicated problem is easier if you split it up. So you take your big, complicated problem and break it up into many smaller, simpler problems that you can solve. Example
2
The diagram below shows a backyard with a swimming pool. What area of the backyard is grass? 80
15
Don’t forget: There will often be different ways to break the problem up. Try to look for the way that will make things easiest for you.
40
All lengths are in yards.
15 Solution — Part 1
You’ve actually seen a similar kind of problem already — in Section 2.3. What you did then was to break the big, complicated problem up into smaller, easier-to-solve problems. That’s all you’re doing here too. To find the area of the grass, you can: • Find the total area of the backyard. • Find the area of the swimming pool. • Subtract the area of the swimming pool from the area of the backyard.
Always Show How You’re Solving the Problem When you solve a problem in math, you should always explain how you got your answer. Use any method that explains your reasoning — for example words, numbers, symbols, charts, tables, graphs... Example
2 (continued)
Look again at the problem in Example 2 above. Solution — Part 2
Check it out: Doing this will help your teacher follow your work. But it will also help you when you check your work.
You should always write down exactly how you’re going to find your answer. So here, you could write: Area of grass = Total area of backyard – Area of swimming pool Or you could draw a diagram:
80 15 15
80 40
=
40
–
15 15
Section 2.4 — Problem Solving
105
Always Show Your Work Even if you use a calculator to do the actual calculations, it’s best to write down all the different answers you found on the way. Example
2 (continued)
Check it out:
Look again at the problem in Example 2 on the previous page.
If you write all your intermediate answers down, then it’s easier to spot mistakes when you check your work.
Solution — Part 3
Total area of backyard = 80 × 40 = 3200 Area of swimming pool = 15 × 15 = 225 So area of grass = 3200 – 225 = 2975
Write Your Answer Clearly, and Remember Units There’s still something missing from the above answer. As it stands, it’s not clear if the answer is 2975 m2, 2975 in2, 2975 yd2, or something entirely different. Example
2 (continued)
Look again at the problem in Example 2 on the previous page. Solution — Part 4
Always check the question to see if you should include units. Area of grass = 3200 – 225 = 2975 yd2.
Check Your Answers You’ve found an answer. But you should always check it — it’s easy to press the wrong button on a calculator, for example. Example
2 (continued)
Look again at the problem in Example 2 on the previous page. Solution — Part 5
Suppose your calculator gave you an answer of 31,775 yd2. You can quickly check if it’s reasonable. Don’t forget: You can use rounding to check your answer — see Section 1.4.
106
The area of grass has to be less than the total area of the backyard. The total area of the backyard was 80 × 40 yd2 = 3200 yd2, so the answer has to be less than this. So the answer of 31,775 must be wrong. It looks like you pressed the calculator’s zero button too many times, giving 32,000 – 225 instead of 3200 – 225.
Section 2.4 — Problem Solving
Guided Practice 3. Tyler and Ashlee both tried to multiply 1.3 by 2 and both got the wrong answer. Their work is shown below. Identify the error each made, if possible. Tyler: 1.3 × 2 = 9.3 Ashlee: 1.3 × 2 = 2 × (1 + 0.3) = (2 × 1) + (2 × 0.3) = 2 + 0.5 = 2.5 4. At the amusement park, the roller coaster ride takes 2 minutes and the merry-go-round takes 3.5 minutes. Gabriela rode the roller coaster two times and the merry-go-round three times. What is the total amount of time that she spent on rides?
Check it out: Check your answers to Exercises 6–9 by substituting your answer back into the original equation.
5. A half-gallon container of orange juice sells for $3.75, and a half-gallon container of apple juice sells for $2.85. Mr. Gonzales buys three containers of orange juice, and Ms. Chung buys two containers of apple juice. How much more than Ms. Chung did Mr. Gonzales spend? Solve the equations given in Exercises 6–9 and check your answer. 6. y ÷ 3 = –16 7. r × 9 = 72 8. p × 11 = –66 9. q ÷ 6 = 6
Independent Practice Find the white shaded areas of the shapes given in Exercises 1–3. All measurements are given in inches. 1. Now try these: Lesson 2.4.3 additional questions — p438
3
2.
2 7
3. 10
7
11 45
4
10 10
4. By making a table for small exponents, find the last digit of 720. 5. Jim is selling necklaces at $1 per necklace. Erin buys 4 and Olivia buys 6. Madison buys half of Erin’s for $1.50 per necklace. How much more did Olivia spend than Madison? 6. Manuel is trying to find out if 174 467 093 477 549 819 816 can be divided exactly by 4. He notices that 12, 112, 212, 312, and 412 can all be divided exactly by 4. What pattern has he noticed? Find a pattern to help decide whether the large number can be divided exactly by 4.
Round Up So you see... most of this is common sense. Always show your work, check your answers, and so on. But it’s small things like this that really help you to get more questions right, more of the time, and that can’t be a bad thing. Section 2.4 — Problem Solving
107
Chapter 2 Investigation
Design a House
Write an expression for the area of each of the 5 rooms in the house. Things to think about:
x
y
Bedroom #1
Bathroom
Part 1:
15 feet
You’re an architect. Your clients know the layout they want in their new home, but they haven’t yet decided on the exact measurements of all the rooms. So some variables have been used in the floorplan below.
Kitchen
Combined Hall
4 feet
15 feet
and
• To check your answers, think about what these areas should add up to.
Living room Bedroom #2
Part 2:
45 feet
On the right is a price list for carpets and floor tiles. The bedrooms and hall/living room are to be carpeted, but the bathroom and kitchen will be tiled. Write an expression for the total cost of carpets and floor tiles.
Carpet: $2 per square foot Tiles: $3 per square foot Start by writing an expression for each room.
Extension 1) Write an expression for the perimeter of each room. 2) If the height of all the rooms is 9 feet, and all the house’s windows and doors have a total area of 162 ft2, write an expression for the total internal wall area. 3) All the internal walls are to be painted, and paint costs $0.20 for each square foot of wall. How much will the paint cost for all the internal walls? 4) The clients decide that the floor area of the bedrooms is to be 270 square feet each. Use this information to find x. Can you find the final costs of paint and floor coverings? Open-ended Extension Explore different floor layouts for a house. Start with a one-room house. You want your house to have a floor area of 400 square feet. However, you want to do as little building as possible, so you want the total length of walls to be as small as possible. 1) What shape should your house be? 2) Would you get a different answer if the walls didn’t have to be straight? 3) What if you want your house to have three rooms, each with an area of 400 square feet? What is the best layout if you want the total length of walls to be as short as possible?
Round Up Remember... If you don’t know the value of something, you can just give it a variable name instead. You might be able to work it out later, when you’ve found out some more information. pter 2 In vestig a tion — Design a House Chapter Inv estiga 108 Cha
Chapter 3 Fractions and Percentages Section 3.1
Understanding Fractions ....................................................... 110
Section 3.2
Exploration — Multiplying Fractions: an Area Model............. 124 Multiplying Fractions .............................................................. 125
Section 3.3
Dividing Fractions .................................................................. 136
Section 3.4
Exploration — Adding Fractions: an Area Model .................. 144 Adding and Subtracting Fractions ......................................... 145
Section 3.5
Exploration — Percents with a Double Number Line ............ 167 Percents ................................................................................ 168
Chapter Investigation — Wildlife Trails ......................................................... 193
109
Lesson
Section 3.1
3.1.1
Understanding Fractions
California Standard:
You’ve worked with fractions in earlier grades and so you should already have a good idea of what they represent. This Lesson reviews some of those concepts and introduces the idea of negative fractions.
Number Sense 1.1 Compare and order positive and negative fractions, decimals, and mixed numbers and place them on a number line.
What it means for you:
Fractions Can Show a Part of the Whole A fraction can represent a part of a single object. For example: The spinner has been divided into seven equal parts (sevenths), and one of these parts is green.
You’ll first review what fractions show. Then you’ll mark positive and negative fractions that fall between –1 and +1 on a number line.
Key words: • • • •
1 7 3 7
(one-seventh) of a spinner is green. (three-sevenths) of the spinner is red.
A fraction can also represent a certain number of a set of objects. For example: 3 10
denominator equivalent fraction numerator negative
(three-tenths) of a set of
ten bowling pins is still standing. 7 10
(seven-tenths) of the set has
been knocked down.
Don’t forget: You can think of the number on the bottom as how many parts there are in all. Then the number on the top is how many of those parts you are talking about.
Fractions Can Be Expressed in Different Ways 2 8
of this circle is shaded. You can see that this is the same 1
as one quarter ( 4 ).
2 8
and
1 4
are equivalent fractions.
Equivalent fractions have the same value. If you have 30 students in your math class and 15 are boys, you could say that
15 30
of the students are boys. However, you could also say that
the students are boys.
15 30
and
1 2
are equivalent fractions.
Guided Practice 1. What fraction of the spinner above is blue? 2. 17 out of the 24 students in a class have a pet dog. Express the number that have a dog as a fraction. 3. What fraction of this triangle is blue? Give a fraction that is equivalent to this. 4. You invite 8 friends to a party, but two of them can’t come. What fraction of your 8 friends can’t come? Give a fraction that is equivalent to this. 110
Section 3.1 — Fractions
1 2
of
Don’t forget: The numerator is on the top of the fraction. The denominator is on the bottom.
3 4
Two-Halves and Three-Thirds Both Equal 1 Whole If the numerator and the denominator of a fraction are the same, the fraction is equal to 1.
2 3 6 10 54 = = = = =1 2 3 6 10 54
So, for example:
numerator denominator
The circle below is divided into thirds. Each segment is circle. Two segments make up segments make
3 3
2 3
1 3
of the whole
of the whole circle, and all three of the
of the circle — which is equivalent to the whole circle.
1 3
whole circle
3 3
2 3
= whole circle
Fractions Can Be Shown on a Number Line Just like integers, fractions can be shown at positions on the number line. Think of 1 on the number line as representing one whole. Positive fractions with a numerator smaller than the denominator, like
1 2
or
4 , 5
represent parts of a whole. This means they are between 0
and 1 on the number line. Example Show
1 10
1
and
3 10
on a number line.
Solution
The number 1 on the number line represents 1 whole. It could also be written as
10 . 10
So divide the distance between 0 and 1 into 10 equal parts (tenths). 1 1 1 10 10 10
1
0
-1 1 10
is one-tenth of
1 3 10 10
the way from 0 to 1.
3 10
2
10 10
is three-tenths of the way from 0 to 1.
Guided Practice Mark the fractions in Exercises 5–9 on a number line. 5.
1 4
6.
1 5
7.
1 8
8.
2 10
9.
8 10
10. Which of the two fractions above are at exactly the same position? Section 3.1 — Fractions
111
Fractions Can Be Negative Too °C
Imagine at 9 p.m. the temperature is 1 °C. 1 2 1 is 2
It then falls by At 10 p.m. it
–1 2
°C, then at 11 p.m. it is at 0 °C.
Over the next hour it falls by another This takes it to
1
1 –2
1 2
1 2 °C
1 2
a degree every hour until midnight.
1 2 °C
0
1 2 °C
–1
degree.
°C at midnight.
This illustrates that just like integers, fractions can be negative. 1
51
So you can have values such as – 2 , – 4 , or – 73 . 5
Negative Fractions Can Be Shown on the Number Line Negative fractions are hard to picture in the same way as positive fractions. They can be shown on the number line though. Like negative integers, they are positioned to the left of zero. A negative fraction with a numerator smaller than the denominator will always be between 0 and –1 on the number line. Negative fractions Positive fractions Part of this number line has been divided into fourths. 1 2 3 − , − , and − are all 4 4 4 between 0 and –1.
–1 4
-1
–3 4
There’s lots more information about positive and negative integers in Section 1.1.
–1 4
0
–1 4
–2 4
–4 4
Don’t forget:
–1 4
–1 4
1
Negative integers decrease in value the further to the left you go. The same is true of negative fractions. So the further to the right on the number line a negative fraction lies, the greater it is. 1
3
On the number line above, you can see that – 4 is greater than – 4 , as it lies further to the right. You can write Example 1
1 3 − >− , 4 4
or
3 1 − <− . 4 4
2 5 6
Show – 6 and – on a number line. Which is greater? Solution
Divide the distance between 0 and –1 into 6 equal parts (sixths). –1 6
–
5 is five-sixths of 6
the way from 0 to –1.
–1
–5 6
–1 6
–1 6
–1 6
–1 6
–1 6
1
0 –
1 is one-sixth of the way from 0 to –1. 6
– 1 lies further to the right, so it is greater, or 6
112
Section 3.1 — Fractions
1 5 − >− . 6 6
Guided Practice Mark each pair of fractions in Exercises 11–13 on a number line. Say which of each pair is greater. 11. – 4 , – 1 5
12. – 1 , – 5
5
8
13. –
8
1 , 10
–
7 10
14. What do you notice about the positions on the number line of the following pairs of fractions: – 1 and 4
Check it out: Absolute value is the size of a number, ignoring any minus signs. For example, the absolute value of 4 is 4, but the absolute value of –4 is also 4. The absolute value of both 0.5 and –0.5 is 0.5.
1 , 4
– 4 and 5
4 ? 5
Negative Fractions Behave Like Negative Integers When you add together a positive fraction and a negative fraction with the same absolute value, you get back to zero. Example
3
Use a number line to find
1 2
1
+ (– 2 ).
Solution
Start at
1 . 2
When you added a negative integer you moved to the left.
You add a negative fraction in the same way. So move
1 2
a space to
1 2
the left. –1
1 2
0
1
2
The solution is 0.
Guided Practice Use a number line to find the values of the following: 15. – 4 + 5
4 5
16. – 1 + 8
1 8
17.
7 10
+ (–
7 ) 10
Independent Practice 1. Draw a rectangle. Shade and label Now try these: Lesson 3.1.1 additional questions — p438
3 5
of it.
2. Tim has visited 13 out of the 50 states. Express this as a fraction. What fraction of the states has he not visited? 3. 26 out of 52 playing cards are black. Express this as a fraction. What simpler fraction is this equal to? Mark the fractions in Exercises 4–6 on number lines. 4.
2 3
2
and – 3
5.
2 5
2
and – 5
6.
1 9
1
and – 9
Round Up The fractions you’ve looked at in this Lesson are mainly proper fractions — where the numerator is smaller than the denominator. In the next Lesson you’ll look at improper fractions. In these, the numerator is bigger than (or equal to) the denominator. Section 3.1 — Fractions
113
Lesson
3.1.2
Improper Fractions
California Standard:
In the last Lesson, the positive fractions you looked at mainly lay between 0 and +1 on the number line — they had numerators that were smaller than their denominators. In this Lesson, you’ll look at improper fractions. In these, the numerator is bigger than or equal to the denominator.
Number Sense 1.1 Compare and order positive and negative fractions, decimals, and mixed numbers and place them on a number line.
Improper Fractions — They’re “Top-Heavy” What it means for you: You’ll develop techniques for dealing with improper fractions. This will give you the skills needed to place such fractions on the number line.
Key words: • • • • •
denominator improper fraction numerator proper fraction mixed number
An improper fraction is one whose numerator is greater than or equal to the denominator.
This is only true for positive numbers.
Some examples of improper fractions are shown below: 5 2
(five-halves)
circles are shaded. 4 3
(four-thirds)
triangles are shaded.
6 6
(six-sixths) of a
hexagon is shaded.
Don’t forget: You met fractions in which the numerator is equal to the denominator in the last 6
Lesson (like ) — they are 6 always equivalent to 1.
Mixed Numbers — Whole Numbers Plus Fractions A mixed number is one that shows the sum of a whole number and a fraction. Here are some examples of mixed numbers: 3
1 4
(three and one-quarter)
squares are shaded. 1
There are 2 3 (two and one-third) glasses of water.
1
1 (one and one-eighth) 8 large cubes are shaded.
114
Section 3.1 — Fractions
Mixed Numbers Have Equivalent Improper Fractions The diagrams shown on the previous page can be represented by both mixed numbers and improper fractions. For example: 5 2
shaded circles is the same as saying 2 1 shaded circles. 2
2 whole circles and one half-circle are shaded.
1 4
13 4
3 squares are shaded is the same as saying
squares are shaded. Thirteen quarter-squares are shaded.
Guided Practice Represent the shaded fraction of these groups of shapes using a mixed number and an improper fraction. 2.
1.
Converting Mixed Numbers to Improper Fractions Suppose you need to convert 2 1 to an improper fraction. You could draw 4
a diagram. The mixed number 2 1 can be represented by this diagram: 4
Check it out: In this method, you’re working out the number of fourths in 1 4
2 . There are 2 × 4 = 8 fourths in two. 1 4
1 4
1 4
1 4
1 4
1 4
1 4
1 4
Then there is also one extra fourth that was expressed as the fraction part.
The diagram shows that 2 1 consists of a total of 9 quarters, or 4
But you don’t really need a diagram — you can work out the numerator and the denominator without one. The new denominator in the improper fraction is always the same as in the equivalent mixed number. A quick way of calculating the new numerator is to multiply the whole number part by the denominator and add the old numerator.
1 4
8 fourths, plus 1 fourth equals 9 fourths, also written as the improper fraction,
9 . 4
Mixed number
1
( 2 × 4) + 1
4
4
2 =
=
9
Improper fraction
4
9 . 4
the denominator remains the same.
Section 3.1 — Fractions
115
Example
Don’t forget: There are 4 × 6 = 24 sixths in four.
1
5
Write 4 6 as an improper fraction. Solution
1 6
1 6
1 1 6 6 1 1 6 6 1 1 6 6 1 1 6 6
1 6
1 6
1 1 6 6
1 6
1 1 6 6 1 1 6 6
1 6
1 1 6 6
1 6
1 1 6 6 1 6
( 4 × 6) + 5
6
6
4 = =
1 6
Then there are also an extra five sixths that were expressed as a fraction before. 1 6
5
29 6
To find the improper fraction’s numerator, multiply the whole number part by the denominator, then add the original numerator to this.
29
Keep the denominator the same.
6 5
is an improper fraction equal to 4 6 .
Guided Practice 1 6
So there are (4 × 6) + 5 = 29 sixths in total.
Write the following mixed numbers as improper fractions. 1
3
3. 5 4
3
4. 4 5
2
4
7. 8 3
1
5. 1 8
6. 10 10
3
8. 7 5
7
9. 2 7
10. 9 8
Converting Improper Fractions to Mixed Numbers Now suppose you need to convert 7 3
7 3
to a mixed number. The improper
fraction is seven-thirds. But to make a whole, you only need three-thirds. Seven-thirds can be split up like this: three-thirds + three-thirds + one-third = seven-thirds
From the diagram, you can see that
7 3
1
is the same as 2 3 .
A faster way of doing this is by division. To get the whole number part, you need to find out how many times the denominator of the improper fraction is contained in the numerator of the improper fraction. The remainder (the number of parts left over) becomes the numerator of the fraction in the mixed number. This is the whole number part of the mixed number.
7 3 116
Section 3.1 — Fractions
Divide the numerator by the denominator.
The remainder is the numerator in the mixed number.
2 R1 3 7 –6 1
)
2 The denominator remains the same.
1 3
Example 59 4
Write
2
as a mixed number.
Solution
14 R 3 59 → 4 59 4 –4 19 – 16 3
)
→ 14
Divide the numerator by the denominator.
3 4
59 3 is written as 14 4 as a mixed number. 4
Guided Practice Write the following improper fractions as mixed numbers. 11.
8 5
12.
9 4
13.
11 8
14.
12 11
15.
16 13
16.
21 2
17.
23 7
18.
91 8
Independent Practice Represent the shaded fraction of the groups of shapes in Exercises 1–2 by both a mixed number and an improper fraction. 1.
2.
Convert the mixed numbers in Exercises 3–10 to improper fractions. 5
1
3. 2 7
4. 4 6 1
Now try these: Lesson 3.1.2 additional questions — p438
7. 27 2
6
8. 12 7
7
5. 1 11 2
9. 19 3
9
6. 8 13 11
10. 6 14
Convert the improper fractions in Exercises 11–18 to mixed numbers. 11.
87 43
12.
12 7
13.
37 8
14.
53 4
15.
53 6
16.
61 9
17.
30 7
18.
72 5
Round Up After this Lesson, you should be able to convert improper fractions to mixed numbers. This is important for placing fractions on the number line — you’ll see this in the next Lesson. You should also be able to convert mixed numbers to improper fractions. You’ll use this skill in adding, subtracting, multiplying, and dividing fractions. Section 3.1 — Fractions
117
Lesson
3.1.3
More on Fractions
California Standard:
In this Lesson, you’ll practice putting improper fractions and mixed numbers on the number line. You’ll also look at negative improper fractions and place them on the number line.
Number Sense 1.1 Compare and order positive and negative fractions, decimals, and mixed numbers and place them on a number line.
What it means for you: You’ll mark positive and negative fractions on the number line using the techniques you practiced for dealing with improper fractions.
Placing Positive Mixed Numbers on a Number Line When placing a positive mixed number on a number line, first find the position of the whole number. Then move the value of the fraction to the right. This is more easily shown with an example. Example 1
Place 2 4 on the number line. Solution
Key words: • • • • •
1 Split the distance between 2 and 3 into four equal parts. These are quarters.
2 is the whole number
denominator improper fraction numerator proper fraction mixed number
1
1 1 , so 2 4 4
1 4
3
2
1
0
-1
24 =2+
2
2 41
is placed one-quarter to the right of 2.
It’s a good idea to check that your answer is sensible. 1
Don’t forget: Numbers further to the right on the number line are greater. For example: 2
1 4
is greater
than 2 so it is placed further to the right.
Convert Improper Fractions to Mixed Numbers First 11
7
It’s very difficult to place an improper fraction, like 3 or 4 , on the number line. The best thing to do is convert the improper fraction to a mixed number and then place it on the number line.
Don’t forget: Convert an improper fraction to a mixed number by dividing.
2 4 is greater than 2 but less than 3. It is also closer to 2 than to 3. This matches the position marked on the number line.
3 R2 3 11
)
−9 2
Example Place
11 3
2
on the number line. Convert to a mixed number, then place it on the number line and check.
Solution
2 3
3
11 3
=
2 33,
so... 0
1
2
4
3
3 32 = 11 3
Guided Practice Place the following fractions on the number line. 1
1. 5 4 4. 118
Section 3.1 — Fractions
8 3
3
2. 4 5 5.
7 5
3
3. 1 8 6.
23 4
Placing Negative Mixed Numbers on a Number Line Like all negative numbers, negative mixed numbers are placed to the left of zero on the number line. However, it’s not always obvious exactly where they should be placed. Example
3
7
Place –3 10 on the number line. Solution
Check it out:
7
7
–3 10 can be thought of as being
7 10
“more negative” 7
than –3, or as –3 + (– 10 ). So it’s
7 10
further left on the
number line than –3.
7
–3 10 lies the same distance to the left of zero as 3 10 lies to the right. So to place it on the number line, you do exactly as you would for a positive number, but to the left instead of the right. Split the distance between –3 and –4 into tenths.
–3 is the whole number
–3
– 107 –4
–3
1
0
–1
–2
7 –3 10 7
Then a quick check: –3 10 is between –3 and –4. It’s closer to –4 than to –3. This matches the position marked on the number line.
Guided Practice Place the following mixed numbers on the number line. 1
1
3
8. –4 3
7. –2 2
1
10. –10 5
9. –1 8 3
11. –4 4
4
12. –1 5
Change Negative Improper Fractions to Mixed Numbers First
Just like with positive improper fractions, it is easiest to convert negative improper fractions to mixed numbers before you place them on the number line.
You convert negative improper fractions to mixed numbers in exactly the same way as positive improper fractions, but remember the negative sign. The example on the next page shows this.
Section 3.1 — Fractions
119
Example −
Write
47 3
4
as a mixed number and place it on a number line.
Solution
First find
47 3
15 47 → 3 47 3 3 17 15 2
)
by dividing the
numerator 47 by the denominator 3. Check it out: 47 − can also be written as 3 47 −47 or . −3 3
But remember the negative sign. So
−
47 3
2
is –15 3 as a mixed number.
R2
→ 15
2 3
Put it on the number line as usual. –2 3
–16 –47 3 =
–17
–15
–15 –15 32
–13
–14
Guided Practice Write the following improper fractions as mixed numbers and place them on a number line. 39
7
13. – 4
33
14. – 5
17
15. – 8
21
66
16. – 6
17. – 10
18. – 2
Independent Practice Convert the improper fractions in Exercises 1–3 to mixed numbers. 1.
27
13 5
2. − 4
3. −
36 7
State which one of each pair of fractions in Exercises 4–6 is greater. Now try these Lesson 3.1.3 additional questions — p439
4. −4 3 , − 4 6 7
7
5.
6. –12 7 ,
28 29 , 6 6
8
−
101 8
Show the fractions in Exercises 7–9 on a number line. 5
1
7. 18 6 , 18 6
14
8. − 3
,
−
13 3
5
9. −17 9 , −
153 9
Round Up After this Lesson, you should be able to place any positive or negative fraction or mixed number on the number line. Placing fractions on the number line is one way of comparing fractions with different denominators, but in the next Lesson you’ll learn a quicker method, involving converting them to decimals. 120
Section 3.1 — Fractions
Lesson
3.1.4
Fractions and Decimals
California Standard:
In this Lesson, you’ll learn how to convert fractions to their equivalent decimals. This allows you to decide whether a fraction or a decimal is greater.
Number Sense 1.1 Compare and order positive and negative fractions, decimals, and mixed numbers and place them on a number line.
What it means for you: You’ll learn how to compare a fraction with a decimal by first converting the fraction to a decimal.
Every Fraction Has an Equivalent Decimal Every fraction has an equivalent decimal. A fraction and its equivalent decimal lie at the exact same point on the number line. Mixed numbers also have equivalent decimals. For example: – 1 1 = –1.2 5
– 1 = –0.125 8
1 = 0.5 2
1
1 4 = 1.25
Key words: • • • • • • •
decimal denominator equivalent numerator terminating decimal repeating decimal improper fraction
Don’t forget: 1.3 means “1 plus 3 tenths” or 1
3 . 10
See Section 1.3 for
–2
–1
– 1 3 = –1.6 5
2
1
3
1 4 = 1.75
Fractions represent division. You can write a fraction as a decimal by dividing the numerator by the denominator. For example:
Example Make sure you do the division the correct way around. You are finding out how many times the bottom number goes into the top number.
3 = 0.3 10
Convert Fractions to Decimals by Division
more information.
Don’t forget:
0
Write
5 8
1 2
= 1 ÷ 2 = 0.5
7 4
= 7 ÷ 4 = 1.75
1
as a decimal.
Solution
Divide the numerator by the denominator. 0.625 8 goes into 50 six times... Æ 8 5.000 8 6 × 8 = 48... –48 20 8 goes into 20 twice... 2 × 8 = 16... – 16 40 8 goes into 40 five times... 5 × 8 = 40... – 40 there’s no remainder, so the division is finished. 0 5
Don’t forget: Doing the division on a calculator will give the same answer.
5 8
is 0.625 as a decimal. Section 3.1 — Fractions
121
Convert Improper Fractions in the Same Way This example shows that improper fractions are converted to decimals in the same way. Example Don’t forget: When you convert an improper fraction to a decimal, you get a decimal greater than 1 or less than –1.
Write
9 5
2
as a decimal.
Solution
Divide the numerator by the denominator. 1.8 5 goes into 9 once... 9 → 5 9.0 5 –5 1 × 5 = 5... 5 goes into 40 eight times... 40 –40 5 × 8 = 40... 0 there’s no remainder, so the division is finished.
)
9 5
Check it out: To convert a mixed number to a decimal, just convert the fraction part and add it to the whole number part. For example: 1
22 = 2 +
1 2
is 1.8 as a decimal.
Guided Practice Convert these fractions to decimals without using a calculator. 1.
1 4
= 2 + 0.5 = 2.5
2.
1 8
3. 7
3
5. 2 4
8 5
4. 6
6. 4 8
7. 9 15
11 10
6
8. 4 10
A Fraction Might Become a Repeating Decimal Check it out: When you divide 1 by 7 on your calculator, you see 0.1428571 on the screen. This doesn’t look like a repeating decimal, but it is. The digits 142857 repeat over and over again, so you could write
1 7
= 0.142857 .
Check it out: A fraction can be converted to either: i) a terminating decimal ii) a repeating decimal There are no other possibilities.
122
Section 3.1 — Fractions
The fractions you have used so far this Lesson all have a terminating decimal equivalent. The decimals stop after a certain number of digits. Other fractions have repeating decimal equivalents. These go on forever, repeating the same digits. For example:
1 3
= 0.3333333...,
1 11
= 0.09090909...
Repeating decimals are often written with a bar over the repeated digits. So 0.333333... can be written as 0.3 , and 0.09090909... as 0.09 .
Guided Practice Convert these fractions into decimals using a calculator. 9.
2 3
10.
7 9
11.
6 11
12.
5 6
Comparing a Fraction with a Decimal It is difficult to compare a fraction with a decimal directly. It’s best to convert the fraction to a decimal and then compare the two decimals. This might be necessary if you need to plot two numbers on a number line. Check it out: Sometimes you don’t need to do a division to compare an improper fraction with a decimal. Converting the improper fraction to a mixed number may be enough. For example: Which is greater: 13 5
13 5
or 1.9? 3 5
is equivalent to 2 .
You can easily see that this is greater than 1.9.
Example
3 6 5
Which is greater:
or 1.15?
Solution
1.2 → 5 6.0 5 –5 10 – 10 0
Convert 5 into a decimal.
1.2 is greater than 1.15.
Compare the decimal produced with the decimal in the question.
)
6
6
Guided Practice Which is the greater in each of these pairs? 13.
7 8
or 0.85
14.
8 5
or 1.7
15.
6 32
or 0.2
Independent Practice Convert the fractions in Exercises 1–4 to decimals without using a calculator. 1.
3 5
2. 3 7 8
3.
3 10
4. 2 4
5
Convert the fractions in Exercises 5–8 to decimals using a calculator. 5. Now try these: Lesson 3.1.4 additional questions — p439
3
3 11
6. 2 9
1
7. 3 6
8.
8 11
Put the sets of numbers in Exercises 9–10 in order, from least to greatest. 9. 4.32,
39 , 8
4
45
2
5
10. 6.7, 6 8 , 6 3
11. Miranda needs a piece of pipe 6.81 meters long. 3
7
13
The store stocks the following sizes: 6 4 m, 6 8 m, 6 16 m. Which size should she buy so that she has the least waste?
Round Up After this Lesson, you should be able to convert fractions to terminating decimals using long division. This is really handy if you’re doing a calculation that involves both a fraction and a decimal. Section 3.1 — Fractions
123
Section 3.2 introduction — an exploration into:
Multipl ying F ea Model Multiplying Frractions: an Ar Area Multiplying fractions has a lot in common with multiplying integers. For example, 3 × ½ means “3 groups of ½,” in the same way that 3 × 4 means “3 groups of 4.” You can show this using graph paper. In fact, using graph paper you can work out all kinds of fraction multiplications. 1 You can show fractions on graph paper. Here’s one fourth, or . 1 4 4 To multiply this, you need to make groups. You can use graph paper. 1
Example 1 Use graph paper to multiply 3× . 4 Solution
1 Draw and shade rectangles to show 3 groups of . 4 1 3 Using the picture, you can see that 3× = . 4 4
=
3 groups of
1 4
=
3 4
Exercises 1. Do these multiplications by drawing rectangles on graph paper to represent the fractions. 2 1 a. 2× b. 3× 7 3 You can even multiply two fractions together using graph paper. Example Use graph paper to multiply Solution
1 3 × . 2 5
2 and 5 are the fractions’ denominators.
Draw a rectangle on graph paper that is 2 squares wide by 5 squares tall. 3 Shade in 3 of the 5 rows to show of the whole rectangle. 5
3 shaded 5
3 3 1 of shaded = 10 5 2
1 3 1 3 Now shade 1 of the 2 shaded columns in a different color to show of , or × . 2 5 2 5 1 3 3 1 3 The part shaded both colors shows × . You can see × = . 2 5 10 2 5
Exercises 2. Do these multiplications by drawing rectangles on graph paper to represent the fractions. 5 3 2 1 a. × b. × 6 4 5 3
Round Up Using graph paper like this is a good way to see what it actually means when you multiply fractions. In the next few Lessons, you’ll learn a way to multiply fractions more quickly. a tion — Multiplying Fractions: an Area Model Explora 124 Section 3.2 Explor
Section 3.2
Lesson
Multipl ying F Multiplying Frractions by Inte ger s Integ ers
3.2.1
California Standard: Number Sense 2.1 volving Solve prob oblems inv Solv e pr ob lems in addition, subtraction, multiplica tion ultiplication tion, and division of positi ve fr actions and positiv fractions explain why a particular operation was used for a given situation.
What it means for you: You’ll learn the skills needed to multiply fractions by positive and negative integers.
You’ve already practiced placing fractions in their correct positions on number lines. In this Lesson you’ll use the number line to see how multiplying fractions by integers relates to multiplying two integers.
Multipl ying F ger s on the Number Line Multiplying Frractions and Inte Integ ers To multiply the fraction
integer fraction numerator denominator proper fraction improper fraction mixed number
1
by the integer 2, think of it as “2 groups of 3 .”
You can then model the calculation on a number line: 1 3
2×
1 3
1 3
2 3
0
Key words: • • • • • • •
1 3
From the number line you can see that 2 × In a similar way, 3 ×
2 can be modeled: 5 2 5
3×
2 5
0
Don’t forget: Integers are numbers like: 0, 1, 2, 3, 4..., or –1, –2, –3, –4...
2 5
1
2
1 2 = . 3 3 2 5
1
6 5 or (1 51 )
Again, using the number line you can see that 3 ×
2
2 6 = . 5 5
In each example, the numerator of the fraction has been multiplied by the integer, while the denominator has stayed the same.
To multiply a fraction by an integer: multiply the numerator by the integer and keep the denominator the same. Don’t forget: To convert an improper fraction to a mixed number, divide the top number by the bottom number using long division. 15 For example: 4 3R3 4 15 The remainder – 12 becomes the 3 numerator of the fraction.
)
3 So 15 is equivalent to 3 .
4
4
Example
1
What is 5 ×
3 ? 4
Solution
Multiply the numerator 3 by 5, and keep the denominator as 4. 5×
5× 3 3 = 4 4 =
15 3 or 3 4 4 Section 3.2 — Multiplying Fractions
125
Example What is
It doesn’t matter in which order you multiply two values. You always get the same answer — this is the commutative property of multiplication. So,
5
4 × 10? 5
Solution
Don’t forget:
Multiply the numerator 4 by 10, and keep the denominator as 5. 4 4 ×10 × 10 = 5 5
=
4
4
× 10 = 10 ×
2
5
.
40 =8 5
Simplify yyour our ans wer answ
5 divides into 40 exactly 8 times. There is no remainder, so an integer rather than a mixed number.
40 5
is equivalent to
Guided Practice Evaluate the following. 1. 6 ×
1 7
2.
2 ×4 9
3. 12 ×
1 6
4. 15 ×
2 5
Multipl y Fraction s by Ne ga ti ves in the Same Way Multiply actions Neg tiv You saw in Chapter 1 that multiplying a positive number by a negative number gives a negative result. The same rule holds when one of the numbers is a fraction. For example, take the following multiplication: –5 ×
3 4
3 , you multiply the numerator of the 4 fraction (3) by the integer (5), and keep the denominator the same.
You’ve seen that to find 5 ×
3 15 3 = = 3 . 4 4 4 Then using the “negative × positive = negative” rule,
This gives 5 ×
–5 ×
15 3 3 =– = –3 . 4 4 4
Multiplying a positive fraction by a negative integer gives a negative result.
126
Section 3.2 — Multiplying Fractions
Example
3 2
What is –5 × 7 ? 8 Solution
Don’t forget: Use the same method to convert negative improper fractions to mixed numbers as you used for positive improper fractions — remember the minus sign, though.
Multiply the numerator by 5, keep the denominator as 8, and remember the negative sign. –5 ×
5× 7 7 =– 8 8 35 3 =– or – 4 8 8
Guided Practice Evaluate the following. 5. –5 ×
1 6
6.
2 × –1 9
7. –8 ×
2 23
5 8. 16 × –3
Evaluate the following. Give your answer as an integer or mixed number. 9. –12 ×
2 3
10. –8 ×
1 11. 6 × –24
2 5
12. 13 ×
3 8
Multipl ying Mix ed Number s b y Inte ger s Multiplying Mixed Numbers by Integ ers Multiplying mixed numbers by integers is a very similar process — you just need to write the mixed number as an improper fraction before you multiply the numerator by the integer. Example
4 2
Evaluate 4 × 1 3 . Solution 2
4 × 13 = 4 × =4× =
(1× 3) + 2 3
Con ver ed n umber to an Conv ertt mix mixed number impr oper fr action improper fraction
5 3
Multipl y inte ger b y n umer a tor Multiply integ by numer umera
20 2 or 6 3 3
Section 3.2 — Multiplying Fractions
127
Multiplying mixed numbers by negative integers works in the same way. Example
5
3
Evaluate 2 5 × (–10). Solution 3
2 5 × (–10) = = Don’t forget:
( 2 × 5) + 3 × (–10) 5
Con ver ed n umber Conv ertt mix mixed number to an impr oper fr action improper fraction
13 × (–10) 5
Multipl y inte ger b y n umer a tor by numer umera Multiply integ
130 5 = –26
Simplify
=–
5 divides into 130 exactly 26 times — there’s no remainder, 130
so 5 simplifies to just 26.
Guided Practice Evaluate the following. Give your answers as mixed numbers or integers. 3
13. –5 × 2 4
3
1
14. 1 15 × (–1)
2
15. 3 × 3 8
16. 4 3 × (–3)
Independent Practice Multiply the fractions given in Exercises 1–4. 1. 3 ×
2 2. 5 × 2
3 16
3. –1 ×
1 6
4. –5 ×
3 16
Evaluate the expressions given in Exercises 5–12. Leave your answer as an integer or mixed number.
Now try these: Lesson 3.2.1 additional questions — p439
4 ×3 3
5. 10 ×
1 2
6.
9. –7 ×
5 33
10.
7. –4 ×
2 ×8 5
3 8
11. –13 ×
4 5
8. –40 ×
5 8
12. –5 ×
14 67
Evaluate the expressions given in Exercises 13–21. 1
13. –2 × 2 3 16. –4 ×
6 5
2
15. –1 × 7 2
17. –1 3 × 5
18. 8 × 1 11
4
5
19. –2 × 1 9
1
14. 1 5 × 3
1
20. 3 × 3 7
1
3
21. 7 × 1 10
Round Up So that’s the basic method for multiplying fractions by integers. Many word problems that you’ll come across will make you put this skill to use — there are some in the next Lesson. It’s also important for simplifying or solving algebraic equations. 128
Section 3.2 — Multiplying Fractions
Lesson
Mor e on Multipl ying More Multiplying Fractions b y Inte ger s by Integ ers
3.2.2
California Standards: Number Sense 2.1 volving Solve prob oblems inv Solv e pr ob lems in addition, subtraction, multiplica tion ultiplication tion, and division of positi ve fr actions and positiv fractions explain w hy a par ticular wh particular oper a tion w as used ffor or a opera was gi ven situa tion. giv situation. Number Sense 2.2 Explain the meaning of multiplica tion and division of ultiplication positi ve fr actions and positiv fractions perf or m the calcula tions perfor orm calculations (e .g ., (e.g .g.,
5 8
÷
15 16
=
5 8
×
16 15
=
2 ). 3
What it means for you: You’ll review the skills needed to multiply fractions by integers. You’ll also develop an understanding of how multiplication relates to problems involving fractions.
Key words: • • • • •
integer fraction denominator numerator mixed number
Don’t forget: When you multiply a fraction by an integer, the denominator stays the same.
In this Lesson, you’ll start by reviewing the skills learned last Lesson. Then you’ll see how multiplication of fractions and integers can be used in real-life problems.
Solving Pr ob lems b y Multipl ying F y Inte ger s Prob oblems by Multiplying Frractions b by Integ ers Think about the problem “How many toes do 5 people have altogether, assuming they each have 10 toes?” The question is asking how many there are in 5 sets of 10. To solve it, you multiply 5 by 10 to get 50. You can do problems of this type that contain fractions in the same way. Example
1
Three people each ate
2 5
of a pizza at a buffet.
How much pizza did they eat altogether? Solution
Multiply 3 by the fraction 3×
2 3× 2 = 5 5
2 . 5
Multipl y the inte ger b y the n umer a tor Multiply integ by numer umera
6 1 Conv ver ed n umber ertt to a mix mixed number = 1 5 Con 5 1 They ate 1 5 pizzas altogether. =
You might come across a similar problem involving mixed numbers. Example Example
4 2 3
Pedro is making paper chains for a party. He needs 1 8 packs of paper Don’t forget: Make sure that you read what the question is asking. If it had asked, “How many packs will Pedro have to buy?”, the answer would be 10 packs. This is because the store is 5 unlikely to sell him of a 8 pack, so he’d have to round the number up.
strips for each chain and needs to make 7 chains. How many packs of paper strips will he use? Solution 3
3
Multiply 7 by 1 8 , since the question is asking you to find 7 sets of 1 8 . 3
7 × 18 = 7 ×
11 8
=
7 ×11 8
=
77 5 = 98 8
Con ver ed n umber to an Conv ertt the mix mixed number impr oper fr action improper fraction a tor Multipl y the inte ger b y the n umer integ by numer umera Multiply 5
He uses 9 8 packs of paper strips. Section 3.2 — Multiplying Fractions
129
Guided Practice 1. Some motorbikes are going to be transported in a trailer. Each motorbike weighs 2. My dog is
3 10
1 3
of a ton. What do four motorbikes weigh?
of a meter tall.
I’m five times taller than my dog. How tall am I? 3. In a relay race, four athletes each run around a track once. The track is 87
1 4
meters long. What is the total distance run?
There’s another type of problem that can be solved by multiplying fractions and integers together. It’s less obvious that this type can be solved by multiplication, so it’s really important that you learn to spot them.
Find a F ger b y Multipl ying Frraction of an Inte Integ by Multiplying If you’re asked to find a fraction of an integer, you multiply the fraction and the integer together. Example 4 3 4 A rectangular garden border is 5 m long by 1 m wide. Three-quarters of the border has grass planted on it. 5m What is the area of the grass? 1m Solution
Find the area of the rectangular border using the formula: area = length × width. Check it out: Area is covered in more detail in Section 2.3. For now, you just need to understand what it means when something is multiplied by a fraction.
Check it out: When you have to find a fraction of something, you can replace “of” with a 3 multiplication. So of 5 4 3 × 5, and so on. means 4
So the area of the border is 5 × 1 = 5 m2 “
3 4
of ” this area is 3 4
×5 = =
3 4
× 5 m2.
3× 5 2 m 4 15 4
3
m2 = 3 4 m2
It’s easier to visualize why multiplying works for a problem like this by using a diagram. Imagine that the area planted with grass was along one side. 5m 3 4
m
1 m2 1 m2 1 m2 1 m2 1 m2
The grass area is
3 4
Section 3.2 — Multiplying Fractions
1m
of 5 m2. You could work it out using the area formula:
area = length × width. So 130
3
So, the area of the grass = 3 4 m2
3 4
of 5 m2 =
3 4
× 5 m2.
Another Way to Look a ... att This his... 1 3
From the previous page, you know that
× 2 is the same as
1 3
of 2.
You can also show this on a number line: 1 ×2 3
1 3
1 of 2 3
1 3
0
1 1 3
1 3
of 2
of 2
2 1 3
of 2
The examples below ask you to find a fraction of a number. Example
4
Shatika, Jose, Kim, and Dan have 5 meters of beading wire. They want to cut it up so that they all have equal lengths. How many meters do they each get? Give your answer as a mixed number. Solution
They each get 1 4
Don’t forget:
×5 = =
You need to include the correct units in your answer. In this example, it’s meters.
1 4
of 5 meters. So multiply
1 4
by 5.
1× 5 4
Multipl y the n umer a tor b y the inte ger Multiply numer umera by integ
5 4
Con ver ed n umber Conv ertt to a mix mixed number 1
= 1 4 meters
Example
5 2
When comparing test scores, Michael noticed that his score was
7 8
of
Janelle’s score. If Janelle’s score was 88, what was Michael’s score?
Check it out: To find
7 8
of a number you
could instead find 1 of it and
Solution
You need to find out
8
multiply it by 7. 1 8
of 88 is 88 ÷ 8 = 11.
7 8
of 88 is 7 × 11 = 77.
7 8
× 88 = =
7 ×88 8 616 8
7 8
of 88. So multiply
7 8
by 88.
umer a tor b y the inte ger Multipl y the n umera by integ Multiply numer Simplify the ans wer answ
= 77
Section 3.2 — Multiplying Fractions
131
Guided Practice 4. One palm tree is 25 feet tall. Another tree is
3 4
of its height.
Find the height of the shorter tree. 5. Amber is baking bread. The recipe calls for 800 g of flour, but Amber wants to make her loaf
5 8
the size of the one in the recipe.
How much flour should she use?
Independent Practice Evaluate Exercises 1–3. 1
1
1. 4 × 5 7
2. 2 13 × –6
1
3. 10 × 1 10
4. Aisha is painting a wall. After the first hour, she has painted
2 15
of
the wall. How much of the wall will she have painted after 4 hours, assuming she continues painting at the same hourly rate? 5. A summer camp knows that each camper will eat sandwiches made 3 16
from an average of
of a loaf of bread. How many loaves of bread
should be used to make sandwiches for 121 campers? 1
6. Adult guinea pigs consume approximately 2 10 ounces 2
of dried food and 6 3 tablespoons of water each day. How much dried food and water would you expect three adult guinea pigs to consume each day? Don’t forget:
7. Juan and Kate are each knitting a scarf. Juan’s scarf is 24 inches long
You need to include units for the answers to Exercise 6–9.
so far. If Kate’s scarf is
3 5
of the length of Juan’s, how long is it?
8. Connor has 25 liters of water for a six-day camping trip. He wants to drink
1 6
of the water each day. How much is this?
Now try these:
9. Chang weighed 7 pounds when he was born. His younger brother
Lesson 3.2.2 additional questions — p440
weighed
7 9
of this. How much did Chang’s brother weigh?
10. Paul’s top pinball score is 87,000. Juanita tells him she scored
3 8
of his score. What was Juanita’s score?
Round Up In earlier grades you spotted repeated-addition problems and solved them by multiplication. Now you have to also watch out for “a fraction of a number” type problems and solve them by multiplication too. Next Lesson, you’ll multiply fractions by other fractions and see how this applies to real life. 132
Section 3.2 — Multiplying Fractions
Lesson
Multipl ying F Multiplying Frractions by F Frractions
3.2.3
California Standards: Number Sense 2.1 volving Solve prob oblems inv Solv e pr ob lems in addition, subtraction, multiplica tion ultiplication tion, and division of positi ve fr actions and positiv fractions explain why a particular operation was used for a given situation. Number Sense 2.2 Explain the meaning of multiplica tion and division of ultiplication positi ve fr actions and positiv fractions perf or m the calcula tions perfor orm calculations (e .g ., (e.g .g.,
5 8
÷
15 16
=
5 8
×
16 15
=
2 ). 3
What it means for you: You’ll learn the skills needed to multiply fractions by other fractions. You’ll also solve some real-life problems that involve doing this.
This Lesson is about multiplying one fraction by another. Not only do you need to know how to do this, but you also have to be able to understand when a problem can be solved by this method.
Multipl y Both Tops and Bottoms of F Multiply Frractions To multiply one fraction by another, you just multiply the numerators together, then multiply the denominators together. This gives you the numerator and denominator of the product. For example: 2 × 5 = 2 × 5 = 10 3 7 3× 7 21
When multiplying two fractions:
Example Evaluate
Key words: • • • • • •
45 iii) 5 9 5× 9 × = = 7 11 7×11 77
Example
So multiplying a fraction by an integer can also be thought of as a fraction-by-fraction multiplication.
Solution
3 4 3 × 4 12 × = = . 5 1× 5 1 5
This is what you were doing in Lessons 3.2.1 and 3.2.2.
iii)
5 9 × 7 11
Multiply both of the n umer a tor s umera tors to g ether , then together, multiply both of the s denomina tor denominator tors to g ether ether.. tog
If you’re asked to multiply a mixed number, just convert it to an improper fraction first.
Evaluate
=
2 4 × 3 7
8 ii) 2 4 2× 4 × = = 21 3 7 3× 7
5 For example, 3 = 3 , and 5 = . 1 1
4 5
ii)
35 i) 7 5 7 × 5 × = = 48 8 6 8×6
A product is the result of multiplying numbers together.
So, 3 ×
7 5 × 8 6
Solution
Don’t forget:
Any integer divided by 1 is equivalent to the integer itself.
1 i)
integer fraction denominator numerator improper fraction mixed number
Check it out:
a c a×c × = b d b×d
multiply both numerators together and both denominators together.
2
3 1 × 24. 5
3 ( 2 × 4) + 1 3 1 × 24 = × 4 5 5 =
3 9 3× 9 × = 4 5 5× 4
=
27 7 , or 1 20 20
Con ver ed n umber Conv ertt the mix mixed number oper fr action. improper fraction. to an impr Multipl y the n umer a tor s to gether Multiply numer umera tors tog and the denomina tor s to gether denominator tors tog ether..
Section 3.2 — Multiplying Fractions
133
Guided Practice Evaluate the following. 1.
3 4
4. 2
× 2 3
1 7
×
5 7
11 10
2.
4 9
×
7 13
3.
5.
7 2
×
7 5
6. 1 3 × 1 13
×
7 9
1
2
Multipl y to Find a F Multiply Frraction of a F Frraction In the previous Lesson, you found a fraction of an integer by multiplying the fraction and integer together. You can use multiplication in the same way to find a fraction of a fraction. Example
3
You have half a pie left and want to split it into thirds to share with your two friends. What fraction of the original whole pie do you each get? Solution
This diagram shows you what’s going on in this problem: Check it out: This works the same with sets of objects as with shapes. For example, if
1 2
1 6
1 3
the original pie
of the
1 2
pie =
1 of the original pie 6
of your class 1
went on a canoe trip, and of 3 these students got wet, 1 1 × 2 3
1 2
= of then your class got wet on the trip.
But you don’t need to draw a diagram. You can find what fraction each person gets, by multiplying 1 3
×
1 2
=
1×1 3× 2
=
1 6
So you each get
1 6
1 3
by
1 . 2
Multipl y the n umer a tor s to gether Multiply numer umera tors tog and the denomina tor s to gether tors tog denominator
of the original pie.
To find a fraction of a mixed number, just change the mixed number to an improper fraction first.
134
Section 3.2 — Multiplying Fractions
Example
4
You have two and a half grapefruits and are making four fruit salads. You want to put one-quarter of the total amount of grapefuit you have into each fruit salad. How much grapefruit should you put in each? Solution 1
1 4
of 2 2 . So multiply these numbers together.
=
1 4
=
1× 5 4×2
=
5 8
So you should put
5 8
You want to find 1 4
1
× 22
×
5 2
Con ver ed n umber to an Conv ertt the mix mixed number impr oper fr action improper fraction Multipl y the n umer a tor s to gether Multiply numer umera tors tog and the denomina tor s to gether denominator tors tog
of a grapefruit in each fruit salad.
Guided Practice 7. It takes half an hour for Elise to practice her musical instrument. She spends one-quarter of that time warming up. What fraction of an hour does she spend warming up? 8. Marsha's creative writing story was one and a half pages long. David's story was two-thirds of the length of Marsha's story. How long was David's story?
Independent Practice Evaluate Exercises 1–6. 1.
3 8
× 1
Now try these: Lesson 3.2.3 additional questions — p440
3 4
4. 2 5 ×
1 4
7. Lola drank
3 4
2.
2 5
×
5.
4 7
× 17
4 31 4
3.
2 9
6.
8 15
×
2 3 7
× 18
1
of a cup of tea. The cup of tea contained 1 4
teaspoons of sugar. How much sugar did Lola consume? 1 3
8. Arthur finished his assignment in 4 hours. His classmate finished in half the time that Arthur did. How much time did it take his classmate to finish? 3
9. A sponsored walk is 9 4 miles. There are refreshments provided two-thirds of the way along the route. After how many miles is this?
Round Up Multiplying fractions by fractions is a skill that you’ll use time and time again in solving real-life problems and in simplifying algebraic expressions — so it’s worth making sure you know how to do it. Section 3.2 — Multiplying Fractions
135
Section 3.3
Lesson
3.3.1
Dividing by Fractions
California Standards:
The last few Lessons were all about multiplying fractions. Dividing by fractions is really closely related to this — to be able to divide by fractions, you actually use the skills you learned for multiplication. Also, before you start dividing by fractions, you need to know about reciprocals, so that’s where this Lesson starts.
Number Sense 2.1 Solve problems involving addition, subtraction, multiplication, and division of positive fractions and explain why a particular operation was used for a given situation. Number Sense 2.2 Explain the meaning of multiplication and division of positive fractions and perform the calculations (e.g.,
5 8
÷
15 16
=
5 8
×
16 15
=
Reciprocals Multiply Together to Give 1 Two numbers that multiply together to give 1 are known as reciprocals of each other. Another name for a reciprocal is a multiplicative inverse. Here are some examples of multiplicative inverses:
2 ). 3
2 3
and
3 2
2 3 2×3 6 × = = = 1 3 2 3× 2 6
4 5
and
5 4
4 5 4 ×5 20 = = 1 × = 5 4 5× 4 20
What it means for you: You’ll learn what it means when you divide by a fraction, and why you can use multiplication to do this. You’ll also learn what reciprocals are, because you need to use them to divide by fractions.
You might have noticed that you “flip” or “invert” a fraction to get its reciprocal. The numerator becomes the denominator and the denominator becomes the numerator.
4 5
Key words: • • • • •
denominator divide multiply numerator reciprocal
The reciprocal of
An integer can be written as a fraction with a denominator of 1. 7
7 1
Similarly, the reciprocal of 12 is
Check it out: To find the reciprocal of a mixed number, first convert it to an improper fraction. For example: =
25 11
So the reciprocal of 2
136
1
For example, 7 can be written as 1 . So its reciprocal is 7 .
7 1 7 ×1 7 × = = =1 1 7 1× 7 7
=
b a is , for a ≠ 0 and b ≠ 0. a b
Write Integers as Fractions — Then Find the Reciprocals
Check your reciprocals by multiplying. You should always get a product of 1:
(2×11) + 3 11
3 2
2 3
So, in mathematical terms:
Don’t forget:
3 2 11
5 4
3 11
is
11 . 25
1 7
1 1 , and the reciprocal of 5 is . 12 5
Guided Practice Write down the reciprocals of the following. Show that they are reciprocals by multiplying. 1.
7 8
Section 3.3 — Dividing Fractions
2.
9 7
3. 8
4. 14
1
5. 1 3
3
6. 3 4
To Divide by Any Fraction, Multiply by the Reciprocal The general rule for dividing by a fraction is to multiply by the reciprocal of the fraction. So, in math language:
Example
1
What is 2 ÷
1 ? 4
Solution
1 , 4
To divide 2 by 2÷
Check it out: Interpret the division in the same way that you might if it contained integers. For instance: 9 ÷ 3 can be interpreted as asking how many 3s there are in 9.
c d a a ÷ = × d c b b
multiply 2 by the reciprocal of
1 =2×4 4 =8
The division 2 ÷
1 4
The reciprocal of
1 4
1 . 4
is 4.
is asking how many quarters there are in 2.
Each circle represents one whole unit.
1 4
1 4
1 4
1 4
1 4
1 4
1 4
1 4
There are 4 quarters in each whole unit, so in 2 whole units there are 2 × 4 = 8 quarters. Example
2
1
What is 4 2 ÷
3 ? Draw a diagram to show what this problem means. 4
Solution 3 4
1
42 ÷
3 4
3 4
3 4
3 4
3 4
3 4
1
3
means how many 4 s there are in 4 2 .
The diagram shows that there are 6. Don’t forget: When you multiply a mixed number, change it to an improper fraction first. So, 4
1 = 2
( 4 × 2) + 1 2
1
3
You can also get this answer by multiplying 4 2 by the reciprocal of 4 : 1
42 ÷
4 4 1 3 9 =4 × = × 3 3 2 4 2
Reciprocal of
3 4
9× 4 = 2×3 =
Change the division to multiplication by the reciprocal. Multiply the fractions.
36 =6 6 Section 3.3 — Dividing Fractions
137
Guided Practice Draw a diagram to represent each of the divisions in Exercises 7–10. Then find the answers. 7. 5 ÷
5 9
1
8. 3 2 ÷
1 4
1
9. 2 4 ÷
3 8
5 6
1
10. 2 2 ÷
The Solution May Be a Fraction When you divide by a fraction, you might get a bit left over. 1
For example, think about the division 2 2 ÷
21 2
÷
3 5 = ÷ 4 2 5 = × 2 20 = 6 10 = 3 1 = 3 3
3 4 4 3
1 4 1
3 s, 4
What is
and also
3
1 2 ÷ ? 3 2
Solution
by
something bigger than
1 2
3 2 1 1 ÷ = × 3 2 2 2
— so the answer is less than 1.
=
Multiply by the reciprocal of the fraction you are dividing by.
1× 3 3 = 2×2 4
Divide a fraction by an integer in exactly the same way — multiply by its reciprocal. Check it out: 1
1 4
Example
÷ 4 could be pictured as
sharing out
1 1 4
pies between
four people. Each person gets 5 16
of a pie.
5 16
138
4
1
What is 1 4 ÷ 4? Solution
5 1 14 ÷ 4 = 4 ÷ 4
5 16 5 16
1 3
of 1 3
3 4
of
3 . 4
You could, of course, have done this division by multiplying by the reciprocal. The examples below are done in this way.
Check it out: 1 2
left over — that’s
The diagram shows that 2 2 contains three complete 3 1 1 So 2 2 ÷ = 3 3 . 4
Example
You’re dividing
3 4
3 4
3 4
Check it out:
3 : 4
=
1 5 × 4 4
=
5×1 5 = 4 × 4 16
5 16
Section 3.3 — Dividing Fractions
Convert the mixed number to an improper fraction. Multiply by the reciprocal of the integer you are dividing by.
Guided Practice Draw diagrams to show the divisions in Exercises 11–13.
1 1 1 ÷3 12. ÷2 13. 1 3 ÷ 3 4 2 Use multiplication to find the solutions to Exercises 14–19. 11.
1 2 15. 15 ÷ 7
1 1 14. 3 ÷ 2 17.
2 1 ÷ 22 7
2
7
18. 3 3 ÷ 3 8
16.
7 3 ÷ 8 7
19.
1 ÷2 15
Independent Practice Give the reciprocals of the numbers given in Exercises 1–8. 1. Check it out: Make sure you don’t multiply the reciprocals of both fractions together — it’s only the divisor that becomes its reciprocal.
7 11
2.
7 3 3
5. 13
6. 3 4
8 9
1 6 1
8. 7 6
2 1 1 1 10. 6 ÷ 11. 2 4 ÷ 5 3 8 Use multiplication to find the solutions to Exercises 12–17.
9. 4 ÷
1 5
3 3 15. 4 ÷ 8
Lesson 3.3.1 additional questions — p440
7. 1
4.
Draw diagrams to represent the problems in Exercises 9–11.
12. 8 ÷
Now try these:
3. 3
13. 12 ÷ 1
2 3
2
14. 1 3 ÷
1
16. 5 3 ÷ 1 2
17.
5 6
3 ÷4 10
18. Is it possible to divide a mixed number by an integer to produce: a) a number greater than 1? b) 1? c) a number less than 1? Explain your reasoning and give an example of each where possible. 19. Write the expression y ÷ z as a product. 20. When Nancy divided a fraction by 2, she got
1 16
for her answer.
What fraction was she dividing? 21. When Ken divided
7 8
by an integer, he got
7 . 32
What integer was he dividing by?
Round Up After this Lesson, you should be able to understand what divisions involving fractions mean — this’ll help you recognize when you should divide by a fraction in real-life problems. You’ll meet some of these in the next Lesson. Section 3.3 — Dividing Fractions
139
Lesson
3.3.2
Solving Problems by Dividing Fractions
California Standard: Number Sense 2.1 Solve problems involving addition, subtraction, multiplication, and division of positive fractions and explain why a particular operation was used for a given situation.
What it means for you: You’ll practice solving lots of real-life problems that involve dividing fractions.
In the last Lesson, you practiced doing divisions involving fractions and mixed numbers. In this Lesson, you’ll use these skills to tackle real-life word problems. It’s really important that you learn to recognize the types of problems that use division of fractions.
Division Problems Often Involve Partitioning Something If you come across a problem that involves splitting something up into equal parts, it often means you have to use division to solve it. Example
1
1
A pipe 7 2 feet long is divided into sections Key words:
How many sections are made from the pipe?
• divide • multiply • reciprocal
Solution
3 4
1
Divide 7 2 by
ft
3 4
ft
3 4
ft
3 4
ft
3 4
3 4
of a foot long. 3 4
ft
ft
3 4
ft
3 4
ft
3 4
ft
3 4
3 : 4 1
1
72 ÷
3 4
Check it out: Don’t be scared by the fractions in these problems. If these problems had contained whole numbers, you’d solve them in exactly the same way — by division. For instance, imagine Example 1 had involved an 8-foot pipe being divided into sections 2 feet long. You’d work out the number of sections you’d get by calculating 8 ÷ 2.
=
( 7 × 2) + 1 2
=
15 2
÷
3 4
=
15 2
×
4 3
=
15× 4 2×3
=
÷
7 2 feet
3 4
Convert the mixed number to an improper fraction.
Multiply by the reciprocal of the fraction you are dividing by. 60 6
= 10
So you get 10 sections of pipe. Example
2
7
Ben has 1 8 quarts of motor oil. He puts half of it in his car. How much oil did he put in his car?
OIL
2Q
uar
ts
Solution 7
You have to find out what half of 1 8 quarts is. So you divide it by 2. 7
18 ÷ 2 =
140
Section 3.3 — Dividing Fractions
(1×8) + 7 8
=
15 8
÷2
=
15 8
×
1 2
=
15×1 8× 2
=
÷2
Convert the mixed number to an improper fraction. Multiply by the reciprocal of the integer you are dividing by.
15 16
quarts
ft
Guided Practice 1. Dolores had
2 3
of a gallon of apple juice.
She shares it equally between herself and two friends. What fraction of a gallon does each person get? 2. TJ is knitting squares that will be sewn together to make blankets for 3
an overseas charity. He has set aside 18 hours over the next month to 4 1 knit squares. It takes him 1 hours to knit one square. 4
How many squares will he have time to knit?
Division Can Be Used for Repeated Subtraction Too Another type of division problem is the repeated subtraction type. Example
3
A large water tank contains 600 gallons of water. The tank has a leak and loses
2 3
of a gallon of water each day.
After how many days will the tank be empty? Solution 2
You need to figure out how many “ 3 of a gallon” there are in the tank. So divide the number of gallons in the tank by 600 ÷
2 3
= 600 × =
Check it out: “Repeated subtraction” and “partitioning” problems are very similar. Exactly the same methods are used to solve them.
2 . 3
Multiply by the reciprocal of the fraction you are dividing by.
3 2
600 × 3 2
= 900 The tank will be empty after 900 days.
Guided Practice 3. A bowl contains 4 liters of fruit punch. 1
How many students can fill 5 -liter cups from the bowl? Check it out: Think carefully about the answer to Guided Practice Exercise 4. The question asks, “How many cabins?” It wouldn’t make sense to have a fraction of a cabin in your answer.
4. The campers in each cabin at a summer camp want to make a 1
welcome sign. There is a roll of paper with 14 2 meters of paper on it. 1
How many 1 5 -meter-long signs can be made from the roll?
Section 3.3 — Dividing Fractions
141
Don’t Forget — Division Is the Inverse of Multiplication Some problems might be tricky to spot as divisions immediately. It could be easier to write a multiplication equation containing a variable, and then solve it. Example
4
1
After 7 3 miles on a sponsored dog walk, there is a stand giving out dog cookies. This is
3 4
of the way along the route.
How long is the whole route? Solution — Method 1
Let d represent the length of the whole route. Don’t forget: To find a fraction of a number, you multiply the fraction by the number.
3 4
1
of d = 7 3 miles 3 4
So,
1
× d = 73
To solve this equation, you divide both sides by 1
d = 73 ÷
3 4
d=
(7 × 3) + 1 3
d=
22 3
×
d=
88 9
= 99
3 4
to get d by itself.
Get the variable d by itself.
÷
3 4
=
22 3
÷
Convert the mixed number to an improper fraction.
3 4
4 3
Multiply by the reciprocal of the fraction you are dividing by.
7
7
The route is 9 9 miles long. 1
Another way to do this would be to find out how many miles 4 of the route is, then multiply this distance by four to find the length of the whole route.
Solution — Method 2 1
7 3 miles is equal to So Don’t forget: This method involves multiplying a fraction by an integer. You do this by multiplying the numerator by the integer and keeping the denominator the same.
142
1 4
3 4
of the route.
1
of the route = 7 3 ÷ 3 =
22 3
÷3
=
22 3
×
1 3
=
22 9
miles
The whole route is equal to four-quarters. So the whole route =
Section 3.3 — Dividing Fractions
22 9
×4=
22 × 4 9
=
88 9
7
= 9 9 miles
Guided Practice 5. After canoeing 19 miles down the river, I was disappointed to find out that I was only
4 9
of the way to my destination. How far was I
expecting to canoe in total? 1
6. A fish tank has 13 3 gallons of water emptied into it. This makes it
3 8
full. How many gallons of water does the tank hold altogether?
Independent Practice Which of the problems in Exercises 1–2 would you solve by division? Explain your answers. 1. Mrs. Garcia assigned 8 math problems for homework. 1
Each problem takes about 2 2 minutes to solve. How long might it take for a student to complete all eight problems? 1
2. The art teacher has 8 2 pounds of clay that he wants to distribute evenly to 12 students. How much clay will each student get? 3. A very tall building has an elevator which is 41 of a mile above the ground when it is at the top floor. What fraction of a mile above the ground is the elevator when it is halfway up the building? 1
4. Judy fills 2 2 buckets with sand. It takes a total of 30 shovel-scoops of sand to do this. What fraction of a bucketful of sand does her shovel hold? 2
5. Mr. Ruiz has exactly 1 3 cups of coffee. He drank the entire amount in 19 sips. What was the average amount of coffee in each sip? 3
Now try these: Lesson 3.3.2 additional questions — p441
6. A wall is 4 feet long. The wall is to be divided up into 4 3 equal sections that will each be painted a different color. How wide will each section be? 1
7. DeAndre was running in a race. After he had run 4 2 miles, he was 3 8
of the way to the finish. How long was the race?
Round Up So that’s how division of fractions can be used to solve some types of real-life problems. You’ve now finished looking at multiplication and division of fractions. In the next few Lessons, you’ll learn the techniques for adding and subtracting fractions and how to give answers in their simplest forms. Section 3.3 — Dividing Fractions
143
Section 3.4 introduction — an exploration into:
Ad ding F ea Model Adding Frractions: an Ar Area Graph paper is very useful when you are doing fraction addition. The squares on the graph paper let you make realistic “models” of fractions, which you can then add by counting squares. It’s a good way to see what it actually means to add fractions. To add two fractions, start by drawing three identical rectangles: — the heights of the rectangles should be equal to one of the denominators, — the widths should be equal to the other denominator.
The denominator is the number on the bottom of a fraction.
Example Use graph paper to add
2 1 + . 5 3
Solution
+
=
• Draw three rectangles on graph paper, 5 squares high and 3 squares wide. 2 2 1 • Show in the first rectangle by shading 2 of the 5 rows. 5 5 3 1 6 5 11 + = • Show in the next rectangle by shading 1 of the 3 columns. 15 15 15 3 • Count the total number of shaded boxes in the first two rectangles. Shade this number of boxes in the third rectangle. number of shaded boxes • Rewrite each of the three fractions using the following formula: total number of boxes You can use exactly the same method to add fractions that sum to more than 1. Example Use graph paper to add
3 2 + . 7 3
Solution
Do everything exactly as before, but draw an extra rectangle if your answer does not fit. You can write your answer as: 23 , • an improper fraction — such as 21 2 • a mixed number — such as 1 .
21
= squares
+
shaded
3 7 9 21
+
2 3 14 21
=
23 or 21
+2 extra squares shaded
1 212
21
Exercises 1. Do these additions by drawing rectangles on graph paper to represent the fractions. 1 2 3 2 5 1 1 2 a. + b. + c. + d. + 2 5 4 5 6 3 4 3
Round Up Drawing pictures of fractions on graph paper is a good way to add them. Think about these pictures while you’re learning about the ideas and methods in this Section. a tion — Adding Fractions: an Area Model Explora 144 Section 3.4 Explor
Lesson
Section 3.4
3.4.1
Making Equivalent Fractions
California Standard:
You’ve met equivalent fractions before — they’re the ones that look different but sit at the same place on the number line. This Lesson is all about how you can make new equivalent fractions.
Number Sense 2.1 Solve problems involving addition, subtraction, multiplication, and division of positive fractions and explain why a particular operation was used for a given situation.
What it means for you: You’ll learn how to make new equivalent fractions — including ones that are simpler.
Make Equivalent Fractions by Multiplying by 1 Equivalent fractions look different but have the same value, like
• • • • •
equivalent fraction simpler factor cancel prime factorization
and
3 6
You form equivalent fractions by multiplying by a fraction equal to 1 — that is, by a fraction whose numerator and denominator are equal. Example
1
By multiplying by a suitable fraction, show that Key words:
1 2
3 6
is equivalent to
1 2
.
Solution
Any fraction with the same numerator and denominator is equal to 1. 1 3 This means you can multiply 2 by 3 and get a fraction with equal value. 1 3 1×3 3 × = = 2 3 2× 3 6
So the two fractions are equivalent.
Example
2
Find three fractions equivalent to
3 . 4
Solution
Don’t forget: You can check the value of a fraction by doing a division — either on a calculator or by using long division. 6 = 6 ÷ 8 = 0.75 8 9 = 9 ÷ 12 = 0.75 12 24 = 24 ÷ 32 = 0.75 32
Pick any three fractions equivalent to 1, for example 3 Then multiply 4 by each of them in turn. 1.
3 2 3× 2 6 × = = 4 2 4× 2 8
2.
3 3 3×3 9 × = = 4 3 4 ×3 12
3.
3 8 3×8 24 × = = 4 8 4 ×8 32
2 3 , , 2 3
8
and 8 .
Section 3.4 — Adding and Subtracting Fractions
145
.
You sometimes have to choose your fractions very carefully. Example
3
Fill in the missing numbers in these fractions:
1 a 32 = = 2 8 b
Solution
To get 8 as the denominator in 1 4 1× 4 4 × = = , so a = 4. 2 4 2× 4 8
a 1 , 8 2
32
To get 32 as the numerator in b , 1 32 1×32 32 × = = , so b = 64. 2 32 2×32 64 So
4
must have been multiplied by 4 .
1 2
must have been multiplied by
32 . 32
1 4 32 = = 2 8 64
Guided Practice 3
1. Find three fractions equivalent to 5 . 2. By multiplying by a suitable fraction, show that
1 5
and
7 35
are equal.
Fill in the missing numbers in Exercises 3–4. 3.
2 4 ? ? 10 12 ? = = = = = = 3 6 9 12 ? ? 900
4.
4 8 ? 48 ? 480 = = = = = 5 10 15 ? 300 ?
Make Fractions Simpler by Canceling Canceling is a way of making equivalent fractions that are simpler. Instead of multiplying the top and bottom lines by the same number, canceling involves dividing them. Example Check it out: When you cancel a fraction, the numbers get smaller in size and easier to use.
Don’t forget: A factor of a number divides evenly into that number. For example, 5 is a factor of 10.
146
4
Form a simpler fraction equal to
10 . 15
Solution
To cancel a fraction, you need to look for numbers that are factors of both the numerator and denominator. 10 2×5 Here, 5 divides into the numerator and denominator: = 15 3×5 2×5 2 You can cancel common factors — so = 3×5 3
Section 3.4 — Adding and Subtracting Fractions
Guided Practice 6
5. By canceling, find a simpler fraction equivalent to 8 . Cancel the fractions in Exercises 6–11 to find simpler forms. 6.
14 16
7.
15 35
9.
14 21
10.
8.
9 15
44 55
11.
26 39
You Can Cancel More Than One Common Factor Don’t forget: A prime factorization is when you write a number as a product of prime factors, for example: 10 = 5 × 2 8 = 2 × 2 × 2 = 23
You can cancel as many common factors as you can find. Example
5
Simplify the fraction
24 . 30
Solution
There are different ways to do this. Check it out: 12 In fact, the fraction is a 15 24 simpler form of , and so is 30 a possible answer in Example 5. But it’s not the simplest form — and usually, the simpler the fraction, the better. There’s more about the simplest form in Lesson 3.4.2.
• You can look for the biggest common factor, then cancel it.
24 4 ×6 4 = = 30 5×6 5
• You can write down the prime factorizations of the numerator and denominator and cancel all the common factors.
24 2× 2× 2×3 4 = = 30 5× 2×3 5
24 12× 2 12 • You can do it in stages. = = You can cancel one common factor first. 30 15× 2 15 12 4 ×3 4 Then you can cancel another common factor. = = 15 5×3 5
All three methods give the same answer.
Guided Practice 12. Find two simpler fractions that are equivalent to 13. By canceling, find a simpler fraction equivalent
8 . 48 21 to 56 .
Fill in the missing numbers in Exercises 14–18. 14.
80 40 ? 4 = = = 60 30 15 ?
15.
150 ? 15 5 = = = 270 135 ? ? Section 3.4 — Adding and Subtracting Fractions
147
16.
108 36 12 ? = = = 126 ? ? 7
17.
420 60 ? ? = = = 490 ? 14 7
18.
180 60 ? 4 = = = 225 ? 25 ?
Independent Practice In Exercises 1–8, find two fractions equivalent to the one given. Now try these: Lesson 3.4.1 additional questions — p441
1.
1 9
2.
3 4
3.
6 7
4.
1 3
5.
2 3
6.
4 7
7.
1 11
8.
5 8
Fill in the missing numbers in Exercises 9–15. 9.
7 14 21 ? ? = = = = 12 ? ? 48 60
10.
? 2 ? 36 ? = = = = 8 4 2 ? 80
11.
18 6 9 ? = = = 72 ? ? 4
12.
6 ? 18 2 54 = = = = 9 18 ? ? ?
13.
? 10 50 2 = = = 120 ? ? 3
14.
40 ? 8 ? ? 32 128 96 = = = = = = = ? 18 9 63 27 ? ? ?
15.
? ? 20 95 10 ? 15 5 = = = = = = = 63 42 ? ? ? 35 ? 7
Round Up There are two tricks to finding equivalent fractions. The first is easy — pick any integer and multiply the top and bottom lines of the fraction by it. The second is harder — you have to find something that both parts of the fraction can be divided by. But if you can, then you’ll be rewarded with a simpler fraction. 148
Section 3.4 — Adding and Subtracting Fractions
Lesson
3.4.2
Finding the Simplest Form
California Standard:
Last Lesson you saw how to make equivalent fractions and how some fractions can be made simpler. This Lesson is all about taking “simpler fractions” as far as you possibly can — to find the simplest form.
Number Sense 2.4 Determine the least common multiple and the greatest common divisor of whole numbers; use them to solve problems with fractions (e.g., to find a common denominator to add two fractions or to find the reduced form for a fraction).
What it means for you: You’ll learn about the simplest form of a fraction and see a couple of ways of finding it.
Fractions Have a Simplest Form A fraction like
1 2
can’t be made any simpler because there isn’t anything
that both 1 and 2 can be divided by. When a fraction can’t be made any simpler, it is in its simplest form. One way to find the simplest form is to keep making a fraction simpler until you can’t go any further. Example
1
Key words:
Find the simplest form of
• simplify • greatest common divisor • prime factorization
Solution
66 . 99
Both 66 and 99 have a factor of 3, so
But 22 and 33 both divide by 11, so Check it out: Some people say “reduced form” instead of “simplest form.”
66 22×3 22 = = 99 33×3 33
22 2×11 2 = = 33 3×11 3
Since nothing divides both 2 and 3, the simplest form of Example
66 99
2
is 3 .
2
Find the simplest form of
84 . 126
Solution
Just keep looking for factors of both the numerator and denominator. 84 42× 2 42 14 ×3 14 2× 7 2 = = = = = = 126 63× 2 63 21×3 21 3× 7 3
Guided Practice Find the simplest form of the fractions given in Exercises 1–8. 1.
4 6
2.
20 35
3.
9 81
4.
12 30
5.
75 90
6.
32 176
7.
135 195
8.
19 133
Section 3.4 — Adding and Subtracting Fractions
149
Greatest Common Divisor Is a Faster Way to Simplify The greatest common divisor of two numbers is the greatest number that they can both be divided by. Don’t forget: A prime factorization is when you “break a number down” into its prime factors. For example, 15 can be broken down like this: 15 = 5 × 3. But it can’t be broken down any further because both 5 and 3 are prime — nothing divides into 5 except 5 and 1, and nothing divides into 3 except 3 and 1.
To find the greatest common divisor, turn both numbers into prime factorizations. Then identify any numbers that are in both factorizations. 24 = 2 × 2 × 2 × 3 84 = 7 × 2 × 2 × 3
24 and 84 both have factors of 2 (twice) and 3
Multiply the common factors together to get the greatest common divisor. So the greatest common divisor of 24 and 84 is 2 × 2 × 3 = 12. Example
3
Find the greatest common divisor of 30 and 42. Solution
30 = 2 × 3 × 5 42 = 2 × 3 × 7
30 and 42 both have factors of 2 and 3
So the greatest common divisor of 30 and 42 is 2 × 3 = 6.
Dividing the numerator and denominator of a fraction by their greatest common divisor is the same as canceling all the common factors at once. Check it out:
This means you can reduce a fraction to its simplest form in a single step.
After you divide two numbers by their greatest common divisor, there is nothing else you can divide both by.
From above, 12 is the greatest common divisor of 24 and 84. So
24 2×12 2 = = , and this must be the simplest form of 84 7×12 7
Example
24 . 84
4
Find the simplest form of: a)
30 270 , b) . 720 42
Solution
a) From above, the greatest common divisor of 30 and 42 is 6. So
30 30 5×6 5 = = has simplest form: 42 42 7×6 7
b) First find the greatest common divisor of 270 and 720. and
270 2 × 3× 3× 3× 5 = 720 2 × 2 × 2 × 2 × 3× 3× 5
270 and 720 both have factors of 2, 3 (twice), and 5.
So the greatest common divisor is 2 × 3 × 3 × 5 = 90. Therefore 150
270 3× 90 3 270 = = has simplest form: 720 720 8× 90 8
Section 3.4 — Adding and Subtracting Fractions
Guided Practice Find the greatest common divisor for each pair of numbers given in Exercises 9–16. 9. 25, 15 10. 38, 42 11. 60, 42 12. 49, 35 13. 27, 99 14. 66, 15 15. 55, 45 16. 47, 37 Find the simplest form of the fractions given in Exercises 17–24. 17.
2 12
18.
14 21
19.
18 24
20.
42 70
21.
33 15
22.
24 30
23.
147 231
24.
30 105
Independent Practice Now try these: Lesson 3.4.2 additional questions — p441
Find the greatest common divisor for each pair of numbers given in Exercises 1–8. 1. 33, 42 2. 38, 57 3. 42, 70 4. 39, 65 5. 8, 52 6. 36, 90 7. 18, 27 8. 40, 16 9. Simplify the following fractions. Then arrange them in order of size from least to greatest. 10 4 8 30 15 18 20 165 10. Which fraction shown below is in its simplest form? 12 11 3 16 38 14 21 46 Write each of the fractions given in Exercises 11–18 in its simplest form. 11.
8 28
12.
14 30
13.
16 32
14.
9 33
15.
68 110
16.
2 186
17.
58 78
18.
120 180
19. Juan scored 22 out of 26 possible points on his history quiz. What is his score as a reduced fraction?
Round Up All fractions have a simplest form, and one way to find it is with the greatest common divisor. Finding a fraction’s simplest form can seem like a lot of trouble. However, one you’ve done it, the numbers are smaller. That makes it easier to use in other calculations. Section 3.4 — Adding and Subtracting Fractions
151
Lesson
3.4.3
Fraction Sums
California Standard:
Adding and subtracting fractions can involve a little more work than multiplying them. This Lesson is about adding and subtracting fractions with the same denominator.
Number Sense 2.1 Solve problems involving addition, subtraction, multiplication, and division of positive fractions and explain why a particular operation was used for a given situation.
What it means for you: As an introduction to adding and subtracting fractions you’ll see how to add and subtract fractions with the same denominator.
Add or Subtract Numerators If Denominators Are Equal The denominator of a fraction is similar to the units in a measurement — it tells you what type of quantity you are dealing with. Then the numerator tells you how many you have. For example, 2 means: (i) you are dealing with fifths, and (ii) you have two of them. 5 Treating fractions like this can make adding and subtracting them easier. Example
Key words: • • • •
fraction add subtract denominator
Check it out: If you’re finding it difficult to understand the logic here, imagine the question was different — for example: “If you have two pens and you add an extra one pen, then how many pens do you end up with?” You’d have 3 pens, just like you have 3 fifths in the example.
Find
1
2 1 + . 5 5
Solution 2
1
The sum + is asking: “If you have two fifths and you add an extra 5 5 one fifth, then how many fifths do you end up with?” 3 The answer is , as you can see in the diagram. 5 +
=
You can write this as math:
2 1 3 + = 5 5 5
You add or subtract fractions with the same denominator by adding or subtracting the numerators, while keeping the denominator the same. For example, Example
4 1 4 +1 5 4 1 4 −1 3 + = = and − = = . 7 7 7 7 7 7 7 7 2
Find: (i) the sum
6 2 6 2 + , and (ii) the difference − . 11 11 11 11
Solution
(i)
152
6 2 6+2 8 + = = 11 11 11 11
Section 3.4 — Adding and Subtracting Fractions
(ii)
6 2 6−2 4 − = = 11 11 11 11
Guided Practice Work out the sums and differences in Exercises 1–8.
Check it out: Do additions and subtractions from left to right — just like with integers. See Lesson 2.1.4 for more information.
1.
2 1 + 7 7
2.
19 11 − 29 29
3.
50 15 − 6 6
4.
6 9 + 17 17
5.
26 15 + 62 62
6.
31 16 − 14 14
7.
29 13 1 − − 4 4 4
8.
6 6 6 − + 7 7 7
You May Need to Simplify Afterwards Don’t forget: See Lesson 3.4.2 to remind yourself about how to simplify fractions.
Even if you add two fractions that are in their simplest form, you might be able to simplify the answer. Example Find
3
1 1 + . Give your answer in its simplest form. 4 4
Solution
Add the fractions: Simplify the result:
Example
1 1 1 +1 2 + = = 4 4 4 4 2 1× 2 1 = = 4 2× 2 2
4
33 37 + ? Simplify your answer by finding the greatest 105 105 common divisor of the resulting fraction’s numerator and denominator.
What is
Solution
Add the numerators and keep the denominator the same. 33 37 33 + 37 70 + = = 105 105 105 105 Don’t forget: See Lesson 3.4.2 for more about greatest common divisors.
To simplify, find the greatest common divisor of 70 and 105. Since 70 = 2 × 5 × 7 and 105 = 3 × 5 × 7, the greatest common divisor of 70 and 105 is 5 × 7 = 35. So divide the numerator and denominator by 35 to find the simplest form. 70 2×35 2 = = 105 3×35 3
Section 3.4 — Adding and Subtracting Fractions
153
Guided Practice Work out the sums and differences in Exercises 9–20, giving your answers in their simplest form. 9.
7 5 − 8 8
10.
1 1 + 2 2
11.
5 10 + 20 20
12.
5 4 − 11 11
13.
5 1 − 6 6
14.
13 3 − 14 14
15.
7 3 − 4 4
16.
2 4 + 9 9
17.
9 5 + 7 7
18.
8 3 − 15 15
19.
9 27 + 18 18
20.
7 8 + 25 25
Independent Practice Now try these:
Work out the sums and differences in Exercises 1–6.
Lesson 3.4.3 additional questions — p442
1.
1 2 + 4 4
4.
126 119 − 3 3
2.
7 5 − 3 3
5.
3.
6 9 + 14 14
2 3 + 11 11
6.
9 25 + 35 35
Work out the sums and differences in Exercises 7–18. Give your answers in their simplest form. 7.
5 9 + 42 42
8.
12 15 + 66 66
9.
97 52 − 20 20
10.
15 5 − 15 15
11.
20 12 − 18 18
12.
21 1 − 7 7
13.
14 10 − 12 12
14.
8 4 + 9 9
15.
9 5 + 21 21
16.
10 1 3 + − 2 2 2
17.
1 2 3 + + 3 3 3
18.
19 10 6 − + 8 8 8
Round Up Adding and subtracting fractions with the same denominator is all about understanding the question.
2 5 + 3 3
just means “two-thirds and then five-thirds more,” so the result must be
The result won’t always be in its simplest form, but you’ve already seen how to fix that. 154
Section 3.4 — Adding and Subtracting Fractions
7 . 3
Lesson
3.4.4
Fractions with Dif ent Denomina tor s Difffer erent Denominator tors
California Standards: Number Sense 1.1 Compar e and order positi ve Compare positiv and ne ga ti ve fr actions neg tiv fractions actions,, decimals, and mixed numbers and place them on a number line.
After the last Lesson on adding fractions with the same denominator, this one is about adding fractions with different denominators. The idea here is to turn them both into fractions with the same denominator and then do exactly what you did in the last Lesson...
Number Sense 2.1 Solv e pr ob lems in volving Solve prob oblems inv ad dition, subtr action, addition, subtraction, multiplication, and division of positi ve fr actions and positiv fractions explain why a particular operation was used for a given situation.
Ev er yP air of Number s Has a Common Multiple Ever ery Pair Numbers
What it means for you: In this Lesson, you’ll learn about one way of adding and subtracting fractions with different denominators.
Key words: • • • •
equivalent fraction denominator common multiple cross-multiply
A common multiple of two numbers is a number that is a multiple of both of them. Take the numbers 2 and 3, for example: Multiples of 2: 2, 4, 6, 8, 10, 12, 14... Multiples of 3: 3, 6, 9, 12, 15... The numbers 6 and 12 appear in both lists, so 6 and 12 are both common multiples of 2 and 3. Example
1
Which of the following numbers are common multiples of 4 and 5? 8, 15, 20, 30, 32, 100 Solution
Common multiples will be multiples of 4 and multiples of 5. From the list, the multiples of 4 are: 8, 20, 32, and 100. From the list, the multiples of 5 are: 15, 20, 30, and 100. So the common multiples of 4 and 5 are 20 and 100.
Don’t forget: A multiple of a number is that number multiplied by an integer.
A quick way to find a common multiple of two numbers is to multiply them together. So 2 × 3 = 6 must be a common multiple of 2 and 3. Example
2
Find a common multiple of 11 and 13. Solution
One common multiple is given by 11 × 13 = 143.
Multiplying numbers together like this will always produce a common multiple. But it might not be the smallest common multiple. Check it out: You’ll learn a method for finding the least common multiple in the next Lesson.
Example
3
Find a common multiple of 12 and 24. Solution
One common multiple is given by 12 × 24 = 288. However, 24 is already a multiple of 12, so another possible answer is 24. Section 3.4 — Adding and Subtracting Fractions
155
Guided Practice Find a common multiple of each pair of numbers in Exercises 1–8. Try to find the smallest common multiples you can. 1. 1, 3 2. 5, 8 3. 10, 2 4. 6, 7 5. 11, 5 6. 3, 4 7. 5, 7 8. 8, 4
Common Multiples Mak e Common Denomina tor s Make Denominator tors Look at the fraction sum
1 4 + . 3 7
This time, you can’t just add the
Check it out:
numerators because the two fractions have different denominators.
When two fractions have the same denominator, you can say that they have a common denominator.
So you need to find two fractions with a common denominator — one 1 4 fraction that is equivalent to 3 and another that is equivalent to 7 . This common denominator has to be a common multiple of 3 and 7. Example
Don’t forget: To find an equivalent fraction, multiply by a fraction equivalent to 1. See Lesson 3.4.1 for a reminder.
4
Find fractions equivalent to
1 3
and
4 7
Use your equivalent fractions to find
that share a common denominator. 1 4 + 3 7
.
Solution
You need to find any common multiple of 3 and 7. The easiest one to use is 3 × 7 = 21. Now find fractions equivalent to
1 3
and
1 1 7 1× 7 7 = × = = 3 3 7 3× 7 21 Then add these equivalent fractions:
4 7
that have a denominator of 21. 4 4 3 4 × 3 12 = × = = 7 7 3 7 × 3 21
1 4 7 12 7 + 12 19 + = + = = 3 7 21 21 21 21
Example Find
1 1 + 2 3
5
.
Solution
1
1
First, make fractions equivalent to 2 and 3 that share a denominator. Use 2 × 3 = 6 as the common denominator. 1 1 3 3 1 1 2 2 = × = = × = 2 2 3 6 3 3 2 6 1 1 3 2 3+ 2 5 Then add the new fractions: + = + = = 2 3 6 6 6 6
156
Section 3.4 — Adding and Subtracting Fractions
Use exactly the same method for subtracting fractions. Example What is
6
3 1 − 4 9
?
Solution
Use a common denominator of 4 × 9 = 36. Your equivalent fractions are: 3 9 27 × = 4 9 36
1 4 4 × = 9 4 36
3 1 27 4 23 − = − = . 4 9 36 36 36
So
Guided Practice Don’t forget: “Finding a difference” means doing a subtraction.
Work out the sums and differences in Exercises 9–17. Give your answers in their simplest form. 9.
Check it out: The results of sums with different denominators may need simplifying just like sums with same denominators.
1 1 + 6 5
10.
1 1 − 5 6
11.
2 1 + 3 4
12.
1 1 + 9 3
13.
1 1 − 3 10
14.
1 1 − 2 4
15.
5 4 + 6 20
16.
7 6 + 9 11
17.
6 2 − 13 9
A Common Denomina tor Allo ws Comparison Denominator Allows Check it out: It’s difficult to say which fraction is greater if the denominators are different.
You can compare two fractions with a common denominator by comparing their numerators. Example
7
Which is greater:
7 9
5
or 6 ?
Solution
Find equivalent fractions with a common denominator. Use a common denominator of 9 × 6 = 54. Then:
7 6 42 × = 9 6 54
Check it out You could use a common denominator of 18.
5 15 7 14 = and = . 9 18 6 18
5 9 45 × = 6 9 54
These fractions have the same denominator, so compare the numerators. 42 < 45, so
42 45 < . 54 54
That means that
7 5 < 9 6
.
Section 3.4 — Adding and Subtracting Fractions
157
Check it out: You’ll use cross-multiplication in another context later in the book. Cross-multiplication is also used to solve proportions (as you’ll see in Lesson 4.1.4).
Cross-multiplication is also a way to compare fractions. It gives exactly the same results as you saw in the previous example, but it’s possibly slightly easier to remember. Look at the fractions in the previous example again:
7 9
5
and 6 .
Cross-multiplication involves multiplying the numerator of each fraction by the denominator of the other.
7 9
42 = 7 × 6
5 6
9 × 5 = 45
You can see that these two numbers are the same as the numerators in the fractions you found in Example 7. 7 7×6 42 = = 9 9×6 54
Don’t forget: Think of cross-multiplication as an easy way to remember which numbers to multiply together when you put fractions over a common denominator. It’s not really a brand new method.
5 5× 9 45 = = 6 6 × 9 54
and
So to compare two fractions, you can cross-multiply, and compare the results. Example
8
Which is greater:
3 5
or
2 3
?
Solution
Cross-multiply:
9=3×3
3 5
2 3
5 × 2 = 10
These are the numerators of equivalent fractions, which both have a common denominator of 5 × 3 = 15. So
3 9 2 10 = . And = . 5 15 3 15
This means
Example
2 3 > . 3 5
9
Which is greater:
6 7
or
13 15
?
Solution
Cross-multiply: 90 = 6 × 15 This means: Don’t forget: “>” means “is greater than.” “<” means “is less than.”
158
6 7
13 15
7 × 13 = 91
91 6 6 ×15 90 and 13 13× 7 = = = = 15 15× 7 105 7 7×15 105
So 13 > 6 or 6 < 13 . 15 7 7 15
Section 3.4 — Adding and Subtracting Fractions
Guided Practice Say which of the two numbers given in each of Exercises 18–25 is greater. 18.
19 17 , 10 20
19.
3 2 , 5 3
20.
5 6 , 9 11
21.
17 4 , 13 3
22.
4 5 , 15 19
23.
5 11 , 7 15
24.
9 4 , 21 9
25.
15 8 , 9 5
Independent Practice Find two common multiples of the numbers in Exercises 1–4. 1. 4 and 9
2. 3 and 12
3. 20 and 6
4. 7 and 1
5. Find a common multiple of 5 and 9 by multiplying these numbers together. 1
4
6. Write 5 and 9 as equivalent fractions, using your answer to Exercise 5 as a common denominator. 1 4 7. Using your answer to Exercise 6, find + . 5 9 Work out the sums and differences in Exercises 8–15. Give your answers in their simplest form. 8.
4 2 + 3 6
9.
1 5 + 5 9
10.
5 8 + 7 5
11.
4 2 − 3 6
1 3 7 2 1 1 5 2 + 13. − 14. − 15. − 2 8 8 3 6 9 6 3 Say which of the two numbers given in each of Exercises 16–19 is greater. 17 8 6 9 5 2 10 3 , , , 16. 17. , 18. 19. 19 9 13 19 21 8 23 7
12.
Now try these: Lesson 3.4.4 additional questions — p442
7
3
20. Andre and Amy both try to work out 1 + 2 , but come up with 8 4 different answers. Identify who got it right and find the mistake that the other made. Andre Amy 7 3 1 +2 8 4 15 11 = + 8 4 15 11 = + 8 8 26 = 8 13 = 4
7 3 1 +2 8 4 15 11 = + 8 4 15 22 = + 8 8 37 = 8
Round Up You can sidestep the problem of adding fractions with different denominators by turning them into fractions with the same denominator. Of course, you have to turn them into equivalent fractions, or you’ll end up answering a different question entirely. Section 3.4 — Adding and Subtracting Fractions
159
Lesson
3.4.5
Least Common Multiples
California Standards:
The last Lesson showed you how to add and subtract fractions with different denominators. This Lesson is about the same thing, but this time using the least common multiple. It’s pretty similar to the method in the last Lesson, but it means you don’t end up having to multiply really big numbers if you don’t need to.
Number Sense 2.1 Solve problems involving addition, subtraction, multiplication, and division of positive fractions and explain why a particular operation was used for a given situation. Number Sense 2.4 Determine the least common multiple and the greatest common divisor of whole numbers; use them to solve problems with fractions (e.g., to find a common denominator to add two fractions or to find the reduced form for a fraction).
The Least Common Multiple Is Less Than the Others You can always find a common multiple of two numbers by multiplying them together. But this method doesn’t always give you the least common multiple (LCM). One way to find the least common multiple is to compare lists of multiples and find the first number in both lists. Take 6 and 9, for example. Some multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48... Some multiples of 9: 9, 18, 27, 36, 45, 54... So the least common multiple of 6 and 9 is 18.
What it means for you: You’ll learn how to find the lowest number that two numbers both divide into exactly. You’ll also see how this can help you with adding and subtracting fractions.
Key words: • least common multiple • greatest common divisor
Check it out: The least common multiple of 16 and 28 is 112. This is a lot less than their product, which is 16 × 28 = 448.
Example
1
Find the least common multiple of 16 and 28. Solution
Multiples of 16: 16, 32, 48, 64, 80, 96, 112, 128... Multiples of 28: 28, 56, 84, 112... So the least common multiple of 16 and 28 is 112. The least common multiple isn’t always less than the result of multiplying the numbers together. Example
2
Find the least common multiple of 9 and 7. Solution
Check it out: Notice how you don’t have to continue either of these lists beyond 63. This is because you know that 9 × 7 = 63 is a common multiple, so you’re only using these lists to see if you can find one that’s smaller.
Multiples of 9: Multiples of 7:
9, 18, 27, 36, 45, 54, 63... 7, 14, 21, 28, 35, 42, 49, 56, 63...
So the least common multiple of 9 and 7 is 63.
Guided Practice Find the least common multiple of the pairs of numbers given in Exercises 1–12. 1. 2, 3 2. 4, 3 3. 12, 9 4. 10, 4 5. 6, 8 6. 15, 35 7. 30, 45 8. 4, 12 9. 14, 21 10. 25, 30 11. 6, 7 12. 55, 33
160
Section 3.4 — Adding and Subtracting Fractions
Check it out:
Prime Factorization Can Get the LCM More Quickly
Sometimes GCD is used for “greatest common divisor.”
You can find the least common multiple of two numbers by multiplying them together, then dividing by the greatest common divisor. Example
3
Find the least common multiple of 12 and 44. Solution
Don’t forget: For more information about prime factorization and finding the greatest common divisor, see Lesson 3.4.2.
First use prime factorization to find the greatest common divisor. 12 = 2 × 2 × 3 44 = 2 × 2 × 11 Greatest common divisor = 2 × 2 = 4. So the least common multiple of 12 and 44 is: 12 × 44 ÷ 4 = 132
Guided Practice Use prime factorization to find the LCM of the pairs of numbers given in Exercises 13–20. 13. 6, 8 14. 9, 12 15. 4, 6 16. 10, 15 17. 39, 15 18. 34, 32 19. 60, 45 20. 27, 45
The LCM Is a Good Choice for Common Denominator Remember, when you need to add or subtract fractions with different denominators, you must put both fractions over a common denominator. This means you need to find a common multiple of the denominators. Using the least common multiple means that you can work with smaller numbers. Don’t forget: See Lesson 3.4.4 to remind yourself about using common multiples when adding fractions.
Example What is
4
1 1 + 12 30
?
Solution
Check it out: If you’d used a common multiple of 12 × 30 = 360, the numbers you’d need to use would be much bigger, and much easier to make mistakes with. And you’d definitely need to simplify your answer. 1 1 30 12 42 + = + = 12 30 360 360 360 7 = 60
• 12 = 2 × 2 × 3 30 = 2 × 3 × 5 So the greatest common divisor of 12 and 30 = 2 × 3 = 6. • Least common multiple of 12 and 30 is 12 × 30 ÷ 6 = 60. • Now find fractions equivalent to denominator of 60. 1 1 5 5 = × = 12 12 5 60
1 12
and
1 30
which have a common
1 1 2 2 = × = 30 30 2 60
Now you can add the fractions together. 1 1 5 2 5+ 2 7 + = + = = 12 30 60 60 60 60 Section 3.4 — Adding and Subtracting Fractions
161
Guided Practice Work out the sums and differences in Exercises 21–29. Give your answers in their simplest form. Check it out: Leave any fractions greater than 1 as improper fractions.
21.
13 9 + 10 15
22.
4 5 + 9 6
23.
1 5 + 6 10
24.
5 1 − 6 8
25.
3 2 + 15 20
26.
13 5 + 42 14
27.
4 2 − 6 8
28.
7 1 − 24 16
29.
2 7 − 9 33
Independent Practice Find the least common multiple of the pairs of numbers given in Exercises 1–12. 1. 2, 9 2. 5, 7 3. 19, 3 4. 6, 10 5. 5, 35 6. 15, 21 7. 6, 33 8. 35, 55 9. 30, 66 10. 42, 28 11. 39, 12 12. 24, 16
Now try these: Lesson 3.4.5 additional questions — p442
Check it out: Leave any fractions greater than 1 as improper fractions.
13. Find the prime factorizations of 6 and 4. 14. Calculate the LCM of 6 and 4 5 1 15. Write and , using the LCM of 6 and 4 as a common 6 4 denominator. 5 1 16. Find − 6 4 Work out the sums in Exercises 17–31. Give your answers in their simplest form. 17.
2 1 − 3 4
18.
3 1 − 4 2
19.
1 2 + 9 7
20.
2 15 + 10 25
21.
8 5 − 9 6
22.
13 3 − 12 4
23.
8 18 + 11 44
24.
13 12 + 15 18
25.
9 9 − 10 14
26.
7 15 + 75 35
27.
4 1 − 27 45
28.
1 15 + 24 56
29.
10 6 + 49 21
30.
7 1 + 9 30
31.
31 10 − 42 56
Round Up Pausing for a second to find the least common multiple before trying to add or subtract two fractions with different denominators is a good way to avoid having to deal with really big numbers. Smaller numbers also make simplifying at the end easier. 162
Section 3.4 — Adding and Subtracting Fractions
Lesson
3.4.6
Mixed Numbers and Word Questions
California Standards: Number Sense 2.1 Solve problems involving addition, subtraction, multiplication, and division of positive fractions and explain why a particular operation was used for a given situation. Number Sense 2.4 Determine the least common multiple and the greatest common divisor of whole numbers; use them to solve problems with fractions (e.g., to find a common denominator to add two fractions or to find the reduced form for a fraction).
What it means for you: You’ll apply all of the skills you’ve learned so far in this Section to mixed number questions and questions written in words.
Key words • mixed number • improper fraction • fact
This Lesson combines two different parts — the first is about how to deal with mixed number sums and the second is about how to make sense of sums written in words.
Mixed Numbers Are Like Sums 3
Mixed numbers are like sums. 1 means “one and three-fifths,” and so is 5 3 the same as 1 + . One way to deal with questions that use mixed 5 numbers is just to replace all the mixed numbers with this kind of sum. Example
1
3 5
1 5
What is 1 + 2 ? Solution 3 5
1 5
3 5
1 5
1 + 2 = (1 + ) + ( 2 + ) 3 5
= 1+ 2 + + = 3+ =3
1 5
4 5
4 5
This method works quite well for simple additions. But there’s a better way of doing it for more complicated sums and differences.
Don’t forget: You can do additions in any order because of the commutative and associative properties of addition. See Lesson 2.3.1 to remind yourself about the commutative property of addition.
Mixed MixedNumber NumberSums SumsCan can Be be Done Done with WithFractions Fractions A more reliable way of doing mixed number sums is to convert all the mixed numbers to improper fractions before you start. Then you can use the addition and subtraction methods covered in the previous Lessons. Using improper fractions, the above example would become: Example
Don’t forget: To convert a mixed number to an improper fraction, multiply the whole number part by the denominator and add it to the numerator. See Lesson 3.1.2 for more details.
3 5
2 1 5
What is 1 + 2 ? Solution
8 11 8 + 11 19 3 1 4 1 +2 = + = = =3 5 5 5 5 5 5 5
Section 3.4 — Adding and Subtracting Fractions
163
If the fraction parts of the mixed numbers have different denominators, you’ll need to find a common denominator. Example 3 7
3 2 3
Find 4 − 1 . Solution
• First form improper fractions. ( 4 × 7) + 3 28 + 3 31 3 4 = = = 7 7 7 7
2 3
1 =
3+ 2 5 = 3 3
• Now you need to find a common denominator, so you have to find a common multiple of 7 and 3. Use 7 × 3 = 21. 31 31 3 93 5 5 7 35 = × = = × = 7 7 3 21 3 3 7 21 • Finally, you can subtract the fractions. 3 7
2 3
4 −1 =
Example 1 4
93 35 93 − 35 58 − = = 21 21 21 21
4 1 3
What is 4 + ? Give your answer as a mixed number. Solution 1 4
4 =
16 + 1 17 17 1 1 1 = , so you can rewrite the sum as 4 + = + . 4 3 4 4 4 3
The least common denominator of 3 and 4 is 12. 51 4 55 1 1 17 1 So 4 + = + = + = 4 3 4 3 12 12 12
And as a mixed number this is 4
7 . 12
Don’t forget: There’s more information about improper fractions and mixed numbers in Lesson 3.1.2.
Guided Practice Work out the sums in Exercises 1–12. 1 3
1. 1 +
2 3
1 2
4. 4 + 5 7. 8 − 3 1 5
1 9
1 3
10. 6 − 3 164
2. 2 + 3
1 3
1 3
3. 1 + 6
5. 7 + 1
3 4
3 5
6. 5 −
2 3
1 2
9. 6 − 6
8. 7 − 5 4 5
Section 3.4 — Adding and Subtracting Fractions
11. 26
19 10 −2 30 12
2 3
2 3
4 5
12. 8
2 3
1 3 1 6
11 7 −3 20 16
Word Questions Can Be Broken into Facts Some math questions look complicated but become easier if you break them down into pieces. Example
5
Jose and Stephanie are climbing a mountain. Jose climbs up 1 Stephanie climbs up
1 2 4
miles but then climbs back down
2 3
1 4
miles.
of a mile.
Who is further up the mountain? Solution
First separate out the facts you’ve been given.
Jose and Stephanie are climbing a mountain. Jose climbs
up 1
1 4
1 4
miles. Stephanie climbs up 2 miles but then climbs
back down
2 3
of a mile. Who is further up the mountain?
Then put the facts together again as math. Jose and Stephanie are climbing a mountain.
Jose climbs up 1
1 4
miles.
1 4
1 miles
Stephanie climbs up 2
1 4
1 2 4
miles.
But then climbs back down 2 3
miles
2 3
of a mile.
of a mile
Who is further up the mountain? If you turn the mountain on its side then the edge is like a number line. 1 1 4
So the question is asking whether 2 − Don’t forget: When fractions have the same denominator, you can compare them by comparing their numerators. See Lesson 3.4.4 to remind yourself about comparing fractions.
1 4
2 3
9 4
2 3
is greater than
15 , 12
2 − = − = 19 12
27 8 − 12 12
=
19 12
2 3
1 4
1 2 4 3 1 1 . 4
2 −
is greater than 1 4
5 4
1 = =
15 12
so:
Stephanie is further up the mountain than Jose. Section 3.4 — Adding and Subtracting Fractions
165
Guided Practice 1
13. A recipe requires 2 cups of flour, 1 cup of milk, and 4 cup of vegetable oil. How many cups of ingredients are needed in total? 14. Two years ago, Eric was 4
1 6
feet tall. Now, he is 5
1 3
feet tall.
How much has he grown? 15. Cody has 2
1 2
pounds of bananas, but sells 1
Elizabeth buys a further
2 3
1 4
pounds to Elizabeth.
pounds from the store. How many pounds
of bananas do Cody and Elizabeth end up with? 16. Carlos is doing a science experiment, and puts a beaker containing 1 24
puts
1 8
1 5
liter of water into
liter of vinegar. He mixes them together and
liter into a test tube. How much is left in the beaker?
Independent Practice Worktry outthese the sums in Exercises 1–9. Now Give your Page x, Qu x-xanswers as mixed numbers. 1 2
1. 1 − 4 6
1 2
4. 5 + 2 7. 12
1 2
2. 1 + 1 9
5. 3
1 6 − 11 6 15
1 2
1 2
3. 2 + 3
1 8 −1 3 21
8. 19
1 5 − 10 16 8
6. 7
2 3
1 17 −6 4 18
9. 12
1 3 + 16 11 17
10. Several students are having a race. Each contestant has to run 100 meters then cycle 1000 meters. Their times are shown below. How long did each student take to finish, and who finished first?
Running
Now try these:
Cycling
Lesson 3.4.6 additional questions — p443
Karina
T.J.
Emily
Kyle
1 4 2 3 3
1 3 9 2 10
3 10 3 2 5
1 5 2 3 9
(all times measured in minutes) 11. Laura and Juan are at a water fountain. Laura drinks water, and Juan drinks 1 5
3 10
1 8
gallon of
gallon. Laura is still thirsty, so she drinks
gallon of fruit juice. Who drank more? How much more did that
person drink?
Round Up Sometimes it can be difficult to make sense of long word problems, but just remember the key steps — break up the question and separate all the facts from one another. Then take each one individually and figure out what it means in math. 166
Section 3.4 — Adding and Subtracting Fractions
Section 3.5 introduction — an exploration into:
Per cents with a Doub le Number Line ercents Double Percents can seem complicated. This is because they are an alternative way to describe an amount. For example, suppose you have 20 apples and then eat half of them. You’ve eaten 10 apples, but this is the same as 50%. One way to see the link is to use a double number line. A double number line shows percents next to actual values. You can use it to estimate solutions. Example Use a double number line to estimate 30% of 40. Solution
Draw a double number line. Label one side of the number line from 0% to 100%, and the other side from 0 to 40.
0%
100%
0
40
You can use some easy-to-find points on the double number line to estimate the value of x. • 50% is halfway between 0% and 100%. 30% Its value is half of 40, which is 20. 0% 25% 50% 100% • 25% is halfway between 0% and 50%. Its value is half of 20, which is 10.
10
0
30% is a bit more than 25%. You can see that 30% is between 10 and 20, but closer to 10. In fact, it looks like 30% of 40 is about 12.
x
20
40
Here, the unknown amount has been labelled x.
Example 160 is what percent of 180? Use a double number line to find an estimate. This time, the unknown is the percent percent.
Solution
Draw a double number line, and mark on 180 (= 100%). Now mark in some other important percents. • 50% is halfway between 0% and 100%. Its value is half of 180, which is 90.
x% 0%
50%
75%
0
90
135
• 75% is halfway between 50% and 100%. Its value is halfway between 90 and 180, or 135. You can see that x is between 75% and 100%. In fact, it looks about 85-90%.
100% 180 160
Exercises 1. Draw a double number line to estimate the solution to these percent problems. a. What is 40% of 80? b. What is 60% of 250? c. 105 is what percent of 150? d. 64 is 20% of what number?
Round Up This method will only give you an estimate of the answer — it probably won’t be exact. But estimates can be really useful for checking your work, or getting a rough idea of the answer. Section 3.5 Explor a tion — Percents with a Double Number Line 167 Explora
Lesson
Section 3.5
3.5.1
Fractions and P er cents Per ercents
California Standard:
Percents are another way of writing fractions, and they’re used a lot in real life. You might have percents as test scores, or on a sale poster in a store. In this Lesson, you’ll see how to turn a fraction into a percent.
Number Sense 1.4 te gi ven Calculate giv Calcula per centa ges of quantities percenta centag and solve problems involving discounts at sales, interest earned, and tips.
What it means for you: You’ll learn about percents as another way to write fractions.
Per cents Ar e a Dif ent Way to Write F ercents Are Difffer erent Frractions Percents are a way to write a fraction as a single number. One percent is the same as the fraction
Percents are written using the symbol %. For example:
Key words: • percent • fraction • numerator • denominator • equivalent fraction • greatest common divisor
1 . 100
23 percent
is written
23% 23 . 100
and means Example
1
An auto dealership has 100 cars for sale. 32 of the cars are red. What percent of the cars are red? Solution
The fraction of the cars that are red is
32 . 100
Written as a percent, that is 32%. Of course, fractions don’t always come in hundredths. To turn a fraction into a percent, you can use equivalent fractions. Example Write
4 5
2
as a percent.
Solution
Rewrite Don’t forget: The numerator is on the top of the fraction. The denominator is on the bottom. For example:
3 4
numerator denominator
4 5
in hundredths:
80 4 8 = = = 80% 5 10 100
You can find the same answer using algebra. You need to multiply a fraction equivalent to 1 to make the denominator 100. Multiply
4 5
by
n . n
Denominator of the result = 5 × n = 100 So
n = 100 ÷ 5 = 20
So to make the denominator 100, multiply
4 5
by
20 : 20
80 4 4 20 4 × 20 = × = = = 80% 5 5 20 5 × 20 100 168
Section 3.5 — Percents
4 5
by
Guided Practice In Exercises 1–6, write each fraction as a percent. 1.
65 100
2.
1 4
3.
21 50
4.
3 25
5.
11 20
6.
9 25
7. A bag contains 50 marbles, 27 of which are green. What percent of the marbles are green? 8. There are 20 students riding the bus, and 14 of them are girls. What percent of the students are girls?
Ther e Is Another Way to Con ver er cents here Conv ertt F Frractions to P Per ercents There is an alternative method for converting a fraction to a percent. You can just do the division shown by the fraction, then multiply by 100. Example Write
1 8
3
as a percent.
Solution
Do the division shown by the fraction (here, 1 ÷ 8), and then multiply by 100. So 1 ÷ 8 = 0.125. Then multiply by 100 to get: 0.125 × 100 = 12.5, so 0.125 = 12.5%. Whether you convert a fraction to hundredths, or do the division and then multiply by 100, the answer should come out the same. Example
4
Main Street Park has 64 trees. Of those, 28 are pine trees. What percent of the trees are pine? Solution
Write the answer as a fraction: Don’t forget:
28 64
of the trees are pine.
n 28 n ? × = : n 64 n 100 Looking at the denominators, you can see that 64 × n = 100. This means n = 100 ÷ 64 = 1.5625. 28 ×1.5625 28 28 1.5625 43.75 So = × = = = 43.75%. 64 ×1.5625 64 64 1.5625 100
1. You can convert this to hundredths by multiplying by
You need to understand both methods, but you can use whichever method you prefer.
2. Or you can find: 28 ÷ 64 = 0.4375. Then multiply by 100: 0.4375 × 100 = 43.75, so 0.4375 = 43.75%.
Guided Practice In Exercises 9–12, write each fraction as a percent. 9.
929 1000
10.
23 200
11.
17 40
12.
3 8
Section 3.5 — Percents
169
You Can Tur naP er cent into a F urn Per ercent Frraction It’s straightforward to turn a percent into a fraction. Don’t forget to rewrite the fraction in its simplest form. Don’t forget: For a reminder about finding the greatest common divisor and the simplest form of a fraction, see Lesson 3.4.2.
Example
5
Write 40% as a fraction in simplest terms. Solution
40% is the same as
40 . 100
The greatest common divisor of 40 and 100 is 20. 40 100
=
2 × 20 5 × 20
=
2 5
, so 40% =
2 5
Guided Practice In Exercises 13–20, write the percent as a fraction in its simplest form. 13. 39% 14. 38% 15. 50% 16. 75% 17. 5% 18. 98% 19. 35% 20. 16%
Independent Practice In Exercises 1–6, write each fraction as a percent. 1.
77 100
2.
1 10
3.
16 25
4.
7 8
5.
23 40
6.
111 500
Write each of the following percents as a fraction in its simplest form. 7. 9% 8. 20% 9. 70% 10. 86% 11. 95% 12. 92% 13. Cora buys 16 apples at the store. She buys 4 red and 12 green apples. What percent of Cora’s apples are green? 14. DaMarcus recorded the weather on 80 days last winter. It snowed on 4 of those days. On what percent of the days did it snow? Now try these: Lesson 3.5.1 additional questions — p443
15. Out of 250 grade 6 students, 150 have brown eyes. What percent of the students have brown eyes? 16. Seth is practicing his baseball skills. He hits 880 out of 1000 pitches. What percent of the balls does he hit? Miguel and Claire go to the park to play tennis. There are 25 tennis courts at the park. When Miguel and Claire get there, 5 courts are available. 17. What percent of the tennis courts are available? 18. What percent of the tennis courts are not available?
Round Up Percents are often easier to work with than fractions. For example, comparing percents is usually more straightforward than comparing fractions. A percent is one number, so there aren’t any numerators and denominators to worry about. 170
Section 3.5 — Percents
Lesson
3.5.2
Per cents and Decimals ercents
California Standard:
You can rewrite fractions as percents to make them easier to compare and use. Percents can be converted to decimals too, and this sometimes helps when you’re using them in calculations.
Number Sense 1.4 ven Calculate giv Calcula te gi per centa ges of quantities percenta centag and solve problems involving discounts at sales, interest earned, and tips.
What it means for you: You’ll see how to rewrite percents as decimals, and decimals as percents.
Key words: • • • •
percent decimal hundredth order
Per cents Can Be Tur ned into Decimals ercents urned You’ve seen that percents are basically the same as hundredths. You’ve also seen that one-hundredth is the same as the decimal 0.01. So
1 100
= 1% = 0.01.
This means that you can rewrite any percent as a decimal by dividing by 100. Example
1
Write 5% and 10% as decimals. Solution
5% =
5 100
= 5 ÷ 100 = 0.05
10% =
10 100
= 10 ÷ 100 = 0.1
Guided Practice Don’t forget: You can leave out any extra zeros on the right-hand side of a decimal number. So 0.10 is written as 0.1.
In Exercises 1–8, write each percent as a decimal. 1. 18% 2. 99% 3. 9% 5. 20% 6. 50% 7. 23.2%
4. 2% 8. 61.78%
Decimals Can Be Tur ned into P er cents urned Per ercents It’s also straightforward to turn a decimal into a percent. Example
2
Write 0.04 and 0.82 as percents. Solution
0.04 = 4 hundredths =
4 100
= 4%
0.82 = 82 hundredths =
82 100
= 82%
Notice that this time, you’ve multiplied by 100.
Guided Practice In Exercises 9–16, write each decimal as a percent. 9. 0.35 10. 0.81 11. 0.9 12. 0.06 13. 0.6 14. 1.0 15. 0.195 16. 0.001 Section 3.5 — Percents
171
You Should Become F amiliar with Common P er cents Familiar Per ercents The table below shows some fractions, decimals, and percents that are equal to each other. The table will be useful for solving some questions. You might already be familiar with some of these values. Table of Common Percents and their Equivalents
Fraction
Decimal
Percent
1 100
0.01
1%
1 10
0.1
10%
1 5
0. 2
20%
1 4
0.25
25%
1 2
0.5
50%
3 4
0.75
75%
1
1
100%
Check it out: This table will help you to answer some of the questions you’ll see in the next few Lessons.
Example
3
There are 32 students in Mr. Baker's art class, and 8 of them are in the school play. What percent of Mr. Baker's art students are in the play? Solution
First, you can represent this as a fraction: Writing in simplest terms,
8 32
=
1 4
8 32
Looking at the table above, you can see that
1 4
is equal to 25%.
So, 25% of Mr. Baker's students are in the play.
Guided Practice In Exercises 17–20, use the table above to find the right percent. 17. In a class of 30 students, 3 have pet cats. What percent of the class have cats? 18. Angela has 20 pens. 5 of them are blue. What percent of Angela’s pens are blue? 19. Tony travels 15 miles to get home. He goes 12 miles by bus, then walks 3 miles. What percent of the journey does Tony walk? 20. Donna buys 13 apples and 13 oranges. What percent of Donna’s fruit is apples? 172
Section 3.5 — Percents
Per cents Mak e It Easier to Place Number s in Or der ercents Make Numbers Order Sometimes you can’t tell right away which of two numbers is greater. They might be fractions with different denominators, or you might have one fraction and one decimal. Changing everything into percents makes things easier to put in order. Example
4
Order the following from least to greatest: 2 5
0.5
35%
Solution
The table on the previous page tells you that So Check it out: You may find decimals easier to understand than percents. If you do, it’s okay to answer this type of question by turning all the numbers into decimals instead.
2 5
=2×
1 5
1 5
= 20%.
= 2 × 20% = 40%.
And 0.5 = 50 hundredths =
50 100
= 50%.
Now you can see that 35% < 40% < 50%. So 35% <
2 5
< 0.5.
Guided Practice In Exercises 21–24, order each set of numbers from least to greatest. 21. 72%,
7 , 10
0.82
23. 45%,
0.4,
12 , 25
1 , 20
3%,
0.09
24. 0.9,
17 , 20
88%,
22. 23 50
3 4
Independent Practice In Exercises 1–4, write each percent as a decimal. 1. 54% 2. 3% 3. 70%
4. 2.5%
In Exercises 5–8, write each decimal as a percent. 5. 0.84 6. 0.3 7. 0.08
8. 0.172
In Exercises 9–12, order each set of numbers from least to greatest. 9. 0.2, 18%, Now try these: Lesson 3.5.2 additional questions — p443
11.
3 5,
33 50 ,
1 4
0.65,
10. 63%
11 20 ,
12. 39%,
54%, 9 25 ,
0.5 21 50 ,
0.35
13. A box contains 50 red pens and 150 blue pens. What percent of the pens in the box are blue? 14. There are 4 red and 16 blue marbles in a bag. What percent of the marbles are blue?
Round Up The table showing the common percents, fractions, and decimals is really useful. It will help you answer percent questions more quickly. You should try to memorize it if you can. Section 3.5 — Percents
173
Lesson
3.5.3
Per cents of Number s ercents Numbers
California Standard:
You saw earlier in this Chapter how to multiply whole numbers by fractions. A percent is another way to write a fraction, so you can multiply whole numbers by percents too. This is called finding a percent of a number.
Number Sense 1.4 te gi ven Calculate giv Calcula per centa ges of quantities percenta centag and solve problems involving discounts at sales, interest earned, and tips.
Multipl y to Find a P er cent of a Number Multiply Per ercent
What it means for you:
You’ve already practiced multiplying fractions by integers.
You’ll learn how to find a certain percent of a number.
Finding a percent of a number is similar because, for example, “1% of 250” means 1% × 250 = 0.01 × 250 =
Key words: • • • •
percent multiply fraction decimal
1 100
× 250.
So it usually helps to convert the percent to a fraction or decimal first.
Example
1
What is 25% of 140? Solution
There are a number of ways you can solve this problem. Check it out:
Written as a fraction, 25% =
Use a fraction:
You can find the fraction and decimal equivalents of 25% in the table of common percents and equivalents from Lesson 3.5.2.
1 4
× 140 =
1 . 4
140 4
= 35 Written as a decimal, 25% = 0.25.
Use a decimal:
0.25 × 140 = 35 1% of 140 = 140 ÷ 100 = 1.4
Find 1% first: So
25% of 140 = 25 × (1% of 140) = 25 × 1.4 = 35
Guided Practice In Exercises 1–12, find the following percents: 1. 10% of 100 2. 2% of 50 3. 20% of 3000 4. 80% of 50 5. 12% of 750 6. 44% of 800 7. 95% of 80 8. 4% of 950 9. 35% of 80 10. 15% of 240 11. 85% of 130 12. 29% of 250
174
Section 3.5 — Percents
Real-lif e Situa tions Can In volv e P er cents of Number s eal-life Situations Inv olve Per ercents Numbers You can often find a percent of a number in real-world situations. Check it out: You can solve this type of question using either fractions or decimals. You can normally solve the questions more quickly using the decimal method with a calculator. But you should get some practice with the fraction method too.
Example
2
In a survey of 200 people leaving a movie theater, 65% said that they enjoyed the movie. How many people in the survey enjoyed the movie? Solution
The problem is asking you to find 65% of 200. 65% of 200 = 0.65 × 200 = 130 So 130 of the people in the survey enjoyed the movie.
Guided Practice 13. Carla has 12 friends at school, and 50% of them ride on the same bus as she does. How many friends ride on the same bus as Carla? 14. A mixture of 30 liters of soap and water contains 10% soap. How many liters of the mixture is soap? 15. A school football team has won 70% of their last 60 games. How many of those games did they win? 16. A parking lot contains 150 cars, and 18% of them are blue. How many blue cars are in the parking lot?
Some Questions Wor k the Other Way Ar ound ork Around If you know a percent of a number you can find the number itself. Example
3
25% of the animals in a pet store are dogs. There are 10 dogs. How many animals are there in the store? Solution
25% of the number of animals is 10.
Check it out: You can use this algebraic approach using decimals: 25% = 0.25, and 0.25 × n = 10, so n = 10 ÷ 0.25 = 40.
You can write this as: 25% × the number of animals = 10 Or: 0.25 × the number of animals = 10 So: the number of animals = 10 ÷ 0.25 = 40. Or you can do this using algebra... 1
Call the number of animals n. Since 25% = 4 , you know This means n = 10 ÷
1 4
1 4
× n = 10.
= 10 × 4 = 40. There are 40 animals.
Section 3.5 — Percents
175
Guided Practice Solve Exercises 17–21, using either fractions or decimals. 17. 50% of a number is 13. What is the number? 18. Nancy has 27 green marbles. If 75% of Nancy’s marbles are green, how many marbles does Nancy have in total? 19. 4% of the trees in a park are oak trees. If there are 7 oak trees in the park, how many trees are there altogether? 20. 25% of a number is 30. What is 75% of the number? 21. In a survey of favorite animals, 2% of people answered ducks, and 10% said elephants. 14 people said their favorite animals are ducks. How many said their favorite animals are elephants?
Independent Practice In Exercises 1–4, find the following percents: 1. 70% of 20 2. 3% of 600 3. 25% of 1200 4. 45% of 220 5. Darnell has 200 baseball cards. He gives 20% of his collection to his friend Kenny. How many cards does Darnell give to Kenny? 6. There are 25 pet cats living on 7th Street. Mrs. Delgado owns 16% of them. How many cats does Mrs. Delgado own? 7. Mr. Jones plants 80 daffodil bulbs in his garden. 85% of them grow successfully. How many daffodil plants does Mr. Jones end up with? 8. A zoo has 425 animals, and 4% of them are penguins. How many penguins does the zoo have? 9. The school soccer team scored 75 goals last season. Luisa scored 36% of the goals. How many goals did Luisa score last season? Use either fractions or decimals to solve Exercises 10–14. 10. 10% of what number is 12? 11. 220 is 80% of what number? 12. 50% of a number is 8. Calculate 25% of the number. Now try these: Lesson 3.5.3 additional questions — p444
13. In the class elections, 20% of the students voted for Elena. If Elena received 54 votes, how many students voted in total? 14. Phoebe is counting the vehicles that pass her house. 10% of them are motorcycles and 50% are cars. If Phoebe counts 6 motorcycles, how many cars does she count?
Round Up It’s really important that you understand this Lesson. In the rest of this Section, you’ll learn about circle graphs, percent increase and decrease, and simple interest. All of these will involve finding percents of numbers. 176
Section 3.5 — Percents
Lesson
3.5.4
Cir c le Gr a phs and P er cents Circ Gra Per ercents
California Standard:
You’ll have seen circle graphs before in earlier grades. They’re often used to compare different percents. Circle graphs are useful because they show clearly how the size of one group relates to another.
Number Sense 1.4 ven Calculate giv Calcula te gi per centa ges of quantities percenta centag and solve problems involving discounts at sales, interest earned, and tips.
100% is Equal to the Whole Amount 100% of something is all of it.
What it means for you: You’ll use circle graphs to compare different percents.
It’s important to remember this when you’re looking at questions about percents. Example
Key words: • • • •
percent circle graph 100% compare
1
Yesenia has a number of marbles. 65% of the marbles are red. The rest are all blue. What percent of Yesenia’s marbles are blue? Solution
All the marbles are either red or blue. So
(percent of red marbles) + (percent of blue marbles) = all marbles 65%
+
percent of blue marbles = percent of blue marbles = =
100% 100% – 65% 35%
So 35% of Yesenia’s marbles are blue.
Guided Practice 1. 47% of students in a school are boys. What percent are girls? 2. A store took a survey of customers’ ages. 29% said they were under 15. What percent were aged 15 or over? 3. 18% of a class of 6th graders have a pet dog. What percent of the class don’t have a dog? 4. A bag contains a number of colored counters. All the counters are either blue, green, or yellow. 30% of the counters are blue, and 40% of the counters are green. What percent of the counters are yellow? 5. Mrs. Goldman’s garden has only red, white, and blue flowers. 37% of the flowers are red, and 39% of the flowers are white. What percent of the flowers are blue? Action
27%
Comedy
41%
Romance
?
6. Visitors to a movie theater were asked which of 3 types of movie they liked best. This table shows the results. What is the missing percent? Section 3.5 — Percents
177
Cir c le Gr a phs Ar e Often Di vided into P er cents Circ Gra Are Divided Per ercents Circle graphs show how a total splits into different parts. This graph represents a math class split into boys and girls. The whole circle represents the whole class. The two sections represent the boys and the girls.
Boys
Girls
The girls section is larger. This means the class has more girls than boys. When a circle graph shows percents, the whole circle represents 100%. A section that represents a certain percent fills that percent of the circle. Example
2
This circle graph shows the results of a survey to find which out of apples, bananas, and oranges students liked best. What percent of the students like bananas best? Solution
Apples Bananas 25% ?% Oranges 35%
The whole circle represents 100%, so the total value of all the sections must be 100%. Call the percent of students who like bananas best b%. Then
25 + 35 + b = 100 60 + b = 100 b = 100 – 60 = 40
The total of all sections is 100% 25 + 35 = 60 Subtr act 60 fr om both sides Subtract from
So 40% of the students like bananas best.
Guided Practice In Exercises 7–10, find the missing value in each circle graph. 7. 8. 9. 10. 30%
?%
?% 34%
?%
50% 55%
?%
31%
21% 24%
Don’t forget: In Exercises 11–15, the numbers on the circle graphs aren’t percentages, but actual quantities.
178
Section 3.5 — Percents
19%
A survey asked people which of three drinks they prefer. Juice This graph shows the number of people who said they 20 prefer each drink. Water 10 11. How many people answered the survey? Milk 20 12. What fraction of these people preferred each drink? 13. What percent of these people preferred each drink? This graph shows the number of insects a scientist Wasps Crickets 75 counted for a study. 125 Bees 14. How many insects were there in total? Butterflies 125 15. What percent of each type of insect were there? 175
You Can Tur nP er cents on Cir c le Gr a phs into Number s urn Per ercents Circ Gra Numbers If you know how many units make up the full 100% of a circle graph, then you can work out how many each section represents. Example
3
Chris, Martina, and D’Andre each ran for student body president. A total of 150 students voted in the election, and the outcome of the election is shown in the circle graph. How many students voted for Martina?
D’Andre 38% Chris 22%
Martina 40%
Solution
The circle graph tells you that 40% of the students voted for Martina. You know that the whole circle graph represents 150 students. So the number of students who voted for Martina is 40% of 150 = 0.4 × 150 = 60 A total of 60 students voted for Martina.
Guided Practice Use the graphs provided to answer Exercises 16–19. 16. This graph shows the results of a survey of students’ favorite school subjects. 80 students were surveyed. Find the number of students that liked each subject best.
Won 60%
Lost 25% Tied 15%
17. The percents of games won, lost, and tied by a soccer team out of their last 40 games are shown on this graph. How many of the games did they win, lose, and tie?
18. This graph shows the percent of each color of balloon at a party. There were a total of 200 balloons at the party. Find how many of each color of balloon there were at the party. Rat Goldfish 18% 24% Cat 28%
Dog 30%
Science 25% English Math 30% 45%
Red 48% Blue 25%
White 27%
19. 300 people were surveyed as to which of 4 types of animals they liked the best. This graph shows the results. How many people in the survey liked each type of animal?
Section 3.5 — Percents
179
Independent Practice 1. A group of sixth-grade students each go to one of 5 clubs after school on Tuesdays. The table shows what percent of the students go to each club. What percent of the students go to the reading club? Explain your answer.
Acting Club Reading Club
?
Debate Club
10%
Music Club
40 %
Recycling Club
10%
2. A group of people were asked which of 3 vegetables they like best. This graph shows the results. Calculate the missing percent. 3. 300 people were questioned. Find the number of people who said they like each vegetable. Washington 36% Lincoln Cleveland 26% ?%
Cabbage ?% Carrots 33% Spinach 42%
4. A town held a vote to decide who to build a statue of. The results are shown on this graph. Find the missing percent. 5. 250 people voted. Figure out how many people voted for each choice.
6. This graph shows the colors of cars in the school parking lot. What percent of the cars are red? 7. If there are 175 cars in the lot, how many of each color are there? Winter Spring ?% 28% Fall Summer 16% 32%
15%
Blue 24%
Red ?%
Other Silver 20% 28%
8. This graph shows the results of a survey asking people about their favorite season. Figure out the missing percent. 9. The survey questioned 325 people. How many said they like each season the best?
Mr. Benson has 300 acres of land on his farm. He grows corn, lettuce, and tomatoes. There are 60 acres of lettuce and 90 acres of tomatoes. 10. How many acres of corn are there? The graph shows how the land on the farm is divided A between the three crops. C 11. Which section of the graph represents each crop? B Explain your answer. 12. What percent of the land is used for each crop? Now try these: Lesson 3.5.4 additional questions — p444
Swimming 175 Running Tennis 150 75 Cycling ?
500 sixth-graders were asked which of four sports they liked best. This graph shows the results. 13. How many students said they liked cycling best? 14. Write down what fraction of the sixth-graders preferred each sport. 15. What percent of the students preferred each sport?
Round Up Circle graphs are useful for giving out information. People who don’t know much about math can still understand what it means when one part of the circle is bigger than another. 180
Section 3.5 — Percents
Lesson
3.5.5
Per cent Incr ease ercent Increase
California Standard:
One real-world use of percents is to describe how certain numbers have changed. People say something has increased or decreased by a certain percent. In this Lesson, you’ll learn about percent increase.
Number Sense 1.4 ven Calculate giv Calcula te gi per centa ges of quantities percenta centag and solv e pr ob lems solve prob oblems in volving discounts at sales, inv interest earned, and tips tips.
What it means for you: You’ll solve problems where percents are used to describe how much a number has increased.
You Can Incr ease a Number b y a Cer tain P er cent Increase by Certain Per ercent When you add to a number, percent increase is a way to describe how much it’s changed. The percent increase is the amount of the increase, written as a percent of the original number. Example
1
Increase 200 by 25%. Solution
Key words: • • • • •
percent increase add tips sales tax
“Increase by 25%” means “add on 25% of the original number.” 25%
+
+
25%
100%
25%
+
25%
=
+
25%
=
100%
125%
The amount of the increase is 200 × 25% = 200 × 0.25 = 50 So to increase 200 by 25%, add 50. 200 + 50 = 250 Don’t forget: You convert a percent to a decimal by dividing by 100.
There’s another, slightly quicker way to find this answer. The answer is 125% of the original amount, so you want 200 × 125% = 200 × 1.25 = 250
Guided Practice Don’t forget: Check that all your answers are reasonable.
In Exercises 1–6, calculate the total after the increase. 1. Increase 100 by 12%. 2. Increase 50 by 20%. 3. Increase 160 by 25%. 4. Increase 60 by 5%. 5. Increase 185 by 60%. 6. Increase 275 by 30%.
Section 3.5 — Percents
181
Tips Ar e Calcula ted Using P er cents Are Calculated Per ercents
Check it out: In real life, people often estimate the right percent to tip, or round the amount up. But in math you should work out the exact tip, and only round to the nearest cent, unless an exercise tells you to do otherwise.
When people eat at a restaurant, they often tip the person who waited on them, or served their food. People often work out how much to tip by finding a percent of the bill. Example
2
Victor goes out to lunch. The bill comes to $14.00, and Victor wants to leave a 20% tip for the waiter. How much tip should he leave? Solution
Victor wants to leave a 20% tip, so he should multiply the dollar amount by 20%. 20% × $14.00 = 0.2 × $14.00 = $2.80 So Victor should leave a tip of $2.80.
Example
3
Aisha buys a meal in a restaurant. The meal costs $25.00, and Aisha decides to leave an 18% tip. How much does Aisha pay in total? Solution
The amount of the increase is
$25.00 × 18% = $25.00 × 0.18 = $4.50.
So the total Aisha pays is $25.00 + $4.50 = $29.50.
Guided Practice 7. Rick is at a cafe. The bill for his meal is $10. If Rick leaves a 20% tip, how much does he pay in total? 8. Dolores is buying breakfast. It costs $21, and Dolores leaves a 19% tip. How much does she leave as a tip? 9. The Jackson family goes out for a meal. The bill comes to $150, and Mrs. Jackson leaves a 15% tip. How much does she leave as a tip? 10. Wayne’s restaurant bill is $120. If Wayne leaves a tip of 25%, how much does he pay in total? 11. Francine is a waitress. A customer adds a 22% tip to his $42 bill. How much of a tip does he leave for Francine? 12. The cost of Phil’s meal is $64. Phil wants to leave a tip of 18%. How much should he pay in total?
182
Section 3.5 — Percents
You Can Figur e Out the P er cent Incr ease Figure Per ercent Increase If you know the original number, and you know what it’s been increased to, you can find the percent increase. Example
4
50 is increased to 58. What is the percent increase? Solution
You could solve this by guess and check: 50 × 110% = 50 × 1.1 = 55 50 × 120% = 50 × 1.2 = 60 50 × 115% = 50 × 1.15 = 57.5 50 × 116% = 50 × 1.16 = 58
Try 10% Try 20% Try 15% Try 16%
— too small — too large — too small — correct
Increasing 50 to 58 is an increase of 16%. You can also use algebra: original amount × percent increase = amount of increase The original amount is 50, and the amount of the increase is 58 – 50 = 8. So
50 × p = 8 p = 8 ÷ 50
Call the per cent incr ease p percent increase Di vide both sides b y 50 Divide by
= 0.16 = 16% So increasing 50 to 58 is an increase of 16%.
Example
5
Lucy’s restaurant bill came to $200. She paid $238. What percent tip did Lucy leave? Solution
The amount of the increase is
$238 – $200 = $38
So
Call the per cent incr ease p percent increase
200 × p = 38 p = 38 ÷ 200
Di vide both sides b y 200 Divide by
= 0.19 = 19% So Lucy left a tip of 19%.
Section 3.5 — Percents
183
Guided Practice In Exercises 13–18, work out the percent increase. 13. 100 increased to 142. 14. 25 increased to 30. 15. 160 increased to 184. 16. 36 increased to 63. 17. 45 increased to 49.5. 18. 260 increased to 377. 19. Wanda buys lunch for $15.00 plus a tip. If the total she pays is $18.00, what percent tip has she left? 20. Guillermo pays $41.30 for his meal, including a tip. The original bill was $35. What percent did Guillermo tip? Check it out:
21. Diana is paying for a meal. The bill is $25.00, and Diana wants to tip at least 18%. If she pays $30, has she left enough as a tip?
Sales tax is calculated in the same way as tips – as a percent of the price before the tax is added.
22. AJ spends $59.40 on a pair of sneakers. The price without sales tax is $55. What percent sales tax does AJ pay?
Independent Practice In Exercises 1–4, find the total after the increase. 1. Increase 50 by 40%. 2. Increase 220 by 5%. 3. Increase 70 by 17%. 4. Increase 56 by 65%. In Exercises 5–8, work out the percent increase. 5. 110 increased to 121. 6. 60 increased to 75. 7. 350 increased to 427. 8. 12 increased to 14.4. 9. Corey’s bill for lunch is $25. He decides to tip 16%. How much does Corey leave as a tip? 10. After sales tax, Mr. Johnson paid $535 for a dishwasher that cost $500 before sales tax. What percent did he pay in sales tax? 11. Ms. Connor gave the hairstylist $50. Her haircut cost $40. What percent of the bill did Ms. Connor leave as a tip for the stylist? 12. Justin is buying a football. The cost before sales tax is $30. If the sales tax is 8%, how much does Justin pay in total? 13. With sales tax, the cost of Gavin’s new hat is $21.50. If the sales tax is 7.5%, what is the cost of the hat, not including tax? Now try these: Lesson 3.5.5 additional questions — p444
14. Mrs. Gomez takes a cab to the airport. The fare is $85.47, and she pays $100. What percent tip does Mrs. Gomez give the cab driver? 15. Zina, Adriana, and Raymond all buy dinner in the same restaurant. Zina’s bill is $28.50, and she leaves a 20% tip. Adriana’s bill is $30. She leaves a tip of 18%. Raymond adds a 22% tip to his bill of $27. How much do they each pay in total? Who pays the most?
Round Up Percent increases are one-half of the story. Next Lesson, you’ll learn about percent decreases. The math you’ll need for that is almost identical to what you’ve learned in this Lesson. 184
Section 3.5 — Percents
Lesson
3.5.6
Per cent Decr ease ercent Decrease
California Standard:
In the last Lesson, you saw how to use percents to describe the change when you add to a number. When you subtract from a number, you can describe the change in a similar way. This is percent decrease.
Number Sense 1.4 ven Calculate giv Calcula te gi per centa ges of quantities percenta centag and solv e pr ob lems solve prob oblems in volving discounts a inv att sales sales, interest earned, and tips.
What it means for you: You’ll solve problems where percents are used to describe how much a number has decreased.
You Can Decr ease a Number b y a Cer tain P er cent Decrease by Certain Per ercent You can use percents to describe what happens when you decrease a number. Percent decrease is the amount of change, described as a percent of the original number. Example
1
Key words:
Decrease 200 by 25%.
• • • •
Solution
percent decrease subtract discount
The amount of the decrease is 200 × 25% = 200 × 0.25 = 50 So to decrease 200 by 25%, take away 50. 200 – 50 = 150
Check it out: The way you calculate percent decrease is just like what you saw last Lesson for percent increase. The only difference is that this time the original number is getting smaller, not bigger.
You can also say that if you take away 25%, then you’re left with 100% – 25% = 75% of the original number. 75% of 200 = 75% × 200 = 0.75 × 200 = 150
Guided Practice In Exercises 1–6, calculate the total after the decrease. 1. Decrease 100 by 15%. 2. Decrease 150 by 50%. 3. Decrease 80 by 20%. 4. Decrease 132 by 25%. 5. Decrease 300 by 35%. 6. Decrease 125 by 6%. 7. A number is decreased by 44%. What percent of the number is left? 8. A number is decreased by 75%. What fraction of the number is left?
Section 3.5 — Percents
185
Per cent Decr ease Is Used to Describe Discounts ercent Decrease Percent decrease is often used to describe the discount when the price of an item that’s for sale is lowered. Example
2
A store discounts a coat by 20%. The coat originally cost $90. What is the discounted price of the coat? Check it out: You could also work this out using the fact that if the discount is 20%, the new price is 100% – 20% = 80% of the old price. You can work this out as: $90 × 80% = $90 × 0.8 = $72.
Solution
The amount of the decrease is $90 × 20% = $90 × 0.2 = $18 So the discounted price is $90 – $18 = $72
Guided Practice 9. Kitty buys some pens. They were originally priced at $3, but now there is a 50% discount. How much does Kitty pay for the pens? 10. This table shows some items on sale at a sporting goods store. Calculate the sale price of each item. Which item has the lowest price after the discount?
Item
Original price
Percent discount
Basketball
$50
18%
Hockey stick
$60
30%
Skateboard
$ 70
40%
11. A travel agent offers Mrs. Winters a discount of 17% on a vacation. If the original price was $3500, how much does Mrs. Winters pay? 12. Juliana buys a clock that originally cost $24, but now has a 5% discount. How much does Juliana pay for the clock? Three cars are shown below with the original price and the discount available for each car. 13. Find the price of each car after the discount. 14. Which car has the lowest price after the discount?
A.
,000
0
$20
B.
15% off
C.
,00
8 $1
$19,000
10% dis
count
186
Section 3.5 — Percents
ount
Disc
5%
You Can Figur e Out the P er cent Decr ease Figure Per ercent Decrease Finding the percent decrease is like finding the percent increase. Example
3
If 45 is decreased to 27, what is the percent decrease? Solution
original amount × percent decrease = amount of decrease The original amount is 45. The amount of the decrease is 45 – 27 = 18. So
45 × p = 18 p = 18 ÷ 45
Call the per cent decr ease p percent decrease Di vide both sides b y 45 Divide by
= 0.4 = 40% So decreasing 45 to 27 is a decrease of 40%.
Example
4
A table originally priced at $210 has been discounted to $189. What is the percent discount? Solution
The amount of the decrease is $210 – $189 = $21 Now find what percent of $210 is equal to $21. Call that percent p. 210 × p = 21 p = 21 ÷ 210 = 0.1 = 10% So the discount is 10%.
Guided Practice In Exercises 15–20, work out the percent decrease. 15. 200 decreased to 180. 16. 120 decreased to 90. 17. 55 decreased to 11. 18. 160 decreased to 152. 19. 500 decreased to 60. 20. 325 decreased to 286. 21. Denzel goes to the bookstore. He buys a book for $15 that has been discounted from $20. What is the percent discount? 22. Brian buys a sweater that has been reduced from $20 to $11. What is the percent discount?
Section 3.5 — Percents
187
Independent Practice 1. The stationery store is selling everything for 40% off. Michael buys a pencil sharpener that originally cost $1. How much does he pay? 2. Mykelti buys a pair of pants in a sale. They have been discounted from $22 to $11. What percent discount is this? 3. Lauren buys a pair of boots which originally cost $68. The store gives a discount of 25%. How much does Lauren pay? 4. Ms. Dominguez is buying a set of golf clubs. They have been reduced from $400 to $320. What is the percent discount? 5. Donny’s new bike has a 35% discount on it. If the original price was $260, what is the discounted price? 6. Tina wants to buy a new scarf. She sees one that is reduced by 40% from $16. What is the new price? 7. Kia is buying a couch for her apartment. The furniture store is offering a discount on one couch from $800 to $700. What percent discount is this? Luther is eating at a restaurant. His bill comes to $50, and he uses a coupon to get a discount of 20%. 8. What is the cost of the meal after the discount has been taken off? 9. Luther decides to leave a tip of 25% of the bill, after the discount has been taken off. How much does he pay in total? 10. These three watches have all been reduced. The labels show the original prices and the percent discount. Find the discounted price of each watch, and say which costs the least after the discount.
$100
A.
Discount 35% $120
20% off
$150 40%
C.
$1200
B.
A.
t
iscoun
20% d
$1350
12% off
Now try these: Lesson 3.5.6 additional questions — p445
C.
D.
$1150
Discount
10%
30% off
B.
$1500
disc
oun
t
11. The labels on these couches show the original prices and the percent discounts. Find the discounted price of each couch, and say which has the lowest price after the discount.
Round Up The topics of percent increase and percent decrease are very similar. Once you’ve got the hang of one, the other should make sense too. Practice plenty of both types of questions until you’re comfortable with them. 188
Section 3.5 — Percents
Lesson
3.5.7
Simple Inter est Interest
California Standard:
Banks pay people when they put their money into savings accounts. But if someone borrows money, they usually have to pay back some extra on top of what they borrow. These are both examples of interest. In this Lesson, you’ll learn about one type of interest.
Number Sense 1.4 ven Calculate giv Calcula te gi per centa ges of quantities percenta centag and solv e pr ob lems solve prob oblems in volving discounts at sales, inv inter est ear ned interest earned ned, and tips.
What it means for you: You’ll learn about simple interest, which is a common use of percents.
Key words: • • • •
percent simple interest principal rate
Inter est Is Ad ded to Loans or Sa vings Interest Added Savings If you borrow money, you usually have to repay more than you borrowed. The extra money you pay is interest. If you save money in the bank, the bank pays you to save your money with them. The extra money you receive is also called interest. Example
1
Pablo borrowed $1000 to buy a car. One year later, he repaid $1050. How much interest did Pablo pay? Solution
The interest is the difference between what Pablo borrowed and what he paid back. So Pablo paid $1050 – $1000 = $50 interest.
When you’re dealing with interest, the time is usually given in years. The original amount that is borrowed or saved is called the principal.
Guided Practice Sara puts $200 in a savings account. After 1 year, the interest has increased it to $210. 1. How much interest did Sara’s money earn? 2. What was the principal? Matthew borrows $500. He repays the loan over 24 months. The total he pays back is $560. 3. What was the principal? 4. How many years did it take Matthew to repay his loan? 5. How much interest did Matthew pay? Omar puts $400 into a savings account. He leaves the money in the account for 48 months. The account pays $25 interest each year. 6. What is the principal? 7. How many years did Omar leave his money in the account for? 8. How much interest did he earn in this time? Section 3.5 — Percents
189
Simple Inter est Is One Type of Inter est Interest Interest Simple interest is when the interest paid each year is a certain percent of the principal. The percent that’s added is called the interest rate. Example
2
Lakeesha puts $300 into an account with a simple interest rate of 8%. How much interest will the bank pay in one year? Solution
The amount of simple interest the bank will add on each year is 8% of the principal of $300 that Lakeesha has paid in. 8% of $300 = 0.08 × 300 = 24 The bank will pay $24 into the account in one year.
Check it out: Annual (as in annual interest rate) is another word for yearly.
The amount of simple interest paid in a certain length of time = (the amount of interest paid in one year) × (the number of years) = (the principal) × (the interest rate) × (the number of years) You can rewrite this as a mathematical formula:
I = Prt Where
I = interest (the amount of interest paid) P = principal (the original amount borrowed or saved) r = rate (the annual interest rate, written as a decimal) t = time (the time in years you’re calculating interest for)
Example Don’t forget: For a reminder of how to work with expressions, see Chapter 2.
3
Patrick borrows $350 at a simple interest rate of 4%. He repays the loan after 6 months. How much interest will he pay? Solution
You can use I = Prt:
So
P = $350 r = 4% = 0.04 t = 6 months = 0.5 years
I = Prt = 350 × 0.04 × 0.5 = 7
Patrick will pay $7 in simple interest at the end of 6 months.
190
Section 3.5 — Percents
Guided Practice Don’t forget: To work out the total someone owes or the balance in their savings account, add the interest to the principal.
Exercises 9–16 are about simple interest. Use the formula I = Prt to answer them. 9. Christina puts $100 in a 5% savings account for 3 years. How much interest does she earn? 10. Dan borrows $125 for 2 years. The interest rate is 20%. How much will he have to pay in total? 11. Bonnie saves $600 in an account with an interest rate of 4%. After 4 years, how much interest will she have earned?
Check it out: When you use I = Prt, remember that t is always the number of years. So if you’re dealing with a period of 6 months, put that into the equation as half a year, or 0.5 years.
12. Hernan puts $350 into a bank account paying 6% interest. What will be the total in his account in 2 years? 13. Sally borrows $400, with an annual interest rate of 8%. She repays the loan after 3 months. How much interest does she pay? 14. Keisha puts $200 into a savings account. How much interest will she earn in 6 months if the interest rate is 10%? 15. Steve takes out a loan of $150. The interest rate is 12%. How much does Steve owe in total after 4 months? 16. Tyreese saves $240. If his account pays 15% interest, how much interest will he receive in 3 years?
Rear e I = Prt to Solv e Other Pr ob lems earrrang ange Solve Prob oblems By rearranging I = Prt, you can solve other problems involving simple interest. Example
4
Dawn paid $200 into a savings account. After 3 years, she had earned $30 simple interest. What is the yearly interest rate? Solution
You can use I = Prt. You know that I = 30, P = 200, and t = 3. Now you want to find r. I = Prt
Write out the ffor or mula orm
30 = 200 × r × 3
Put in vvalues alues ffor or I,, P, and t
30 = 600r
200 × 3 = 600
r =
30 = 0.05 = 5% 600
Di vide both sides b y 600 Divide by
The annual interest rate on Dawn’s savings account is 5%.
Section 3.5 — Percents
191
Guided Practice Use the formula I = Prt to answer Exercises 17–22. 17. Joanna puts $100 into a savings account. In 4 years, she receives $28 interest. What is the interest rate on Joanna’s account? 18. Cleavon saves $200, with an interest rate of 6%. How long does it take Cleavon to earn $60 interest? 19. Simon borrows some money with a 10% interest rate. After 3 years, he has repaid the loan with $36 interest. How much did Simon borrow? 20. Rosa borrows $50 at an interest rate of 12%. She repays the loan with $12 interest. How long does it take Rosa to repay the loan? 21. Kate’s savings account has a 10% interest rate. In 6 months, she receives $55 interest. How much money did Kate put in the account? 22. John puts $250 in his savings account. In 2 years, he earns $25 interest. What is the interest rate?
Independent Practice Exercises 1–8 are all about simple interest. 1. Winnie borrows $700, at an interest rate of 13%. How much interest does she owe after 2 years? 2. Jorge puts $1700 in a savings account. If the interest rate is 5%, how much interest does Jorge get in 3 years? 3. Aleesha saves $1200. Her bank pays her an interest rate of 7%. How much interest does Aleesha receive in 4 months? 4. Martin borrows $245 for 6 months. How much interest does he pay, if the interest rate is 10%? 5. Chloe borrows $300, and pays 15% interest on the loan. She owes $180 interest after how many years? 6. In 1 year, Aurelia earns $14 interest on her savings. If the interest rate is 8%, how much did she save? 7. Caleb’s savings of $80 earn $36 interest over 5 years. What is the interest rate on Caleb’s account? Now try these: Lesson 3.5.7 additional questions — p445
8. Austin put $260 in a savings account at a simple interest rate of 8.5% for 4 years. Joe put $325 in a savings account at a simple interest rate of 8% for 4 years. Who earned the most in interest at the end of 4 years?
Round Up In real life, it’s useful to be able to understand interest. People who understand interest rates can make sure they get a good deal when they want to save or borrow money. Banks often use a more complicated type of interest, called compound interest. You won’t be learning about that in grade 6, but you might see it in math lessons in later grades. 192
Section 3.5 — Percents
Chapter 3 Investigation
Wildlif e Trails ildlife You’re a park ranger responsible for the trails at a nature park. A colleague has suggested organizing a Park Fun Run, and you need to help her plan it. Use the map and the additional information below.
921 miles Adam’s Bluff
2 21 miles 5 miles
Echo Canyon
Deer Run
Bird Park
Cornerstone
334 miles
Part 1: How long are the trails from: • Bird Park to Deer Run? • Deer Run to Cornerstone?
Additional Informatio n: • The distance from De er Run to Bird Park is half a mile shorter th an the distance from Bird Park to Corne rstone. • The distance from De er Run to Cornerstone is one-thir d of the distance from Bird Park to Corne rstone. • The distance from Ec ho Canyon to Deer Run is 5% shorter than the distance from Echo Canyon to Adam 's Bluff.
• Echo Canyon to Deer Run?
Part 2: Your colleague wants the Park Fun Run to cover all the trails once. • Is this possible? Where could it start and finish? • How long would the Park Fun Run be? Extensions • Mark wants to hike from Adam's Bluff to Bird Park along the shortest route. What route should he take? What is the length of this route in miles? • A new trail is being planned between Echo Canyon and Bird Park. 1 The trail will be exactly 2 2 times the length of the trail from Deer Run to Cornerstone. (i) What is the length, in miles, of the new trail? (ii) With this new trail, will it still be possible to organize a Park Fun Run that covers all the trails once? If possible, describe a suitable new route. Open-ended Extension • The organizers of the Park Fun Run need to provide drinks for the runners. Drinks stations can be placed at the above locations, and should be as evenly spaced as possible. Find the best locations for them (assuming no new trails have yet been built). • An additional trail from Deer Run to Cornerstone has been suggested. How will this affect the route of the Park Fun Run?
Round Up Being able to use fractions is an incredibly useful skill. Buy two drinks for a dollar and a half each, and you’ll be adding fractions to work out the total. Cha pter 3 In vestig a tion — Wildlife Trails 193 Chapter Inv estiga
Chapter 4 Ratio, Proportion and Rate Section 4.1
Exploration — Billboard Ratios ................................. 195 Ratio and Proportion ................................................. 196
Section 4.2
Proportions in Geometry ........................................... 209
Section 4.3
Exploration — Plotting Conversions .......................... 221 Converting Units ........................................................ 222
Section 4.4
Exploration — Running Rates ................................... 236 Rates ....................................................................... 237
Chapter Investigation — Sunshine and Shadows ............................. 253
194
Section 4.1 introduction — an exploration into:
Billboar d R a tios Billboard Ra Photographs don’t usually show objects at their actual size. However, some things are the same on a photograph as they are in real life. For example, if a person’s shoulders are actually 3 times wider than their head, then this will also be true in a photo. This is what you are going to explore... Body measurement height
You need a recent photograph of yourself. With a partner measure: (i) yourself, and (ii) the picture of yourself.
head width arm span
The measurements you need are shown in this table. Make a copy of the table to record your measurements.
2
Picture
Real life
12 1.5
160 20
1 2
5 3
3
leg length
4
shoulder width
5
1 4
Use centimeters for all your measurements. Also, make the measurements in real life and on the picture in the same way (for example, measure your arm span fingertip to fingertip). Example Find the ratios of the measurements on the picture to the real-life measurements. Solution
Add an extra column to your table. To work out the ratios, divide each measurement on the picture by the corresponding real-life measurement.
Ratio picture ÷ real life
0.075 = 12 ÷ 160
Exercises 1. What do you notice about your ratios? How do you explain this? Example Your picture is used on a billboard poster. In this picture you are 10 meters (1000 cm) tall. Use ratios to determine the width of your head on the poster. Solution
billboard height 1000 = = 6.25 real-life height 160 billboard head width billboard head width This is the same for all measurements, so: = = 6.25 real-life head width 20 This tells you that on the billboard, your head would be 6.25 × 20 cm = 125 cm wide. The ratio of height on the billboard to real-life height is:
Exercises 2. Calculate each of your measurements for a billboard on which you will be 10 meters tall. 3. Calculate your measurements if your picture were put on a postage stamp. On the postage stamp, you will be only 1.5 cm tall.
Round Up This was all about ratios. You saw that the ratio of measurements on a picture to measurements in real life is the same, no matter what you measure. Very useful... as you’ll see later in the Section. Section 4.1 Explor a tion — Billboard Ratios 195 Explora
Lesson
Section 4.1
4.1.1
Ratios
California Standard:
You’ve probably come across ratios before — for example, making orange drink by diluting about one part concentrated drink with about four parts water. This way of stating how the amount of one thing compares with the amount of another thing is called a ratio.
Number Sense 1.2 Interpret and use ratios in different contexts (e.g., batting averages, miles per hour) to show the relative sizes of two quantities, using appropriate notations (a/b, a to b, a:b).
What it means for you: You’ll learn what ratios are and the different ways you can represent them.
Ratios Are a Way of Comparing Amounts of Things A ratio is the amount of one thing compared with the amount of another thing. In the diagram, the ratio of rabbits to cats is 5 to 3. This is because there are 5 rabbits but only 3 cats.
Key words: • ratio • fraction
Example
1
What is the ratio of carrots to hamsters in the picture below? And what is the ratio of hamsters to carrots?
Solution
Count the number of carrots. There are 3 carrots. Count the number of hamsters. There are 2 hamsters. Make sure you get the order of the numbers right. The ratio of carrots to hamsters has the number of carrots first. The ratio of carrots to hamsters is 3 to 2. In the same way, the ratio of hamsters to carrots is 2 to 3.
Guided Practice 1. What is the ratio of tennis rackets to tennis balls in the picture opposite? 2. What is the ratio of tennis balls to tennis rackets in the picture? 3. There are ten people on a basketball court and two baskets. What is the ratio of people to baskets on the court?
196
Section 4.1 — Ratio and Proportion
There Are Three Ways of Writing a Ratio The previous examples used words like “5 to 3” to write a ratio. But there are other ways to write ratios, and they all mean the same thing. There are three ways you need to know about. Check it out: A ratio is really just another name for a fraction.
The c olon “:: ” is another way o ” in ratios. of writing “ tto You’d still say this “5 to 3.”
Example
This fraction tells you that there are 5 of one thing for every 3 of another.
5 3
5 to 3 5:3
2
The ratio of dogs to cats in the picture below is 4 to 3. How else could you write this ratio? Write the ratio of cats to dogs in three different ways.
Check it out: Notice that forming a ratio basically involves dividing one number by another. Writing a ratio as a fraction shows this most clearly.
Solution
There are three ways of writing ratios. 4 You could also write the ratio of dogs to cats as 4 : 3 or . 3 3 The ratio of cats to dogs is 3 to 4, or 3 : 4, or . 4
Guided Practice 4. The ratio of cars to people on a street is 1 : 2. How else can you represent this ratio? 2
5. The ratio of boys to girls in a class is . 3 Express the ratio of girls to boys in three different ways. 6. What is the ratio of the width of the first rectangle shown below to the second?
4 yards
3 yards
Section 4.1 — Ratio and Proportion
197
Ratios Don’t Have Any Units Ratios don’t have any units. A ratio is just one number compared with another, such as “3 to 2.” Usually, ratios compare things measured in the same units. Then, the units cancel out when you divide the two quantities. Example
3
What is the ratio of the area of the big square on the right to the smaller one?
Area 2 = 4 cm
Solution
Don’t forget: Ratios can be written as fractions.
To form the ratio of the areas, divide one by the other. 7 cm 2 7 = Ratio of big area to small area = 4 cm 2 4
Area 2 = 7 cm
But if the units are different, the description of what the ratio shows is usually a bit more detailed. Example Don’t forget: See also Lesson 4.4.1 — where rates are covered.
4
A walker hikes 7 miles in 3 hours. What is the ratio of the distance walked in miles to the time taken in hours? Distance walked in miles
Solution
Ratio of the distance walked in miles to the time taken in hours = Time taken in hours
7 . 3
Guided Practice Jess makes purple paint using 4 cans of blue paint to 1 can of red. It takes her 3 hours to paint her bedroom, and she uses all her paint. 7. What is the ratio of cans of blue paint to cans of red paint? 8. What is the ratio of cans of paint used to the time taken in hours?
Independent Practice In Exercises 1–4, write the ratio shown in two equivalent ways. 11 1. 7 : 6 2. 3 to 8 3. 4. 7 : 11 7
Now try these: Lesson 4.1.1 additional questions — p445
Exercises 5–7 are based on the the circle graph on the right. It shows the professions of 24 former pupils of a school.
Working, but profession unknown: 3
Teachers: 6 Civil servants: 4
Unemployed: 2 Business people: 9
5. What is the ratio of people working in business to civil servants? 6. What is the ratio of unemployed people to people in business? 7. What is the ratio of people who are working and whose professions are known, to working people whose professions are unknown?
Round Up You’ve now met all of the basics of ratios. Next, you’ll see how to use ratios in real life and how to simplify them to make your calculations easier. 198
Section 4.1 — Ratio and Proportion
Lesson
4.1.2
Equivalent Ratios
California Standard:
In real life, you might have to scale ratios up or down — like when you change a cake recipe from serving 4 people to serving 8 people. In this Lesson you’ll learn how to find equivalent ratios that allow you to scale quantities. It’s very similar to making equivalent fractions, so some of this might seem familiar to you.
Number Sense 1.2 Interpret and use ratios in different contexts (e.g., batting averages, miles per hour) to show the relative sizes of two quantities, using appropriate notations (a/b, a to b, a:b).
What it means for you: You’ll learn how to form different ratios with the same value, and to find the simplest form of a ratio.
Equivalent Ratios Show the Same Thing 2
Equivalent ratios are the same kind of thing. Example
Key words: • simplifying • equivalent
1
Fractions like 4 and 2 are different, but they represent the same thing. They’re called equivalent fractions.
1
What is the ratio of red squares to blue squares in the pattern on the right? Solution
There are different answers to this question. Altogether, there are 12 red squares and 24 blue squares, so one answer is 12 : 24. But you can also say that the ratio of red to blue squares is 1 : 2. This is because there is 1 red square for every 2 blue squares. The ratios 12 : 24 and 1 : 2 are equivalent ratios. Don’t forget: You simplify ratios in exactly the same way as you simplify fractions. For example,
12 1×12 1 = . = 24 2 ×12 2 Look back to Lessons 3.4.1 and 3.4.2 for more information.
If you have two equivalent ratios, then you can say that the one that uses smaller numbers is simpler. So 1 : 2 and 12 : 24 are equivalent ratios, but 1 : 2 is simpler. In fact, 1 : 2 is in its simplest form, since no other equivalent ratio uses smaller numbers. To simplify a ratio, divide both parts by the same number. For example, to simplify 12 : 24, divide both sides by 12 to get 1 : 2. Example
2
Andrew has a bag of marbles. The bag contains 4 blue and 14 black marbles. What is the ratio of blue to black marbles in its simplest form? Solution
There are 4 blue marbles, and 14 black marbles. So the ratio of blue to black marbles is 4 : 14. But you can simplify the ratio 4 : 14 to 2 : 7.
Section 4.1 — Ratio and Proportion
199
Guided Practice Simplify the ratios in Exercises 1–6. 1. 7 to 14 2. 3 : 9 3. 5 : 50 3 4. 5. 16 : 18 6. 24 to 15 18 7. There are 12 socks in Ryan’s drawer. There are 9 black socks, and the rest are white. What is the ratio of black to white socks? Give your answer in its simplest form. 8. Chad works in a grocery store. He sells 22 apples in the same time that he sells 8 oranges. What is the ratio of apples to oranges sold? Give your answer in its simplest form.
You Can Simplify Ratios in Stages Don’t forget: A common factor is a number that divides evenly into two (or more) other numbers. For example, 12 is a common factor of 24 and 36.
You can find the simplest form of a ratio in different ways. One way to do this is to make the ratio simpler in stages. Example
3
Find the simplest form of the ratio 105 : 70. Solution
Just keep looking for numbers that divide into both sides of the ratio (common factors). Don’t forget: You must remember to do the same thing to both sides of the ratio.
105 and 70 both divide evenly by 5, so 105 : 70 is equivalent to 21 : 14. 21 and 14 both divide evenly by 7, so 21 : 14 is equivalent to 3 : 2. Nothing divides into both 3 and 2, so 3 : 2 is in its simplest form. So the simplest form of 105 : 70 is 3 : 2.
Don’t forget: You can find the greatest common divisor (GCD) of two numbers by finding their prime factorizations. 105 = 3 × 5 × 7 70 = 2 × 5 × 7 Then multiply the common factors (5 and 7) together to find the greatest common divisor. So the GCD of 105 and 70 is 5 × 7 = 35.
Another way is to find the greatest common divisor of the numbers in the ratio. Example
4
Find the simplest form of the ratio 105 : 70. Solution
The greatest common divisor of 105 and 70 is 35. So divide both sides of the ratio by 35. This means that the simplest form of 105 : 70 is 3 : 2.
Guided Practice Find the simplest form of the ratios in Exercises 9–14. 9. 18 to 12 10. 6 : 9 11. 25 : 75 24 12. 13. 72 : 56 14. 70 to 98 96 200
Section 4.1 — Ratio and Proportion
Ratios Can Contain Variables Ratios can include variables as well as numbers. Example
5
Find a number, x, so that the ratio 7 : 2 is equivalent to x : 4 Solution
You need to compare the ratios 7 : 2 and x : 4. If you write the ratios as fractions, it’s easier to compare them directly. The two ratios are equivalent, so the fractions must be equal. 7 x x stands for any number. You have to find what x is. = 2 4 The denominator of the second ratio is twice as big as the denominator of the first ratio. Since the fractions are equal, the numerator of the second ratio must also be twice as big as the numerator of the first. So x = 2 × 7 = 14. This means 7 : 2 is equivalent to 14 : 4.
Guided Practice 15. Find the number x, so that the ratio 9 : 3 is equivalent to 3 : x. 16. Find the number y, so that the ratio 14 : 44 is equivalent to y : 66.
Independent Practice 1. Mr. Jefferson has 7 boys in his science class, and 19 girls. What is the ratio of boys to girls? 2. There are 10 sculptures and 16 paintings in a room at the art museum. What is the ratio of sculptures to paintings? Give your answer in its simplest form. Now try these: Lesson 4.1.2 additional questions — p446
In Exercises 3–6, write each ratio in its simplest form. 18 6. 48 to 33 3. 5 to 10 4. 80 : 60 5. 20 7. Find a number y, so that the ratio 1 : 9 is equivalent to y : 18. 8. Find a number z, so that the ratio 5 : 6 is equivalent to z : 30. For Exercises 9–10, give your answers in the simplest form. A bag contains 40 beads, 25 coins, and 10 keys. 9. What is the ratio of beads to coins? 10. What is the ratio of keys to other objects in the bag?
Round Up So now you know how to find equivalent ratios, and how to simplify ratios. The next step is to write proportions, which is what you’ll learn all about in the next Lesson. Section 4.1 — Ratio and Proportion
201
Lesson
4.1.3
Proportions
California Standard:
Equivalent ratios are ones that can be simplified to be the same thing. In this Lesson, you’ll learn that if two ratios are equivalent ratios, then you can write them as a proportion.
Number Sense 1.3 Use proportions to solve problems (e.g., determine the value of N if 4/7 = N/ 21, find the length of a side of a polygon similar to a known polygon). Use cross-multiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse.
What it means for you:
A Proportion Is an Equation of Equivalent Ratios A proportion is an equation showing two equivalent ratios. For example, the ratios 1 : 2 and 3 : 6 are equivalent. 1 3 So you can write the proportion: = 2 6 Example
1
You’ll learn what proportions are, how to write them in math, and how to solve them.
To paint one wall, Gabrielle uses three cans of blue paint, and one can of white paint. To cover the room’s four walls, she uses twelve cans of blue paint and four cans of white paint.
Key words:
Write a proportion arising from this situation.
• proportion • equivalent ratios
Solution
Check it out: You can check whether your proportion is correct by simplifying both fractions until they are in their simplest form, and checking that they are equal.
To write a proportion, you need two equivalent ratios. The first ratio comes from the paint Gabrielle used to paint one wall. She used 1 can of white and 3 cans of blue. 1 This gives a ratio of white to blue of 1 : 3, or . 3 The other ratio comes from the paint used to paint the entire room. 4 . The ratio of white to blue for four walls is 4 : 12, or 12 1 4 You can write these two equivalent ratios as a proportion: = 3 12 Example
2
Pattern 2 consists of four tiles like Pattern 1.
1
2
Write a proportion involving the numbers of black dots and red dots in Patterns 1 and 2. Solution
Pattern 1 contains 2 black dots and 5 red dots. This gives a ratio of
2 . 5
Pattern 2 must contain 4 × 2 = 8 black dots and 4 × 5 = 20 red dots. 8 This gives a ratio of . 20 2 8 These ratios must be equivalent, so you can write the proportion = . 5 20 202
Section 4.1 — Ratio and Proportion
Guided Practice Don’t forget: Remember that you can write a proportion for any two ratios that are equivalent. You can’t write proportions if the ratios aren’t equivalent.
1. The ratio of boys to girls in a class is 5 : 6. The ratio of boys to girls in the whole school is 100 : 120. Can these two ratios be written as a proportion? Explain your answer. 2. Can the ratios 3 : 7 and 5 : 21 be written as a proportion? Explain your answer. 3. The second pattern below is made up of three tiles like the first pattern. Write a proportion using the numbers of black dots and red dots in the two patterns.
Proportions Can Involve Variables In the last Lesson, you saw that ratios can involve variables. The same is true of proportions. Example
3
The ratios 3 : 4 and x : 12 are equivalent. Write a proportion involving x. Solution
Just treat x like you would any other number. 3 x So you can write the proportion = . 4 12 You may have to make a proportion describing a word problem. Example
4
Meredith likes to mix her apple juice with a little water. She uses a ratio of 1 part water to 4 parts apple juice. If x is the number of cups of apple juice she would need to mix with 2 cups of water, write a proportion involving x. Solution
Don’t forget: The ratios “1 : 4,” “1 to 4,” and 1
“ 4 ” all mean the same thing.
You need two ratios to write a proportion. Since Meredith needs to mix 2 cups of water with x cups of apple juice, one ratio of water to apple juice you can use is 2 to x. You also know Meredith wants a ratio of water to apple juice of 1 : 4. 2 1 Use these two ratios to write a proportion: = x 4
Guided Practice To make smoothies, Joshua uses a ratio of oranges to strawberries of 1 : 6. One morning, Joshua makes a smoothie using 3 oranges. 4. Call the number of strawberries Joshua uses x. Write a proportion involving x. Section 4.1 — Ratio and Proportion
203
If you have a proportion containing a variable, you can solve it. Check it out: This is exactly the same as Example 4 on the previous page, but this time you have to find x — not just write a proportion.
Example
5
Meredith likes to mix her apple juice with a little water. She uses a ratio of 1 part water to 4 parts apple juice. If x is the number of cups of apple juice she would need to mix with 2 cups of water, find x. Solution
There are different ways to solve this... (i) You can reason it out. If the ratio of water to apple juice is 1 : 4, then there will be 4 times as much apple juice as water. Since there are 2 cups of water, there will be 8 cups of apple juice. 2 1 = . x 4 The numerator on the right is half the numerator on the left. So the denominator on the right must be half the denominator on the left. This means x = 4 × 2 = 8, and Meredith needs 8 cups of apple juice.
(ii) Or you can use your proportion:
Guided Practice 5. Write and solve a proportion for the equivalent ratios x : 6 and 4 : 12. To make smoothies, Joshua uses a ratio of oranges to strawberries of 1 : 6. One morning, Joshua makes a smoothie using 3 oranges. 6. Solve your proportion from Exercise 4 to find x.
Independent Practice In Exercises 1–6, decide whether the ratios are equivalent. For those ratios that are equivalent, write a proportion. 1. Now try these: Lesson 4.1.3 additional questions — p446
2 4 and 3 6
4. 5 : 9 and 25 : 45
2.
3 5 and 5 3
5. 4 : 2 and 2 : 1
3.
4 and 2 : 9 21
6. 9 : 12 and 81 : 108
A recipe for carrot cake needs a ratio of carrots to walnuts of 2 : 3. 7. If 6 cups of walnuts are used with c cups of carrots, write a proportion involving c. 8. Solve your proportion to find c.
Round Up Proportions can be useful in everyday life — you’ll come across them countless times when you’re cooking and painting, and so on. Next Lesson, you’ll see another way to solve them to find unknowns. 204
Section 4.1 — Ratio and Proportion
Lesson
4.1.4
Proportions and Cross-Multiplication
California Standard: Number Sense 1.3 Use proportions to solve problems (e.g., determine the value of N if 4/7 = N/21, find the length of a side of a polygon similar to a known polygon). Use cross-multiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse.
What it means for you: You’ll learn about the cross-multiplication method for solving problems involving proportions with a missing variable.
Key words: • proportion • cross-multiplication • multiplicative inverse • reciprocal • ratio
In this Lesson, you’ll learn a good way to solve proportions — the method of cross-multiplication. Most of the ideas in this Lesson apply to fraction equations too, so it’s worth concentrating extra hard during this Lesson.
Multiply the Diagonals in Cross-Multiplication 2 6 = . 3 9 If you multiply the numerator of each fraction by the denominator of the other, you get two products. (This is called cross-multiplication.) Take this proportion:
2 6 = 3 9
3 × 6 = 18 2 × 9 = 18
For any proportion, these two products are always equal. Example
1
2 8 Use cross-multiplication to check that the ratios and are 5 20 equivalent. Solution
If the two ratios are equivalent, then you can write them as a proportion. Then you can check that the products of the two diagonals are equal. 2 8 5 × 8 = 40 = 2 × 20 = 40 5 20 Both products are equal, so the ratios are equivalent.
Cross-multiplication gives:
Example
2
The ratios 1 : 9 and 4 : a are equivalent. Use cross-multiplication to find the value of a. Solution
1 4 = 9 a Now cross-multiply, and remember that the products will be equal. 1 4 9 × 4 = 36 = 1×a=a 9 a So a = 36.
The two ratios are equivalent, so write them as a proportion:
Section 4.1 — Ratio and Proportion
205
Guided Practice Don’t forget: If cross-multiplication gives you two different numbers, then the ratios are not equivalent.
In Exercises 1–6, use cross-multiplication to decide whether the ratios are equivalent. 4 20 2 8 1. and 2. and 3. 9 : 12 and 12 : 16 5 25 3 14 4 7 4. 8 to 9, and 9 to 8 5. 15 : 18 and 20 : 24 6. and 3 5 In Exercises 7–9, use cross-multiplication to find the variable in the proportions. 1 9 z 7 3 1 7. = 8. = 9. = 9 g 6 1 c 8
Use Cross-Multiplication to Solve Any Proportion Sometimes cross-multiplication is just the first step on the way to finding the value of an unknown. Example
3
Solve the proportion
2 3 = for y. y 12
Solution
Cross-multiply to find: 2 × 12 = 3 × y. You can write this more simply as 3y = 24. This is an equation like you saw in Lesson 2.2.4. So divide both sides by 3 to find y. 3 y = 24 3 y ÷ 3 = 24 ÷ 3 y=8 Example
4
Solve the proportion
5 b = for b. 16 8
Solution
Cross-multiply to find: 5 × 8 = 16 × b, or more simply, 16b = 40. Now divide both sides by 16 to find b. 16b = 40 b = 2.5
Guided Practice Find the missing variable in Exercises 10–13. 10.
206
3 9 = y 18
Section 4.1 — Ratio and Proportion
11.
4 1 = 24 x
12.
8 c = 15 90
13.
16 h = 8 2
You Can Solve Real-Life Problems Using This Method Example
5
The ratio of the mass of sand to cement in a particular type of concrete is 4.8 : 2. If 6 kg of sand are used, how much cement is needed? Solution
You need to write a proportion, so you need two ratios of sand : cement. The first ratio is given in the question, 4.8 : 2. The second ratio will involve a variable, c, which represents the mass of cement needed. So the other ratio is 6 : c. 4.8 6 = Write a proportion: 2 c Cross-multiply: 4.8 × c = 2 × 6, or more simply, 4.8c = 12. Solve your equation for c: c = 12 ÷ 4.8, which gives c = 2.5. So 2.5 kg of cement are needed.
You can see why cross-multiplication actually works by using multiplicative inverses. Example
6
3 a = , then 3b = 4a. 4 b Show every step of your work. Do not use cross-multiplication. Show that if
Don’t forget: There’s more information in Lesson 3.3.1 about multiplicative inverses (also called reciprocals). The important fact is that when you multiply a number by its multiplicative inverse, you get 1.
Solution
This involves a little bit of algebra... a 1 3 1 Remember that = 3× , and = a × . (See Lesson 3.2.1.) b b 4 4 1 1 So the original equation is the same as 3× = a × . 4 b • Multiply both sides of this equation by 4 (the multiplicative inverse of
1 1 So 4 × =1 and b × = 1 . 4 b
Don’t forget: Rearranging the order of things that are multiplied together uses the associative and commutative properties of multiplication. See Lesson 2.3.2 for more information
• •
1 ) 4
and b (the multiplicative inverse of
1 ). b
1 1 This gives: 4 × b ×3× = 4 × b × a × . 4 b 1 1 You can rearrange this as: 4 × × b ×3 = b × × 4 × a 4 b 1 1 But 4 × = 1 and b × = 1 , so this simplifies to b × 3 = 4 × a. 4 b
And that’s it... you’ve shown that 4a = 3b.
Section 4.1 — Ratio and Proportion
207
Guided Practice 14. Lois is mixing up purple paint in the ratio 5 parts blue to 3 parts pink. If Lois uses 15 cans of pink paint, how many cans of blue paint will she need to mix the correct purple? 15. A new school is due to open. In total, it will need to employ 2 staff members for every 28 pupils. The school is predicted to have 2744 pupils. How many staff members will it need to employ?
Independent Practice In Exercises 1–4, say which ratios are equivalent by using cross-multiplication. 3 48 1. 5 : 8 and 20 : 32 2. and 4 60 7 49 3. and 4. 8 to 3, and 6 to 2 5 35 In Exercises 5–12, solve the proportions using cross-multiplication. Now try these: Lesson 4.1.4 additional questions — p446
6 x = 1 2 9 3 = 9. 117 x
5.
4 16 = y 4 5 10 = 10. 35 x
6.
x 6 = 7 21 2 4 11. y = 76
7.
2 9 = x 36 y 1 12. = 6 3
8.
13. Roberta is going on vacation. She needs to take 3 skirts for each week’s vacation. Is she right in taking 8 skirts for a 3-week vacation? 14. Eva notices that she is taught by 3 male teachers and 4 female. If this ratio is true for the whole school, how many male teachers would there be if there are 52 female teachers altogether? Exercises 15–17 apply to Jose, who is making a fruit salad for his friends, using 7 oranges, 14 pears, 14 apples, 98 grapes, and 28 cherries. An average portion of fruit salad contains 7 grapes. 15. How many pears are in an average portion? 16. How many oranges are in an average portion? 17. How many cherries are in an average portion? a c 18. If = , then what can you say about ad and bc? b d Show every step of your work. Do not use cross-multiplication.
Round Up In this Lesson, you’ve learned how to solve proportion equations using cross-multiplication. The next Section uses this to solve proportion problems in geometry. 208
Section 4.1 — Ratio and Proportion
Section 4.2
Lesson
4.2.1
Similarity
California Standard:
This Lesson is about similarity in geometry. If two figures are similar, they are the same shape, though they don’t have to be the same size. You’re going to use proportions to find out more about similar shapes.
Number Sense 1.3 Use proportions to solve problems (e.g., determine the value of N if 4/7 = N/21, find the length of a side of a polygon similar to a known polygon). Use cross-multiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse.
Similar Figures Are the Same Shape Figures that have exactly the same shape (but could be different sizes) are called similar figures. Example
1
Which of the shapes below are similar to Figure A? 2
What it means for you:
3
1
You’ll learn about similar shapes and be able to express relationships in geometry using ratios.
4 5
Figure A
6
7
Solution
Key words: • corresponding sides/angles • similar • proportion • ratio
Check it out: If two figures are the same shape, and the same size, then they are called congruent.
Shapes 2, 3, and 7 don’t have 3 sides, so they’re not similar to Figure A. Shape 6 doesn’t have a right angle, so that’s not similar to Figure A. Shape 4 has two sides the same length, so that’s not similar to Figure A. Shape 1 is smaller than Figure A, but is the same shape. Shape 5 is bigger than Figure A and has been rotated, but it is also the same shape. Shapes 1 and 5 are similar to Figure A. To say whether two shapes are similar, you need to look at their angles and side lengths. Example
2
Which one of the shapes on the right is similar to Figure B below? Check it out: Any two triangles whose angles have the same measures are similar. However, this doesn’t work for other shapes — all three shapes below have the same angles, for example. 120° 120° 60°
60° 120° 60°
70°
70° 110° Figure B
110° 1
110°
117°
70° 2
70° 63°
3
110° 4
Solution
Shape 3 has different angles from Figure B, so that isn’t similar. Shape 1 has the same angles, but it’s been stretched in only one direction, making it a different shape, and not similar to Figure B. Shape 4 has the same angles, but the two right angles are opposite each other — different from Figure B. So Figure B and Shape 4 are not similar. Shape 2 has the same angles and the same side lengths. It could be cut out and placed exactly over Figure B. Shape 2 is similar to Figure B. Section 4.2 — Proportion in Geometry
209
Guided Practice 1. Which one of the shapes 1–4 below is similar to Figure C? Explain your answer. 1 2 30°
4
3 60°
60°
60°
30°
Figure C
Check it out: Basically, similar shapes are enlargements or reductions of each other (but can be reflected and rotated too).
Sides of Similar Shapes Give the Same Ratios The two triangles below are similar. Look at the ratios of the lengths of corresponding sides.
3 cm 1 = , Shortest sides: 6 cm 2
Check it out: This ratio of
1 2
means that the
lengths of the sides in Triangle 1 are
1 2
as big as the
corresponding sides in Triangle 2.
medium sides:
Triangle 2 Triangle 1 5 cm
8 cm
10 cm
3 cm
4 cm 6 cm
4 cm 1 5 cm 1 = , and longest sides: = . 8 cm 2 10 cm 2
The ratios of the lengths of corresponding sides are equal. This is true for all similar shapes. Example
3
All these triangles are similar. Find the ratios of the lengths of 2.4 in. corresponding sides between 1.5 in. 1 in. 1 3 Figure D and: 1.2 in. 2 in. 1.8 in. 2 in. (i) Shape 1 4 in. 6 in. (ii) Shape 2 (iii) Shape 3 2 3 in. Figure D
4.5 in.
3 in.
Solution
Don’t forget: An equation showing that two ratios are equal is a proportion. Here, you have lots of ratios equal to each other, so there are lots of proportions. 4 in. 5 = is a For example, 2.4 in. 3 proportion.
210
Since all the shapes are similar, the ratios of all corresponding side lengths will be equal. (i) For Shape 1:
2 in. 3 in. 4 in. = = =2 1 in. 1.5 in. 2 in.
(ii) For Shape 2:
2 in. 3 in. 4 in. 2 = = = 3 in. 4.5 in. 6 in. 3
(iii) For Shape 3:
2 in. 3 in. 4 in. 5 = = = 1.2 in. 1.8 in. 2.4 in. 3
Section 4.2 — Proportion in Geometry
Example Check it out: This time, you can’t use the fact that similar shapes have identical angles. You have to use what you know about corresponding sides of similar shapes — that they all give the same ratio.
4 4 in.
Are the two shapes below similar? Explain your answer. 2.5 in.
3.9 in.
2 in. 2.4 in.
1.6 in. 2 in. 1.6 in.
Solution
Work out the ratios of the lengths of corresponding sides. If the shapes are similar, all the ratios will be equal. Shortest sides:
1.6 in. = 0.8 2 in.
Longest sides:
2.5 in. = 0.625 4 in.
You don’t have to work out the ratio for the other sides — these two ratios are different, so you know the shapes aren’t similar.
The ratios of the lengths of the sides are not all equal, so the shapes are not similar.
Guided Practice All these triangles are similar. Find the ratios of the lengths of corresponding sides between Figure E and: 2. Shape 1 3. Shape 2 2 in.
5 in.
2.4 in.
0.8 in. 2 in. 2 1.6 in.
6 in. 1
4 in. Figure E
4.8 in.
4. Which two shapes below are similar? Explain your answer. 1.32 cm 2
3 cm 2.2 cm
1
1.2 cm
1.8 cm 3.6 cm
2 cm 2.4 cm
3
2.5 cm
Section 4.2 — Proportion in Geometry
211
Example Don’t forget: If you find it easier to think about pictures, draw some. A quick sketch of two triangles with the lengths of the sides marked in can be much easier to think about than a description.
5
Two triangles are similar. The first triangle has sides of length 7 in., 8 in., and 9 in. The second triangle has sides of length 14 in. and 16 in., but the length of the longest side is unknown. Calculate the ratio between the lengths of corresponding sides. Use this ratio to find the length of the missing side. Solution
Work out the side-length ratios of the bigger triangle to the smaller triangle. Using the shortest sides:
14 in. =2 7 in.
So sides in the bigger triangle are twice as long as in the smaller triangle. So the length of the longest side of the bigger triangle must be 2 × 9 in. = 18 inches.
Guided Practice In Exercises 5–6, the shapes in each pair are similar. Find the missing lengths by working out the ratios of the corresponding side lengths. 10 cm 5. 6. 16 cm 16 cm cm 4 cm 4 4 in. 2 in. 4 in. 2 in. 4 cm 4 cm 1.3 in. 16 cm 16 cm 10 cm
Independent Practice Now try these:
1. Show that triangles 2 and 4 are 17 in. 12 in. similar to each other, but that triangles 1 1 and 3 are not similar to any others. 12 in. 28 cm
Lesson 4.2.1 additional questions — p447
20 cm
a°
124° d
9 in.
b°
c°
103° 24 cm
e
14 cm 124° 134° 105° 14 cm 12 cm 74° 103° 22 cm
44 cm
12 in. 2 15 in.
Now you know how to recognize similar figures, and work out missing angles and lengths. Next Lesson, you’ll use proportions to solve more problems with similar shapes. Section 4.2 — Proportion in Geometry
5 in.
4 4 in. 3 in.
2. Fill in the missing lengths and angles in the similar pentagons on the left.
Round Up
212
17 in. 6 in. 3 9 in.
Lesson
4.2.2 California Standard: Number Sense 1.3 Use proportions to solve problems (e.g., determine the value of N if 4/7 = N/21, find the length of a side of a polygon similar to a known polygon). Use cross-multiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse.
What it means for you: You’ll use proportions to find missing lengths in similar shapes.
Key words: • proportion • similar shapes • ratio • corresponding sides
Proportions and Similarity Last Lesson, you learned about similar shapes. You saw that if you take two similar triangles and work out the ratio of corresponding sides, you always get the same answer. This Lesson, you’re going to use proportions to find missing lengths. But first, a bit about notation...
You Can Use Proportions to Find Missing Lengths S Look at these similar blue and yellow shapes. C To name a particular side, you write the 2 in. B 10 in. P letters at either end with a bar over A 5 in. them. So the longest side of the yellow 2 in. 4 in. D Q shape is CD , while the corresponding 6 in. side of the blue shape is RS . R The lengths of sides are written in the same way, but without the bar. So the length of CD is written CD.
You’ve seen that for similar shapes, the ratio of the lengths of any two corresponding sides is equal. You can use this to write a proportion. Example
1
Look at the blue and yellow shapes above. Write a proportion using the ratios AB : PQ and BC : QR. Solve your proportion to find the missing length BC.
Don’t forget: These ratios can be written in different ways. (i) AB : PQ (ii) AB to PQ AB (iii) PQ
Don’t forget: The ratio of the length of corresponding sides in similar triangles will always be the same. But you must make sure you get the numbers in the right order. In Example 1, the yellow triangle’s sides are always on the top.
Check it out: This is the same method as was used in Example 5 of the previous Lesson, only written down using proportions.
Solution
These are similar shapes, so the ratios of the lengths are the same. AB BC = This means . PQ QR 2 in. z in. = If the length of the missing side is z inches, then , 4 in. 6 in. 2 z or more simply, = . 4 6 Cross-multiplication then gives 4z = 6 × 2, or 4z = 12. Now you can divide both sides by 4 to get z = 12 ÷ 3 = 4. The length of side BC is 4 inches.
Guided Practice In Exercises 1–2, the shapes in each pair are similar. Find the missing lengths using proportions. F Q 1. 2. B 8 cm A
x
12 cm
D 3c
7 cm
m C
7.5 cm
15 ft X
P
H E
b 67.5 ft
12 ft
z
Y
y
G
a
R
63 ft
Section 4.2 — Proportion in Geometry
Z
213
There are other ratios you can use to find missing lengths in similar shapes. Example
2 S
Look at the shapes on the right. Calculate the ratios
AB CD
and
PQ RS
C
2 in. B A 2 in. D
.
What do you notice? Solution
10 in.
4 in. Q
Now look at the blue shape:
6 in.
R
Here, you’re working out ratios of two sides on the same shape. First look at the yellow shape:
P
5 in.
AB 2 in. 2 = = CD 5 in. 5 PQ 4 in. 4 2 = = = RS 10 in. 10 5
The two ratios are equal.
This is always true for similar shapes. If you work out the ratio of the lengths of two sides in one shape, then the ratio of the corresponding pair of sides in any similar shape will be the same. Example
3
Look at these similar shapes. Write a proportion using the ratios AB : CD and WX : YZ. Solve your proportion to find p.
A 4 cm
D
Z
W
7 cm 6 cm
p cm
B C
X Y
Solution
4 cm 4 = . This is the ratio of the lengths of the The ratio AB : CD is 7 cm 7 parallel sides on the green trapezoid.
Check it out: Similar shapes are sometimes described as being “in proportion.”
Since the purple shape is similar, the ratio of the lengths of its parallel 6 4 4 sides will also equal 7 . So = . p 7 Now solve your proportion using cross-multiplication: 4p = 42, which gives p = 10.5.
Guided Practice For Exercises 3–6, use the similar shapes in the diagram below. A D W 3. What is the ratio AB : BC? 7 in. 4. What is the ratio YZ : XY? 5. Use your answers from Exercises B 9 in. 3 and 4 to set up a proportion. C 6. Solve your proportion to find the length YZ. X
214
Section 4.2 — Proportion in Geometry
Z d Y 17 in.
In questions about lengths, make sure you remember about units. Example
4
The two shortest sides of a right triangle are in the ratio 4 : 3. How long is the shortest side of a similar triangle, if its second shortest side is 17.5 miles long? Solution
Here, the question doesn’t give you a picture. In cases like this, it’s a good idea to draw one yourself. Here’s a right triangle whose two shortest sides are in the ratio 4 : 3. You know that the second shortest side of a similar triangle is 17.5 miles long. Call the length you need to find z miles.
4 3
17.5 miles
z miles Now you can set up a proportion. You have a choice which ratios to use, but the math is the same. 17.5 z (i) You can either use corresponding sides: = 4 3 Cross-multiply: 3 × 17.5 = 4z Divide both sides by 4 to get: z = 52.5 ÷ 4 = 13.125
(ii) Or you can use the ratio 17.5 4 = of the two short sides in each triangle: z 3 Cross-multiply: 3 × 17.5 = 4z Divide both sides by 4 as before to get: z = 13.125 Don’t forget the units. Your final answer is that the shortest side of the triangle has length 13.125 miles.
Guided Practice An architect designs a building with a triangular wall. The ratio of the lengths of the two shortest sides of the triangle is 2 : 5. On the model of the building, the shortest side has length 8 cm. Don’t forget: If there’s no picture in the question, you should probably draw one yourself.
7. What is the length of the second shortest side on the model? 8. On the actual building, the length of the shortest side of the triangular wall is 4 m. Write and solve a proportion to find the length of the second shortest side of the wall.
Section 4.2 — Proportion in Geometry
215
Don’t forget: Use what you know about one triangle to help you find out about the other.
Don’t forget: If you’re not sure where to start a question, write down some information about the triangles. Include a diagram, plus ratios of corresponding sides. Remember, if shapes are similar, there will be some equal ratios.
Independent Practice Exercises 1–3 are about the two similar triangles below. 1. Find the angle at S. K 2. Find the length SR. C 9 in. 3. Find the length CE. 10 in. 62° E
4. The ratio of the corresponding sides of these two squares is 4 : 3. Find the length XY.
28° 8 in. A
D S
R
12 cm
B
D
Now try these:
C
These two triangles are similar. 5. Find the length of the missing side x. 6. Determine the measure of the angle at C. 7. What is the measure of the angle at M?
W
X
Z
Y T
A
3 cm
Lesson 4.2.2 additional questions — p447
15 in.
M
x cm 3 cm
2 cm
C
R
70° 6 cm
S
8. The ratio of corresponding sides of two similar shapes is 7 : 10. If the smaller shape has a side of length 21 in., find the length of the corresponding side in the larger shape. 9. One sheet of paper is 8.5 inches by 11 inches. Another sheet of paper is 16 inches by 22 inches. Are the two sheets of paper similar? 10. Alisha has a picture that is 12 cm wide and 18 cm long. She makes a poster that is similar to the original picture, but has a width of 30 cm. What is the length of the poster? 11. A quadrilateral has a perimeter of 7 inches. The ratio of the length of the sides in this quadrilateral and a similar one is 14 : 37. What is the perimeter of the second quadrilateral?
Round Up That’s it for similar shapes. Next, you’ll be looking at scale drawings, which will use what you know about similar shapes and proportion and link it to real-life things such as maps. 216
Section 4.2 — Proportion in Geometry
Lesson
4.2.3
Scale Drawings
California Standards:
In this Lesson, you’ll learn about scale drawings. Scale drawings are really just similar shapes. Any kind of object, from a huge airplane to a tiny mechanical component, can be drawn to scale. This is useful when an object is just too big or too small to draw life-size.
Number Sense 1.2 Interpret and use ratios in different contexts (e.g., batting averages, miles per hour) to show the relative sizes of two quantities, using appropriate notations (a/b, a to b, a:b ). Number Sense 1.3 Use proportions to solve problems (e.g., determine the value of N if 4/7 = N/21, find the length of a side of a polygon similar to a known polygon). Use cross-multiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse.
What it means for you: You’ll learn how to make a scale drawing, and use scales to find real lengths from a scale drawing, and vice versa.
Scale Is a Kind of Ratio Scale drawings are used when an object is too large or too small to draw full-size. The scale is the ratio of any length in the drawing to the corresponding length in real life. So the scale tells you how much bigger or smaller the drawing is than the real object. Sometimes the scale can be given using units. Example
1
An architect made a scale drawing of a recreation center, using a scale of 1 cm : 3 meters. In the drawing, the swimming pool is 2 cm wide. How wide is the actual pool? Solution
The scale is 1 cm : 3 meters. So 1 cm on the drawing represents 3 meters in real life. This means that if the pool is 2 cm wide on the drawing, the real pool will be 6 meters wide.
Key words: • scale • ratio • proportion • map • dimensions
2 cm
Sometimes the ratio involves just numbers. Example
2
The same architect has drawn a scale drawing of another recreation center. This time, she used a scale of 1 : 500. The center’s pool is 50 m long. How long is the pool in the drawing? The pool is tiled. If the actual length of each tile is 15 inches, how long would each tile be drawn? Solution
This time the scale is 1 : 500. So 1 cm on the drawing represents 500 cm in real life, and so on. If the actual pool is 50 meters long, then the drawing will be: (50 ÷ 500) meters = 0.1 m = 10 cm long. The scale of 1 : 500 also means 1 inch on the drawing represents 500 inches in real life. So if a tile is 15 in. long, then in the drawing they would be: (15 ÷ 500) inches = 0.03 inches long.
Section 4.2 — Proportion in Geometry
217
Guided Practice Lenny has a drawing of a boat, made with a scale of 1 inch : 15 feet. 1. The length of the boat in the drawing is 4 inches. What is the length of the real boat? 2. The real boat is 30 feet wide. How wide is the boat in the drawing? 3. The real boat’s mast is 20 feet high. How high is the mast in the drawing?
Check it out: In Exercise 6, the scale drawing is bigger than the actual component. Remember, a scale is written in the order “drawing : real-life.”
Mr. Michael’s house is drawn using a scale of 1 : 144. In the drawing, a rectangular window is 0.25 in. by 0.5 in. 4. What are the dimensions of the actual window in inches? 5. How tall is an actual door if it’s drawn 0.05 feet tall? 6. A computer contains a tiny component 0.1 mm long. If a drawing is made of this component using a scale of 100 : 1, how long will the component be in the drawing? Just like with other ratios, you can use proportions to find missing lengths. You have to be really careful with units though. One way to deal with units is to multiply and divide by them as though they are numbers. Example
3
This is a scale drawing of the side of a house. It is drawn using a scale of 2.5 inches : 24 feet. Find the actual width of the side of the house. (Each square on the paper is 0.25 inches wide.) 1.78 inches
Solution
Call the actual width of the house w.
Don’t forget: You solved proportions last Section. If you can’t remember the cross-multiplication method, go back to 4.1.3 and recap.
To make a proportion, you need two ratios. 2.5 inches . You can always use the scale for one of these ratios: 24 feet For the other ratio, you need to use any measurement on the drawing, and the corresponding measurement in real life. Here, you’re interested in the width of the house, so that’s the measurement to use. Ratio of width in drawing to width in real life is Use your two ratios to write a proportion:
1.78 inches . w
1.78 inches 2.5 inches = w 24 feet
Now cross-multiply, treating the units as though they are numbers. w × 2.5 inches = 1.78 inches × 24 feet w × 2.5 = 1.78 × 24 feet Divide both sides by “inches” 2.5w = 42.72 feet Multiply out 1.78 × 24 w = 42.72 feet ÷ 2.5 Divide both sides by 2.5 w = 17.088 feet 218
Section 4.2 — Proportion in Geometry
Guided Practice 7. Find the actual height of the house from Example 3. Miguel makes a drawing of an aircraft for a school project. The scale of the drawing is 3 inches : 10 feet. The length of the aircraft in Miguel’s drawing is 14.7 inches. 8. Let the length of the real aircraft be z. Write a proportion involving z. 9. Solve your proportion to find z. 10. If the actual aircraft is 44.5 feet wide, use a proportion to find how wide the aircraft will be in the drawing. Don’t forget: To use a proportion to find a missing length, you need two equivalent ratios. Use the scale as one ratio.
Sarah has made a scale drawing of a house using a scale of 1 : s. The height of the house in the drawing is 4 inches. The height of the actual house is 28 feet. 11. Find the number s. 12. If the actual height of the front door is 7 feet, use a proportion to calculate how tall the front door is in the drawing.
A Map Is a Scale Drawing of an Area A map is another example of a scale drawing. Example
4
Write and solve a proportion to find the distance between the library and the hospital.
Library
Hospital
The scale of the map is 1 grid square : 4 miles.
4 miles
Solution
To write a proportion, you need two ratios. From the scale,
1 grid square — use this as one of your ratios. 4 miles
Call the actual distance d. Then Your proportion is then:
2.5 grid squares is an equivalent ratio. d
1 grid square 2.5 grid squares = 4 miles d
d × 1 grid square = 4 miles × 2.5 grid squares Cross-multiply Divide both sides by the units “grid square” d × 1 = 4 miles × 2.5 Simplify the equation d = 10 miles The distance between the library and the hospital is 10 miles.
Section 4.2 — Proportion in Geometry
219
Guided Practice Exercises 13–15 are about the same map. On this map, the distance between town A and town B is 4 inches. The actual distance between the towns is 8 miles. 13. What is the scale of the map in the form 1 in. : x miles? 14. If the real distance between towns C and D is 17 miles, how far apart would they be on the map? 15. If the distance between towns E and F on the map is 0.5 inches, how far apart are they in real life? 16. The actual distance from Green City to Blue River is 160 miles. The distance between these cities on a map is 2.5 inches. If the distance on the same map between Blue River and Yellow Grove is 3 inches, what is the actual distance between these two places?
Independent Practice 1. Miguel makes a model of an aircraft for a school project. The scale of the model is 1 inch : 10 feet. If the length of Miguel’s model is 20 inches, what is the length of the actual aircraft? Now try these: Lesson 4.2.3 additional questions — p448
Jenna is taking a car trip from Forest Bear Lake Forest Park Park to Bear Lake. The grid squares on the map measure 1 inch × 1 inch. 2. How many inches on the map is Forest Park from Bear Lake? 1 inch = 4 miles Give the straight-line distance. 3. What is the straight-line distance of Bear Lake from Forest Park in real life? 4. The actual length of the road between the two places is 37 miles. On the map, how long is the line representing the road? Use a proportion to calculate your answer. The length of a tiny organism is 0.4 mm. The length of a scale drawing of the same organism is 3.2 cm. 5. The scale of the drawing is p : 1. Find p. 6. If the actual organism has a width of 0.2 mm, what is the width of the organism on the scale drawing? 7. Harriet has two maps of the same city. The first map has a scale of 1 inch : 2 miles, and the second map has a scale of 1 inch : 3 miles. The distance from City Hall to the Lake on the first map is 2.5 inches. What is this distance on the second map?
Round Up This is all useful information. You’ll almost certainly need to use these skills again — whenever you’re reading a map, for instance. Next Lesson, you’ll look at using ratios to convert between different units. 220
Section 4.2 — Proportion in Geometry
Section 4.3 introduction — an exploration into:
Plotting Con ver sions Conv ersions How tall is your textbook? There are different answers. You could say that it’s about 8.5 inches. Your friend might say that it’s almost 22 centimeters. Someone else might say that it’s about 0.00013 miles. That’s what this Exploration is all about — measurements and units. Choose five objects in the classroom that are no more than 3 feet long.
Length in inches 8.6 5.3 28
Object
Measure the exact length of each object twice: (i) in inches, and (ii) in centimeters. Record your measurements in a chart like this one.
Textbook Pen Height of desk
Length in centimeters 22 13.5 71
90
Now draw a coordinate grid like this one. Make sure that your axes are long enough to show your measurements.
80
(28, 71)
centimeters
×
70
Plot each object’s measurements as a point on the grid. • Plot the number of inches along the x-axis (the horizontal axis). • Plot the number of centimeters up the y-axis (the vertical axis).
60 50 40 30
×
You should be able to draw a straight line though the points.
20
0 0
×
10
(8.6, 22) (5.3, 13.5) 5
10
15
20
25
30
35
40
Now you can use your graph to convert measurements in inches to centimeters, and measurements in centimeters to inches.
inches
Example (i) 30 cm to inches, (ii) 5 inches to centimeters.
Solution
(i) First find 30 on the centimeters axis (the vertical axis). Go across to your plotted line, and down to the inches axis. This tells you that 30 centimeters is about 12 inches. (ii) This time find 5 on the inches axis (the horizontal axis). Go up to your plotted line, and across to the centimeters axis. This tells you that 5 inches is about 12.5 centimeters.
(i) 30
centimeters
Use your graph to convert:
20
12.510 0 0
(ii) 5
10
inches 12
Exercises 1. Use your graph to convert these measurements from inches to centimeters, and from centimeters to inches. a. 45 cm b. 34 inches c. 10 cm d. 10 inches
Round Up You can do the same thing for other units too. For example, you could find the mass of a few different objects in both grams and ounces, plot the results on a graph, and then use your graph to convert other masses from one set of units to the other. You’ll see lots about units in this Section. Section 4.3 Explor a tion — Plotting Conversions 221 Explora
Lesson
Section 4.3
4.3.1
Customary and Metric Units
California Standards:
This Lesson, you’ll learn how to convert between units in either the customary or the metric system. So you take a length in inches, say, and convert it to feet. Or maybe you know a length in centimeters and you need to convert it to kilometers. That kind of thing...
Algebra and Functions 2.1 Convert one unit of measurement to another (e.g., from feet to miles, from centimeters to inches). Mathematical Reasoning 3.1 Evaluate the reasonableness of the solution in the context of the original situation.
What it means for you: You’ll see how to convert between different units within the customary or metric systems. For example, feet to inches, or meters to kilometers.
Customary Lengths Are Inches, Feet, Yards, and Miles The customary system of lengths is the system that uses inches, feet, yards, and miles. This conversion table shows how many of one unit make up another kind of unit.
Customary Lengths 1 foot (ft) = 12 inches (in.) 1 yard (yd) = 3 feet (ft) 1 mile (mi) = 1760 yards (yd)
You can use it to convert a measurement in one unit to a measurement in another unit. The numbers in a conversion table (12, 3, and 1760 in the above table) are sometimes called conversion factors. Example
1
A vase is 1 yard tall. What is its height in: (i) feet? (ii) inches? 1 yd
Solution
(ii) If 1 foot is 12 inches, then 3 feet must be 3 × 12 = 36 inches. Notice that when you convert to a smaller unit, the number gets bigger. This makes sense... the smaller each unit, the more of them you need to cover the same distance.
1 inch
Check it out: Since 1 yard = 36 inches, the conversion factor between inches and yards is 36.
Check it out: You should memorize the conversion factors in the table above.
1 yard
• conversion • customary system • metric system • unit • conversion table • conversion factor
1 foot 1 foot 1 foot
Key words:
(i) Using the conversion table, you can see that 1 yard is 3 feet.
In fact, if your conversion factor is greater than 1: • you multiply by it to convert to a smaller unit, since this gives you a bigger number. • you divide by it to convert to a bigger unit, since this gives you a smaller number. Example
2
Convert: (i) 3 miles to yards, (ii) 7920 yards to miles. Solution
There are 1760 yards in a mile, so use 1760 as your conversion factor. (i) Yards are smaller than miles, so 3 miles = 3 × 1760 = 5280 yd (ii) Miles are bigger than yards: 7920 yd = 7920 ÷ 1760 = 4.5 miles 222
Section 4.3 — Converting Units
This diagram summarizes how to convert between customary lengths. Don’t forget: Always check that your answers are sensible. If you’re converting 7 miles to yards, say, you’d expect your answer to be a very big number. You could use estimation to check your answers (see Section 1.4 for more information). For example, if you’re converting 7 miles to yards, you’d expect an answer bigger than 7 × 1000 = 7000, but less than 7 × 2000 = 14,000.
Check it out: Dekameters and hectometers aren’t used very much at all. You also don’t see decimeters very often.
To get from miles to inches, first × by 1760, then × by 3, then × by 12.
× 12
INCH
×3
FOOT
÷ 12
× 1760
÷3
YARD
To get from inches to miles, first ÷ by 12, then ÷ by 3, then ÷ by 1760.
÷ 1760
MILE
Guided Practice Convert: 1. 7 miles to yards 3. 72 inches to feet 5. 5 feet to inches 7. 1 mile to feet 9. 2 miles to inches
2. 8800 yards to miles 4. 12 feet to yards 6. 9 yards to inches 8. 8 yards to inches 10. 100,000 inches to miles
11. If Letitia cycles 3 miles, how many feet has she cycled? 12. Mount Whitney is 14,494 feet tall. How many miles is this?
Check it out: In the metric system, you can work out how big a unit is by its prefix. The prefix “kilo-” means thousand, so 1 kilometer is 1000 meters. “deka-” means ten, “hecto-” means hundred, “kilo-” means thousand, “deci-” means tenth, “centi-” means hundredth, “milli-” means thousandth. This system works for other kinds of metric units too, not just for units of length. For example, 1 kilogram is 1000 grams. You should memorize these prefixes and what they mean — especially: “kilo-”, “centi-”, and “milli-”.
The Metric System Is Based on the Meter The metric system for lengths is based on the meter. All metric units for length contain the word “meter” — millimeter, centimeter, decimeter... Here’s a conversion table. Metric Lengths 1 dekameter (dam) = 10 meters (m) 1 hectometer (hm) = 100 meters 1 kilometer (km) = 1000 meters
1 decimeter (dm) = 0.1 meters 1 centimeter (cm) = 0.01 meters 1 millimeter (mm) = 0.001 meters
Dekameters, hectometers, and kilometers are all bigger than meters.
Decimeters, centimeters, and millimeters are all smaller than meters.
Notice in this table that some of the conversion factors are less than one. Example
3
Convert 120 meters to millimeters. Solution
Don’t forget: 1 1000
Dividing by 0.001 (= ) is the same as multiplying by 1000. See Lesson 3.3.1 for more information.
You’re converting from a big unit (meters) to a smaller one (millimeters), so the number needs to be bigger. The conversion factor from the table is 0.001. This is less than 1, so to make a bigger number, you need to divide by the conversion factor. 120 m = 120 ÷ 0.001 = 120 × 1000 = 120,000 mm Section 4.3 — Converting Units
223
Example
4
A pencil is 10 cm long. How long is the pencil in kilometers? Solution
Don’t forget: A bigger unit means a smaller number.
First, convert 10 cm to meters by multiplying by 0.01. This gives a smaller number: 10 centimeters = 10 × 0.01 = 0.1 m Next, convert 0.1 m to kilometers by dividing by 1000. This also gives a smaller number: 0.1 m = 0.1 ÷ 1000 = 0.0001 km
This diagram summarizes how to convert between metric lengths.
MILLIMETER
Check it out: Notice that: (i) there are always two conversion factors. This is because 1 cm = 0.01 m (giving a conversion factor of 0.01), but 1 m = 100 cm (giving a conversion factor of 100), and (ii) these conversion factors always multiply together to give 1 (for example, 0.01 × 100 =1).
Multiply by 10 to convert one step smaller. Multiply by 100 to convert two steps smaller, etc...
× 10
CENTIMETER
÷ 10
× 10
DECIMETER
× 10
÷ 10 Divide by 10 to convert one step bigger. Divide by 100 to convert two steps bigger, etc...
÷ 10
METER ÷ 10
÷ 10 This means that, to convert meters to centimeters, you can either: (i) multiply by 100, or (ii) divide by 0.01.
DEKAMETER
÷ 10
HECTOMETER KILOMETER
× 10
× 10 × 10
Don’t forget: Check that all your answers are sensible.
Guided Practice Convert: 13. 7 kilometers to meters 15. 85 mm to cm 17. 180 m to km 19. 0.1 km to m
14. 8700 meters to kilometers 16. 120 cm to mm 18. 15,000 m to km 20. 160 mm to cm
21. A snail moves 3 m. How many cm is this?
224
Section 4.3 — Converting Units
Independent Practice There are 12 inches in 1 foot, meaning the conversion factor is 12. 1. Would you divide by 12 or multiply by 12 when converting from feet to inches? 2. Would you divide by 12 or multiply by 12 when converting from inches to feet? 3. How would you convert from yards to miles? In Exercises 4–9, find the missing values in the conversions. 4. 36 in. = ? yd 5. 144 in. = ? yd 6. 27 ft = ? yd 7. 100 yd = ? ft 8. 99 ft = ? yd 9. 0.25 mi = ? ft 10. TJ’s guitar is 3.5 ft long. What is the length of the guitar in inches? 11. Would you divide by 100 or multiply by 100 when converting from meters to centimeters? 12. There are 1000 millimeters in a meter, and 100 meters in a hectometer. What is the conversion factor when converting millimeters to hectometers? 13. Would you multiply or divide by this conversion factor when converting millimeters to hectometers? Find the missing value in Exercises 14–19. 14. 30 m = ? cm 15. 2 km = ? m 17. 0.5 cm = ? m 18. 4 dam = ? dm
16. 250 mm = ? cm 19. 0.37 dm = ? mm
20. An Olympic swimming pool has a length of 50 meters. What is this distance in kilometers? The length of the wingspan of an airplane is 40 m. 21. Find the wingspan of this airplane in centimeters. 22. Find the wingspan of this airplane in kilometers. A compact disc has a diameter of 120 millimeters. 23. What is the diameter of a compact disc in centimeters? 24. What is the diameter of a compact disc in decimeters? 25. Ying and Carol jog exactly 2 miles together every Saturday. They both guess how many inches their jog is. Ying guessed 100,000 inches and Carol guessed 10,000 inches. Whose guess is closer to the actual amount? Now try these: Lesson 4.3.1 additional questions — p448
Micrometers are metric units that can be used to measure very small objects. In fact, 1 millimeter is equal to 1000 micrometers. 26. How many micrometers are in 0.25 millimeters? 27. How many micrometers are in 1 kilometer?
Round Up So to convert something from one unit to another, you need to make the number either bigger (if you’re converting to a smaller unit) or smaller (if you’re converting to a bigger unit). Conversion factors are useful, but you have to use them sensibly. There’s more on conversions next Lesson. Section 4.3 — Converting Units
225
Lesson
4.3.2 California Standard: Algebra and Functions 2.1 Convert one unit of measurement to another (e.g., from feet to miles, from centimeters to inches).
What it means for you: You’ll convert measurements from one unit to another using proportions.
Key words: • proportion • conversion • ratio
Conversions and Proportions Last Lesson, you saw that to convert between different units you can multiply or divide by a conversion factor. But you can also think about conversion factors as ratios. And where there are ratios, proportions can’t be far behind. This Lesson is all about doing conversions using proportions.
A Conversion Table Is a Set of Ratios A ratio is a way of comparing two quantities. But you’ve seen that ratios can also be used for converting quantities from one measuring system to another (think back to scale drawings, for example, where you saw things like “1 centimeter represents 10 meters”). In fact, you can think of the conversion tables you saw last Lesson as a table of ratios. For example, you can say the ratio of inches to feet is 12 : 1. Example
1
What is the ratio of: (i) feet to yards? (ii) yards to feet? Solution
There are 3 feet in a yard. (i) This means the ratio of feet to yards is 3 : 1. (ii) Remember... the order of the quantities in a ratio is important. If the ratio of feet to yards is 3 : 1, then the ratio of yards to feet must be 1 : 3.
Don’t forget: You could also write these ratios as “3 feet : 1 yard,” “1 m : 100 cm,” and so on.
Example
2
What is the ratio of: (i) meters to centimeters? (ii) centimeters to meters? Solution
There are 100 centimeters in a meter. (i) The ratio of meters to centimeters is 1 : 100. (ii) The ratio of centimeters to meters is 100 : 1.
Guided Practice What is the ratio of: 1. meters to kilometers? 3. inches to yards? 5. millimeters to centimeters? 7. miles to feet?
226
Section 4.3 — Converting Units
2. kilometers to meters? 4. yards to inches? 6. centimeters to millimeters? 8. feet to miles?
You Can Use Proportions to Convert Between Units You can use proportions to solve problems involving conversions. Check it out: The advantage of using proportions rather than multiplying or dividing by a conversion factor is that the method is exactly the same whether you’re converting to a smaller unit or a larger unit.
The method is exactly the same as the method you’ve seen in earlier Lessons. You find two equivalent ratios, write a proportion, then solve it using cross-multiplication. Example
3
The length of a bird is 8.5 cm. Use a proportion to find the length of the bird in millimeters. Solution
8.5 cm
You need two ratios to write a proportion. • Don’t forget: You must write both ratios the same way around — either both cm : mm, or both mm : cm.
Check it out: Notice that the effect is that you multiply the number of centimeters by 10 — just like in the previous Lesson.
•
The first ratio is the ratio of centimeters to millimeters. 1 This is 1 : 10, or . 10 The second ratio involves the length of the bird. The length in centimeters is 8.5 cm. Call its length in millimeters d. 8.5 . Then your second ratio is 8.5 : d, or d
Now you can write and solve your proportion: d × 1 = 8.5 × 10 d = 85
1 8.5 = 10 d
Cross-multiply Simplify
This means that the bird is 85 mm long. Use exactly the same method for converting millimeters to centimeters. Example
Check it out: This method is an alternative method to the method you used last Lesson, but you’ll see that you end up with exactly the same answers.
4
Convert 125.7 mm to centimeters. Solution
As always, find two ratios. • •
1 . 10 The second ratio involves the length you’re converting. Call the length in centimeters d. d Then your second ratio is d : 125.7, or . 125.7 The first ratio is the ratio of cm : mm, which is
Now write and solve a proportion:
1 d = 10 125.7
Cross-multiply d × 10 = 1 × 125.7 10d = 125.7 Simplify d = 12.57 Divide both sides of the equation by 10 So 125.7 mm = 12.57 cm. Section 4.3 — Converting Units
227
Guided Practice Check it out: You could even use d cm here as your 125.7 mm second ratio. Then when you write your proportion and cross-multiply, you get:
1 cm d cm = 10 mm 125.7 mm or d cm ×10 mm = 1 cm ×125.7 mm Now you can divide both sides by “cm” and “mm” to find: d × 10 = 1 × 125.7, or d = 12.57 The only difference is that you’ve canceled the units and so you have to add them back in when you write your final answer. The point is... as long as you don’t do something really bad (like write one ratio with cm on top and the other with mm on top, or forget how to cross-multiply), you should get the correct answer.
Use proportions to carry out the conversions in Exercises 9–11. 9. What is 48 inches in feet? 10. What is 500 km in centimeters? 11. Convert 14 inches into feet. In Examples 3 and 4, the ratios were written without units. But if you prefer, you can include units in your ratios, just like you saw with scale drawings. The method works exactly the same. Example
5
Convert 125.7 mm to centimeters. Solution
Your first ratio is the ratio of centimeters to millimeters:
1 cm 10 mm
Call the distance you need to find d. d Then your second ratio is: 125.7 mm 1 cm d = This gives you a proportion: 10 mm 125.7 mm Solve by cross-multiplication in the usual way. Cross-multiply d × 10 mm = 125.7 mm × 1 cm d × 10 = 125.7 × 1 cm Divide both sides of the equation by “mm” 10d = 125.7 cm Simplify d = 12.57 cm Divide both sides of the equation by 10
Guided Practice 12. What is the ratio of feet to miles? 13. Convert 9000 feet into miles using your ratio from Exercise 12.
Independent Practice Now try these: Lesson 4.3.2 additional questions — p448
1. What is the ratio of yards to miles? 2. What is the ratio of miles to yards? 3. What is the ratio of centimeters to meters? 4. What is the ratio of meters to centimeters? Use proportions to find the answers to Exercises 5–8. 5. Convert 7515 yards to miles. 6. Find 0.006 kilometers in millimeters. 7. Jonny needs 69 yards of fencing for his garden. What is this in feet? 8. An Egyptian camel trek is 8.75 km. How far is this in meters?
Round Up In this Lesson, you’ve learned to convert between units using proportions. You can use either method from the last two Lessons to solve conversion problems — you should get the same answer. 228
Section 4.3 — Converting Units
Lesson
4.3.3 California Standard: Algebra and Functions 2.1 Convert one unit of measurement to another (e.g., from feet to miles, from centimeters to inches).
Converting Between Unit Systems So far, you’ve converted metric units to other metric units, and customary units to other customary units. This Lesson, you’ll use conversion tables for converting between the two unit systems. You’ll see that all the techniques from the previous Lessons work in exactly the same way in this Lesson too.
What it means for you: You’ll convert between the customary and metric unit systems.
Key words: • customary system • metric system • conversion
Check it out: Notice that the numbers in one table are the reciprocals of the numbers in the other table.
Use the Conversion Tables to Help You The tables below show conversion factors you can use to convert between customary and metric units. Customary to Metric 1 inch (in.) = 2.54 cm 1 foot (ft) = 30.48 cm 1 yard (yd) = 0.91 m 1 mile (mi) = 1.61 km
Metric to Customary 1 cm = 0.39 inches (in.) 1 cm = 0.033 feet (ft) 1 m = 1.09 yards (yd) 1 km = 0.62 miles (mi)
You can convert feet to centimeters by multiplying by 30.48 (to give you a bigger number, since feet are bigger than centimeters). But you can also convert feet to centimeters by dividing by 0.033 (which also gives you a bigger number, since 0.033 is less than 1). So you only really need one of the above tables. If you know the conversion factors in one table, you can use them to convert from metric to customary or from customary to metric.
Don’t forget: Dividing by a number gives the same result as multiplying by its reciprocal. And multiplying by a number gives the same result as dividing by its reciprocal. See Lesson 3.3.1 for more information.
Check it out: This may seem more complicated, but there’s no new math here. The only difference between this stuff and the calculations between units from the past two Lessons, is that you’re using a different conversion table.
Example
1
Convert 10 km to miles. Solution
There are two ways you could use the tables to get your answer. (i) Multiply by 0.62 (to get a smaller number, since miles are a bigger unit than kilometers). So 10 km = 10 × 0.62 miles = 6.2 miles. (ii) Divide by 1.61. So 10 km = 10 ÷ 1.61 miles = 6.21 miles (to 2 decimal places) Example
2
Convert 1 mile to meters. Solution
From the table, you can see that 1 mile is 1.61 km. And you’ve already seen that to convert kilometers to meters you multiply by 1000. So 1 mile = 1.61 × 1000 meters = 1610 meters.
Section 4.3 — Converting Units
229
The diagram below might be helpful for some conversion questions. × 10
MILLIMETER × 2.54 or ÷ 0.36
CENTIMETER
÷ 10
× 12
INCH ÷ 2.54 or × 0.36 ÷ 12 ÷ 1.09 or × 0.91
FOOT ÷3
÷ 1000
YARD ×3
METER
÷ 1.61 or × 0.62
KILOMETER MILE
× 1.09 or ÷ 0.91 × 1000
× 1.61 or ÷ 0.62
Most of the conversion factors for converting between metric and customary systems are only approximations. Most of them are only given to two decimal places. This means your answer won’t always be exact. You can sometimes get slightly different answers if you do the question in different ways. Example
Check it out: 1 inch = 2.54 cm is actually an exact conversion, so 0.9144 meters is the correct answer in this Example.
3
Convert 1 yard into meters by: (i) converting yards to inches, inches to centimeters, and then centimeters to meters, (ii) using the conversion factor 0.91, (iii) using the conversion factor 1.09. Comment on your answers. Solution
(i) Do the conversion in three stages. 1) yards to inches: 1 yard = 36 inches 2) inches to cm: 36 inches = 36 × 2.54 cm = 91.44 cm 3) centimeters to meters: 91.44 cm = 91.44 ÷ 100 m = 0.9144 m (ii) 1 meter is slightly bigger than a yard, so multiply by 0.91 to make your number smaller. 1 yard = 1 × 0.91 m = 0.91 m Check it out: The various answers in Example 3 are different, but only slightly different.
230
(iii) Divide by 1.09 to make the number bigger. 1 yard = 1 ÷ 1.09 m = 0.91743... m The answers are slightly different. This is because the conversion factors used are only approximations.
Section 4.3 — Converting Units
Guided Practice Don’t forget: Depending on how you do these conversions, you might get slightly different answers.
Convert the following: 1. 6 meters to yards 3. 3 feet to centimeters 5. 2 kilometers to yards 7. 9 yards to centimeters
2. 18 yards to meters 4. 2 miles to meters 6. 6 inches to millimeters 8. 24 feet to dekameters
9. Josh runs a marathon of 26 miles and 385 yards. How far is this in kilometers? 10. A boat race is 4 miles and 374 yards. How long is the race in meters?
You Can Use Proportions Too The examples so far have been done by reasoning whether to multiply or divide by the conversion factor. But you could do them using proportions if you prefer. Just use the numbers in the conversion tables as ratios. Example
4
Convert 10 km to miles. Don’t forget:
Solution
You could include units in your ratios too, if you prefer. See Lesson 4.3.2 for some examples.
This is the same as Example 1, but this time it’s done with proportions. As always when using proportions, you need two ratios. •
The first ratio of miles to kilometers comes from the table. 1 This is 1 : 1.61, or . 1.61 • The second ratio involves the measurement you want to convert. Call the converted distance d miles. Then your second ratio is d d : 10, or . 10 1 d = . Now write a proportion and solve by cross-multiplication: 1.61 10 Cross-multiply 1.61d = 10. Divide both sides by 1.61 d = 10 ÷ 1.61 d = 6.21 miles (to 2 decimal places)
Guided Practice Convert the following using proportions: 11. 10 meters to yards 12. 3 yards to meters 13. 3.7 feet to centimeters 14. 5.1 miles to meters
Section 4.3 — Converting Units
231
Convert Weight and Time in Exactly the Same Way So far, you’ve only looked at converting lengths. But you can convert time and weight (and most other kinds of quantities) in exactly the same way. You just need to know the conversion factor. Example
5
Convert 1 hour to seconds. Solution
Do this in two stages — hours to minutes, and then minutes to seconds. There are 60 minutes in an hour, and 60 seconds in a minute, so the conversion factor for both stages is 60. 1) Minutes are a smaller unit than hours, so multiply by 60 to get a bigger number: 1 hour = 1 × 60 = 60 minutes 2) Seconds are a smaller unit than minutes, so again multiply by 60: 60 minutes = 60 × 60 = 3600 seconds
Guided Practice Use the information below to convert the quantities that follow. 1 kg = 2.2 lb 1 gallon = 3.79 liters 15. 3 kg to lb 16. 16 lb to kilograms 17. 7 gallons to liters 18. 44 liters to gallons
Independent Practice Don’t forget: Depending on how you do these conversions, you might get slightly different answers.
In Exercises 1–6, find the missing length. Give your answers to 2 decimal places. 1. 4.5 in. = ? cm 2. 100 mm = ? in. 4. 0.5 m = ? in. 5. 500 cm = ? yd
3. 14 yd = ? m 6. 45 ft = ? m
7. The length of a model plane is 56 inches. How long is the model in meters? Now try these: Lesson 4.3.3 additional questions — p449
8. Michaela and Ricky are each trying to guess how many feet are in a kilometer. Michaela guessed 3000 and Ricky guessed 3500. Whose guess was closest to the correct number of feet? 10. The dimensions of Zak’s bedroom are 5 ft × 8 ft. What are the dimensions of his room in meters?
5 ft
9. How many kilometers are there in 215,820 inches?
8 ft
Round Up You’ve now seen how to convert between different unit systems of length. And if you can do that, you can also convert pretty much anything else you want. For example, you might want to convert dollars to some other currency if you travel overseas. 232
Section 4.3 — Converting Units
Lesson
4.3.4
Other Conversions
California Standard:
You’ve seen now how to convert all sorts of things between different units. And you’ve seen two different techniques for carrying out these conversions — multiplying or dividing by a conversion factor, or solving a proportion. But these methods don’t work for all conversions, unfortunately.
Key words: • conversion • unit • conversion table • proportional
Some Units Are More Complicated to Convert The methods you’ve seen so far will work for most conversions you might meet, but not all of them. For example, converting degrees Fahrenheit (°F) to degrees Celsius (°C) is a little more complicated. Looking at some graphs should help you see the difference. These first three graphs show various quantities you’ve already seen.
Check it out:
inches
You can use this kind of graph to do conversions. You find a length on one scale (1.5 feet, say). Then you go up to the line and across to the other scale. For example: 24 1.5 feet 18 is the same as 12 18 inches. 6 0 0 0.5 1 1.5 2 feet
12 0 0
km against miles
200
24
Check it out: For all the conversions you’ve seen in previous Lessons, zero using one scale (inches, say) was always zero using the other scale too. This is why the first three graphs go through the point (0, 0). For example, 0 km = 0 miles, 0 cm = 0 km, and so on.
cm against m
inches against feet
1 feet
2
3.2 kilometers
You’ll see how to convert between different units where you can’t use a simple conversion factor. There’s also some information on using calculators.
centimeters
What it means for you:
inches
Algebra and Functions 2.1 Convert one unit of measurement to another (e.g., from feet to miles, from centimeters to inches).
100 00
1 2 meters
1.6 0 0
1
2 miles
They are all slightly different, but they are all straight lines and they all go through the origin, the point (0, 0). The math term for this is to say that the quantities on each graph are proportional. Now look at the graph showing degrees Fahrenheit (°F) and degrees Celsius (°C). This graph is a straight line, but it doesn’t go through the point (0, 0). And this means you can’t use a simple conversion factor, or proportions, to do the conversion. In other words, °F and °C are not proportional.
°F 35.6 33.8 32 0 0
1
2
°C
Guided Practice Using the graphs in Exercises 1–4, say which pairs of quantities are proportional.
x
4. s
3. n
2. q
1. y
p
m
Section 4.3 — Converting Units
r
233
Check it out: You don’t have to learn the formulas for converting Fahrenheit to Celsius, and vice versa.
To convert Celsius to Fahrenheit (or Fahrenheit to Celsius), you have to use a formula. F = 1.8C + 32 C = (F – 32) ÷ 1.8
If F is a temperature in Fahrenheit, and C is a temperature in Celsius, then:
Notice how you don’t just multiply or divide by a conversion factor. The formula for F involves a multiplication and an addition. The formula for C involves a division and a subtraction.
Check it out: You can write the conversion factors you’ve seen before as a formula. For example, if m is a length in miles and y is the same length in yards, then y = 1760m, where 1760 is the conversion factor.
Example
1
Convert 98.6 °F to degrees Celsius. Solution
The formula you need to use is: Substitute in your value for F:
C = (F – 32) ÷ 1.8. C = (98.6 – 32) ÷ 1.8.
When you have a formula with more than one operation, you do anything in parentheses first. So C = (98.6 – 32) ÷ 1.8 = 66.6 ÷ 1.8 = 37 Check it out: Remember PEMDAS in Lesson 2.1.4. 1. Parentheses 2. Exponents 3. Multiplication / Division 4. Addition / Subtraction
This means that 98.6 °F is equal to 37 °C.
Example
2
Convert 240 °C to degrees Fahrenheit. Solution
The formula you need to use is: Substitute in your value for C:
F = 1.8C + 32. F = 1.8 × 240 + 32.
When you have a formula with multiplication and addition, you do the multiplication first. So F = (1.8 × 240) + 32 = 432 + 32 = 464 This means that 240 °C is equal to 464 °F.
Guided Practice Convert the following: 5. 32 °F to °C 7. 50 °F to °C 9. 45.2 °C to °F
234
Section 4.3 — Converting Units
6. 100 °C to °F 8. –40 °C to °F 10. –428.8 °F to °C
The final part of this Lesson (about calculators) is relevant to everything in this book — not just converting between units.
Be Car eful If You’ tor Careful ou’rre Using a Calcula Calculator You can use a calculator for expressions like the ones on the previous page. However, you have to be really careful — different calculators do operations in different orders. Don’t forget: You have to work out the things in parentheses first. See Lesson 2.1.4 for more information.
For example, if you press the buttons shown on the right, some calculators will do operations in the 8 + 2 × 5 = order you type them (and give the answer 50). But other calculators do operations in the PEMDAS order (and give the answer 18). When in doubt about what order your calculator will do operations, use parentheses buttons to force your calculator to do what you want. Example
3
Show how to use parentheses to find 12.5 + 8.2 × 8 on a calculator. Solution
1 2
·
5 + (
8
·
2 × 8 (
=
Don’t forget: To enter a negative number, –11 for example, you normally need to press
(–( 1 1 .
But on some calculators you need to press
1 1 +– .
Independent Practice Which of these pairs of quantities on these graphs are proportional? 1. b
2. y
3. g
4. t s
Make sure you know how to do this on your calculator.
a
Now try these: Lesson 4.3.4 additional questions — p449
Convert the following: 5. 1445 °F to °C 7. 111 °C to °F
x
f
6. 855 °C to °F 8. 16 °F to °C
Evaluate Exercises 9–14 using a calculator. 9. 8g + 3.3, when g = 9.2 10. 52 × 7 + f, when f = 0.251 11. 1 – 11(k + 2.5), when k = 1.7 12. 24 – (–8 + 6m), when m = 3 13. 6n × (11 + 7n) + 2n, when n = 4 14. –12.6h – (–11.9h – h2)2, when h = –3 15. Joseph tries to calculate 18 ÷ 9 + 4 × 4 on a borrowed calculator. He types 1 8 ÷ 9 + 4 × 4 . Why can he not be sure that the calculator will give the correct result? What should he have typed?
Round Up This Lesson covered the idea of proportionality, which shows when you can use the conversion methods from previous Lessons. It also reviewed some of the work from earlier in the book about substituting values into formulas, and how to evaluate formulas using a calculator. Section 4.3 — Converting Units
235
Section 4.4 introduction — an exploration into:
Running R a tes Ra You’re going to work out how fast the different members of your group can move — either walking, running, hopping, skipping, or crawling. Then you’re going to work out how far they could go in different amounts of time, or how long it would take them to cover a particular distance. Work in groups of 4 or 5. Each person chooses a different method to move from place to place. You can choose from: walking, running, hopping, skipping, or crawling. Now measure and mark out distances of: (i) 20 feet, (ii) 50 feet, (iii) 70 feet. Each person is going to cover all three distances in their chosen way, while someone times them. Record your results in a chart like this one.
Time in seconds taken to move: Person Method 20 feet 50 feet 70 feet Tim Running 2.1 5.6 8.0 3.4 8.5 12.0 Carleen Walking
After each person’s 20-foot “run,” predict how long their 50-foot and 70-foot “runs” will take. Now you can use your results to make a graph. +
70
Example 60
Label each line clearly with the person’s name and their method of traveling traveling.
Who traveled faster? How can you tell this from the graph? Solution
For Tim, plot the three points (2.1, 20), (5.6, 50), (8.0, 70). For Carleen, plot the points (3.4, 20), (8.5, 50), (12, 70). Then draw best-fit lines for each person through (0, 0) and the three points.
+
+
50
Distance in feet
Plot the results from the table above, and draw best-fit lines.
+
Tim: running
40
Carleen: walking
30
+ +
20 10
The lines go through (0, 0), since they both
0 0
2
4 6 8 Time in seconds
10
12
Using the graph, you can see that Tim traveled faster, as his line is steeper.
Exercises 1. Use your graph to answer the following questions about the people in your group. a. Who moved the fastest and who moved the slowest? How does your graph show this? b. Use your graph to predict how long each person would need to go 30 feet. c. How far could each person go in 6 seconds? 2. After each person in your group had done their first “runs” of 20 feet, how did you make your predictions about the time they would need to go 50 feet and 70 feet?
Round Up Distance, time, and speed are closely linked. The slope of your graphs showed speed, and you could use those graphs to work out how long someone would need to cover a certain distance, or how far they would get in a particular time. There’s more about speed in this Section... a tion — Running Rates Explora 236 Section 4.4 Explor
Lesson
Section 4.4
4.4.1
Ra tes
California Standard:
You’ve already seen ratios in Section 4.1. A rate is like a ratio in some ways — it compares two quantities. But unlike ratios, rates have units too — this is the big difference between ratios and rates.
Alg ebr a and Functions 2.2 Alge bra Demonstr a te an Demonstra under standing tha a te is a understanding thatt rra measur e of one quantity measure per unit vvalue alue of another quantity quantity..
What it means for you: You’ll learn about what rates are, and how you can use them to compare two quantities.
Key words: • rate • per • ratio
AR a te Compar es Two Quantities Ra Compares In Section 4.1, you saw that a ratio compares two quantities, and you formed a ratio by dividing. A rate is a special kind of ratio. It also compares two quantities, and it’s also formed by dividing. But a rate always has units, because the quantities you divide are measured in different units. Speed is a good example of a rate. It’s often measured in units of “meters per second” or “miles per hour.” A rate can be written as a whole number, a decimal, or a fraction. Example
Check it out: The word “per” is a division word. It tells you to divide the thing before it by the thing after it.
Check it out: If you walked 8 miles in 2 hours, you could say your rate is “8 miles per 2 hours.” However, it’s far more useful (and usual) to say that your rate is “4 miles per hour” — this is an example of a unit rate. Unit rates are probably the most useful kind of rates. They tell you how much one quantity changes when the other changes by one unit (1 hour, say). Unit rates are easier to compare. For example, it’s clear that “8 miles per hour” is faster than “7 miles per hour.” But it’s not so obvious that “64 miles per 8 hours” is faster than “35 miles per 5 hours.”
1
Mike wants to know the rate at which his pinwheel turns in a gentle wind. He counts that it does 18 turns in 9 seconds. What rate is the pinwheel turning at? Solution
To find the rate that the pinwheel turns at, you need to divide the number of turns it completes by the time that it does them in. Rate = 18 turns ÷ 9 seconds The “/” is pronounced “per”. It means “each.” 18 turns per second = 9 = 2 turns per second (or 2 turns/second)
Rates are useful for comparing how quickly something is happening. Example
2
Nicole read 25 pages in 50 minutes, while Santos read 20 pages in 32 minutes. Who read faster? Solution
First work out both of their reading rates. Nicole: 25 pages ÷ 50 minutes = 0.5 pages/minute Santos: 20 pages ÷ 32 pages = 0.625 pages/minute You can see that Santos read more pages per minute. So Santos read faster than Nicole.
Section 4.4 — Rates
237
Guided Practice Find each of the specified rates in Exercises 1–3. 1. Find the “number of gallons of water per tank” if there are 25 gallons in 5 tanks. 2. Find the “distance traveled per gallon of fuel” if a car travels 120 miles using 8 gallons of fuel. 3. Find the “distance traveled per minute” if a cyclist travels 3 km in 10 minutes. Vikram paid $2.44 for four pens. Bob paid $3.90 for six pens. 4. Calculate “how many cents per pen” Vikram paid. 5. Calculate “how many cents per pen” Bob paid. 6. Who got the better deal, assuming all the pens are identical?
The R a te De pends on Whic h Way Ar ound You Di vide Ra Depends hich Around Divide When you work out a rate, you always divide one quantity by another. This means you can make two different rates using the same quantities. It depends which way around you do the division. Check it out: You can do something similar using speed. If you took 4 hours to walk 8 miles: • The rate you walked at 8 would be 4 miles per hour, or 2 miles/hour. • But you could also say that the rate you walked at was 4 8
hours per mile, or 0.5 hours/mile. Both answers are correct. One tells you how far you walk in each hour, the other tells you how long it takes to walk each mile.
Example
3
There are US $9.40, to every British £5. Work out the exchange rate in: (i) the number of dollars per pound (ii) the number of pounds per dollar Solution
(i) To work out the exchange rate in dollars per pound, divide the number of dollars by the number of pounds. $9.40 dollars per pound = = 1.88 $/£ £5 You do the same to the units as to the numbers. Here, you divided dollars by pounds, so the units of the rate are dollars per pound ($/£). (ii) This time, you need the exchange rate in pounds per dollar. So divide the number of pounds by the number of dollars. £5 pounds per dollar = = 0.53 £/$ $9.40 Notice that this time the units are pounds per dollar (£/$). Example
Check it out: “Weight per unit cost” means you divide the weight by the cost to find the weight per dollar, or weight per cent.
238
Section 4.4 — Rates
4
At the movie theater, Sanjay buys a 15 oz bag of popcorn for $4. Find: (i) the number of ounces of popcorn per dollar, (ii) the number of dollars per ounce of popcorn. Solution
These are the two ways of expressing this rate. (i) weight per unit cost: 15 oz ÷ $4 = 3.75 oz/$ (ii) cost per unit weight: $4 ÷ 15 oz = 0.27 $/oz (to 2 decimal places)
Don’t forget: The two different rates must be reciprocals. You use the same two quantities to work them out, but the numerator of one rate is the denominator of the other, and vice versa. For example: pounds pounds per dollar = dollars but dollars dollars per pound = pounds There’s more information about reciprocals in Lesson 3.3.1.
The two ways of expressing a rate are reciprocals of one another. 1 1 = 0.53 £/$, and = 1.88 $/£ For example 1.88 $ / £ 0.53 £ / $ Notice how taking a reciprocal of a quantity swaps its units around too. Example
5
Car A does 18 miles per gallon, while Car B does 0.05 gallons per mile. Which car is more efficient? Solution
To convert gallons per mile into miles per gallon, form the reciprocal. 1 miles per gallon, or 20 miles per gallon 0.05 gallons per mile = 0.05 Now you can compare the two quantities: Car B is more efficient, as it drove more miles per gallon of fuel used.
Guided Practice Check it out: If time is involved in a rate, it is usually the denominator. Think about it — miles/hour, words/minute, $/hour. Also, if you are comparing two times (for example the number of hours you are at school every week), the larger unit usually goes on the bottom — hours/week, for example.
Find the rates in Exercises 7–10. Explain what each of the quantities actually means. 7. 27 boats ÷ 3 lakes 8. 10 meters ÷ 10 seconds 9. 16 oranges ÷ 8 people 10. 150 words ÷ 3 minutes Express the rates in Exercises 11–14 in two different ways. You must include units in each of your answers. 11. A woman uses 2 gallons of gas driving 50 miles. 12. A store sells 5 ounces of parsley for $2. 13. A man takes 1 hour to drive 30 miles. 14. A rabbit hops 10 meters in 4 seconds.
Independent Practice Now try these: Lesson 4.4.1 additional questions — p449
In Exercises 1–4, work out two unit rates using the given quantities. 1. 12 ducks on 3 ponds. 2. 150 gallons in 50 seconds. 3. 5 apples between 2 people. 4. 1360 km in 20 hours. In Exercises 5–8, say what units the rates will have. 5. 4 miles ÷ 8 minutes. 6. 3 dollars ÷ 7 ounces. 7. 2 liters ÷ 5 glasses. 8. 21 carrots ÷ 7 rabbits. 9. Kevin walked 6 feet in 4 seconds, while Aretha walked 10 feet in 5 seconds. Who was walking faster? 10. Cedro bought 10 ounces of Cereal A for $2.25. Then he bought 15 ounces of Cereal B for $3.30. Which cereal is less expensive?
Round Up Both rates and ratios involve dividing one quantity by another, but with rates you have to remember to include the units as well. You see rates all the time in real life — they’re incredibly important. Section 4.4 — Rates
239
Lesson
4.4.2
Using R a tes Ra
California Standards:
In the last Lesson, you saw how to form a rate by dividing two quantities. In this Lesson, you’re going to see how you can use a rate to work out what one of those two quantities must have been.
Alg ebr a and Functions 2.2 Alge bra Demonstr a te an Demonstra under standing tha a te is a understanding thatt rra measur e of one quantity measure per unit vvalue alue of another quantity quantity.. Alg ebr a and Functions 2.3 Alge bra Solv e pr ob lems in volving Solve prob oblems inv ra tes ver a ge speed, tes,, a av era distance distance,, and time time..
What it means for you: You’ll learn about how to use rates to find other quantities.
It’s like doing the last Lesson again, only backward.
Multipl yb y the R a te to Find One of the Quantities Multiply by Ra A rate explains the connection between two quantities. So if you know a rate and one of the quantities, you can work out the value of the other. For example, if you know that a car can travel 150 miles on 3 gallons of fuel, you can work out its fuel efficiency in miles per gallon. Miles per gallon =
Miles traveled 150 miles = = 50 miles per gallon Gallons used 3 gallons
Key words: • rate • per
Now suppose the same car did another journey, and used 6 gallons of fuel. For every 1 gallon of fuel, it would travel 50 miles. So if it used 6 gallons of fuel, it must have traveled 6 × 50 = 300 miles.
Check it out: You’ll see this idea again in later Lessons in connection with speed.
Check it out: value of X . value of Y This means: value of X = value of Y ×( X per Y )
1 gallon used 50 miles
1 gallon used 50 miles
1 gallon used 50 miles
Number of Miles =
1 gallon used 50 miles
1 gallon used 50 miles
1 gallon used 50 miles
Miles × Number of Gallons Gallons
The same idea works for all rates. If you multiply a rate “X per Y” by a value of Y, you find the corresponding value of X.
X per Y =
To see why, multiply both sides of the first equation by the value of Y.
Check it out: If the rate is in “dollars per ounce,” then you multiply by the number of ounces to find the cost in dollars.
240
Section 4.4 — Rates
Example
1
Thalia goes to the health-food store to buy 10 ounces of granola. Granola costs $0.25 per ounce. How much will the granola cost? Solution
number of dollars number of ounces So to find the cost, you need to multiply the rate “$0.25 per oz” by the weight of granola that Thalia is buying “10 oz”.
This rate means:
0.25 dollars per ounce =
Number of dollars = 0.25 dollars/ounce × 10 ounces = 2.5 dollars The granola will cost her $2.50.
Example Check it out: Remember... value of X = rate (X per Y) × value of Y Here, the rate is “days per room,” meaning: X is “days,” and Y is “rooms.” So you have to multiply the rate by number of rooms to get the time in days.
2
A painter paints at a rate of 2 days per room. How long will she need to paint 9 rooms? Solution
This rate means:
2 days per room =
number of days number of rooms
So to find the number of days she will need, multiply “2 days per room” by the number of rooms. Number of days = 2 days/room × 9 rooms = 18 days She will need 18 days.
Guided Practice 1. The math teacher gives out 2 exercise books per student. If she has 24 students, how many exercise books will she need? 2. The manufacturing cost of baseballs is 32¢ per ball. How much would it cost to manufacture 88 baseballs? Check it out: For Exercise 4, remember to use the number of seconds, because the rate is given in “joules per second.” (A “joule” is a unit of energy.)
3. A particular kind of carpet costs $9 per square yard. What would it cost to carpet a room with a floor area of 16 yd2? 4. A car travels 11 miles per liter of fuel. How far will the car be able to travel on 7 liters of fuel? 5. An electric heater gives out heat at the rate of 3000 joules per second. If the heater is on for 30 minutes, how many joules of heat will be given out altogether?
Di vide b y the R a te to Find the Other Quantity Divide by Ra Think back to the example of a car with a fuel efficiency of 50 miles per gallon. If that car now goes on a journey of 250 miles, you can use its fuel efficiency to find the amount of fuel used. Check it out: value of X . value of Y This means: value of Y = value of X ÷ ( X per Y ) X per Y =
To see why, multiply both sides of the first equation by the “value of Y,” and then divide by the rate “X per Y.”
50 miles 1 gallon used
50 miles 1 gallon used
50 miles 1 gallon used
50 miles 1 gallon used
50 miles 1 gallon used
Number of Gallons = Number of Miles ÷
Miles Gallons
The same idea works for all rates. If you divide a value of X by a rate “X per Y,” you find the corresponding value of Y. Section 4.4 — Rates
241
Example Check it out: Remember... value of Y = value of X ÷ rate (X per Y) Here, the rate is “dollars per ounce,” meaning: X is “dollars,” and Y is “ounces.” So you have to divide the number of dollars by this rate to find the weight in ounces.
3
Thalia goes to the health-food store with $4. Granola costs $0.25 per ounce. How much granola can she buy? Solution
number of dollars number of ounces So to find the number of ounces of granola, you need to divide the number of dollars “$4” by the rate “$0.25 per oz.”
This rate means:
0.25 dollars per ounce =
Number of ounces = 4 dollars ÷ 0.25 dollars/ounce = 16 ounces Thalia can buy 16 oz of granola.
Example
4
A painter paints at a rate of 2 days per room. How many rooms will she be able to paint in 38 days? Solution
This rate means:
2 days per room =
number of days number of rooms
So to find the number of rooms she can paint, divide the number of days by the rate “2 days per room.” Number of rooms = 38 days ÷ 2 days/room = 19 rooms She will be able to paint 19 rooms.
Guided Practice 6. The math teacher gives out 3 exercise books per student. If she has 39 books, how many students will she have enough books for? 7. The manufacturing cost of basketballs is $1.14 per ball. How many balls could be manufactured for $136.80? 8. A particular kind of carpet costs $11 per square yard. What area of floor could be carpeted for $264? 9. A car travels 12 miles per liter of fuel. How much fuel would be used on a journey of 192 miles? 10. An electric heater gives out heat at the rate of 3500 joules per second. If 4,200,000 joules of heat have been given out, for how long has the heater been on?
242
Section 4.4 — Rates
Don’t forget: You can treat units like numbers. If there is a unit on the top line of a fraction, and the same one on the bottom line of it, you can cancel them out, just like numbers.
Independent Practice 1. Joe types at a rate of 150 words per minute. If he has typed 1500 words, how long has he been typing for? 2. Terrence buys some soil for his garden. It costs $90/m3. He spends $225. How many m3 of soil did he buy? 3. The city bus uses gasoline at a rate of 15 miles/gallon. How far can the bus go on 3 gallons of fuel?
Check it out: You need to think carefully about whether you’re being asked for the higher or lower unit rate. Words like “faster” and “most” are usually telling you to look for the higher unit rate. Words like “slower” and “least” are usually telling you to look for the lower unit rate.
4. Jen drinks 8 glasses of water per day. How many days will it take her to drink 28 glasses of water? 5. Letitia is selling her 28 baseball cards. Todd offers her $0.25 per card. Daria offers $1.50 per 7 cards. Who is offering to pay more? 6. Find the distance traveled per hour if a person runs 8 miles in 2.5 hours. 7. The rate at which water comes out of a faucet is 6 liters per minute. How long would it take to fill a bath with a capacity of 130 liters? 8. Water comes out of a faucet at a rate of 0.85 liters per second. How much water will be supplied if the faucet is left on for 3 minutes? 9. An aircraft is traveling at 900 miles per hour. The pilot sees a mountain when he is 50 miles away. How many minutes will it be before the aircraft reaches the mountain? 10. Mr. Clark’s car holds 12 gallons of gasoline when the tank is full. Starting with a full tank, he drove 153 miles, and used 25% of the gas. How many miles to the gallon does his car get? 11. At a health-food store, breakfast cereal is sold by the weight. If you can normally buy 2 lb of cereal for $3, then how much could you buy for $4 if there is a sale and the price of everything in the store is reduced to 50% of its normal price?
Now try these: Lesson 4.4.2 additional questions — p450
12. At 50 miles per hour, a car can travel 600 miles on a full tank of 15 gallons of fuel. At 30 miles per hour, the car can travel 30% further for each gallon of fuel. If the car travels 250 miles at 30 miles per hour, how many gallons of fuel would it use? And how long would the journey take?
Round Up Now you know how to use a rate to calculate another quantity. But you have to be sure you use the rate in the correct way. In the last Lesson, you saw that rates were important in real life. Well... in the next Lessons, you’ll be thinking about one rate in particular — speed. And speed is one of the most common rates you’ll ever come across. You might even have to use it every day. Section 4.4 — Rates
243
Lesson
4.4.3
Finding Speed
California Standards:
The rate that you’ll come across most often in everyday situations is speed. Speed is a rate because to find it you divide a distance by a time. It’s a measure of the rate that an object moves at.
Alg ebr a and Functions 2.2 Alge bra Demonstr a te an Demonstra under standing tha a te is a understanding thatt rra measur e of one quantity measure per unit vvalue alue of another quantity quantity.. Alg ebr a and Functions 2.3 Alge bra Solv e pr ob lems in volving Solve prob oblems inv ra tes ver a ge speed, tes,, a av era distance distance,, and time time..
What it means for you: You’ll see how to use measurements of distances and times to calculate speeds.
Speed Is a R a te Ra A rate compares two quantities. You’ve seen in the last two Lessons that to find a rate you divide one quantity by another. Think about a car — you would usually talk about the speed that the car was going in miles per hour. That’s a rate, because you’re dividing the number of miles the car has gone by the number of hours it took. Speed = 30 miles per hour 30 miles
Key words: • • • • •
rate speed distance time average
Check it out: A speed in miles/hour is a measure of the distance that the car would cover if it carried on traveling exactly at that speed for a whole hour.
Speed =
Distance Time
Use the F or mula to Calcula te Speeds For orm Calculate You can use the formula Speed = Distance ÷ Time to calculate the speed at which an object is traveling. Example
1
Will walked for 2 hours at a constant speed up a mountain trail that is 6 miles long. What was Will’s hiking speed? Solution
Distance = 6 miles Time = 2 hours
Speed =
6 miles Distance = = 3 miles/hour 2 hours Time
Guided Practice 1. Feo’s solar-powered lawn mower is set to travel at a constant speed. If it covers 1 mile in 2 hours, what speed is it traveling at? 2. A satellite is orbiting the earth at a constant speed. It takes 2 hours and 20 minutes to complete an orbit of length 56,280 km. What is its speed in km/min? 3. A plane travels at a constant airspeed. It covers a distance of 375 km in 3 hours. What is the plane’s airspeed? 4. Light travels at a constant speed. Calculate the speed of light, given that it travels 74,948,114.5 m in 0.25 seconds. 244
Section 4.4 — Rates
Ther e Ar e Man y Dif ent Units of Speed here Are Many Difffer erent There are a number of common units of speed, and some less common ones. To compare speeds, you need them to be in the same units. Example
2
Joshua is driving at a speed of 40 miles/hour. His friend is driving at a speed of 17 yards/second. Who is traveling more quickly? Solution
Don’t forget: You need to know a few conversion factors for questions like this. 1 mile = 1760 yards 1 mile = 1.61 km 1 km = 1000 meters 1 hour = 60 × 60 = 3600 s. You saw some of these in Lesson 4.3.3
To compare the speeds, you need them to be in the same units. Joshua’s friend is driving at 17 yards/second, so he travels 17 yards each second. So in 1 minute he travels 17 × 60 = 1020 yards. And in 1 hour he travels 1020 × 60 = 61,200 yards. So you could say his speed is 61,200 yards per hour. But 61,200 yards = 61,200 ÷ 1760 = 34.77 miles. So you could also say his speed is 34.77 miles per hour. Now that both speeds have the same units, you can compare them directly. Joshua is traveling at 40 miles/hour, while his friend is traveling at 34.77 miles/hour, so Joshua is travelling more quickly.
Guided Practice The fastest bird in the world is the peregrine falcon. This bird can reach speeds of 300 kilometers/hour when it swoops. 5. What is this in miles/hour? 6. What is this in meters/second? The peregrine falcon’s cruising speed is 24.6 meters/second. 7. What is this speed in miles per hour? 8. How far could it fly in 3 hours? 9. The fastest fish in the world is the sailfish. Its top speed is about 68 miles/hour. How fast is this in meters/second?
This F or mula Gi ves You the Aver a ge Speed For orm Giv era Suppose a car goes on a journey. Its speed will vary all the time. It may stop at a crosswalk or traffic lights, but then it will go faster on open roads. Check it out: If an object is traveling at a constant speed, then its speed at any moment will be exactly the same as its average speed.
If you work out the car’s speed by dividing the total distance it went by the time it took, you are calculating its average speed over the whole journey. This doesn’t mean the car was always traveling at that average speed. It just means that if it did travel at that speed for the journey time, it would cover the same distance as the actual journey.
Section 4.4 — Rates
245
The car’s speedometer tells the driver how fast it is traveling at any instant in time — not the average speed.
The blue line shows speedometer readings. The red line shows the average speed.
Speed
Check it out:
This graph shows the actual and average speeds for a car journey.
Actual speed Average speed
Time
Guided Practice 10. A fish swims from one end of a lake to the other in 2 hours. The lake is 12 km long. What is the average swimming speed of the fish?
Check it out: There’s more information about average speeds in Lesson 4.4.5.
A skydiver jumped out of a plane at 10,800 feet. He fell for 50 seconds before his parachute opened at 3000 feet. It took a further 200 seconds for him to reach the ground. 11. What was his average speed with his parachute closed? 12. What was his average speed with his parachute open? 13. What was his average speed over the whole drop?
Independent Practice 1. A sprinter completes the 100 m dash in 10 seconds. What is his average speed over the course of the race? 2. Shona is riding her scooter. She covers 4 blocks in 10 minutes. How fast is she traveling? Give your answer in blocks/minute. Don’t forget: Always remember that the units of an answer should match the units in the question. For a reminder, see Lesson 4.4.1.
3. A hockey player shoots the puck a distance of 51 m in 1.5 s. What is the average speed of the puck? 4. A cheetah runs one mile at constant speed. It takes 1 minute. What is the cheetah’s speed in miles per hour? 5. Ryan rides his bike along the edge of the park, taking the route shown in the diagram. It takes him 4 hours. What is his average speed?
15 km 5 km
6. A sailing boat travels a distance of 2 miles in 15 minutes. What is its average speed in meters/second? Now try these: Lesson 4.4.3 additional questions — p450
Laura is standing 255 m away from Ervon. When she rings a bell, he hears the sound of the bell 0.75 seconds later. 7. What is the speed of sound in air in meters per second? 8. What is the speed of sound in air in miles per hour? 9. A raindrop forms in a cloud at an altitude of 1750 m. It takes 250 seconds to fall to Earth. What is its average speed?
Round Up You can work out an object’s speed by dividing the distance it traveled by the time it took. Unless the speed of the object is constant, that speed will be an average speed over the whole journey. In the next Lesson, you’ll see how to rearrange the formula that you’ve learned — and use speeds to work out times and distances. 246
Section 4.4 — Rates
Lesson
4.4.4
Find ing Time And Distance Finding
California Standards:
In the last Lesson, you saw how to use the formula “Speed = Distance ÷ Time” to work out the speed of an object. In this Lesson, you’ll see how to use the formula to work out the distance an object has traveled or the time it took when you’re given its speed. It’s really just a “special case” of what you saw in Lesson 4.4.2.
Alg ebr a and Functions 2.2 Alge bra Demonstr a te an Demonstra under standing tha a te is a understanding thatt rra measur e of one quantity measure per unit vvalue alue of another quantity quantity.. Alg ebr a and Functions 2.3 Alge bra Solv e pr ob lems in volving Solve prob oblems inv ra tes ver a ge speed, tes,, a av era distance distance,, and time time..
What it means for you: You’ll see how to use speeds to calculate distances and times.
Use Speed and Time to Find Distances An ordinary garden snail moves at an average rate of 40 meters per hour. Suppose you want to find out how far it would travel in 2 hours. 2 hours
Speed = 40 meters per hour
Key words: • • • •
rate speed distance time
?m
Check it out: A rate that is describing how far an object moves in a certain time is a speed.
In one hour, it will move 40 m. So in two hours it will move 2 × 40 = 80 m. The rate has been multiplied by the time it took. If you know the rate at which an object is traveling and the time that it is traveling for, you can work out the distance it travels using this formula:
Distance = Speed × Time
Check it out: This is a rearrangement of the formula “Speed = Distance ÷ Time.” Speed = Distance ÷ Time. multiply both sides by “Time” Speed × Time = Distance.
Example
1
Mrs. Jackson walks at an average speed of 2 miles per hour. If she walks for 3 hours, how far will she go? Solution
Distance = Speed × Time = 2
Example
miles × 3 hours = 6 miles hours
2
Mr. Jackson walks at an average speed of 0.4 meters per second. If he walks for 9850 seconds, how far will he go? Solution
Distance = Speed × Time = 0.4
meters × 9850 seconds = 3940 meters second
These two Examples show that the units of the final answer depend on the units of the other quantities. If you use a speed in meters per second and a time in seconds, the distance you find would be in meters. Section 4.4 — Rates
247
Guided Practice 1. Raymon drives his car at an average speed of 40 miles/hour for 3 hours. What distance will he cover? 2. Alice wins a race on field day in a time of 50 seconds. Her average speed is 8 m/second. Which length race did she win? 3. A sunflower grows at a rate of half a centimeter a day. How tall will it be after 30 days? 4. A stone falls off a cliff. It hits the ground after 3 seconds. The stone traveled at an average speed of 14.7 m/second. How high is the cliff?
Use Speed and Distance to Find Times Think about the snail again. This time you know that it’s gone 80 meters, but you don’t know how long it took. ? hours
Speed = 40 meters per hour
80 m
To find the time that it took the snail to travel 80 m, you can divide the distance it traveled by the speed it moves at. Time taken = 80 ÷ 40 = 2 hours. Check it out:
If you know the speed at which an object is traveling, and the time that it is traveling for, you can work out the distance that it travels using this formula:
This is a rearrangement of the formula “Speed = Distance ÷ Time.”
Time =
Distance Speed
Speed = Distance ÷ Time. multiply both sides by “Time” Speed × Time = Distance divide both sides by “Speed” Time = Distance ÷ Speed
Example
3
A pro baseball pitcher pitched a baseball at a speed of 151.1 feet per second. How long did it take the ball to travel the 60.5 feet from the pitcher’s mound to home plate? Solution
Time =
distance speed
feet = 60.5 feet ÷ 151.1 second
= 0.400 seconds (to 3 decimal places)
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Section 4.4 — Rates
Pitcher’s Mound 60.5 feet Home Plate
Guided Practice 5. A bird flies at an average speed of 22 miles/hour. It covers a distance of 11 miles. How long was it flying for? 6. Eddie’s house is 3.4 km away from his school. He cycles to school every morning. His average speed is 0.2 km/minute. How long will it take him to get to school? 7. A caterpillar eats leaves at a rate of 11 leaves/hour. How long will it take it to eat 55 leaves? 8. A train travels a distance of 581 miles at an average speed of 83 miles per hour. How long will it take to do this?
Independent Practice Don’t forget: “meters per second” is often written as “m/s.”
Don’t forget: These average speeds are unit rates. For a reminder on comparing unit rates, see Lesson 4.4.1.
1. Caryne, Reese, and Katia ran a 100 m race. Caryne’s average speed was 8.52 m/s, Reese’s was 9.11 m/s, and Katia’s was 9.02 m/s. Who won the race? 2. A horse trots at a speed of 9 miles/hour. How much ground will the horse cover in 3 hours? 3. A hot-air balloon goes up at a speed of 50 feet/minute. It rises to a height of 1100 feet. How long will it take to reach this height? 4. Fayard watches a raindrop run down a window. The window is 150 cm tall. The drop runs at an average speed of 2 cm/second. How long will it take to reach the bottom of the window? 5. Dulcie has a model plane that flies at a constant speed of 1.5 m/s. If it is airborne for 30 seconds, how far will it go? 6. A roller-coaster ride lasts 90 seconds. The roller-coaster train travels at an average speed of 50 km/hour. How far does it travel? 7. A train leaves the city at 10 a.m. It travels at an average speed of 93 miles/hour. The next stop is 62 miles away. At what time will it reach the next stop? 8. Town A is 4 miles from Town B. A bus leaves Town A for Town B
Now try these:
traveling at an average speed of
Lesson 4.4.4 additional questions — p450
Town B. He lives speed of
1 28
1 4
1 2
mile/minute. Mr. Jones lives in
mile from the bus stop, and walks at an average
mile/minute. Mr. Jones leaves his house to catch the bus at
the same time the bus leaves Town A. Will he catch the bus or miss it?
Round Up In the same way that you can use a time and a distance to find a speed, you can use a speed to find a time or distance. It’s just a case of rearranging the formula and putting the numbers in.
Section 4.4 — Rates
249
Lesson
4.4.5
Aver a ge R a tes era Ra
California Standards:
You’ve already seen the formulas linking average speed, distance, and time. In this Lesson, you’ll get a lot of practice using them.
Alg ebr a and Functions 2.2 Alge bra Demonstr a te an Demonstra under standing tha a te is a understanding thatt rra measur e of one quantity measure per unit vvalue alue of another quantity quantity.. Alg ebr a and Functions 2.3 Alge bra Solv e pr ob lems in volving Solve prob oblems inv ra tes ver a ge speed, tes,, a av era distance distance,, and time time..
Al ways Use Your Thr ee F or mulas Alw hree For orm A F aster Travels Fur therdistance, P er Hour Faster Further Per You’ve seen Object three formulas linking time, and speed. Speed =
Key words: • • • • •
rate speed distance time average
Time =
Distance Speed
Distance = Speed × Time
What it means for you: You’ll see how to work out average speeds over a whole journey.
Distance Time
You need to use these formulas for all questions involving time, distance, and speed. Don’t try to take any shortcuts. Example
1
A cyclist travels at 18 miles per hour for one hour, and for another hour at 32 miles per hour. What is her average speed for the whole journey? Solution
To find her average speed for the whole journey, you need the total distance and the total time. The question tells you that the total time is 2 hours.
Check it out: The answer to Example 1 is the average of the two speeds. But you should never assume that things will be that easy. Look at Example 2.
To find the total distance, break the journey into two parts, and use one of the above formulas for each part. For the first part: distance = 18 miles per hour × 1 hour = 18 miles. For the second part: distance = 32 miles per hour × 1 hour = 32 miles. So the total distance = 18 + 32 = 50 miles. Now find the average speed for the whole journey. Average speed = 50 miles ÷ 2 hours = 25 miles per hour Now look at this next example. Example
2
Joshua walks for 1 mile at 4 miles per hour, and for another mile at 2 miles per hour. What is his average speed for the whole journey? Solution
Check it out: The answer to Example 2 is not the average of the two speeds. The “obvious” answer of “3 miles per hour” would have been wrong.
Do the question in two parts again. This time, the total distance is 2 miles. For the first part: time = 1 mile ÷ 4 mph = 0.25 hours. For the second part: time = 1 mile ÷ 2 mph = 0.5 hours. So the total time = 0.25 + 0.5 = 0.75 hours. This means the average speed is 2 ÷ 0.75 = 2.67 miles per hour.
250
Section 4.4 — Rates
Guided Practice Work out the average speed of the entire journey in Exercises 1–4. 1. Eloise rides her bike for 2 hours. For the first hour, her average speed is 10 km/h. For the second hour, her average speed is 8 km/h. 2. Orlin rows upstream for 20 minutes at a speed of 0.1 km/min. Then for 10 minutes he rows downstream at 0.4 km/min. 3. Vanessa rides on an elephant for 2 hours (average speed 4.5 miles/hour) and a camel for 1 hour (average speed 3 miles/hour). 4. Brian pulls his sled up a hill in 120 s at a speed of 0.6 m/s. Then he slides back down in 30 s at a speed of 2.4 m/s.
Al ways R el y on Those Same Thr ee F or mulas Alw Rel ely hree For orm You can use those formulas to solve some very complex problems. You just have to take things one step at a time. Example
3
A ball is dropped vertically from a platform 45 meters above the ground. It falls for 3 seconds before it hits the ground. The ball then bounces vertically, and rises for 2 seconds at an average speed of 10 m/s. What is the ball’s average speed as it falls? How high does the ball rise? What is the ball’s average speed over the two parts of the journey? Solution
Don’t forget: This example looks very complex. But you just have to keep using those same three formulas. For each part of the journey, there are 3 quantities — distance, time, and speed. Write down the two quantities you know, then use the formula that will tell you the other one.
This looks pretty difficult. Just work slowly and carefully, and it might also help if you draw some simple diagrams. For the first part — as the ball is falling. Distance = 45 meters. Time = 3 seconds. So its average speed as it falls = 45 m ÷ 3 s = 15 m/s For the second part — as the ball is rising. Average speed = 10 m/s. Time = 2 seconds. So the distance it rises = 10 m/s × 2 s = 20 m
45 meters 3 seconds
2 seconds 10 m/s
To find the average speed for the whole journey, you need the total distance traveled and the total time taken. Total distance traveled = 45 m + 20 m = 65 m. Total time taken = 3 s + 2 s = 5 s. So the overall average speed = 65 m ÷ 5 s = 13 m/s.
Section 4.4 — Rates
251
Guided Practice Work out the average speed of the entire journey in Exercises 5–8. 5. A duck swims 50 m at a speed of 5 m/s, then flies 550 m at a speed of 11 m/s. 6. Glover walks 1 mile to the station to catch a train. He walks at 2 miles/hour. He gets the train for 20 miles. Its speed is 80 miles/hour. 7. Diana drives 5 km at a speed of 15 km/h, then 10 km at a speed of 30 km/h. 8. Justin is in a circus parade. He walks 50 m at a speed of 1 m/s. Then he puts on stilts, and walks another 140 m at a speed of 0.7 m/s.
Independent Practice Work out the average speed of the entire journey in Exercises 1–7. 1. Keshila jogs for 2 hours at a speed of 3 miles/hour, and then walks for 2 hours at a speed of 2 miles/hour. 2. Wayan drives for 100 miles at a speed of 50 miles/hour, then for 50 miles at 25 miles/hour. 3. A farmer drives his combine for 5 hours at an average speed of 15 km/h. Then he drives his tractor for 3 hours at an average speed of 10 km/h. Now try these: Lesson 4.4.5 additional questions — p451
4. Hannah is doing the long jump. Her run up is 24 m and her run-up speed is 8 m/s. She jumps 6 m. Her jump speed was 6 m/s. 5. A rose grows at a speed of 0.5 mm/day for 4 days. Then it rains. For the next 6 days, the rose grows at the faster speed of 1 mm/day. 6. A spider climbs upward for 90 cm at a speed of 2 cm/s. Then it drops back down 50 cm at a speed of 10 cm/s. 7. Dan competes in a triathlon. He does the 2 km swim at an average 2
Check it out: Even when a journey is split into more than two parts the method is still the same. Work out the time and distance of each part, add them together, and put them into the formula.
speed of 2 3 km/h. His 40 km bike ride has an average speed of 6
22 7 km/h. His 10 km run is at an average speed of 20 km/h. 8. A skydiver jumps out of a plane and falls for 43 seconds at an average speed of 190 feet per second before his parachute opens. He then drops the next 2850 feet at 15 feet per second, until he lands. How high was the plane when he jumped, and how long did it take him to reach the ground?
Round Up For nearly all “speed questions,” you’ll be given two bits of information and you’ll need to work out the third using one of the formulas at the beginning of this Lesson. The question might look complicated, and it might be in many parts, but the basic idea will be the same. You use two quantities and one of the formulas to find out the other quantity. 252
Section 4.4 — Rates
Chapter 4 Investigation
Sunshine and Shadows Shadows change length and direction as the Sun moves across the sky. The picture below shows a tree and a statue on a sunny day. Their shadows at various times of day are shown. The two red triangles in the diagram are similar, as are the two yellow triangles. Part 1: • At 7 am, the tree casts a shadow that is 41 feet long. Find the length of the shadow of the statue at 7 a.m. 18 feet
• At 10 am, the statue casts a shadow that is 3.7 feet long. Find the length of the shadow of the tree at 10 a.m. 6 feet midday
Things to think about:
7 am 10 am
midday
3.7 feet
41 feet 10 am
7 am
• How can you use proportions with similar triangles?
Part 2: • At midday, the shadows are at their shortest. The ratio of the height of the objects to the length of their shadows is 1 : 0.28. Find the lengths of the shadows of both objects at midday.
Extension • Find the average rate that the length of the statue’s shadow shrinks between 7 a.m. and 10 a.m. Give your answer in feet per hour. • Between the hours of midday and 6 p.m., the tree’s shadow increases in length at an average rate of 9.6 feet per hour. Find the length of the tree’s shadow at 6 p.m. Open-ended Extension • Devise a method for finding the height of a local statue, tree, building, or other landmark, based on similar triangles. You should work out what equipment you would need, and how it would be used (including any preparations that you may need to make). • Once you have devised your method, try it. Does your method work? How accurate was your result? What problems did you encounter?
Round Up This Investigation shows one example of where similar triangles occur in nature. Similar shapes appear elsewhere too — for example, when you make a copy of a picture. Cha pter 4 In vestig a tion — Sunshine and Shadows 253 Chapter Inv estiga
Chapter 5 Data Sets Section 5.1
Exploration — Estimation Line-up ............................. 255 Statistical Measures .................................................. 256
Section 5.2
Adding Extra Data ..................................................... 268
Section 5.3
Claims and Data Displays ......................................... 275
Section 5.4
Exploration — Sampling Survey ............................... 286 Sampling ................................................................... 287
Section 5.5
Statistical Claims ....................................................... 300
Chapter Investigation — Selling Cookies .......................................... 309
254
Section 5.1 introduction — an exploration into:
Estima tion Line-Up Estimation This Exploration is all about jellybeans. Remember it’s math though... so actually you have to analyze a list of guesses about the number of jellybeans in a big jar. You have to estimate how many jellybeans there are in a jar. Write your estimate in large numbers on a sticky note.
It might help to stick all the estimates to a wall.
The whole class should arrange their sticky notes in order — from smallest to biggest. This Exploration is about using your list of numbers to answer these questions: • Are the guesses all quite similar? Or all very different? • Are the guesses mostly too high? Or mostly too low? Example What is the range of this class’s estimates?
208 260 270 310 310
Solution
The RANGE is the difference between the smallest estimate and the largest estimate. So subtract the smallest estimate from the largest. For these numbers, the range = 310 – 208 = 102.
The range tells you how different the guesses are. A large range means that the guesses are very different different. A smaller range means that they are more similar similar.
Example What is the mode of this class’s estimates?
208 260 270 310 310
Solution
The MODE is the number that was guessed most often. For these numbers, the mode = 310. Example What is the median of this class’s estimates?
You might have more than one mode. The mode and median are both “typical values.” They tell you if most typical values guesses were too high high, or too low low.
208 260 270 310 310
Solution
The MEDIAN is the number in the middle of the ordered list. For the numbers above, the median = 270. With an even number of estimates, the median is midway between the middle two. So for the numbers on the right, the median = 265.
205 208 260 270 310 310 Median is midway between 260 and 270 270.
Exercises 1. For your class’s estimates, find the:
a. range, b. median, c. mode
2. Make some statements about your class’s estimates. Use the range, mode, and median.
Round Up These statistics are ways to summarize a whole list of numbers — there’s more in this Section. Section 5.1 Explor a tion — Estimation Line-Up 255 Explora
Lesson
5.1.1 California Standard: Statistics, Data Analysis, and Probability 1.1 Compute the range, mean, median, and mode of data sets.
Section 5.1
Median and Mode The median and mode are both intended to “summarize” a whole data set in a single number. They should show some kind of “most usual” or “middle” value of a set of data. But although they’re similar in some ways, they’re worked out very differently. This Lesson is about how to find them.
What it means for you:
Data Sets Can Contain All Sorts of Values
You’ll learn to find the median and mode for sets of data.
Data sets often contain numerical values. For example, the data set below represents the number of hours eight adults said they slept last night.
Key words: • • • • •
central tendency data median mode values
{12, 8, 7, 8, 8.5, 8, 9, 6}
Braces “{“ and “}” are used to show that values are grouped together in a set.
Data sets can contain other types of information too. For example, the data set below represents the hair color of five students. {blonde, brunette, red, blonde, black}
If there are two items the same, they’re listed twice.
Data sets can contain huge amounts of data, and it’s very likely that most people won’t be interested in reading every single value. So, often a value that represents a typical value for the set is used. These typical values are often referred to as measures of central tendency.
The Median Is a Measure of Central Tendency The Median The median of a data set is a value that divides the set into two equal groups — one group containing values bigger than the median, the other containing smaller values.
Check it out: You can only find the median of data sets where you can put the values in order.
So the median is the middle value when a set of values is put in order. If you have an odd number of values, the median is fairly easy to find. Example
1
Find the median of the following data set: {3, 2, 6, 8, 2, 10, 6, 4, 9} Solution
First, arrange the values in order: {2, 2, 3, 4, 6, 6, 8, 9, 10} There are nine values, so the median is the fifth value. There are four values less than the median...
{2, 2, 3, 4, 6, 6, 8, 9, 10}
The median of the data set is 6.
256
Section 5.1 — Statistical Measures
median
... and four values greater than the median.
If there’s an even number of values, then finding the median is slightly trickier because there are two middle numbers. Here, you find the value exactly midway between the two middle numbers. Example
2
Find the median of this data set: {4.6, 8.9, 9, 10, 10, 14.7} Solution
The values in this data set are already in order from least to greatest. There are six values in the set — the median lies midway between the third and fourth values. Median = 9.5 There are three values below the median...
{4.6, 8.9, 9, 10, 10, 14.7}
... and three values above the median.
The third and fourth values are 9 and 10 — so the median is 9.5.
Guided Practice
Check it out: It’s easy to leave out a value when you’re rewriting the data set in order. It’s safer to copy the data in the original order first, then cross out the values as you rewrite them.
Find the median of each of the following data sets. 1. {12, 8, 10, 19, 21, 7, 14} 2. {$101, $201, $150, $198, $300} 3. {5, 8, 3, 6, 12, 9, 5, 5, 4, 11} 4. {–6, –3, 7, 4, –2, –2, 5, 2} 5. {–2.1, 5.7, 8.1, –10.2, –100, 42.778}
The Mode Is Another Measure of Central Tendency Check it out:
The Mode
The mode is sometimes called the modal value. So in Example 3, the modal value is 2.
The mode of a data set is the value that occurs most often. To find the mode of a data set, look for the value that’s listed more than any other value. Example
Check it out: To find the mode, you don’t have to write the values out in order. However, you might still find it useful to do this, as it makes it easier to see which value is repeated most.
3
Find the mode of the data set: {17, 2, 6, 8, 2, 10, 4, 35, 10, 7, 2} Solution
The number 2 occurs three times — this is more than any other value in the set. {17, 2, 6, 8, 2, 10, 4, 35, 10, 7, 2} So the mode is 2.
Section 5.1 — Statistical Measures
257
Some Data Sets Have No Modes — Others Have Many Data sets don’t always have one mode — as these examples show. Example Check it out: The mode is a good measure of central tendency to use if your data set contains nonnumerical values.
4
Find the mode of this set of data: {brown, blue, green, blue, yellow, brown, orange, white} Solution
Blue and brown both appear twice. No other color appears more often. So the data set has two modes, blue and brown. Example
5
Find the mode of this set of data: {3, 5, 19, 5, 3, 19} Solution
Each number occurs twice — no value occurs more often than the others. So this data set has no modes.
Guided Practice Give the mode(s) of the following data sets. 6. {$12, $8, $7.50, $7.50, $10, $8, $8, $9.50} 2 1 1 3 7. { , , , 1, } 8. {0, 1, –3, 5, 1, 0, –3, –3, 0} 5
2
2
4
Independent Practice Give the median and mode(s) of the data sets in Exercises 1–4. 1. {42, 56, 73, 64, 42} 2. {2, 5, 7, 3, 8, 10, 14} 3. {$16, $28, $20, $15}
4. {0.1, 0.4, 0.7, 0.4, 0.5, 0.7}
5. Write a set of data for which the mode and the median are the same.
Now try these: Lesson 5.1.1 additional questions — p451
Exercises 6–7 are about Rick’s survey of car colors. 6. Rick listed the colors of the 25 cars in a parking lot. The mode for his list is blue. What does this tell you? 7. There are only 5 car colors on Rick’s list: white, red, black, blue, and green. The mode is blue. What can you say about the possible minimum and maximum number of blue cars? Explain your answer. In Exercises 7–10, find a number for each blank so the median is 12. 8. {42, 2, 5, 7, 36, __} 9. {4, 12, 18, __} 10. {2, 5, 14, 12, __} 11. {6, 6, 7, 9, 11, 17, 17, 20, 23, __} Decide whether each statement is true or false. Explain your answers. 12. The median of a data set always equals one of the data values. 13. Not all data sets have a median.
Round Up These typical values beginning with “m” can get confusing. The median is the middle number when they’re arranged in order, and the mode is the most common value. There’s another similar “m” coming up in the next Lesson too — the mean. 258
Section 5.1 — Statistical Measures
Lesson
5.1.2
Mean and Range
California Standard:
The mean (like the mode and median) is another way of finding a “typical value” of a data set.
Statistics, Data Analysis, and Probability 1.1 Compute the range, mean, median, and mode of data sets.
What it means for you: You’ll learn how to find the mean and range for a set of data. Then you’ll see how the mode, mean, and median don’t all make sense in every situation.
The range, on the other hand, gives a different kind of information — it tells you how spread out the data is.
The Mean Is Also a Measure of Central Tendency The mean is often referred to as the average. You have to do a calculation to find it: The Mean The mean of a data set is found by: (i) adding up all the items in a data set, and then (ii) dividing by the number of items in the data set.
Key words: • • • • • •
mean measure of central tendency sum range mode median
Don’t forget: Finding the sum of a set of numbers means adding them all up.
Example
1
Find the mean of the data in the set {2, 7, 4, 8, 10, 24, 57, 39, 8}. Solution
First, find the sum of the numbers in the set: Sum = 2 + 7 + 4 + 8 + 10 + 24 + 57 + 39 + 8 = 159 Next divide the sum by the number of values in the set. Mean = 159 ÷ 9
Don’t forget: There’s more about rounding in Section 1.4.
There are 9 values in the set.
= 17.7 The exact answer was 17.666666... (You often have to round mean values to a sensible level of accuracy.)
Guided Practice Calculate the mean of each data set in Exercises 1–4. Round your answers to the nearest hundredth. 1. {34, 67, 65, 45, 78, 35, 90} 2. {2, 3, 3, 3, 2, 4} 3. {4.2, 4.5, 4.8, 4.9, 4.5} 4. {12, 12, 24, 23, 26, 32, 14, 18, 18}
Check it out: The mean is also used to calculate a student’s GPA (grade point average).
5. In a college course, a student’s final score is the mean of three tests. To earn a B grade, students need a final score of at least 80. A student earns 78, 91, and 73 points on the three tests. Will this student get a B?
Section 5.1 — Statistical Measures
259
Use the Mean to Find Missing Values On its own, the mean doesn’t tell you anything about individual values in a data set. But you can use it to find one missing value if you know the rest. Suppose you’ve taken five tests, but can’t find one result. If you know your mean score, you can use it to figure out the missing grade. Example
2
After taking five tests, Tiffany earns a grade average of 85%. She can only find four of her test papers. On these, she scored 90%, 82%, 87%, and 98%. Use the mean to find out her missing test score. Solution
You have to work backward to solve this sort of problem.
Check it out: You can use a variable such as x to stand for the missing score. 90 + 82 + 87 + 98 + x = 425 So 357 + x = 425, which means x = 425 – 357 = 68. See Lesson 2.2.3 for more information about solving this kind of equation.
You know that the mean is the total of all five test scores divided by the number of tests (5). In other words: mean = sum of all five scores ÷ 5 mean = 85 This means you can multiply 85 by 5 to find the sum of the 5 scores. Sum of all five scores = 85 × 5 = 425 Now compare this sum to the sum of the four test scores she already knows: 90 + 82 + 87 + 98 = 357 357 is 68 less than 425. So the missing test score is 68%.
Guided Practice 6. The mean of two numbers is 76. If one number is 15, what is the second number? 7. Max’s final grade is the mean of four tests. He wants to earn a final grade of 90. He got 83, 95, and 88 on the first three tests. What must he get on the fourth test to earn an average of 90?
The Range: Subtract the Least from the Greatest The range tells you how spread out the data is. A set of data that has a big difference between the highest and lowest values will have a much bigger range than one in which the values are all fairly similar. The Range The range of a data set is the difference between the greatest and least values in the set.
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Example
3
Find the range of heights of these basketball players: Player 1: 195 cm; Player 2: 210 cm; Player 3: 202 cm; Player 4: 180 cm Solution
Subtract the least value from the greatest: 210 cm – 180 cm = 30 cm The range of the heights is 30 cm. Example Check it out: If the highest value is 11 and the lowest is 2, then for the range to stay as 9, the other data values must not be greater than 11 or less than 2.
Check it out: There are many possible answers to Example 4, since you can pick any value to use as the greatest value. So another possible answer is {14, 5, 7, 8, 6, 10, 10}.
4
A data set contains 7 values. The range of these values is 9. Suggest a data set that meets this description. Solution
You know that the difference between the highest and lowest values is 9. Pick any number for the highest value — 11, say. Then the lowest value must be 11 – 9 = 2. But you can pick any other values for the remaining 5 items, as long as none of them is less than 2 or greater than 11. There are many So one possible set of values is {11, 8, 3, 3, 2, 2, 4}. different possibilities.
Guided Practice Calculate the range of the data sets in Exercises 8–9. 8. {90, 120, 80} 9. {1.2, 1.4, 1.1, 2.5, 3, 2.2} 10. The range of the set of ages of people at a party is 12. Suggest possible ages for the youngest person and the oldest.
Independent Practice Calculate the mean and range for each data set in Exercises 1–2. 1. {5, 9, 3, 12} 2. {–5, 7, 18, –1, 6} Now try these: Lesson 5.1.2 additional questions — p451
For Exercises 3–5, use the data set {19, 13, 7, 2, 1, 1, 25, 4}. State how many values in the set are: 3. greater than the mean. 4. less than the mean. 5. equal to the mean. Each set in Exercises 6–9 has a mean of 5. Find the missing values. 6. Set A {1, 9, 7, 8, 2, __} 7. Set B {1, 4, 10, 5, 6, __} 8. Set C {5, 4, 6, 5, 4, __} 9. Set D {10, –2, 0, __} 10. The range of a data set is 12. If the greatest value is 188, what is the least value? 11. Which one data value could be removed from the data set below to give it a range less than 9? {1, 2, 4, 6, 7, 7, 9, 11}
Round Up Now you know about the range and three measures of central tendency — mean, mode, and median. You’ve also seen a measure of spread — the range. There’s more about these next Lesson. Section 5.1 — Statistical Measures
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Lesson
5.1.3 California Standards: Statistics, Data Analysis, and Probability 1.1 Compute the range, mean, median, and mode of data sets. Statistics, Data Analysis, and Probability 1.3 Understand how the inclusion or exclusion of outliers affects these computations. Statistics, Data Analysis, and Probability 1.4 Know why a specific measure of central tendency (mean, median) provides the most useful information in a given context.
What it means for you: You’ll see how extreme values can affect calculations of the mean, mode, and median, and why this is important.
Key words: • • • • • •
mean measure of central tendency sum range mode median
Check it out: The range would also be heavily affected by an outlier.
Extreme Values The last two Lessons have looked at the median, mode, and mean. These are “typical values” — values that fall somewhere in the middle of your data. This Lesson is about values that are “right at the end” of your data — these are called extreme values.
Extreme Values Are Sometimes Called Outliers An outlier is a data value that is much greater than or much less than other values in the data set. For example, the following set of numbers contains an outlier: {9.99, 10.03, 10.08, 10.09, 10.10, 10.12, 10.19, 13.01}. The value 13.01 is a lot higher than all the others. There are various reasons why some values might lie far from the others. (i) Sometimes measurements are made incorrectly (for example, if someone used the wrong end of a ruler to measure something). (ii) Sometimes the measurement is “unfair” in some way, because of a problem, perhaps. For example, the eight numbers in the set above are times of athletes running a 100-meter race. The first seven athletes finished in times very close to 10 seconds. Unfortunately, the final athlete fell over, tore a muscle, and limped to the end of the race. (iii) Sometimes the reading is a genuinely unusual, but correct, value. For example, some people are much taller than average, while others are much shorter. Outliers can have a really dramatic effect when you calculate a mean value. In cases (i) and (ii) above, it would probably be best to ignore them in your calculation. In case (iii) though, you shouldn’t really just ignore them — they’re as genuine as any other value. Example
1
This data set shows the times in seconds that it took for a ball to fall through a tube in a science experiment: {2.1, 2.3, 2.5, 2.5, 2.6, 7} Identify the outlier. Then calculate the mean with and without the outlier, and suggest which would be better to use as a typical value. Solution
The outlier is 7 seconds. It’s much bigger than the others. The mean with the outlier is: (2.1 + 2.3 + 2.5 + 2.5 + 2.6 + 7) ÷ 6 = 19 ÷ 3 = 3.2 seconds The mean without the outlier is: (2.1 + 2.3 + 2.5 + 2.5 + 2.6) ÷ 5 = 12 ÷ 5 = 2.4 seconds Check it out: Always try to find out the possible reason for an outlier before you ignore it.
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Here, the measurement of 7 seconds probably happened because there was either: a problem carrying out the experiment (maybe the ball got stuck), or a problem with the measurement (maybe the stopwatch was started early accidentally). So the mean of 2.4 s is probably more appropriate.
Section 5.1 — Statistical Measures
Example
2
This data set shows the weights in pounds (lb) of six dogs brought into a veterinary practice: {8, 22, 25, 33, 36, 42}. (All the weights are genuine.) Identify the outlier. Then calculate the mean with and without the outlier, and suggest which would be better to use. Check it out: The very low weight might be from a different breed, and so perfectly normal.
Solution
The outlier is 8 lb. It’s much smaller than the others. The mean with the outlier is: 166 ÷ 6 = 27.7 lb The mean without the outlier is: 158 ÷ 5 = 31.6 lb Here, it’s probably best to include the outlier. It’s a small data value, but it’s a genuine weight of one of the dogs. So the mean of 27.7 lb is probably more appropriate.
Guided Practice 1. In which set or sets from A to D might 99 be considered an outlier? A: {0, –3, 8, 99} C: {88, 100, 99, 104, 89, 91} B: {99, 106.4, 99.8, 99.95, 101.6} D: {413, 99, 526, 480, 475} Use the following data set for Exercises 2–5: {12, 15, 19, 20, 200} 2. Find the mean of the data set with the outlier included. 3. Find the mean of the data set without the outlier included. Say which of these answers is appropriate to use if the data shows: 4. the correctly measured lengths of different sea snakes, which are to be used by the makers of snake antivenom. 5. the distances a javelin was thrown by members of a sports club, though the final value was the result of a person typing the result into a computer incorrectly.
Choose the Best Measure for the Situation You’ve seen that outliers can have a big effect on the mean. Sometimes, outliers can make the mean unrepresentative of the rest of the data set. Example
3
A doctor sees 15 patients during the morning. The length of time he spent with each patient is recorded below to the nearest minute. {5, 7, 3, 3, 78, 11, 10, 8, 6, 3, 3, 8, 9, 8, 11} Find the mean. Suggest a problem with using this as a typical time. Solution
Check it out: If a few values are very different from the majority, they’ll often cause the mean value to not be representative.
Mean = Sum of all 15 times ÷ 15 = 173 ÷ 15 = 11.5 minutes. The mean is greater than all but one of the values in the data set. The very high value of 78 “pulls” the mean upward, so it is not very typical of the values in the data set. Section 5.1 — Statistical Measures
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Check it out: There’s more about how the mean is affected by extreme values in Lesson 5.2.2.
This is a problem with using the mean as a typical value — it’s heavily affected by extreme values. One solution is to use a different measure of central tendency. Example
4
Using the same data set as in Example 3, find the median and mode. Choose the best measure(s) of central tendency for this data. Solution
The data set is: {5, 7, 3, 3, 78, 11, 10, 8, 6, 3, 3, 8, 9, 8, 11} Mode: The number 3 occurs more than any other number, so the mode = 3 minutes. Median: the values written in order are: 3, 3, 3, 3, 5, 6, 7, 8, 8, 8, 9, 10, 11, 11, and 78. The median is the middle value, so the median = 8 minutes. The mode is equal to the lowest value, and you’ve already seen that the mean is higher than all but one value, so neither of these is very typical of the data set. The median is the most representative measure of central tendency.
Guided Practice For each data set below, find (where possible) the mean, median, and mode. Then say which would be the most suitable measure of central tendency to use, and explain why. 6. Math test scores: {75, 84, 88, 72, 64, 10, 92, 87}. The outlier was scored by someone new to the town, who had not been to a lesson in this school before. 7. Weekly allowance: {$10, $15, $10, $10, $7, $50}. This data is to be used to see how many weeks it will take a group of students to raise enough money to throw a birthday party. Check it out: If you can’t put the data into any sensible kind of order, the mode is the only measure of central tendency that’s possible.
Now try these: Lesson 5.1.3 additional questions — p452
Independent Practice 1. In which data set(s) below might –4 be considered an outlier? W: {0, –6, 2, 12, –3, –8, 11} Y: {–3.8, –4.2, –3.3, –5.1, –2.7} X: {29.4, 33.7, 39.4, 28.9, 31.1} Z: {3.8, 4.2, 3.3, 5.1, 2.7} For each data set below, find (where possible) the mean, median, and mode. Then say which would be the most suitable measure of central tendency to use, and explain why. 2. Shoe sizes: {5, 7, 6, 7, 5, 5, 6, 6, 7} 3. Age of party guests: {14, 18, 17, 13, 14, 11, 14, 14, 73} 4. Pets owned: {dog, rabbit, dog, cat, gerbil, dog, cat, dog}
Round Up In some cases, you can just ignore outliers — they’re sometimes just “mistakes.” But you shouldn’t just ignore outliers because they’re inconvenient — that’s “fiddling your data.” You can always use the median or the mode if you want a typical value that isn’t heavily affected by extreme values. 264
Section 5.1 — Statistical Measures
Lesson
5.1.4
Comparing Data Sets
California Standard:
You’ve seen how to find the mean, median, mode, and range of a data set. The next step is to see how you can use them to compare data sets.
Statistics, Data Analysis, and Probability 1.1 Compute the range, mean, median, and mode of data sets.
What it means for you:
Two Key Concepts: “Typical Value” and “Spread”
You’ll see how you can use the mean, median, mode, and range to compare data sets.
With any data set, there are two basic things you’ll really need to know: 1) a “typical value,” 2) how spread out the values are.
Key words:
You’ve seen that you can find a typical value in different ways — you can use either the mean, median, or mode.
• • • • • •
measure of central tendency mean mode median spread range
You’ve also seen that you can describe how spread out your data is, using the range. Example
1
Give a description of the data set below. {4, 6, 3, 4, 5, 2, 9, 8, 2, 6, 6} Solution
There are 3 types of typical values — the mean, the median, and the mode. Don’t forget: There’s a lot more about how to work out the mean, median, mode, and range in Lessons 5.1.1 and 5.1.2.
Check it out: The spread of a data set means how spread out (or how close together) all the values are.
4 + 6 + ... + 6 + 6 55 = =5 11 11 Median... put the data in order first: 2, 2, 3, 4, 4, 5, 6, 6, 6, 8, 9. There are 11 values, so the median is the 6th value, which is 5. Mode = 6
Mean =
Then you can describe how spread out your data is by looking at the range. Here, the range = 9 – 2 = 7.
Guided Practice In Exercises 1–5, give a summary of each data set. 1. {8, 4, 7, 4, 5, 4} 2. {21, 11, 22, 14, 17, 13} 3. {–12, –15, –15, –11, –9, –10} 4. {1, 5, 2, 7, 3, 7, 4, 5, 9, 10, 14, 3, 2, 5, 13} 5. {–5, 5, 2, 7, 3, 7, 4, 5, 9, 10, 20, 3, 2, 5, 13} 6. What do you notice about the sets in Exercises 4 and 5?
Section 5.1 — Statistical Measures
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Compare Data Sets Using Typical Value and Spread If you have two sets of data, you may want to compare them. For each set, work out a “typical value,” and how spread out the data is. Example
2
Alfredo and Joel are both on a long jump team. During the course of the season, they each do a series of jumps. Alfredo’s jumps (in meters): 5.1, 5.42, 6.01, 4.46, 5.33, 5.12, 4.42, 6.02. Joel’s jumps (in meters): 5.23, 5.15, 5.3, 5.33, 5.22, 5.18, 5.3. Compare their performances over the course of the season. Solution
Work out the mean, median, mode, and range for both jumpers.
Don’t forget: The mean, median, and range are all different kinds of “typical value.” The range is a measure of spread.
Alfredo first: 5.1 + ... + 6.02 41.88 = = 5.235 meters Mean = 8 8 Median... put the data in order first: 4.42, 4.46, 5.1, 5.12, 5.33, 5.42, 6.01, 6.02 So the median is 5.225 meters. All the values are different, so there is no mode. Range = 6.02 – 4.42 = 1.6 meters. Now Joel: 5.23 + ... + 5.3 36.71 = = 5.244 meters (to 3 decimal places) Mean = 7 7 Median... put the data in order first: 5.15, 5.18, 5.22, 5.23, 5.3, 5.3, 5.33 The median is 5.23 meters. The mode = 5.3 meters. Range = 5.33 – 5.15 = 0.18 meters.
Don’t forget: A large range might mean that all the values are very spread out. Or it might mean that there are one or two outliers. Look at the actual values to check.
You can see that for both jumpers, the mean and the median are very similar. Both have a “typical jump” of about 5.22–5.24 meters. However, the big difference is in the boys’ ranges. Alfredo’s range is much greater than Joel’s. This means that Joel is a lot more consistent than Alfredo, or Alfredo’s jumps are a lot more variable than Joel’s.
Guided Practice Compare the pairs of data sets in Exercises 7–8. 7. {6, 8, 9, 5, 7, 8, 5} and {1, 4, 7, 9, 3, 4} 8. {7, 4, 5, 7, 2, 8} and {28, 34, 33, 29, 33, 34} 9. Sam and Eddy each have a collection of earthworms. Sam’s worms have lengths (in cm) of 5.2, 8.3, 4.4, 6.7, 9.2, and 10.1. Eddy’s worms have lengths (in cm) of 4.6, 6.3, 5.5, 6.6, 6.9, and 7.1. Compare the two collections. 266
Section 5.1 — Statistical Measures
Example
3
Ryan and Hap scored the following points in 10 games of basketball. Ryan: 10, 5, 6, 14, 3, 2, 8, 4, 16, 6 Hap: 8, 7, 8, 10, 6, 8, 9, 4, 10, 8 (i) If the team is behind by 8 points, why might the coach put Hap in? (ii) If the team is behind by 14 points, why might the coach put Ryan in? Solution
Compare the two sets of data. Ryan: mean = 7.4; median = 6; mode = 6; range = 14 Hap: mean = 7.8; median = 8; mode = 8; range = 6
Don’t forget: A small range shows consistency — with no really big or really small values. A bigger range shows there are some much bigger or some much smaller values.
(i) Hap has a median of 8, meaning he has scored 8 in at least half the games he has played. Ryan’s median is 6, which means he scored 6 points or fewer in half of his games. (ii) Hap’s range is much lower, meaning he is more consistent. But his highest score is only 10, showing that he may not be able to score really highly in a single game. Ryan, however, has a range of 14, showing that he has scored at least 14 points in a game before.
Guided Practice One-day first aid courses are taught on Wednesdays and Saturdays. The age of the participants in each course is summarized below. Wednesday class: mean = 39.75; median = 36; mode = 28; range = 42 Saturday class: mean = 23.25; median = 21.5; mode = 13; range = 30 10. Students in school are unable to attend one of the classes. Using the statistics, which would you say students cannot attend? 11. What are the maximum possible ages of the people in each class?
Independent Practice
Now try these: Lesson 5.1.4 additional questions — p452
Give a summary of each data set in Exercises 1–3. 1. {56, 78, 45, 66, 49, 71} 2. {111, 124, 109, 105, 122, 106, 99, 96} 3. {2.31, 2.43, 2.54, 2.12, 2.54, 2.33, 2.65} Compare the data sets in Exercises 4–5. 4. {82, 63, 71, 78, 58} and {152, 133, 141, 148, 128} 5. {1.2, 1.7, 1.4, 1.6} and {0.7, 2.2, 0.9, 2.1} 6. A research company reported that the typical teacher in the U.S. is a 43-year-old female. What measure of central tendency might the company have used to determine that the typical teacher was female? Does this mean there are no male teachers? Explain your answers.
Round Up The mean, median, mode, and range are used all the time in real life to summarize the results of surveys. It’s important you understand what each piece of information tells you. Section 5.1 — Statistical Measures
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Lesson
5.2.1 California Standards: Statistics, Data Analysis, and Probability 1.1 Compute the range, mean, median, and mode of data sets.
Section 5.2
Including Additional Data: Mode, Median, and Range The mode, median, mean, and range of a data set can change as new data is added. Adding just one extra number to the data set might (or might not) be enough to change any of them. This Lesson looks at how including new data can affect the mode, median, and range.
Statistics, Data Analysis, and Probability 1.2
Adding Data Values Can Change the Mode
Understand how additional data added to data sets may affect these computations.
The mode of a data set is the item that occurs most often. New data can cause the most common value to change.
What it means for you:
Example
1
You’ll see how the mode, median, and range of a data set change if extra values are included.
Joe has conducted a survey asking the number of TVs in people’s households. The data set is shown below. {0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 5, 7}
Key words:
The next day, three more completed surveys arrive in the mail. The new responses are {1, 3, 3}.
• • • •
data set mode median range
Find the mode of the data set: (i) not including the extra responses, (ii) including the extra responses. What effect do the extra responses have on the mode?
Check it out:
Solution
If you had a huge data set, containing hundreds and hundreds of values, then it would be far less likely that an extra three responses would change the mode.
(i) The mode of the original data set is 2 (there are eight values of 2). (ii) The mode of the new data set is 3 (there are now nine values of 3, but still only eight of 2). Here, the extra data increased the mode from 2 to 3. If the three extra responses in Example 1 had been different, the mode may not have been affected. For example, the responses {0, 2, 5} would not have changed the mode.
Guided Practice Don’t forget: The mode is sometimes called the modal value.
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1. If 9 is added to the data set {1, 5, 5, 5, 7, 9, 9, 10, 15}, how is the mode affected? The number of actors needed for six short plays are 4, 6, 4, 5, 6, and 6. 2. What is the modal (most common) number of actors? 3. Three more plays are added to the list. The numbers of actors needed are 4, 4, and 5. What is the modal number of actors now? 4. The data set is now {4, 6, 4, 5, 6, 6, 4, 4, 5}. Could a single number be added that would change the mode? Explain your answer.
Section 5.2 — Adding Extra Data
Check it out:
Adding Data Values Can Also Change the Median
If there are an odd number of values in the data set, the median is the actual value in the middle of the ordered data set. If there are an even number of values, then the median is midway between the middle two values. See Lesson 5.1.1 for more information.
The median is the middle value when all the values of a data set are arranged in order. Adding a new value can easily change the median. Example
2
The heights of four friends are 156 cm, 157 cm, 164 cm, and 168 cm. Another friend joins them. His height is 207 cm. How does adding this friend’s height affect the median? Solution
First, you need to find the original median. So arrange the original data set in order. Original data set: {156, 157, 164, 168} Old median = 160.5 cm
Check it out: The extra height is greater than the old median, so the new median is greater than the old one.
The original median is halfway between 157 cm and 164 cm (the 2nd and 3rd values).
Now insert the additional data value into the set, keeping the heights in order. Next, find the new median and compare it to the original. new value
New data set: {156, 157, 164, 168, 207} Check it out: Notice how the increase in the median is only a few centimeters, even though the new friend is much taller than the other friends.
New median = 164 cm
The median is now the 3rd value, 164 cm.
The median increased from 160.5 cm to 164 cm. Example
3
Another friend joins the friends from Example 2. Her height is 146 cm. How is the median affected now? Solution
Check it out:
Now the ordered data set is: {146, 156, 157, 164, 166, 168}.
This time, the new height is less than the previous median of 164 cm, so the median has fallen. Actually, the median has returned to its original value of 160.5 cm. This is because the two new heights were added to “either end” of the data set, meaning the middle value has remained unchanged.
There are 6 values, so the new median is midway between the 3rd and 4th values. The new median is 160.5 cm.
Guided Practice 5. What is the median of the data set {8, 3, 4, 9, 12, 14}? 6. If a value of 1 is added to the data set in Exercise 5, how is the median affected? 7. Choose another number that could have been added for the median to have changed as you described in your answer to Exercise 6? 8. You add two new values to a data set — one less than the old median and one greater than the old median. What happens to the median? Section 5.2 — Adding Extra Data
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Check it out:
New Data Values Can Increase the Range
Notice that adding new data points can only increase the range — never decrease it. The only way to decrease the range would be to remove the greatest or least data point.
The range is the difference between the greatest and least values. If a new data point is either bigger than the greatest value or smaller than the least value, then the range will increase.
Check it out: You can add as many values as you like without changing the range as long as the values are no less than the original least value and no greater than the original greatest value.
Example
4
Would the range of the data set {2, –5, 4, 9, 10, 12} change if: (i) the value 15 were added? (ii) the value 7 were added? Solution
The current range is 12 – (–5) = 17. (i) 15 is greater than the current greatest value, so the range will increase. In fact, the new range will be 15 – (–5) = 20. (ii) 7 is not greater than the current greatest value or less than the current least value, so the range will not change.
Example
5
In a race, the first runner completes the course in 3 hours 10 minutes. The cut-off time, after which no further times are recorded, is 5 hours. Several racers finish at 5 hours and have their times recorded. Find the range of recorded race times. How would the range be affected if there were no cut-off time and other, slower, racers had their finish times recorded? Solution
The range is: 5 hrs – 3 hrs 10 mins = 1 hr 50 mins If there were no cut-off time, and the slowest recorded time was greater than 5 hours, then the range would increase.
The range can also be affected if data sets are combined. Example
6
The results of a shorter race for under 20s are shown below. Age group
Check it out: In Example 6, even though both original ranges were 7 minutes, the range increased to 11 minutes when the data was combined.
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Race times (minutes)
Range
Age 10–14
45
43
50
50 – 43 = 7 minutes
Age 15–19
39
46
—
46 – 39 = 7 minutes
As there were so few runners, race officials decide to combine the runners into one category for the awards. How does the range change? Solution
The new data set is: {45, 43, 50, 39, 46} So the new range = 50 – 39 = 11 minutes.
Section 5.2 — Adding Extra Data
Guided Practice For each set in Exercises 9–12, decide whether adding the value 5 to the set will change the range. If the range does change, explain how. 9. Set A {0, 2, 2, 7} 10. Set B {13, 7, 9} 11. Set C {–6, 2, 5, 3} 12. Set D {11, 9, 14, 18, 9} 13. One data value is added to the set {48, 51, 68, 73} and it doubles the range. Name the two possible values that were added.
Independent Practice For each set in Exercises 1–4, decide whether adding the value 16 will change the mode. If the mode does change, explain how. 1. Set A {4, 10, 16, 11} 2. Set B {400, 30, 30, 16} 3. Set C {6.3, 1.4, 15.6, 12.7, 12} 4. Set D {7, 9, 7, 7, 16} In Exercises 5–8, use the data set {15, 17, 17, 15, 15, 19}. 5. Find the mode of the data set. 6. Name a data value that could be added to the set so that the mode remains the same. 7. How many data values must be added to change the mode to 17? Name the values. 8. What is the least number of data values that must be added to change the mode to 19? Name the values. Determine whether the statements in Exercises 9–12 are true or false. 9. Adding a data value to a set of data will always change the mode. 10. For the mode to remain the same, you must add a data value that is equal to the mode. 11. Adding a value to a set may cause that set to have an extra mode. 12. Adding a value bigger than the greatest value of a data set will change the range. Now try these: Lesson 5.2.1 additional questions — p452
Don’t forget: Values in data sets don’t always have to be integers.
The ages of children on two different trips to the beach are listed below. Trip 1 ages: {7, 7, 8, 10, 10, 10, 11} Trip 2 ages: {7, 7, 9, 11, 12, 12, 12, 13, 13} 13. Find the mode and range for each group. 14. If the two trip groups were combined, what would the new mode and range be? Use the data set {6, 7, 19} for Exercises 15–16. 15. Add the value 1 to the set. How does this change the median? 16. Add the value 7 to the original set. How does the median change? 17. Give an example of a data set with a range of 15 that changes to 21 when three data values are added. Show the original set and the new set that includes three added values.
Round Up Adding numbers to data sets might (or might not) change the mode, median, or range — it depends on what you add. It’s safest to find the measures from scratch using the new data set. Section 5.2 — Adding Extra Data
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Lesson
5.2.2 California Standards: Statistics, Data Analysis, and Probability 1.1 Compute the range, mean, median, and mode of data sets. Statistics, Data Analysis, and Probability 1.2 Understand how additional data added to data sets may affect these computations.
What it means for you: You’ll see how the mean of a data set changes if extra data is included.
Key words: • data set • mean
Including Additional Data: The Mean Last Lesson you saw how the mode, median, and range of a data set changed if new values were added. The next thing to think about is what happens to the mean if you add in extra values to your data set.
Adding Data Points Can Change the Mean Remember... you find the mean by adding all the values of a data set together, and dividing this total by the number of values you have. Adding extra values to a data set usually affects the mean. Example
1
The heights of four friends are 156 cm, 157 cm, 164 cm, and 168 cm. What is their mean height? Another friend who is 158 cm tall joins them. How does his height affect the mean height? Solution
First calculate the original mean: Original mean = (156 + 157 + 164 + 168) ÷ 4 = 645 ÷ 4 = 161.25 cm Don’t forget: The mean is often called the “average.”
Now include the extra height of 158 cm, and calculate the new mean. New mean = (645 + 158) ÷ 5 = 803 ÷ 5 = 160.6 cm You can add the new value to the original sum.
Remember to divide by 5, since there are now 5 heights in the data set.
Adding the new height decreased the mean. This is because the extra height was less than the old mean. New data values will usually affect the mean, but not always. Example
2
Find the mean of the data set {8, 17, 36, 15}. How is the mean affected if an extra value of 19 is added? Solution
The mean of the data set is:
8 + 17 + 36 + 15 76 = = 19 4 4
Now add the new value of 19 and find the new mean. This is:
76 + 19 = 19 5
This time, adding the extra value didn’t affect the mean because the new value was equal to the old mean.
272
Section 5.2 — Adding Extra Data
When you combine data sets, find the new mean carefully. Example Don’t forget: You could find the new mean by adding the old totals, and then dividing by 6: (52 + 36) ÷ 6 = 88 ÷ 6 = 14.67
Check it out: Calculate the new mean properly — by adding up all the values, and dividing by how many there are. Don’t assume the new mean is midway between the two old means.
3
Find the mean of the data set {5, 4, 18, 25}, and the mean of the data set {24, 12}. What is the mean of the combined data set {5, 4, 18, 25, 24, 12}? Solution
5 + 4 + 18 + 25 52 = = 13 4 4 24 + 12 36 The mean of {24, 12} is: = = 18 2 2 Now combine the data sets and find the new mean.
The mean of {5, 4, 18, 25} is:
This is:
5 + 4 + 18 + 25 + 24 + 12 88 = = 14.67 (to 2 decimal places) 6 6
Guided Practice The weights of four friends are 132 lb, 155 lb, 147 lb, and 151 lb. 1. Find the mean weight of the four friends. 2. Find the new mean if a friend weighing 140 lb joins the group. A data set has a mean of 109.2. A new data value is then added. State whether the mean of the data set would increase or decrease if the new data value is: 3. 119 4. 96.7 5. 115 6. 107 7. Find the mean of the data set {125, 165, 133, 129, 100}. 8. This data set is combined with the set {43, 44, 27}. Find the mean of the combined set.
Different Values Affect the Mean to Different Extents Adding data points near the original mean won’t change the mean too much. But adding values that are further away has more effect. Check it out: Notice that when 40 is added, the new mean becomes higher than any of the data set’s original values. The mean is no longer very representative of the data set.
Example
4
How is the mean of the data set {12, 12, 15} affected if you include i) the number 14? ii) the number 40? Solution
The mean of the original data set is (12 + 12 + 15) ÷ 3 = 39 ÷ 3 = 13 Don’t forget: The extra value 40 is an outlier for this data set. Outliers can have a drastic effect on the mean (see Lesson 5.1.3).
i) The new mean = (39 + 14) ÷ 4 = 53 ÷ 4 = 13.25 The mean isn’t changed much, since 14 is close to the old mean. ii) This time, the new mean = (39 + 40) ÷ 4 = 79 ÷ 4 = 19.75 The mean is quite different this time. This is because the extra value is so far away from the old mean (it’s an outlier).
Section 5.2 — Adding Extra Data
273
Means Found Using Many Values Are Affected Less Something else also affects how much a new value will change a set’s mean — the number of values used to work out the mean originally. Example
5
Find the mean of each of the following data sets. How is the mean of these sets affected if an extra value of 13 is added to each one? (i) {7, 9, 13, 11}, (ii) {7, 9, 12, 8, 9, 13, 12, 10, 8, 12, 6, 14, 11, 9}.
Solution
7 + 9 + 13 + 11 40 = = 10 4 4 53 = 10.6 If the value 13 is added, the new mean is: 5 So adding the new value increases the mean by 0.6.
(i) The mean of the first set is:
Check it out: The more values that are used to calculate a mean, the less it will be affected by a single new value.
7 + 9 + ... + 11 + 9 140 = = 10 14 4 153 = 10.2 If the value 13 is added, the new mean is: 15 So this time, adding the new value increases the mean by just 0.2.
(ii) The mean of the second set is:
Guided Practice Use the data set {6, 7, 19} for Exercises 9–10. 9. Find the mean of the original set. 10. Adding which value would affect the mean most: 7, 10, or 11? 11. Leonora asked 3 people how many pets they own. The mean was 2. Esteban did a similar survey of number of pets owned by 40 people, and also found that the mean number was 2. Whose result would be affected more if a person owning 18 pets was included in their data?
Independent Practice
Now try these: Lesson 5.2.2 additional questions — p453
The masses in grams of eight mice were recorded as 22, 25, 24, 22, 26, 25, 21, and 25. 1. What is the mean mass of the mice? 2. A ninth mouse, with a mass of 45 grams, was included in the data set. What is the new mean? 3. A researcher finds data for 100,000 mice from all over the country. The mean was 23 grams. How would including the 45-gram mouse affect this mean?
Round Up Adding values to data sets will usually change the mean. The more extreme the new value is, and the fewer values used to find the mean in the first place, the greater the effect. 274
Section 5.2 — Adding Extra Data
Lesson
Section 5.3
5.3.1
Analyzing Graphs
California Standards:
When you have lots of data in a long list, it’s not always easy to figure out what it means. That’s why it’s often converted into a graph or chart. These can show you what the spread of the data is like, and which value is most common.
Statistics, Data Analysis, and Probability 2.3 Analyze data displays and explain why the way in which the question was asked might have influenced the results obtained and why the way in which the results were displayed might have influenced the conclusions reached.
Data Can Be Recorded on a Frequency Table Frequency is the number of times something happens. A frequency table is a way to record and organize data. This frequency table shows the results of a survey. A group of students was asked how many movies they each saw last month. Number of movies seen
What it means for you: You’ll meet some useful types of graphs. You’ll see how to take information from those graphs, including finding the mode and range of the data set shown.
Key words: • • • • • • •
data set frequency table bar graph line plot pictograph mode range
Check it out: Frequency tables are sometimes called tally charts.
Check it out: Bar graphs are sometimes drawn the other way around, with bars going across, and the scale along the bottom.
As each student is questioned, one tally mark is added in the row that matches their answer.
Tally
Frequency
None
III I
2
One
IIII IIII
9
Two
IIII
4
Three
I
1
Four
IIII
5
Five S ix
er The t otal numb number of students who gave each answer is added up when the survey is complete. This total is the fr equen frequen equencc y .
0
I
1
You can use graphs and charts to show frequency data. On a bar graph, each bar has a label which tells you what group that bar represents. The size of a bar tells you how large that group is. 10 9 8
ale shows that the The sc scale height of the bars represents er of students the numb number students.
Number of students
Statistics, Data Analysis, and Probability 1.1 Compute the range, mean, median, and mode of data sets.
1
7 6 5 4 3 2
2
None The height of the bar labeled “None None” is 2 . vies last month. So 2 students saw no mo movies
3
1 0
ne
No
o e ee our Five Tw Thr On F Number of movies seen
Six
0 1 2 3 4 5 6 7 8
Section 5.3 — Data Displays
275
A line plot uses crosses marked above a number line to represent the data.
X X X X X X X X X X X
For any number on the number line, each X above it represents one student in the survey who saw that number of movies.
ne
No
e
On
5 students saw four m oovies vies vies, so there are 5 Xs above four on the line plot.
X X X X X ree
o Tw Th
X X X X X r
u Fo
X e Fiv
Six
Number of movies seen
A pictograph uses pictures or symbols to show the data.
Each symbol on this pictograph students. represents 2 students
None Number of movies seen
On a pictograph, each symbol often has a value of more than one unit. You need to look at the key before you can read the graph.
One
Only 1 student saw three movies. ol One half-symb half-symbol is used to show 1 student in that row.
Two Three Four Five Six Key:
= 2 students
Guided Practice
Wed Thu Fri Key:
= 10 pumpkins
In a survey, some students were asked which of six colors they like best. This bar graph shows the results. 7. Which color was the favorite of 3 students? 8. Which two colors were the least popular? 9. How many students said their favorite color was either red, yellow, or blue? 276
Section 5.3 — Data Displays
8 7 6 5 4 3 2 1 0 d ra ng e Ye llo w G re en Bl ue Pu rp le
Tue
This pictograph shows the number of pumpkins sold by a grocery store during five days in November. 4. When were the most pumpkins sold? 5. How many pumpkins were sold on Wednesday? 6. On which day were exactly 30 pumpkins sold?
Re
Mon
O
In Exercises 1–3, remember... the number of goals scored is the number underneath the number line. The amount of crosses above each number represents the number of matches in which that many goals were scored.
This line plot shows the number of goals scored in the soccer matches played by a school team this year. X 1. In how many matches did the team X score 0 goals? X X 2. In how many matches did the team X X X score 2 or more goals? X X X X X 3. How many matches did the team play 0 1 2 3 4 5 this year? Number of goals scored
Number of students
Check it out:
A Graph Can Tell You the Range and Mode The mode of a data set is the value that occurs most often. Don’t forget: For a reminder about the mode of a data set, see Lesson 5.1.1. For the range, see Lesson 5.1.2.
So if a data set is made into a line plot, bar graph, or pictograph, the mode is the value with the most crosses, the biggest bar, or the most symbols. Example
1
What is the mode of the data set shown on this line plot?
X X X X X X X X
Solution
X X X X X X X X X X X
The mode is the value on the line plot that has the greatest number of Xs above it. The number 19 has the most Xs above it. So the mode is 19.
15 16 17 18 19 20
The range of a data set is the difference between the greatest and least values in the set. So it’s easier to find the range from a graph than from the raw data set because you can clearly see the least and greatest values. Example
2 6
What is the range of the data set shown on this bar graph?
To work out the range, you need to look at the highest and lowest of the actual values — not how often they occurred.
4 3
Solution
2
The different values in the data set are the labels on the bars of the graph.
1 0
25
26
27
28
29
30
31
So the least value in the data set is 25, and the greatest is 31. 31 – 25 = 6, so the range = 6.
Guided Practice This pictograph shows the ages of the cars parked in a parking lot, to the nearest year. 10. What is the mode of the ages of the cars? 11. Find the range of the ages of the cars.
10 9 8
Age of car in years
Don’t forget:
5
7 6 5 4 3 2 1 0
Key:
= 2 cars
Section 5.3 — Data Displays
277
Number of students
The graph below shows the points out of 20 that the students in one class scored on a 8 7 math quiz. 12. What was the mode for the quiz scores? 13. What is the range of quiz scores?
Don’t forget: Make sure you read from the correct scale. For example, Exercises 12 and 13 are asking for actual scores, not the number of people who achieved that score.
6 5 4 3 2 1 0
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 Quiz score
Independent Practice
0 1 2 3 = 2 dog owners
Key:
1. How many dog owners took their dog to the vet exactly once? 2. How many dog owners were interviewed in total? 3. What is the mode of the data? 4. What is the range?
The bar graph on the right shows the number of lunches sold each day by a restaurant. 5. Find the mode of the data. 6. Find the range of the data. 7. For how many days was this data collected?
Number of days
Number of visits
A number of dog owners were asked how many times they had taken their dog to the veterinarian in the past month. The results are shown on the pictograph below.
6 5 4 3 2 1 0
30
31
32
33
34
35
36
Number of lunches sold
Which of the line plots below have: 8. a range of 8? 9. a mode of 2? X X X X X X
A Now try these: Lesson 5.3.1 additional questions — p453
1
C
2
X X
3
X X X X X X X X X X X X X 4
5
X X
6
7
8
X X
22
9 10
X
0 1 2 3 4 5 6 7 8 9 10 11 12
X X X X X
X
X X X
24
26
X X X X 28
X
B
32
30
X X
X X
D
30 31 32 33 34 35 36 37 38 39 40
10. Explain the advantages of looking at data on a line plot instead of in a data set.
Round Up You should be able to see how useful these graphs can be. They make it much easier to spot the mode and find the range of a data set. Not only that, they organize the data. As you’ll see in the next Lesson, that can help you to find the mean and the median of the data set too. 278
Section 5.3 — Data Displays
Lesson
5.3.2
California Standards:
Finding the Mean and Median fr om Gr a phs from Gra
Sta tistics ta Anal ysis Statistics tistics,, Da Data Analysis ysis,, and Pr oba bility 1.1 Proba obability
In the last Lesson, you saw how to find the mode and the range of a data set from a line plot, bar graph, or pictograph.
Compute the range, mean, ta data median, and mode of da sets sets..
That’s not all you can do with these graphs. You can use them to find the mean and the median too.
Sta tistics ta Anal ysis Statistics tistics,, Da Data Analysis ysis,, and Pr oba bility 2.3 Proba obability
You Can Use a Gr a ph to Find the Median Gra
Anal yz e da ta displa ys and Analyz yze data displays explain why the way in which the question was asked might have influenced the results obtained and why the way in which the results were displayed might have influenced the conclusions reached.
What it means for you: You’ll see how bar graphs, line plots, and pictographs can make it easier for you to find the mean and median of data sets.
The median is the middle value when a set of values is put in order. A graph can give you the information you need to find the median of a data set. Example
1
What is the median of the data set shown on this line plot?
X X X X X X X X
Solution
Remember that each X on the line plot represents one item in the data set.
X X
9 10 11 12 13
So you can turn the plot back into a set of data: Key words: • • • • • •
data set bar graph line plot pictograph mean median
{9, 10, 10, 10, 10, 11, 11, 11, 13, 13} There are ten values in the set, so the median is midway between the fifth and sixth values. The fifth and sixth values are 10 and 11, so the median is 10.5.
Guided Practice Don’t forget:
In Exercises 1–4, find the median of the data set shown on each graph.
For more about the median of a data set, see Lesson 5.1.1.
1.
X X X X X X X X X X X X X
2.
6 5 4 3 2 1
19 20
3.
21 22 23
0
4.
4 5 6 7
Key:
= 2 units
5
6
7
X X X X X X X X X X 0
1
2
8
9
X X X X X X X X X X 3
4
5
6
7
Section 5.3 — Data Displays
279
You Can Use a Graph to Find the Mean
Don’t forget: See Lesson 5.1.2 for more about the mean of a data set.
The mean of a data set is the sum of all the values, divided by the number of items in the data set. There are a couple of ways that you can use a graph to find the mean of a data set. Example
2
X X X X X X X X
What is the mean of the data set shown on this line plot? Check it out: Some graphs organize the data into intervals of values. So a bar graph might have bars representing 1–5, 6–10, 11–15, and so on.
This is the same graph as in Example 1. So the data set is:
X X
9 10 11 12 13
{9, 10, 10, 10, 10, 11, 11, 11, 13, 13} The mean is the sum of these values, divided by the number of values.
4 3
(9 + 10 + 10 + 10 + 10 + 11 + 11 + 11 + 13 + 13) ÷ 10 = 108 ÷ 10 = 10.8
2 1 0
Solution
1–5
6–10 11–15 16–20
You can’t tell from these graphs exactly what the values in the original data set were, so you can’t use them to find the exact median or mean.
Example
3 10
What is the mean of the data set shown on this bar graph?
9 8 7
Solution
6
The scale tells you that there are 5 items of data with a value of 1, 8 items with a value of 2, and so on. You can turn this into a multiplication to find the sum of all the values in the data set.
5 4 3 2 1 0
Check it out: Finding the mean by multiplication is often much quicker than writing out the data set, especially if there are a lot of items in the data set. And it’s easy to make a mistake when you have to write out a long list of values.
The data set has: 5 items with the value 1 8 items with the value 2 5 items with the value 3 9 items with the value 4 3 items with the value 5 30
5×1 8×2 5×3 9×4 3×5
1
2
= = = = =
5 16 15 36 15 87
The sum of the values is 87, and there are 30 values. So the mean is 87 ÷ 30 = 2.9.
280
Section 5.3 — Data Displays
3
4
5
Guided Practice 5. Find the mean of each of the data sets shown on the graphs in Exercises 1–4. In Exercises 6–7, find the mean of the data set shown on each graph. 6.
7.
28
41
29
42
30
43
31
44
32
45 0
1
3
2
5
4
6
Key:
= 2 units
Independent Practice Which of the line plots below have: 1. a median of 32.5? 2. a mean less than 12? X X X X X X X X X X X X X X X X X X X
A 1
C
2
X X
3
4
5
X X
6
7
8
22
9 10
X X
X X X X X
X
0 1 2 3 4 5 6 7 8 9 10 11 12
X
X X X
24
26
X X X X 28
X
B
32
30
X X
X X
D
30 31 32 33 34 35 36 37 38 39 40
All the classes in an elementary school have from 25 to 31 students. This pictograph shows the class sizes.
25 26 27 3. How many classes are there in this school? 28 4. Find the median and mean number of students 29 in each class. 30 31 Key:
Now try these: Lesson 5.3.2 additional questions — p454
= 2 classes
5. Which one of the graphs below represents a data set that has the same mean, median, and mode? A
6 5 4 3 2 1 0
5
B4
C
3 2 1 0 95 96 97 98 99 100 101 102 103 104
4 3 2 1 0
15 16 17 18 19 20
39 40 41 42
Round Up The types of graphs you’ve used to find the mean, median, mode, and range of a data set are all similar types of graphs. Bar graphs, line plots, and pictographs all show the same sort of information. There are other types of graphs that can show data in different ways. You’ll see a couple of these in the next Lesson. Section 5.3 — Data Displays
281
Lesson
5.3.3
Other Types of Graphs
California Standards:
You’ve now seen how to get information from line plots, bar graphs, and pictographs. In this Lesson, you’ll learn about a couple of other types of graphs that can show information in different ways.
Statistics, Data Analysis, and Probability 2.3 Analyze data displays and explain why the way in which the question was asked might have influenced the results obtained and why the way in which the results were displayed might have influenced the conclusions reached.
Circle Graphs Show Proportions Circle graphs divide data into different groups or categories. The whole circle represents all the data in a data set. Circle graphs can show how different groups compare to each other, but they don’t always give detailed numerical information.
What it means for you:
Example
You’ll see some of the information you can get from circle graphs and line graphs, and how these types of graphs can help you to compare two data sets.
1
Three students ran in the election for class president. This circle graph shows what proportion of the votes each candidate received.
Cesar Taisha
List the three candidates in order of the number of votes they received, from most to least. Key words: • • • •
Solution
data set circle graph line graph compare
The circle graph doesn’t tell you how many votes, or even what percent the students got. You can still tell that Taisha got the most votes, because her part of the circle is the largest. Matt’s part of the circle is smallest, so he got the least votes.
Don’t forget:
The result of the election was:
See Lesson 3.5.4 for more about circle graphs.
Check it out:
1st — Taisha 2nd — Cesar 3rd — Matt
Guided Practice
If you didn’t know that the graphs represented the same number of students, you couldn’t do Exercises 3–5, as circle graphs show proportions. However, because the graphs for Class A and Class B both represent the same number of students, each slice represents actual numbers of students too. Think about this... 10% of 1000 students would be quite a small slice of a circle graph, but represents 100 students. But 50% of 20 students would be half a circle graph, and yet only represents 10 students.
282
Matt
Section 5.3 — Data Displays
Class A Baseball Basketball Football
The students in a 6th grade class were asked which of three sports they like best. This circle graph shows the results of the survey. 1. Which sport was most popular with Class A? 2. Which sport was least popular with Class A? Class B
The same number of students in another 6th grade class were asked the same question. This circle graph was drawn using their answers.
Baseball Basketball Football
3. Which class has more students who like baseball best? 4. Which class has more students who like basketball best? 5. Which class has more students who like football best?
Check it out: Don’t mix up line plots and line graphs. The names are similar, but the two types of graphs are very different.
Line graphs show how two measures are related. They’re often used to show how a measure changes as time passes. Example
2
This line graph shows the monthly sales of winter coats from one store over a period of six months. What happened to the sales of winter coats from October to December? What happened to the sales of winter coats from December to March?
160 140
Number of sales
Usually, one of the quantities on a line graph is time. So the line graph would show how some quantity changes as time passes.
A Line Graph Can Show Changes Over Time
120 100 80 60 40 20 0
Oct Nov Dec Jan Feb Mar
Solution
In October, 110 coats were sold. In December, 140 coats were sold. The sales of coats increased from October to December. You can tell this from the graph, because the line is sloping upward as you look from left to right. From December to March, the line on the graph slopes downward, because the sales of winter coats decreased during this period from 140 in December to 30 in March.
Guided Practice The line graph below shows the temperature recorded by a weather station at several points during the same morning.
Temperature in °C
Check it out:
12 11 10 9 8 7 6 5 4 3 2 1 0
3 a.m. 4 a.m. 5 a.m. 6 a.m. 7 a.m. 8 a.m. 9 a.m. 10 a.m. 11 a.m.
Time
6. What was the temperature at 10 a.m.? 7. At what time was the temperature 6 °C? 8. During which hours did the temperature fall? 9. What was the change in temperature between 7 a.m. and 8 a.m.? 10. Describe the overall change in temperature during the time period shown on the graph. Section 5.3 — Data Displays
283
You Can Compare Two Data Sets on One Graph With a line graph, you can show information about two data sets at once. This makes the data easier to compare. Example
3
The line graph below shows the annual sales in one store of two different products over a period of 6 years. Describe the information shown on the graph. 1600
Number of sales
1400
Product A
1200 1000 800 600 400
Product B
200 0
2001
Solution
2002
2003
2004
2005
2006
Year
At the start of the period, the sales of product A were much higher than the sales of product B. Over the 6 years, sales of product A decreased and sales of product B increased. By 2006, the sales figures for the two products were very similar. If the pattern of sales shown here continued, you would expect sales of product B to overtake sales of product A in the next few years.
You could put the same information on a line plot or pictograph, using different colors or symbols for the different products.
1600
Number of sales
You can show information about two data sets on other types of graphs. This double bar chart uses the same data as the line graph in Example 3.
1400
Product A
1200
Product B
1000 800 600 400 200 0
2001 2002 2003 2004 2005 2006
Year
Guided Practice Use the graph shown in Example 3 to answer Exercises 11–14. 11. What was the difference in sales between product A and product B in 2005? 12. Which year had the greatest difference in sales between the two products? 13. In which years were sales of product B double what they had been the year before? 14. In which year were sales of product B the same as they had been the year before? 284
Section 5.3 — Data Displays
Independent Practice 1. Which of the circle graphs below shows the same data set as the bar graph to the right? Explain your answer. A
C
B
D
This circle graph shows results of a survey recording the number of hours worked in one week by a group of volunteers. 2. How many people worked for a total of 17 hours? 16 h 3. What is the lowest number of hours worked? 12 h 4. What is the highest number of hours worked? 25 h 5. What is the range of the data set? 21 h 6. What is the mode of the data set?
18 h
Number of sales
This graph shows how many customers bought frozen yogurt from a particular store during two different weeks.
Now try these: Lesson 5.3.3 additional questions — p454
50 45 40 35 30 25 20 15 10 5 0
Week 1
Week 2
Mo
Tu
We
Th
Fr
Sa
Su
7. How many people bought frozen yogurt on Thursday in week 1? 8. How many customers bought frozen yogurt on Friday in week 2? 9. How many customers in total bought frozen yogurt during the two weeks? 10. What was the range of sales in week 1? 11. What was the range of sales in week 2? 12. In one of the two weeks, the weather started out cool, but got warmer by the end of the week. Which week do you think that was? Explain your answer.
Round Up You should now have an idea of the kind of information you can read or display using different types of graphs. Later in this Chapter, you’ll see how the way a graph is drawn can affect what people think when they see it. Section 5.3 — Data Displays
285
Section 5.4 introduction — an exploration into:
Sampling Sur vey Surv You need to find out something about the students in your school. Think of ways you could try and find out this information. You could try and ask everybody, but that might take a very long time. Another way is to ask just a small selection of people — a sample. Work in groups of four students. Your group must first decide what to find out. You must then write a survey question, such as: • “How many pets do you have?” or • “How many minutes does it take you to get to school?” Do NOT ask EVERYONE in the class (or the school) the survey question. Each person in the group will only ask a selected group of students — this is called sampling. Each person in your group must use a different sampling method. Choose from the methods below: 1
One person should ask for volunteers to participate in the survey. That person must ask the survey question to the first eight students who volunteer.
2
One person should ask the survey question to the eight students sitting closest to your group.
3
One person should ask the survey question to the person in each of the other groups whose first name comes last alphabetically.
4
One person should write the names of every student in the class on a piece of paper. That person must put all the names in a container. They must then choose eight names without looking. They must then ask the question to the eight students whose names they chose.
Do not ask members of your group.
Exercises 1. Were the results of each survey method the same? If not, describe the differences. 2. Which of the four methods best represents the answers from your class? Explain your thoughts to one of your classmates. 3. Think of some reasons why it is sometimes necessary to collect information from a small sample of people, instead of asking everyone in a school/city/country. 4. Now imagine you are going to select a sample of students from the whole school. Which sampling method would you use? Why?
Round Up Organizations everywhere always need information from large groups of people, and this is exactly how they find it — by using small samples. But not all samples are equally good — you probably saw that for yourself. This Section tells you all you need to know about sampling. a tion — Sampling Survey Explora 286 Section 5.4 Explor
Lesson
Section 5.4
5.4.1
Using Samples
California Standard:
You’ve seen lots of ways to analyze data sets, but this Section is about where that data comes from in the first place. For example, to find out how much time people in California spend watching TV, you could (in theory) ask them all. But it’s much cheaper and easier to ask just some of them instead. The difficult thing is to know who to ask.
Statistics, Data Analysis, and Probability 2.1 Compare different samples of a population with the data from the entire population and identify a situation in which it makes sense to use a sample. Statistics, Data Analysis, and Probability 2.4 Identify data that represent sampling errors and explain why the sample (and the display) might be biased.
What it means for you: You’ll learn what a sample is and why you might sometimes need to use one. You’ll also see some things that you have to watch out for when using samples.
Key words: • • • •
population sample bias claim
Data Is Sometimes Collected from a Whole Population A population is the group of people or objects you want to gather information about. It could be everyone in your class, all the cars in California, all children in the U.S.A., or maybe all the animals in the world. If a population is small, you could collect data about every person or object. For example, if you wanted to know how long each student in a class of 35 watched TV for one evening, you could ask every student in your population of 35.
Sometimes a Sample Is Used If the population is big, it would be very time consuming and expensive to examine the whole population. Sometimes the size of a population isn’t even known and it would be impossible to find every member. For example, if you wanted to know how long each student in all of California watched TV for one evening, you couldn’t ask every member of your population — you’d need to find every student in California. What you can do is use a smaller part of the population called a sample. A sample is part of a population. The idea is that the sample will give you the same sort of information that the entire population would have given if you’d asked.
Check it out: To survey the entire population, the bank would need to wait until every single customer had used an ATM machine — which is likely to be a long time. Also, there would be so much data produced that it could be expensive to analyze.
Example
1
A bank would like to find out what new services its 20 million customers would like it to offer. The bank programs its ATM machines one Saturday morning with a questionnaire customers must complete before they can get any money. Identify the population of this survey and the sample used. Solution
The population is all 20 million of the bank’s customers — these are the people the bank wants to know about. The sample consists of the customers who want to withdraw money from ATM machines on that Saturday morning — these are the people the bank actually asks. Section 5.4 — Sampling
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Guided Practice For each situation, identify the population and the sample. 1. A researcher wants information about the giardia present in streams in Orange County. He takes 8 one-liter containers of water from a stream in Santa Ana to conduct some tests on the giardia present. 2. To find out about the behavior of black bears in northern Washington, some were captured and fitted with radio collars. These bears were then released back into the wilderness and their movements tracked by monitoring the radio collars. 3. A person planning a party is looking for a band. To decide whether one particular band is suitable, she listens to 10 songs at random from the 50 songs the band play on occasions such as weddings.
Samples Should Represent the Entire Population Check it out: You have to make sure your sample is big enough too. Generally, bigger is better — a sample of 6 people out of a population of 10,000 won’t give very useful results.
A sample should be representative of the entire population. This means it should contain the same kinds of people or objects as the population. That way, you should get the same information from the sample as you would have gotten from the population. A sample that doesn’t contain certain types of people or objects that do exist in the population is biased. Example
2
An experiment into how children react to a certain vitamin supplement was conducted using 5000 boys from all over the USA. Explain why this study is biased. Solution
The sample contains only boys, but the population contains both boys and girls. The sample is not representative of the population. Example Check it out: The population in this study is all the residents in the town. The sample used is the people who returned their form.
3
A study is carried out to discover whether residents of a town think a road should be widened. A questionnaire was mailed to every household in the town, and 97% of the replies were against the road widening. Explain why this study may be biased. Solution
Check it out: This kind of sample is called self-selecting. The people in the sample have chosen to be in it.
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It’s impossible to tell if the sample is representative of the whole town. People with strong views (such as those who live on the road to be widened) are more likely to reply to this kind of survey. People with less strong views may be less likely to respond, but they are still part of the population, and so their views are still important.
Claims made about the population based on biased samples are invalid. Example
4
A country music radio station in California wants to find out what kind of music Californians prefer to listen to. It asks listeners to call in and vote for their favorite type of music. Later in the day, the radio DJ announces, “Californians’ favorite music is country music.” Explain why the DJ’s claim is not valid. Solution
The population is all Californians. The sample consists of people who are listening to that particular country music station (who are likely to be country music fans). People who dislike country music are probably not represented in the sample, making the sample biased. Results from this sample cannot be applied to all Californians, and so the claim is invalid.
Guided Practice 4. A survey is conducted to find out how Americans spend their leisure time. A sample of 1000 people are telephoned on a Saturday evening. It’s found that the majority of people spend their leisure time watching TV at home. Describe why this sample is biased. 5. A study into the jobs of Californian people was carried out by knocking on all the doors on three Sacramento streets one Monday sometime between 8 a.m. and 5 p.m. and interviewing the householder. Describe why the results of the study would be invalid.
Independent Practice 1. Explain how a sample is different from a population.
Now try these: Lesson 5.4.1 additional questions — p455
In the studies described in Exercises 2–4, explain whether you would collect data from the entire population or a sample. 2. A study to find the average salary of workers in the United States. 3. A study to find the median height of students in a class. 4. A study to find the price of movie tickets in the United States. Identify part(s) of the population underrepresented in these situations. 5. A survey collects responses on an internet website. 6. Data on traveling habits is collected at train and bus stations. 7. To gather opinions about a concert, the first 25 people leaving the hall were interviewed. Explain why claims based on this sample about what the whole audience thought would not be valid.
Round Up If the population is big, it makes sense to use a sample. But you have to take care how you select your sample — there are different ways of doing this, and they’re not all equally good in all situations. Section 5.4 — Sampling
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Lesson
5.4.2
Convenience, Random, and Systematic Sampling
California Standard: Statistics, Data Analysis, and Probability 2.2 Identify different ways of selecting a sample (e.g., convenience sampling, responses to a survey, random sampling) and which method makes a sample more representative for a population.
If your population is too big to gather data about each member, you need to use a sample. The idea of sampling is to learn about a whole population by studying only a part of it. So it’s important that a sample is chosen so that it accurately represents the population.
Convenience Sampling — Not Usually Representative Convenience sampling is used because it’s easy — not because it’s best.
Statistics, Data Analysis, and Probability 2.4 Identify data that represent sampling errors and explain why the sample (and the display) might be biased.
What it means for you: You’ll find out about some methods of selecting a sample of a population, and think about the advantages and disadvantages of each.
Key words: • • • • •
population sample convenience sampling random sampling systematic sampling
Check it out: In Example 1, recording the ages of the first 100 people to board might produce a biased sample too. Certain age groups might be more likely to arrive at the ship in good time and board earlier.
Check it out: Many businesses use convenience sampling because it is easy to sample their own customers. And researchers collecting information from people at the mall often choose the easiest subjects — approaching those that walk nearest them and appear to be friendly.
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Convenience Sampling In convenience sampling, each person or object is chosen because they are easy to access. For example, you might want to find out what people in California think about an issue. One way to get some answers would be to go to your local mall and ask some friendly-looking people some questions. This is convenience sampling. It might be a useful way to get a few opinions, but the sample won’t be representative of people in California. That’s the disadvantage of this method — samples aren’t representative of the population, meaning you can’t apply results to the whole population. Example
1
A tour company wants to know how satisfied passengers on its cruise are. It interviews 100 passengers sitting in the ship’s bar one lunchtime. What kind of sampling is this? Describe the advantages and disadvantages. Solution
This is convenience sampling. The sample will be easy to find but biased, since the bar will be more attractive to certain people — such as those without children. This means the company couldn’t claim that the results apply to the population (all the passengers).
Guided Practice Exercises 1–2 describe convenience sampling. Give one example of how each sample may be biased. 1. A running club wants to know the median age of runners across the USA. It finds the median age of its members. 2. A teacher wishes to find the average height of 6th grade students in a school. She asks the first 20 students to arrive for her class.
Random Sampling — Should Be Representative Random Sampling In random sampling, each person or object in the population has an equal chance of being selected. Random sampling is difficult because, for large populations, it’s very hard to make sure that every member of the population could be selected. For example, one way to get a random sample of people in California would be to get a list of everybody in California. Then you’d need to randomly pick people on that list. Then you’d need to go and talk to them — no matter where they lived. You couldn’t decide not to visit someone because they lived a long way away, or because they didn’t want to talk to you. That would mean the sample wasn’t random. Don’t forget: The size of a sample is important too.
But... the big advantage of random sampling is that as long as your sample is big enough, it should be representative of the population. Example
2
A hotel wants to see if, in general, its customers are satisfied with the service they receive. They devise two possible ways to obtain a sample. Method 1: Interview the guests staying in the rooms nearest the lobby. Method 2: Select guests at random and interview these people. Say whether each method is convenience sampling or random sampling. Which method do you think would be better to use, and why? Solution
Method 1 is convenience sampling. The sample may be biased. The area near the lobby area may be very noisy, for example. This will mean the hotel will get more complaints than is representative of the whole hotel (or fewer, if the area near the lobby is very quiet). Method 2 is random sampling. Each customer has an equal chance of being selected — it doesn’t matter where in the hotel they have stayed. This method would be the better one to use, as the sample should be representative of all the hotel’s guests at that time.
Guided Practice 3. Which one of the following is true of random sampling? A. Random sampling is a quick way to obtain data. B. It should produce samples that are representative of the population. C. It is easy and quick to do. 4. Describe how random sampling could be used to select a representative sample of 50 of a school’s 6th grade students.
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Systematic Sampling Means Using a Set Pattern Systematic Sampling In systematic sampling, people or objects are chosen by a set pattern. An example of systematic sampling would be collecting data from every fifth student entering the auditorium. It’s usually a good way to pick a sample, but it has a big disadvantage... Suppose every 100th item on a factory assembly line is faulty because of a computer glitch. And suppose that the factory’s quality-control team uses systematic sampling, and tests every 300th item produced. There are two possibilities: (i) a faulty item is never checked, or (ii) the only items checked are faulty. Neither of these would give a true picture of the actual situation. But in many situations, systematic sampling is usually easy and effective.
Independent Practice 1. What is the convenience sampling method? 2. What does it mean if a sample is collected using random sampling? 3. The manager of a clothes store wants to know how much the store’s customers spend on clothes each month. He interviews the first 100 shoppers in line for a sale. What kind of sampling is this? 4. Which of the following is NOT one of the reasons why you might want to use convenience sampling? A. The data is easily available. B. It can be less expensive than some other sampling methods. C. Convenience sampling can be a way to quickly obtain data. D. It always produces samples that accurately represent the populations. 5. What is systematic sampling? Now try these: Lesson 5.4.2 additional questions — p455
6. The math teacher has told students to take a sample from a population of 150 items. What type of sampling does each student use? Sarah writes the numbers 1–150 on slips of paper and puts them in a paper bag. She then draws out the numbers 32, 26, 116, 53, and 141, so she uses the 32nd, 26th, 116th, 53rd, and 141st items for her sample. th m in Micah uses every 10 ite mple. the population for her sa
Miguel chooses the first 10 items in the population.
Round Up So now you have seen three methods of selecting a sample from a population — convenience, random, and systematic. There are advantages and disadvantages to each, of course. 292
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Lesson
5.4.3
Samples and Accuracy
California Standards:
When you select a sample, you need it to be as representative as possible of the whole population. A sample that’s representative should have a similar mean, mode, median, and range to the population. Most likely, they won’t be exact matches, of course — but they should be pretty close.
Statistics, Data Analysis, and Probability 2.1 Compare different samples of a population with the data from the entire population and identify a situation in which it makes sense to use a sample. Statistics, Data Analysis, and Probability 2.4 Identify data that represent sampling errors and explain why the sample (and the display) might be biased.
What it means for you: You’ll see that no matter how carefully you select your sample, it’s unlikely to represent the population totally accurately.
Key words: • • • •
population sample mean sampling error
Sampling Errors Should Hopefully Be Small Collecting data from a sample is usually easier and cheaper than collecting data from an entire population. But you need your sample to represent the population as accurately as possible. You can check how accurately your sample represents the population by checking its statistics — for example, the mean, mode, median, and range. A sampling error for the mean is the difference between the mean of the sample and the mean of the population. You can also find sampling errors for measures such as the mode and median. The smaller the sampling error, the more accurately the sample represents the population. Example
1
A state with 50 gymnastics teams reports the number of medals they each won in state competitions this year. Medals won: 1, 12, 3, 16, 20, 9, 8, 5, 2, 7, 36, 7, 7, 9, 19, 7, 8, 27, 2, 6, 4, 9, 7, 8, 7, 6, 8, 6, 9, 10, 8, 19, 12, 8, 9, 9, 2, 5, 8, 10, 7, 2, 6, 6, 7, 6, 29, 8, 5, 5
Don’t forget: You find the mean of a data set by adding up all the values and dividing the total by the number of values. See Lesson 5.1.2 for more information.
A sample of 5 was randomly selected from the population. Find the sampling error for the mean. Sample: {10, 16, 8, 8, 7} Solution
First, you need the population mean. This is
1 + 12 + ... + 5 + 5 456 = = 9.12 medals. 50 50
Next, find the sample mean. This is
10 + 16 + 8 + 8 + 7 49 = = 9.8 medals. 5 5
So the sampling error is 9.8 – 9.12 = 0.68 medals.
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Example
2
Find the sampling error of the median number of medals won (using the data in Example 1) if the following sample is used: {8, 27, 6, 8, 9}. Solution
To find the population median, you have to put all 50 values in order. Then the median will be midway between the 25th and 26th values. In fact, the median for the population is 7.5 medals. Don’t forget: Subtract the population median from the sample median to get the sampling error.
Now you need the sample median. So put the values in order: 6, 8, 8, 9, 27. The median is the middle value — this is 8. So the sampling error for the median is 8 – 7.5 = 0.5 medals.
Guided Practice Below is a list of the distances (in miles) traveled to school by students in a class of 28 students. 1, 8, 3, 5, 2, 5, 3, 3, 2, 7, 4, 2, 4, 3, 5, 6, 1, 1, 5, 3, 2, 3, 2, 1, 4, 1, 2, 1 The following sample is used: {4, 1, 8, 2, 3, 6, 3, 4} 1. What is the population mean? 2. What is the sample mean? 3. What is the sampling error for the mean? 4. What is the population median? 5. What is the sample median? 6. What is the sampling error for the median? 7. Repeat Exercises 2–6 for the sample {5, 4, 2, 6, 1, 3, 2, 4, 1}. Comment on your results.
Different Samples Give Different Sampling Errors If you take different samples from the same population, then you’ll probably get different sampling errors. Example
3
Find the sampling errors for the mean using the data in Example 1, if the following two samples are used. Sample 1: {5, 5, 2, 20, 8}, Sample 2: {9, 8, 6, 8, 27} Solution
Don’t forget: It’s completely normal to get different sampling errors from different samples — it’s not a mistake.
5 + 5 + 2 + 20 + 8 40 = =8. 5 5 This gives a sampling error of 8 – 9.12 = –1.12 medals.
For Sample 1, the sample mean is
9 + 8 + 6 + 8 + 27 58 = = 11.6 . 5 5 This gives a sampling error of 11.6 – 9.12 = 2.48 medals.
For Sample 2, the sample mean is
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Check it out: There are many different factors that can affect the sampling error — the size of the sample is just one of them. You can still have a large sample that’s biased and unrepresentative. However, using a fairly large, randomly selected sample should reduce the risk of this.
Check it out: In Example 4, the larger sample gave a smaller sampling error. But notice that the smaller samples used in Examples 1 and 3 gave smaller sampling errors. That’s statistics for you. If you picked lots of small samples and lots of big samples, then you’d generally expect the bigger samples to have smaller sampling errors. But it’s still possible that you could pick a few small samples with really small sampling errors, or a few big samples with quite big sampling errors.
The aim when you pick a sample is to make your sampling errors as small as possible. Generally, the bigger your sample, the smaller your sampling error is likely to be. (However, the bigger your sample, the more expensive and difficult it is to collect the data.) Example
4
For the data in Example 1, find the sampling errors for the mean if the following randomly generated samples are used. Sample 1: {2, 6, 6, 8, 9}, Sample 2: {2, 2, 2, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 12, 19, 19, 29, 36}. Solution
2 + 6 + 6 + 8 + 9 31 = = 6.2 . 5 5 This gives a sampling error of 6.2 – 9.12 = –2.92 medals.
For Sample 1, the sample mean is
2 + 2 + ... + 29 + 36 211 = = 10.55 . 20 20 This gives a sampling error of 10.55 – 9.12 = 1.43 medals.
For Sample 2, the sample mean is
Notice that the larger sample gave a smaller sampling error.
Guided Practice 8. Find the median of the samples given in Example 4. 9. What is the sampling error for each of these samples? 10. Is this what you would have expected? Explain your answer.
Independent Practice A scientist measures the lengths of 300 fish, before dropping them into a pond. Later, three different samples of the fish are measured (in cm). Sample A: {4, 5, 5, 5.5, 6, 6.5} Sample B: {3, 5, 5.5, 5.5, 6, 6, 7.5, 8, 11} Sample C: {4.5, 5, 5, 5.5, 7, 7.5, 7.5, 8, 10, 11} Now try these: Lesson 5.4.3 additional questions — p455
1. Find the mean of each of the three samples. 2. The mean length of the population of fish is 5.99 cm. Find the sampling error for each sample based on the mean. 3. Which sample most accurately represents the population? Is this what you would have expected? Explain your answer. 4. Another sample of the fish is chosen using the first ten fish the keeper manages to catch. What type of sampling is this? Would you expect this sample to accurately represent the population? Explain.
Round Up Sampling isn’t likely to give you absolutely perfect results, but if it’s done carefully, then it’s good enough for most purposes. In real life, picking a sample is a compromise — a bigger sample should make for a smaller sampling error, but bigger samples are more expensive and difficult to use. Section 5.4 — Sampling
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Lesson
5.4.4
Questionnaire Surveys
California Standard:
You can get lots of data from people by asking them questions, but only if they choose to answer them. This isn’t perfect, but it’s the only way of getting some types of information.
Statistics, Data Analysis, and Probability 2.2 Identify different ways of selecting a sample (e.g., convenience sampling, responses to a survey, random sampling) and which method makes a sample more representative for a population. Statistics, Data Analysis, and Probability 2.3 Analyze data displays and explain why the way in which the question was asked might have influenced the results obtained and why the way in which the results were displayed might have influenced the conclusions reached.
What it means for you: You’ll find out some good and bad points of surveys, and some ways to make your data as useful as possible.
Key words: • • • • • •
population sample survey self-selected sampling convenience sampling random sampling
Check it out: When a survey is sent out in the mail, very few people will probably respond.
Check it out: Surveys can be used to collect data on pretty much anything — such as age, gender, household income, number of cars owned, number of family members...
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People Have to Choose to Answer a Survey A lot of information is collected by contacting people by phone, mail, or in person and asking them to provide some data. This is sometimes called self-selected sampling, because the people must agree to give the information. Self-selected samples aren’t always ideal — a random sample might be better in some ways. But you can’t force people to answer questions if they don’t want to. Even though it’s not a perfect way to gather information, this kind of self-selected sample can still provide a lot of very useful information. Example
1
A company wants to find out which cleaning products are used in California households. It sends out questionnaires in the mail to all the customers listed in their database. Identify the population and the sample used. What are the benefits and problems of collecting data in this way? Solution
Population: All the households in California. Sample: Those people listed in the company’s database who choose to fill in and return the questionnaire. Benefits: The company learns useful information such as which products are most popular, and the location of customers who purchase certain products. It also learns which households are more likely to complete other surveys in the future. Possible problems: Because the sample is self-selecting, and because the only people in the sample are previous customers of the company, the sample is unlikely to be representative of the population. Also, the costs of printing and mailing are likely to be high, and a lot of this money will be wasted, given that most people will probably not respond. Instead of sending out questionnaires, a lot of organizations these days use the internet to gather information. As with any method, there are benefits and disadvantages.
Example
2
The same company as in Example 1 decides to use the internet to gather the information it needs. It sends e-mails to all the customers whose e-mail addresses it has. The e-mail tells customers that if they fill in an online questionnaire, they have a chance to win $1000. Identify the population and the sample used. What are the benefits and problems associated with collecting data in this way? Solution
Population: All the households in California. Sample: The customers who: (i) have made a purchase from the company, (ii) gave their e-mail address, and (iii) who responded to the e-mail. Benefits: The survey provides the company with information at very little cost — there are no mailing costs, and the data can be automatically analyzed by computer. Possible problems: The sample won’t be representative of the population. The sample is self-selecting (so is likely to include a lot of people with strong opinions), and won’t include people who don’t have access to the internet.
Guided Practice A campaign group wants to find out if taxpayers would be willing to pay higher taxes if a new community swimming pool were promised for their area. 1. Describe how information could be gathered from local taxpayers. 2. What would the population and sample be in your example? 3. Describe some advantages and disadvantages of your method(s). A car dealer wants to collect data about when local people are planning to buy a new car. She organizes a prize drawing, which people enter by filling out a form with their name, address, and when they expect to buy a new car. 4. What are the population and sample in this example? 5. Describe some advantages and disadvantages of this method.
Questions Can Be Biased Too Surveys can give inaccurate information if a sample is biased. But information collected by asking people questions can also be biased if the questions encourage the person to give a particular answer. Check it out: Biased questions are also called “leading questions.”
A biased question is one that encourages people to answer in a particular way.
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Check it out: When a claim is based on the responses to biased questions, it is invalid. The claim can’t be trusted, since there is no real, unbiased evidence to back it up. See Lesson 5.5.1 for more information about claims.
People often ask biased questions if they are collecting data to try to back up one of their opinions. For example, suppose someone is in favor of banning smoking from public parks... Example
3
Explain why the following question is biased. Shouldn't smoking stinky cigars and unhealthy cigarettes be banned from all public parks? Solution
The question is worded so that the strong negative opinion about smoking in public makes it difficult for a person to disagree. These are called leading questions. They lead the person answering toward a particular reply. Example
4
Explain why the following questions are biased. (i) Experts say that you should chew your food slowly and take at least 30 minutes for a meal. How much time do you think students should get for lunch break? (ii)
A lot of models and movie stars have a yoga instructor. Don't you think it would be better to offer yoga instead of basketball in middle school?
Solution
(i) A response of at least 30 minutes or more is encouraged by stating a related "fact." (Beware... it may not be a genuine fact. It doesn’t say who the experts are, or if any other people disagree. Only one opinion is given.) (ii) Many people think models and movie stars are attractive and successful. This question suggests that yoga is one of the causes of their success. This may make people more likely to agree that yoga should be offered instead of basketball.
Guided Practice Explain why the questions below are biased. 6. “Don't you think we should brighten up the halls by painting over that boring tan color?” 7. “Painting the school will cost a lot of money that could be spent on more important things. Don't you agree that painting the school can wait for another couple of years?” 298
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Write Survey Questions So They’re Not Biased You need to write survey questions so that they don’t suggest that a certain reply is correct. You shouldn’t reveal your opinions (or anyone else’s) when asking questions. Here are the questions from earlier, written in a less biased way. Are you for or against banning smoking in public parks? How much time do you think students should get for lunch break?
Do you think it would be better to offer yoga or basketball in middle school?
Guided Practice A school committee is surveying students about whether a salad bar should be added to the cafeteria. 8. Write a biased question in favor of adding a new salad bar. 9. Write a biased question against adding a new salad bar. 10. Write an unbiased question to gather opinions on a new salad bar.
Independent Practice For Exercises 1–2, explain why each sample may not be representative of a population consisting of all the adults in a particular town. 1. A researcher asks shoppers entering or leaving a grocery store if they can answer some questions about their purchases. 2. A receipt from a department store gives a website where customers can go online to complete a questionnaire about public transportation.
Now try these: Lesson 5.4.4 additional questions — p456
For Exercises 3–6, decide whether the question is biased. If it is biased, explain why and suggest a better way of asking the question. 3. “Should the driving age in California be raised to 21?” 4. “Should we reduce the number of car accidents by fining drivers who talk on their cell phones while driving?” 5. “A community center would give our children a safe place to play. Can we count on your vote this Tuesday for keeping our children safe?” 6. “Do you think mining companies should be allowed to ruin our wilderness areas and kill innocent animals?” 7. 300 middle school students were asked, “Isn’t it unfair that people under 18 can’t vote for president?” As a result, it was claimed that 95% of middle school students thought the voting age should be lowered. Is this claim valid? Explain.
Round Up Surveys are a good way of collecting data from people. You’ve got to be really careful how you word the questions, though. You also have to make your sample as representative as possible — this isn’t always easy, since the way you carry out your survey means certain people are more likely to provide information than others. But even though it’s difficult, surveys like this can be incredibly useful. Section 5.4 — Sampling
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Lesson
Section 5.5
5.5.1
Evaluating Claims
California Standard:
You’re likely to be bombarded by claims every day. Claims are presented as facts — but they might not be. For instance, advertisements often claim that one product is better than another.
Statistics, Data Analysis, and Probability 2.5 Identify claims based on statistical data and, in simple cases, evaluate the validity of the claims.
What it means for you: You’ll look at data intended to support claims and decide if the claim is valid.
Key words: • • • •
You can only really tell whether a claim is true or not if you examine the evidence.
A Claim Is a Statement Presented as a Fact Claims are often reported in newspapers and on television and radio. These are examples of claims:
The average American eats pizza at least once a week.
claim opinion valid invalid
More people choose this toothpaste than any other. In 100 years, there won’t be any glaciers left on Earth.
Claims are different from opinions. A claim is presented as a true fact. An opinion is just someone saying what they believe, such as “I think this brand of cola tastes best.”
Claims Are Often Supported by Data Data is often used to provide evidence for a claim. You need to look carefully at the data to see if it supports the claim. Example
1
A grocery-store owner claims that $20 is the amount most often withdrawn from the ATM outside his store during one month. Don’t forget: The modal amount (or the mode) is the amount that occurs most frequently. See Lesson 5.1.1 for more information.
Does this circle graph from the bank’s monthly report, which shows data about that ATM, support his claim?
$20 $40
$100 $80 $60
Solution
This circle graph does show that the modal amount withdrawn was $20. So the circle graph does support the store owner’s claim. This doesn’t show that $20 is always the amount withdrawn most often. Next month’s data might be different. So if the store owner had claimed that $20 was always the amount most frequently withdrawn, then you would need to look at more evidence.
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Section 5.5 — Statistical Claims
Example
2
A different store owner claims that $20 is the mean amount withdrawn from the ATM outside her store during the same month.
$20
Does this circle graph from the bank’s monthly report, which shows data about that ATM, support her claim? Don’t forget:
Solution
Remember... In Example 2, values higher than $20 will push the mean up. And there are no values less than $20 to bring the mean back down. See Lesson 5.2.2 for more information.
You need to read the claim very carefully.
$10
$40
0
$60
$80
This time, the claim is that $20 is the mean amount. This can’t be true, since $20 is the smallest amount withdrawn, and all the other amounts will make the mean amount greater than $20. So the circle graph does not support this store owner’s claim.
Guided Practice 1. Is this statement a claim or an opinion? "I think the red team is a better team than the blue team." Explain your answer. 2. It’s claimed that most families using a swimming pool on a Saturday morning have exactly 2 children. Does the line plot on the right support this claim? 3. Give one example of a claim that would be supported by the line plot.
X X X X X X X 0
X X X X X X X X X X
Each cross represents one family.
X X X X X X X
X X X X X X X X X
X
1 2 3 4 5 6 7 Number of children
8
Sometimes the Evidence Is Not Clear Sometimes, some statistics may support a particular claim, but other statistics will not. People trying to convince or persuade someone will very often use just the statistics that support their claim. They may ignore statistics that don’t support their claim, or which make their claim look untrue.
Don’t forget: Measures of central tendency are the mean, median, and mode. See Lessons 5.1.1 and 5.1.2 for more information.
For example, companies make claims to try to persuade you to buy their products. They’ll want to use statistics to make their product look like it’s better than the competitors’. One way of doing this is by using a particular measure of central tendency, depending on what claim you want to make. Maybe they want to make the typical value sound high. Or maybe they want to make it sound low.
Section 5.5 — Statistical Claims
301
Example
3
A newly opened health club wants to attract some new young members. Its current members’ ages are: 25, 25, 37, 45, 53, 54, 55, 57, 63, 67. It wants to advertise, making the claim that the typical age of its current members is low. To achieve this, its advertisement makes the claim that “There are more 25-year-olds at the club than any other age.” Does the data support this claim? Comment on the claim generally. Solution
The modal age is 25, so the data does support the claim.
25 + 25 + ... + 63 + 67 481 = = 48.1 . 10 10 And the median age is 53.5. However... the mean age is
So if the health club had used any other measure of central tendency, the typical age would have sounded a lot higher. Here, the health club wanted to make the typical age of its members sound low in the advertisement, so used the only measure of central tendency that would do this. The club’s claim is not untrue, but it probably is misleading. Statistics aren’t always used in deliberately misleading ways. Sometimes, whether a claim is supported by data can be unclear for different reasons. Example
4
At the swim meet, the team swam in three relay races. The times for each team member in each race are shown below. Lucas claims that he’s the fastest Race 1 Race 2 Race 3 swimmer in the relay swim team. L uc a s 22.5 s 25.7 s 19.5 s His sister, Sam, claims that Lucas is Latoya 23.0 s 19.7 s 22.1 s not the fastest. Whose claim do the results support?
Jorge
27.5 s
26.2 s
27.4 s
Anna
23.1 s
20.2 s
21.4 s
Solution
Don’t forget: See Lesson 5.1.4 for more information about comparing data sets.
It’s hard to tell just by looking at these results if Lucas is the fastest. He swam fastest in Race 1 and Race 3, but not in Race 2. One way to analyze the results is to find each person’s mean time. Lucas: (22.5 + 25.7 + 19.5) ÷ 3 = 22.6 s Latoya: (23.0 + 19.7 + 22.1) ÷ 3 = 21.6 s Jorge: (27.5 + 26.2 + 27.4) ÷ 3 = 27.0 s Anna: (23.1 + 20.2 + 21.4) ÷ 3 = 21.6 s Both Latoya and Anna had faster average times than Lucas. So Sam could say that the results support her claim. On the other hand, no one swam faster than Lucas’s time of 19.5 s. So Lucas could say that the results support his claim.
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Section 5.5 — Statistical Claims
In cases like this, it’s not clear whose claim the data supports. The best thing is probably to say whether you’re interested in the fastest average time, or the fastest single time. Or you could look more closely at the data. Did something happen to Lucas in the second race to slow him down? Or perhaps he was doing a different stroke — maybe Lucas is fastest at one stroke, but not at another.
Guided Practice A babysitting company has six sitters of ages 13, 13, 14, 15, 15, and 45. Use this information for Exercises 4–7. 4. Find the mean, median, mode, and range of the babysitters’ ages. 5. The company claims that the typical age of its babysitters is 19. Which measure of central tendency supports this claim? 6. The claim in Exercise 5 is valid, but why is it misleading? 7. Make two claims that better represent the typical babysitter’s age.
Independent Practice
Now try these: Lesson 5.5.1 additional questions — p456
Use the terms data, valid, invalid, or claim to complete the sentences in Exercises 1–3. 1. A __________ is a statement that is presented as a fact. 2. A claim is considered ______ if the _____ is incorrectly interpreted. 3. A claim is usually _________ if there is no ________ to support it. 4. Someone reading the circle graph on the right made 40-49 this claim: "Most people on the Ages of School 30-39 50-59 school board are aged 40–49." Board Members Explain why this claim is invalid. 20-29 60-69 Suggest why the person may have thought that this claim was true. Here are the ages of the counselors at a summer camp: {19, 20, 22, 23, 20, 55, 20} 5. Find the mode, median, and mean for this data set. Use your statistics from Exercise 5 to say whether the claims in Exercises 6–10 are valid or invalid. For each valid claim, say which statistic supports the claim. Are any of these claims valid but misleading? Explain your answer. 6. Most counselors are aged 24. 7. The typical age of the counselors is over 25. 8. Half the counselors are older than 24. 9. Half the counselors are 20 or younger. 10. More than 14% of the counselors are over 30.
Round Up People often make claims. Often their claims are valid and backed up by data. But sometimes the data doesn’t support the claims or has been misinterpreted. Either way, the claim won’t be valid. Section 5.5 — Statistical Claims
303
Lesson
5.5.2
Evaluating Displays
California Standard:
If people want to convince you of something, then they will probably present their data in such a way that it emphasizes what they want you to believe. One way to do this is with graphs and charts.
What it means for you: You’ll see how data can be displayed in ways that may mislead people.
Key words: • • • • • • •
data set display line plot mean median mode range
Check it out: A “break in the scale” means the numbering on the axis doesn’t start at zero. In the first graph in Example 1, the numbering starts at 70.
A Break in the Axis Can Create a Visual Effect Putting a break in the vertical axis of a graph can change the impression given by a graph. Example
1
At a school-board meeting, a member claims that electricity costs in local high schools are much higher than for elementary and middle schools. He uses the graph below to 78 support his claim. 76 Electricity costs (in thousands of dollars)
Analyze data displays and explain why the way in which the question was asked might have influenced the results obtained and why the way in which the results were displayed might have influenced the conclusions reached.
Why might someone complain that the graph is misleading?
74 72 70 0
Solution
The break in the vertical scale makes the bar on the right appear much longer than the other bars.
Elementary Middle School School
High School
Elementary Middle School School
High School
80 Electricity costs (in thousands of dollars)
Statistics, Data Analysis, and Probability 2.3
If the vertical scale starts at zero, the difference appears to be much less dramatic. The break in the scale on the first graph emphasized the difference in height of the bars.
70 60 50 40 30 20 10 0
The same idea can make things look like they’re rising or falling very quickly.
Data can also be displayed as circle graphs, line graphs, or bar graphs.
Which of these graphs makes it look like electricity costs are rising more quickly?
Graph B
80
78
Electricity costs (in thousands of dollars)
Don’t forget:
2 Electricity costs (in thousands of dollars)
Example
Graph A
76 74 72 70 0
2004
2005
2006
70 60 50 40 30 20 10 0
2004
2005
2006
Solution
Graph A, because the difference in height between the bars is emphasized by the break in the scale. This makes it look like costs are rising quickly, and is possibly misleading.
304
Section 5.5 — Statistical Claims
Using a different scale can achieve the same effect. Example
3
Graph A
76 74 72 70 0
2005
2004
2006
85
Electricity costs (in thousands of dollars)
78
Electricity costs (in thousands of dollars)
Electricity costs (in thousands of dollars)
Which graph makes electricity costs look to be rising most quickly?
Graph B
80 75 70 0
2004
2005
78 76
Graph C
74 72 70 0
2006
2005
2004
2006
Solution
This time, the effect is achieved by using different scales on the vertical axis. (All three graphs have vertical scales starting at 70.) Graph B looks like it’s rising least quickly. This is because its vertical scale is marked in intervals of 5, rather than in intervals of 2 as in Graphs A and C. Graph C looks like it’s rising most quickly, because it is marked in intervals of 2, and it has been stretched vertically.
The next example shows how the spacing on the horizontal axis can be changed to emphasize a particular viewpoint. Example
4
4.00 3.50 3.00 2.50 2.00 1.50 1.00 0.50
Graph A
4.00 3.50 3.00 2.50 2.00 1.50 1.00 0.50
Graph B
2005
2000
1995
1990
1985
1980
2005 2000 1995 1990 1985 1980
Year
Lunch cost (in $)
Lunch cost (in $)
Which of these graphs might someone use to support the claim that school lunch prices have risen steeply in the past 25 years? Explain.
Year
Solution
Graph A — it appears much steeper than Graph B. Graph A gives the impression of a greater rate of increase because the line is steeper. In Graph B the line appears flatter, so the increase in prices does not appear to be so drastic. Both graphs are drawn from the same data, but the years on the horizontal axis of Graph B are spread further apart.
Section 5.5 — Statistical Claims
305
This graph shows the number of bags of popcorn sold by students in grades 6, 7, and 8 to raise money for a charity. The graph was produced by grade 6. 1. Students in grade 7 thought this graph was misleading. Suggest why. 2. Draw a graph to show the same information more clearly.
Number of bags sold
Guided Practice 64 62 60 58 56 54 52 50 0
6th
7th Grade
8th
3. Which graph below makes the price of a school carnival ticket appear to increase more dramatically? Explain why. 4 Price (in $)
Price (in $)
3
Graph A 2 1
Graph B
2006 2005 2004 2003 2002 2001 2000
2006
2005
Year
2004
2003
2002
2001
2000
0
2.00 1.75 1.50 1.25 1.00 0.75 0.50 0.25 0
Year
4. Suggest how someone could avoid getting the wrong impression from a graph.
Things Can Be Emphasized in Other Ways Too You’ve seen circle graphs before. Like any other kind of graph, the way they are presented can emphasize a particular viewpoint. Blue 25%
Red 25%
Other Silver 25% 25%
For example, take this circle graph showing how common different colors of cars are. All the different categories (blue, red, silver, and other) are equally popular.
But if you draw this in three dimensions, things look a little different. Suddenly, silver looks (at first glance, anyway) to be Blue the most popular color, because more of the diagram is Other Red silver than any other color. The three-dimensional Silver effect emphasizes the amount of silver cars.
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Section 5.5 — Statistical Claims
You can do the same thing with bar graphs. You can emphasize some bars but not others by using a three-dimensional graph. 50 40 30 20 10 0
Look at this graph showing how old the students who play basketball at one particular club are. There’s a good mix of ages — you can see this from a quick glance at the graph. 12
13
14
Now look at the same graph drawn in three dimensions. Check it out: The colors of the other bars have also been made less bright. This also helps to emphasize the amount of red on the graph.
50 40 30 20 10 0
Suddenly, red is the dominant color — you can see the whole volume of the red bar, but not of the others.
12
13
14
The grouping of data values can also give a misleading impression. For example, look at the graph below. 800
This graph shows the number of students of different ages who voted in the last election for the student council.
700 600 500 400
It looks like there is a good mix of students of different ages voting — none of the bars are much taller than the others, and none are much shorter.
300 200 100 0
12 15 6-18 101 13-
600
Age
500 400
But look what happens if you now draw one bar for each year of age.
300
It’s now clear that the vote was dominated by 12- and 16-year-olds.
100
200
0
10 11 12 13 14 15 16 17 18 Age
Guided Practice 5. Why might the manufacturer of Brand A want to use a graph like this one to compare how popular different makes of cars are? 6. Why might someone claim that this graph was misleading? 7. Redraw the graph to show the same information in a clearer way.
Brand D Brand B
Brand C
Brand A
Section 5.5 — Statistical Claims
307
Independent Practice 50
Number of Cyclists
For an annual children’s bicycle race, the participants enter by age categories. Graph 1 on the right shows this year’s entries.
Graph 1
40 30 20
Number of Cyclists
1. Which age interval has the 10 greatest number of participants? Age 0 2. Why can’t you tell from the 0-4 5-9 10-14 15-19 graph if any 19-year-olds entered? Graph 2 below shows the same data, but with the ages grouped in a different way. Graph 2
30 20 10 0
0-3
4-6
7-9 10-12 3-15 1
Age
3. How do the new intervals used in Graph 2 change the overall impression of the ages of the competitors, compared to Graph 1?
Graph 1 appeared in the newspaper with the headline “5 through 9 year olds dominate the event.” 4. What headline might have appeared if Graph 2 was printed in the newspaper instead? 5. Give an example of a claim for which you would want to exaggerate the difference between the lengths of the bars of a bar graph. 6. Give an example of a claim which you would want to support by a line graph in which there appears to be less dramatic change over time.
Now try these: Lesson 5.5.2 additional questions — p457
Number of People
Number of People
The graph on the right appeared in the school newspaper in an article about the popularity of spring sports. It shows the attendance at school sporting events during different seasons. 7. Approximately how much more popular do sporting events in the spring seem to be, Fall Winter Spring Seasons compared with sporting events in the fall? 8. How does your answer to this question differ if you use the graph below, which shows actual data values too? 180 Explain why this is. 170 160 9. Redraw the bar graph using a vertical 150 140 scale from 0 to 200 in increments of 50 130 without any break in the scale. 120 110 10. Describe the impression given by the 100 0 Fall Winter Spring graph you have drawn. Seasons
Round Up When you’re looking at a data display, take care — it might be drawn in a misleading way. Always check the scales and think about how the data would appear if the display was drawn differently. 308
Section 5.5 — Statistical Claims
Chapter 5 Investigation
Selling Cookies You work for a supermarket. A team of sales people from Company A are visiting. They say their cookies cost a little more than Company B’s brand, but they sell far better. Below are parts of a report from Company A, and some sales data. Sales Data
Brand Company A Company B
4300
Company A 4250 Company B
4100 4075
4200 4150 4050
Jan
Feb
Mar
Apr
May
Feb 4136 4198
Mar 4208 4106
Apr 4220 4123
May 4240 4100
Jun 4222 4200
• Typical monthly sales for Company A are 100 greater than for Company B. • In every month since March, Company A cookies have sold more than Company B. • According to our survey, 88% of people prefer Company A’s cookies, and feel they are just as healthy as any other brand.
4100 4000
Packs of Cookies sold in 6 months from Jan-Jun Jan
Jun
Part 1: Look at the graph taken from the Company A report and the sales data. The sales people explain that the graph shows Company A’s cookies selling in far greater quantities than Company B’s. What advice would you give to your manager about the graph? Part 2: Now look at the above excerpt from the Company A report, the sales data, and the questions asked in the Company A survey (on the right). Are the claims in the report in any way misleading?
Appendix: Company A’s Survey Questions 1) Would you agree that Company A’s cookies are the tastiest cookies you can buy? 2) Our experts say that Company A’s cookies are just as healthy as any other cookies. Do you agree that Company A’s cookies are no less healthy than any other brand?
Extension Your supermarket wants to do its own research, rather than trust Company A. 1) Describe two ways your company could use sampling to collect this information. Which method would you recommend? Why? 2) Write three unbiased survey questions that could be used. Open-ended Extension Conduct a survey to find out people’s favorite: food OR sport OR type of music. 1) Write an unbiased survey question for your topic. Ask 20 people the same question. 2) Now reword your question to favor a particular response. Ask a different set of 20 people the new question. Did your question affect the responses? 3) Describe the sample of people chosen to answer each question, and the population they represent. Do you think your samples were representative of the population? Explain.
Round Up Statistics can be used accurately, or they can be used in ways that may mislead people. Even if you’d never mislead anyone, it’s good if you can see when other people are trying to mislead you. Cha pter 5 In vestig a tion — Selling Cookies 309 Chapter Inv estiga
Chapter 6 Probability Section 6.1
Outcomes and Diagrams .......................................... 311
Section 6.2
Exploration — Heads or Tails .................................... 320 Theoretical Probability ............................................... 321
Section 6.3
Exploration — Pick a Card ........................................ 339 Dependent and Independent Events ......................... 340
Section 6.4
Exploration — Shooting Baskets ............................... 349 Experimental Probability ........................................... 350
Chapter Investigation — A Game of Chance ................................... 356
310
Lesson
Section 6.1
6.1.1
Listing Possible Outcomes
California Standard:
You might have played probability games or board games at school or at home. If you have, then even if you didn’t realize it at the time, you’ve worked with events and outcomes.
Statistics, Data Analysis, and Probability 3.1 Represent all possible outcomes for compound events in an organized way (e.g., tables, grids, tree diagrams) and express the theoretical probability of each outcome.
What it means for you: You’ll think about actions that can have more than one possible result. You’ll list all the possible outcomes for particular situations.
An Outcome Is the Result of an Action or Experiment Many situations have more than one possible outcome. Example
1
Sarah and her brother Nathaniel are trying to decide who gets to drive the car tonight. They decide to toss a coin. How many different outcomes are possible when they toss the coin? Solution
Key words: • • • • •
outcome event action experiment combine
Possible Outcomes:
Heads is one possible outcome. Tails is one possible outcome.
H
T
These are the only possible results. There are a total of 2 possible outcomes for the action of tossing a coin.
An event is a description that matches one or more possible outcomes. Example
Check it out: Think of an outcome as any possibility, and an event as a set of outcomes matching a particular condition.
2
At a carnival, kids spin the wheel shown to win a prize. How many possible outcomes match the following events? Event A: winning a flashlight Event B: winning a magnet Event C: winning coloring pencils Solution
There are 8 sections on the wheel, so there are 8 possible outcomes. There is 1 section of the wheel that will let you win a flashlight, so 1 outcome matches event A. You can win a magnet by spinning either of 2 sections. So 2 outcomes match event B. There are 3 ways to win coloring pencils, so 3 outcomes match event C. Two of the outcomes (the ball and the yo-yo) don’t correspond to any of events A, B, or C.
Section 6.1 — Outcomes and Diagrams
311
Guided Practice This picture shows a bag containing 5 marbles. Exercises 1–3 are about picking marbles from the bag. How many outcomes match the following events? 1. Picking a red marble 2. Choosing a green marble 3. Choosing a marble that is not green Exercises 4–6 are about the four spinners shown below. 3. 4. 1. 2.
4. Which spinner has the most possible outcomes? 5. Which spinner has the least possible outcomes? 6. Latisha is looking at events that describe spinning a particular color (for example, “spinning red” or “spinning yellow”). Which spinner has the most possible events like this? Which has the least?
An Event Can Match Very Different Outcomes Sometimes very different outcomes match the same event. Example
Check it out: The word “die” is the singular of dice.
3
Eduardo rolls a normal 6-sided die. How many possible outcomes match the event “rolling an even number”? Solution
All possible outcomes: Possible even numbers:
1
2 2
3
4 4
5
There are 6 possible outcomes of rolling the die. Only 3 of those match the event “rolling an even number.”
Guided Practice Exercises 7–12 give possible events for rolling a die. List all the possible outcomes that match each event. 7. Event: rolling an odd number 8. Event: rolling a number less than 3 9. Event: rolling a number greater than 6 10. Event: rolling a number other than 5 11. Event: rolling a multiple of 3 12. Event: rolling a negative number 312
Section 6.1 — Outcomes and Diagrams
6 6
You Can Combine Outcomes of Different Actions Events can be made up of outcomes from different actions. Example Check it out: Once you combine outcomes of different actions, you can have many more events. Events for the combined action in Example 4 might include “heads and any color,” “yellow and any result of the coin toss,” or “yellow and heads.” These types of events are called compound events.
4
Kiona tosses a coin, and her friend Daniel spins the spinner shown. List all the possible outcomes of this combined action. Solution
The outcomes are made up of the result of one toss of the coin, plus the result of one spin of the spinner. There are 6 possible outcomes: Yellow and Heads Blue and Heads Yellow and Tails Blue and Tails
Red and Heads Red and Tails
Guided Practice A two-digit number is chosen by picking one number from Box A and one number from Box B. The digit from Box A is recorded in the tens place. The digit from Box B is recorded in the ones place.
Box B
Box A 8 3 5 7 1
2
3
4
1
13. List all the possible two-digit numbers beginning with a 3 that could be drawn from the boxes using this method. 14. Which of the following is NOT a possible outcome of this method? 53 54 51 57 15. List all the possible outcomes for the event: “drawing a two-digit number less than 20.”
Independent Practice
$100 $2 00
0 50 $1 0 $200
Lesson 6.1.1 additional questions — p457
$900 $100 0
Now try these:
For each of Exercises 1–3, sketch a spinner like this one. Color or label the sections so that the spinner has exactly: 1. 2 possible outcomes matching the event “spinning blue” 2. 3 possible outcomes matching the event “spinning red” 3. 4 possible outcomes matching the event “spinning green”
00 $5 $600 $700 $8 00
$300 $400
This spinner is used on a game show to determine how much money a contestant wins. Name an event using the spinner with exactly 4. 1 possible outcome. 5. 2 possible outcomes. 6. 3 possible outcomes. 7. 12 possible outcomes.
Round Up It’s easy to list all the possible outcomes of most of the examples in this Lesson, because there aren’t many to choose from. For more complicated situations, you’ll need better ways to organize the information. You’ll learn about some ways to do that in the next two Lessons. Section 6.1 — Outcomes and Diagrams
313
Lesson
6.1.2
Tree Diagrams
California Standard:
Making a list of possible outcomes is straightforward for simple actions like rolling dice or spinning a spinner. For more complicated situations, it can be hard to know for sure that you’ve listed all the possible outcomes.
Statistics, Data Analysis, and Probability 3.1 Represent all possible outcomes for compound events in an organized way (e.g., tables, grids, tree diagrams) and express the theoretical probability of each outcome.
What it means for you: You’ll learn about tree diagrams, and how you can use them to find all the possible outcomes of a situation.
Key words: • outcome • event • tree diagram
A tree diagram can help you organize your work and help you check that your list is complete.
Tree Diagrams Show All Possible Outcomes A tree diagram uses branches to show the different possible outcomes of each of a set of actions that take place.
Outcome 1 Outcome 2
To create a tree diagram, think about the possible outcomes for each different stage of the experiment. Use these outcomes to build the branches. Example
Outcome 3
1
Make a tree diagram to show all the possible outcomes for the experiment of tossing a coin twice in a row.
Check it out: Not all tree diagrams are drawn going across the page — sometimes the branches go down the page instead.
Solution
Step 1: Draw a branch for every possible outcome on the first toss of the coin.
Coin toss 1 H
T
H
H
T
T
H
T
Step 2: From each possible outcome of coin toss 1, draw branches for all the possible outcomes of coin toss 2.
Coin toss 1
Coin toss 2 H H T H T
Check it out: A tree diagram can show as many stages of an experiment as you want. The diagram in this example could be extended to show 10, 20, or 100 coin tosses – but it would get very large and complicated if it did.
Check it out: You have to read descriptions of events carefully... notice how there are 2 outcomes that match the event “a head and a tail in any order.”
314
T
Step 3: Follow along each branch, starting at the left and finishing at the right. At the end of each branch, list the outcomes that led to that point.
Coin toss 1
Coin toss 2
Outcomes H
HH
T H
HT TH
T
TT
H
T
From the tree, you can see that there are 4 possible outcomes. There’s 1 outcome that matches the event “two heads.” There’s 1 outcome that matches the event “two tails.” There’s 1 outcome that matches the event “first a head, then a tail,” and 1 outcome that matches the event “first a tail, then a head.”
Section 6.1 — Outcomes and Diagrams
Guided Practice This tree diagram shows the ways lights in three rooms of a house can be either off or on. 1. Copy and finish the tree diagram to represent all the possible outcomes for the lights. 2. Use your tree diagram to list all the possible outcomes for the lights in the three rooms.
Hall
Kitchen
Bedroom
ON ON OFF
ON
ON OFF
Use your tree diagram to list all the possible outcomes for each of the following events: 3. lights are on in exactly two of the rooms 4. lights are off in exactly two of the rooms 5. lights are on in the hall only 6. Explain why for this situation, the outcome of “on–off–off ” is different from “off–off–on.”
You Can Combine Any Actions Using Tree Diagrams You can use a tree diagram to show a combination of different actions. Example Check it out: You could draw the tree diagram for Example 2 by starting with the branches for tossing the coin, then adding the branches for rolling the die. You’d still get the same number of outcomes.
H
2
Guadalupe and Kareem are playing a game. Guadalupe rolls a die, and Kareem tosses a coin. Use a tree diagram to show all the possible outcomes of rolling one die and tossing one coin. Solution
Step 1: Draw a branch for each possible outcome of rolling the die. Step 2: Add branches from each possible outcome of rolling the die, showing the possible outcomes of tossing the coin. Step 3: List the outcomes at the end of each branch.
T
Roll die
Toss coin
Outcomes H
1H
T
1T 2H
H T H T H T H
2T 3H 3T 4H 4T 5H
H
5T 6H
T
6T
T
Section 6.1 — Outcomes and Diagrams
315
Guided Practice Kim is choosing what to wear to school. She has three shirts and two pairs of pants. The shirts are white, blue, and red. The pants are black and brown. 7. Draw a tree diagram to show the different outfits Kim can wear. Begin your tree with the branches for the shirts first. 8. List all the possible outcomes. How many different outfits of one shirt and one pair of pants can Kim make? 9. Redraw the tree diagram from Exercise 7, beginning with the branches for the pants first. 10. Compare the two tree diagrams. How does changing the order affect the number of outcomes?
Independent Practice Three different-color chips are placed in a bag. One is picked, recorded, then put back for the second pick. First Pick Second Pick Use this tree diagram of the possible outcomes to do Exercises 1–5. 1. List all the possible outcomes for picking two chips. (“Blue, red” is considered different from “red, blue.”) 2. List all the possible outcomes for the event “picking a second chip the same color as the first one.” 3. List all the possible outcomes for “picking two chips where the first chip drawn is blue.” 4. List all the possible outcomes for the event “picking two chips where a blue chip is never picked.” 5. Suppose after the second draw, the color is recorded and the chip is replaced in the bag for a third draw. How many outcomes showing different orders of chips would there be? Why? Now try these: Lesson 6.1.2 additional questions — p458
6. A bag contains 3 marbles: one purple, one yellow, and one green. In one experiment, a marble is picked and then placed back in the bag before the second marble is picked. In another experiment, a marble is picked and then kept out of the bag while the second marble is chosen. Use tree diagrams to compare the possible outcomes of drawing two marbles for each experiment.
Round Up Tree diagrams can be very useful for showing all the possible outcomes of a situation. But remember to draw them neatly so you can see what’s going on. Make the first branches quite spread out, so that the later branches don’t get all squashed up. 316
Section 6.1 — Outcomes and Diagrams
Lesson
6.1.3
Tables and Grids
California Standard:
Last Lesson you learned how to use tree diagrams to help you organize your information about the possible outcomes of a situation. In this Lesson, you’ll see some other ways to organize this type of information.
Statistics, Data Analysis, and Probability 3.1 Represent all possible outcomes for compound events in an organized way (e.g., tables, grids, tree diagrams) and express the theoretical probability of each outcome.
Tables Can Help You to Organize Your Information Tables and grids use rows and columns to show all the possible outcomes for a combination of two actions.
What it means for you:
Key words: • • • • • • •
outcome combination table grid cell row column
Check it out: The individual boxes in a grid or table are called cells.
Example
1
Use a table to find the possible outcomes for tossing a coin twice. Solution
1st coin toss
Step 1: Write the possible outcomes for the first toss of the coin in a row along the top of your table. Step 2: Write the possible outcomes for the second toss of the coin in a column down the side of your table.
Heads 2nd coin toss
You’ll see how to use a grid to find all the possible outcomes of a situation.
Ta i l s
Heads Ta i l s
Step 3: Use the rows and columns to make a grid of cells. Step 4: Fill in each cell of the grid by combining the outcomes listed on the row and column of that cell. 1st coin toss
When you fill in a table with letter codes like this, the first letter represents the first action (the first toss of the coin in this case), and the second letter comes from the second action.
Ta i l s
Heads
HH
TH
Ta i l s
HT
TT
This cell represents getting T ails on the first toss and H eads on the second toss, so the cell contains T H .
The grid shows the 4 possible outcomes for tossing a coin twice: HH, HT, TH, TT.
Guided Practice 1st child
1. A family is chosen at random from all the families in California that have 2 children. Copy and complete the table to show all the possible combinations of boys and girls the family could have.
Boy 2nd child
Check it out:
2nd coin toss
Heads
G irl
Boy G irl
Section 6.1 — Outcomes and Diagrams
317
2nd pick
The cells in the table below represent the outcomes for drawing a marble from a bag, recording it, and drawing another marble. 1st pick 2. Copy the table and fill in ? ? ? the color of marble that belongs in place of each G re e n WG GG BG question mark. ? WW GW BW 3. How many possible outcomes match the event ? WB GB BB “picking at least one blue Key: B = blue G = green W = white marble”? 4. How many possible outcomes match the event “not picking any green marbles”?
Tables Are Useful in Situations Involving Numbers It can be especially helpful to use a table when you need to do calculations with numbers in the rows and columns of the table. Example
2
Two dice are rolled. Find all the possible outcomes that match the event “a sum of 7 is rolled.” Solution
Step 1: Find the possible outcomes for the first die: 1, 2, 3, 4, 5, or 6. Write these in a row along the top of your table. Step 2: Find the possible outcomes for the second die: 1, 2, 3, 4, 5, or 6. Write these in a column down the side of your table.
1st die rolled
Step 3: Create a grid of cells inside the table.
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5 You can look at the column and 6 row of each of these 6 cells and write the outcomes with a sum of 7: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1).
6
7
8
9
10
11
7
8
9
10
11
12
Step 4: Fill in each cell of the grid by adding the outcomes listed on that row and column. In this case, the grid shows there are 6 possible outcomes that have a sum of 7.
2nd die rolled
1
(1, 6) means 1 on the first roll, 6 on the second roll. 318
Section 6.1 — Outcomes and Diagrams
Guided Practice Exercises 5–7 are about outcomes for rolling two dice and adding the results together. Use the grid from Example 2 to find the answers. 5. List all the possible outcomes with a sum of 4. 6. How many possible outcomes are there for the event “a sum of 6”? 7. Describe an event that has only one possible outcome. Two dice are rolled. The number on the second die is subtracted from the number on the first die. The result is called n. 8. Draw a grid showing all the possible outcomes of this experiment. 9. How many possible outcomes match the event “n = 2”? 10. How many possible outcomes match the event “n is negative”?
Independent Practice 1. An ice-cream truck sells 4 flavors of ice cream: vanilla, chocolate, mint, and strawberry; and 3 ice cream toppings: marshmallows, sprinkles, and whipped cream. Create a table of the possible outcomes for choosing 1 ice cream flavor with 1 ice cream topping. Exercises 2–3 are about a computer game that allows you to choose the hair and eye color of a character. Hair color choices are: brown, black, and blond. Eye color choices are: brown, green, and blue. 2. Draw a table that shows all the possible hair and eye color outcomes. (Tip: use Br, Bk, and Bd for the hair color abbreviations, and Br, G, and Bl for eye color.) 3. How would adding red hair and hazel eyes to the choices affect the number of possible outcomes? Exercises 4–9 are about a math game. Two dice are rolled and the numbers on the dice are multiplied. 4. Create a table to show all the possible products of the two dice.
Now try these: Lesson 6.1.3 additional questions — p458
Use your table from Exercise 4 to find all the possible outcomes matching each of the following events: 5. A product of 12 6. A product of 4 7. A product greater than 20 Use your table from Exercise 4 to name the following: 8. An event that has exactly 1 outcome. 9. An event that has at least 20 outcomes.
Round Up Tables and grids are most useful for combining two actions that both have a few possible outcomes. A grid can show that sort of situation much more clearly than a tree diagram with lots of branches. But the problem with grids is that they can’t show combinations of more than two actions. Section 6.1 — Outcomes and Diagrams
319
Section 6.2 introduction — an exploration into:
Heads or Tails If you were to toss a coin, what’s the probability that it will land on heads? The answer is 0.5 — but what does that actually mean? That’s what this Exploration is all about. If you flip a coin 24 times, how many times do you predict it will land on heads? Write down your prediction. You’re now going to try it for real... Flip a coin 24 times. Record your results in a chart like this one. After each flip, calculate the fraction of your total flips so far that landed on heads. Write your result in the last column. 1.0
++
0.9
Fraction of heads
0.8 0.7
0.6 0.5
+ +
+
+
+++
+++++
++
++
++++++
Flip # H or T? H 1 H 2 T 3 T 4 H 5 H 6 H 7
Fraction of heads 1 1 =1.0 2 2 =1.0 2 3 =0.67 2 4 =0.5 3 5 =0.6 4 6 =0.67 5 7 =0.71
Divide the number of heads by the total flips so far. Then convert to a decimal using a calculator.
0.4 0.3 0.2 0.1
0.0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Flip #
Show your information on a graph like this one. • Plot the flip number along the horizontal axis. • Plot the fraction of heads on the vertical axis.
Example 1. Use the above graph to say what fraction of the flips were heads after: a. 8 flips, b. 24 flips. 2. What happens to the line on the graph as you move to the right? Solution
1. The fraction of heads was: a. 0.75 after 8 flips, b. 0.53 after 24 flips. 2. The graph seems to be approaching 0.5, and is zig-zagging a lot less than at the start.
Exercises 1. Compare your graph to another student’s. Describe any similarities and differences. 2. After 24 flips, what fraction of your flips were heads? Subtract one from the other. Find the difference between this fraction and 0.5. 3. If you flipped the coin 50 times, do you think your answer to Exercise 2 would change? Explain your answer using your graph and what you know about probability. 4. Now combine your data with the data from the rest of the class. Find: (i) the total number of times a coin was flipped, and (ii) the total number of heads. Find the fraction of heads, and the difference between this fraction and 0.5. How does this compare with your answer to Exercise 2? 5. What would you expect if you flipped a coin 1000 times? 10,000 times?
Round Up For the first few flips, your graph may have zig-zagged up and down a bit. But as you did more flips, things should have settled down somewhere near 0.5, since that’s the theoretical probability. The idea of what probability actually means is tricky, but you’ll see more about it in this Section. a tion — Heads or Tails Explora 320 Section 6.2 Explor
Lesson
Section 6.2
6.2.1
Pr oba bility Proba obability
California Standard:
A lot of the time, you can’t say for sure whether or not one particular event will happen. But you can often say how good the chances are. Probability is a way of using numbers to describe the chance of an event happening.
Sta tistics ta Anal ysis Statistics tistics,, Da Data Analysis ysis,, and Pr oba bility 3.3 Proba obability Repr esent pr oba bilities as present proba obabilities opor tions tios,, pr propor oportions tions,, ra tios decimals betw een 0 and 1, between and per centa ges betw een 0 percenta centag between and 100 and vverify erify tha thatt the pr oba bilities computed ar e proba obabilities are reasona ble easonab le; know that if P is the probability of an event, 1–P is the probability of an event not occurring.
What it means for you: You’ll learn about using probability as a way to describe how likely events are to happen.
Key words: • • • • • •
probability chance likely percent fraction decimal
Some Ev ents Ar e Mor e Lik el y to Ha ppen Than Other s Events Are More Likel ely Happen Others People often talk about things that might happen, using words like “chance,” “likely,” and “probability”: “What is the probability it will snow today?” “How likely is it that the school football team will win its next game?” “What is the chance that you will go to a movie this weekend?” Look at the line below and think about where your answer would be for each of these questions.
Impossible
Very unlikely
Fairly unlikely
Even chance
Quite likely
Certain
Very likely
Guided Practice Decide where you would put the chances of the following events happening on the scale above. 1. Leaves falling from the trees next fall. 2. Finding a live elephant in your bedroom when you get home. 3. Winning a raffle if you have 1 out of 100 tickets. 4. Winning a raffle if you have 99 out of 100 tickets. 5. Winning a raffle if you have 1 out of 1,000,000 tickets. 6. A tossed coin landing on heads. 7. Put the events named in Exercises 1–6 in order, from most likely to least likely.
Pr oba bility Is a Way to Sa y Ho w Lik el y an Ev ent Is Proba obability Say How Likel ely Event In math, probability is a way of describing the chance that an event will occur. Probability can be written using fractions, decimals, or percents. Check it out: The closer the probability gets to 1, the more likely the event is to happen. The closer the probability is to 0, the less likely the event is.
You can replace the words on the line above with numbers that represent how likely an event is to occur. 0% 0
50%
100%
1 2
1
Section 6.2 — Theoretical Probability
321
Check it out:
A probability of 0 (or 0%) means that there is no chance.
If you did a very large number of experiments with 2 outcomes that each have a probability of 50% (tossing a coin, for example), then you would expect that about half the experiments would have one outcome (heads, say), and about half the experiments would have the other outcome (tails).
A probability of 1 (or 100%) means that the event will definitely happen. A probability of
1 2
(or 50%) means that the event might happen, but
there’s an equal chance that it won’t.
Guided Practice Estimate the probability that each of the following things will happen. Write your answers as percents. 8. It will go dark tonight. 9. Your math teacher will turn into a pineapple. 10. A 6th-grader from California chosen at random will be a girl. Use the list of probabilities below to answer Exercises 11–12.
1 5 0 10 2 8 25 10 11. Which of the probabilities above represents an impossible event? 12. Which of the probabilities above represents a certain event?
Don’t forget:
Pr oba bility Is Usuall y Calcula ted Exactl y Proba obability Usually Calculated Exactly
Calculating an exact probability isn’t the same as saying whether an event will definitely (or definitely not) happen. See Lesson 6.4.2 for examples of where you can’t find an exact probability.
There are many situations where you can say exactly what the probability of an event is. Example
1
What is the probability of spinning the color red on this spinner? What is the probability of spinning the color blue? Solution
Spinning the color red is certain, so the probability is 1. Spinning the color blue is impossible, so the probability is 0.
Example
2
What is the probability of spinning the color blue on this spinner? Check it out: By looking at the spinner, you can see that there’s a 1 in 4 chance of spinning the color blue. A chance of 1 in 4 is written mathematically as a probability of 1 = 0.25 = 25%. 4
322
Solution
The blue section is one-fourth of the spinner. So the probability of spinning the color blue is 1 = 0.25 = 25% 4
If you spun the spinner many times, about one-fourth of the spins would land on blue.
Section 6.2 — Theoretical Probability
Guided Practice Exercises 13–15 are about the spinner shown. Find the probability of spinning the colors below. Write your answers as decimals. 13. Blue 14. Yellow 15. Pink In Exercises 16–18, find the probability of spinning the color yellow on each of the following spinners. Write your answers as fractions. 16. 17. 18.
Independent Practice Each set of cards shown below is turned over and shuffled, then one card is picked. For each set, find the probability of picking 1. a triangle card 2. a star card
Don’t forget: For more information about fractions, percents, and decimals, see Chapter 3.
A.
B.
C.
D.
A bag has 1 red, 1 blue, and 2 yellow marbles in it. 3. How many marbles are in the bag? 4. How many marbles are red? What is the probability of drawing a red marble? 5. How many marbles are blue? What is the probability of drawing a blue marble? 6. How many marbles are yellow? What is the probability of drawing a yellow marble? 3
7. The probability of an event occurring is 8 . Which two values below represent this same probability? 0.375
Now try these: Lesson 6.2.1 additional questions — p458
375%
0.375%
37.5%
3 % 8
8. The probability of an event occurring is 55%. Which two values below represent this same probability? 55
5.5
0.55 100
55 100
11 20
9. In a tiled hallway, kids are jumping from one tile to the next. The probability of landing on a green tile is 60%. What fraction of the hallway area is covered with green tiles?
Round Up Probability is useful because you can use it to compare the chances of different events happening. The event with the highest probability is the most likely to occur. Section 6.2 — Theoretical Probability
323
Lesson
6.2.2
Expr essing Pr oba bility Expressing Proba obability
California Standard:
In this Lesson, you’ll learn how to find the probability of an event by looking at all the possible outcomes. There’s a handy formula you can use to do this.
Sta tistics ta Anal ysis Statistics tistics,, Da Data Analysis ysis,, and Pr oba bility 3.3 Proba obability Repr esent pr oba bilities as present proba obabilities opor tions tios,, pr propor oportions tions,, ra tios decimals betw een 0 and 1, between and per centa ges betw een 0 percenta centag between and 100 and vverify erify tha thatt the pr oba bilities computed ar e proba obabilities are r easona ble easonab le; know that if P is the probability of an event, 1–P is the probability of an event not occurring.
What it means for you: You’ll meet and use an important formula that will help you to work out probabilities.
You Can Calcula te Pr oba bilities b y Counting Outcomes Calculate Proba obabilities by There’s some special probability notation you’ll need to know. “P(A)” means “the probability that event A will happen.” So for tossing a coin, you can write “the probability of getting heads is as “P(Heads) =
1 .” 2
You can often find probabilities by thinking about the possible outcomes of the situation. P(event) =
Number of favorable outcomes Number of possible outcomes
Key words: • • • •
probability event outcome favorable
Favorable outcomes are the outcomes that match the event. Example
Another way to think of probability is as a ratio. The probability of an event is the ratio of favorable outcomes to possible outcomes.
Solution
Check it out:
So
This formula only works if all the possible outcomes are equally likely. For the actions in this Section, this is the case (for example, when you roll a die, all the different numbers are equally likely).
Don’t forget: It’s usually a good idea to write fractions in their simplest form. For example, a probability of 40 100
2
can be written as 5 — see Lesson 3.4.2 for more information.
324
1
Find the probability of rolling a 2 on a standard 6-sided die.
Check it out:
1 ” 2
The possible outcomes are: The favorable outcomes are:
1
2
3
4
5
6
2
There are 6 possible outcomes, and 1 favorable outcome. P(2) =
1 6
Guided Practice Write the answers to Exercises 1–7 as fractions. Jaden has a bag containing 100 marbles. There are 50 red marbles, 30 blue marbles, and 20 green marbles. He picks out one marble. Find the probability that the marble is: 1. Red 2. Blue 3. Green There are 50 socks in Jasmine’s drawer. There are 25 black socks, 10 blue socks, 10 orange socks, and 5 striped socks. Jasmine picks a sock without looking. Find the probability that the sock is: 4. Black 5. Blue 6. Orange 7. Striped
Section 6.2 — Theoretical Probability
Read the Description of an Ev ent Ver y Car efull y Event ery Carefull efully Always read the situation carefully. And then think very carefully too. Example Don’t forget: A standard pack contains 52 cards. There are 4 suits: hearts, clubs, spades, and diamonds. Hearts and diamonds are red, while clubs and spades are black. Each suit has 13 cards: the numbers from 1–10 (1 is usually called an “ace”), plus the jack, queen, and king (these are called “face cards”).
Don’t forget: A prime number can’t be divided evenly by any other numbers except itself and 1. (The number 1 isn’t prime.)
2
What is the probability of picking a queen out of a standard pack of 52 playing cards? Solution
There are 52 possible outcomes. The pack of cards contains 4 queens. So
P(queen) =
4 1 = 52 13
Guided Practice Find the probability of the following events when rolling a die once. Give your answers as fractions in their simplest form. 8. Rolling an odd number 9. Rolling an even number 10. Rolling a number more than 2 11. Rolling a prime number Find the probability of the following events when picking one card from a standard pack of 52. 12. Picking the jack of diamonds 13. Picking a seven 14. Picking a club 15. Picking a black card
Pr oba bility Can Gi ve You Inf or ma tion About Outcomes Proba obability Giv Infor orma mation If you know the probability of an event, you might be able to figure out numbers of outcomes. Example
3
The probability of picking a red marble out of a bag is 25%. There are 20 marbles in the bag. How many red marbles are there? Solution
P(red) = 25% =
Number of favorable outcomes 1 = Number of possible outcomes 4 1 Number of favorable outcomes There are 20 = marbles in total 4 20
Number of favorable outcomes = 20 ×
1 4
Multipl y both sides b y 20 Multiply by
=5 So there are 5 red marbles. Section 6.2 — Theoretical Probability
325
Guided Practice Exercises 16–21 are about picking one marble (without looking) from a bag full of marbles. Each exercise is about a different bag. 1
16. P(picking yellow) = 2 , and there are 11 yellow marbles. How many marbles are there altogether? 17. If there are 100 marbles, and P(picking red) = 10%, how many red marbles are there? 18. P(picking blue) = 25%, and there are 8 blue marbles. How many marbles are there in total? 1
19. P(picking black) = 3 . If there are 10 black marbles, how many marbles are there in total? 3
20. There are 80 marbles in the bag in total. P(picking silver) = 4 . How many silver marbles are in the bag? 21. There are 5 green marbles in the bag. What is the total number of marbles if P(picking green) = 20%?
Independent Practice 1. The winning contestant on a game show picks one of 80 boxes to find out what prize they win. 1 box contains $1,000,000 4 boxes contain vacation tickets 10 boxes contain cameras 15 boxes contain $10 20 boxes contain a pair of socks 30 boxes contain signed photos of the game show host Find the probability of winning each prize as a fraction. Sixty-four raffle tickets have been sold. Maria bought 5 of the tickets and her friend Kyle bought 10. Write the probabilities of the following events in fraction form:
Now try these: Lesson 6.2.2 additional questions — p459
2. Maria will win the raffle. 3. Kyle will win the raffle. 4. Someone other than Maria or Kyle will win the raffle. 5. If Maria wanted to have a 25% chance of winning, how many of the 64 tickets would she need to purchase? Destiny buys 20 tickets, so that 84 tickets have now been sold. 6. How does this affect Kyle's and Maria's chances of winning? Use percents rounded to the nearest whole percent to compare the probability of winning before and after Destiny's purchase.
Round Up Counting outcomes is fairly straightforward in simple situations like these. But for more complicated sets of events and outcomes, you’ll need to organize your information. 326
Section 6.2 — Theoretical Probability
Lesson
6.2.3
Counting Outcomes
California Standards:
Last Lesson, you saw how to find probabilities with a formula. But to use it, you need to know how many possible and favorable outcomes there are. Luckily, you learned two ways to do that a few Lessons ago. Now you’ll see just how useful tables, grids, and tree diagrams can be.
Repr esent all possib le present possible or compound outcomes ffor events in an or ganiz ed w ay org anized wa (e .g ., ta bles rids ee (e.g .g., tab les,, g grids rids,, tr tree dia g rams) and e xpr ess the diag expr xpress theor etical pr oba bility of theoretical proba obability eac h outcome each outcome.. Sta tistics ta Anal ysis Statistics tistics,, Da Data Analysis ysis,, and Pr oba bility 3.3 Proba obability Repr esent pr oba bilities as present proba obabilities opor tions tios,, pr propor oportions tions,, ra tios decimals betw een 0 and 1, between een 0 and per centa ges betw percenta centag between and 100 and vverify erify tha thatt the pr oba bilities computed ar e proba obabilities are reasona ble easonab le; know that if P is the probability of an event, 1–P is the probability of an event not occurring.
What it means for you: You’ll see how ways of counting outcomes such as grids and tree diagrams can help you work out probabilities of events.
Key words: • • • • • •
probability outcome event grid table tree diagram
Ta bles and Grids Help to Calcula te Pr oba bilities Calculate Proba obabilities You know that:
P(event) =
Number of favorable outcomes Number of possible outcomes
So if you can make an organized list of possible outcomes, it makes it easier to find probabilities. Example
1
Two dice are rolled. Find the probability that the sum of the numbers rolled is: (i) 8, and (ii) greater than 8. Solution
1st die rolled
(i) This table shows all the possible outcomes for the sum of the numbers rolled on two dice. All you need to do is count how many possible and favorable outcomes there are, and put them into the formula. There are 5 ways to roll 8, and there are 36 possible outcomes altogether. So, P(rolling 8) =
2nd die rolled
Sta tistics ta Anal ysis Statistics tistics,, Da Data Analysis ysis,, and Pr oba bility 3.1 Proba obability
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
5 Number of favorable outcomes = 36 Number of possible outcomes
Don’t forget:
(ii) The favorable outcomes for the event “rolling a sum greater than 8” are the outcomes 9, 10, 11, and 12. In total, this makes 10 outcomes.
The table used here is the same as the one from Example 2 of Lesson 6.1.3.
So, P(rolling over 8) =
5 Number of favorable outcomes 10 = = 36 18 Number of possible outcomes
Guided Practice Don’t forget: To find the probability of rolling a sum less than 6, count how many outcomes of 2, 3, 4, and 5 there are.
Use the table above to find the probability of each of the following events when rolling two dice: 1. Rolling a sum of 9 2. Rolling a sum of 10 3. Rolling a sum of 2 4. Rolling a sum less than 6 5. Rolling a sum of 17 6. Rolling a sum that is a prime number Section 6.2 — Theoretical Probability
327
The table below lists the number of students at a middle school by grade and gender.
Check it out: The easiest way to tackle Exercises 7–16 is to start each answer by writing a ?
fraction of the form 300 . Then you can convert to percents by multiplying by 100.
One student is chosen at random. Determine the probability of choosing each type of student listed in Exercises 7–16. Write each probability as a percent. 7. P(girl) 9. P(6th grader) 12. P(6th grade boy) 15. P(9th grader)
Boys
Gi r l s
To t al s
G r ad e 6
57
51
108
Gr ad e 7
42
57
99
Gr ad e 8
45
48
93
To t al s
144
156
3 00
8. P(boy) 10. P(7th grader) 11. P(8th grader) 13. P(7th grade girl) 14. P(8th grade boy) 16. P(6th, 7th, or 8th grade student)
Tree Dia g rams Can Sho w You All P ossib le Outcomes Diag Show Possib ossible Tree diagrams can help you find probabilities too. Example
2
A coin is tossed three times. Find the probability of getting tails on exactly two of the three tosses. Solution
This tree diagram shows all the possible outcomes for tossing a coin three times.
Coin toss 1 H
T
Coin toss 2 H
H
T
There are 8 possible outcomes.
T
Coin toss 3 H
Outcomes
So
HHH
T
H
HHT HTH
T
H
HTT THH
P(exactly 2 tails) =
T
H
THT TTH
T
TTT
There are 3 outcomes that have exactly two tails: HTT, THT, and TTH.
3 Number of favorable outcomes = 8 Number of possible outcomes
Guided Practice Write your answers to Exercises 17–21 as decimals. Use the tree diagram above to find the probability of each of the following events for tossing a coin three times: 17. Heads on exactly 2 tosses 18. Tails on 2 or more tosses 19. Tails on fewer than 2 tosses 20. The coin landing on the same side all 3 tosses 21. The first and third coins landing on the same side 328
Section 6.2 — Theoretical Probability
Independent Practice Exercises 1–12 are about the color spinner shown below. The spinner is spun once. Find the following probabilities. 1. P(green) 2. P(yellow) 3. P(white) 4. P(not purple) 5. The spinner is spun twice. Make a tree diagram or a table that shows all the possible outcomes. Use the abbreviations R for red, Y for yellow, G for green, and B for blue. Use your diagram from Exercise 5 to answer Exercises 6–12. 6. How many possible outcomes are there? 7. How many ways can you get 2 spins of the same color? Find the following probabilities. Write your answers as fractions in their lowest terms. 8. P(2 spins the same color) 9. P(one blue and one green in any order) 10. P(blue first, then green) 11. P(exactly one spin is red) 12. P(at least one spin is red) Don’t forget: The product of two numbers is the result of multiplying them together.
Contestants on a game show spin a wheel twice. If the product of the numbers is odd, they win a prize. The numbers on the wheel are 1, 2, 4, 5, and 6. 13. Make a table with a grid that shows all the possible products from any two spins of the wheel. 14. What is the probability that a contestant will win? 15. What is the probability that a contestant will lose? The wheel in Exercises 13–15 had two odd numbers and three even numbers on it.
Now try these: Lesson 6.2.3 additional questions — p459
16. Design a new wheel that has an equal number of odd and even numbers on it. 17. Use your new wheel. Determine which probability is greater, P(even product) or P(odd product). 18. If your wheel in Exercise 16 had a different number of integers on it (but half of those integers were still even, with the rest being odd), would your answer to Exercise 17 have been different? Explain your answer.
Round Up The most important use of tree diagrams and tables of outcomes is in finding probabilities. Look back at Lessons 6.1.1–6.1.3 if you need a reminder about anything to do with listing possible outcomes. Section 6.2 — Theoretical Probability
329
Lesson
6.2.4
California Standards: Sta tistics ta Anal ysis Statistics tistics,, Da Data Analysis ysis,, and Pr oba bility 3.3 Proba obability Repr esent pr oba bilities as present proba obabilities opor tions tios,, pr propor oportions tions,, ra tios decimals betw een 0 and 1, between and per centa ges betw een 0 percenta centag between and 100 and vverify erify tha thatt the pr oba bilities computed ar e proba obabilities are reasona ble w tha easonab le; kno know thatt if P is the pr oba bility of an proba obability event, 1– oba bility 1–P is the pr proba obability of an e vent not occur ring ev occurring ring..
What it means for you: You’ll learn how to find the probability that a particular event does not happen.
Pr oba bility of an Ev ent Proba obability Event Not Ha ppening Happening The probabilities you’ve seen so far represent the chances that an event will happen. In this Lesson, you’ll learn how to find the probability that an event doesn’t happen. You’ll be pleased to know that the math you’ll need to do isn’t much different from what you’ve been doing so far in this Section.
Find Pr oba bilities b y Counting Outcomes Proba obabilities by So far you have worked out probabilities of events happening. You can also find the probability that an event doesn’t happen by counting the number of outcomes that don’t match the event. Example
1 Ruben spins this spinner.
Key words: • • • •
Find the probability that Ruben spins the color yellow. What is the probability that Ruben does not spin the color yellow?
probability event outcome favorable
Solution
There are 3 possible outcomes: red, yellow, and blue. 1 3 If he doesn’t spin yellow, Ruben must spin either red or blue. So there are 2 favorable outcomes for not spinning yellow.
So
P(Yellow) =
So
P(Not yellow) = P(Red or blue) =
2 3
Guided Practice Check it out:
In Exercises 1–8, determine each probability for one spin of the spinner shown below. Give your answers as simplified fractions.
In some situations, P(not event A) can also be written as P(event B). For example, for tossing a coin, P(not tails) is the same as saying P(heads).
330
Section 6.2 — Theoretical Probability
1. P(Green) 3. P(Yellow) 5. P(Red) 7. P(Orange)
2. P(Not green) 4. P(Not yellow) 6. P(Not red) 8. P(Not orange)
You Can Find P(not A) b y Counting Outcomes by
Two events are said to be complementary if between them they cover all the possible outcomes and if only one of them can happen. So “A” and “not A” are complementary (since an event has to either happen or not happen). “Not A” is called the complement of “A.”
Example
2
Mario is playing a game using the spinner shown below. He spins it twice, and adds the numbers he spins to get his score.
1
3
7
5
If A is the event “Mario scores 8,” then find: (i) P(A), (ii) P(not A).
What do you notice about P(A) + P(not A)? Check it out: If event A is “Mario scores 8,” then event not A is “Mario does not score 8.”
2nd spin
This table gives the possible outcomes.
1st spin
Check it out:
You can use the probability that event A will happen to find the probability that A won’t happen.
Solution
1
3
5
7
1
2
4
6
8
3
4
6
8
10
5
6
8
10
12
7
8
10
12
14
(i) The table shows there are 16 possible outcomes. And there are 4 possible outcomes where Mario does score 8. 1 4 So P(Mario scores 8) = = (or 25%). 16 4 (ii) But if there are 4 possible outcomes where Mario does score 8, then there must be 16 – 4 = 12 outcomes where Mario does not score 8. 12 3 = (or 75%). So P(Mario does not score 8) = 16 4 Notice that P(A) + P(not A) = 1 (or 100%).
You Can Also Find P(not A) If You Kno w P(A) Know Check it out:
The result about P(A) and P(not A) adding up to 1 is always true. P(event A happening) + P(event A not happening) = 1 = 100%
You can write this rule as: P(A) + P(not A) = 1.
Example
3
The weather channel says there is a 30% chance of rain today. What is the probability that it will not rain today? Solution
so
P(rain) + P(not rain) = 100% 30% + P(not rain) = 100%
This means P(not rain)
= 100% – 30% = 70%
Guided Practice Exercises 9–16 give P(A). Find P(not A) in each case. 9. P(A) =
3 4
13. P(A) = 0.9
10. P(A) = 0.58 11. P(A) = 11% 12. P(A) = 43% 14. P(A) =
11 12
15. P(A) = 1
16. P(A) = 0
Section 6.2 — Theoretical Probability
331
Guided Practice Find the following probabilities for Mario’s game from Example 2. Write your answers as decimals 17. P(Mario scores 14) 18. P(Mario doesn’t score 14) 19. P(Mario scores 6) 20. P(Mario doesn’t score 6) 21. P(Mario scores less than 10) 22. P(Mario doesn’t score less than 10) 23. P(Mario scores 7) 24. P(Mario doesn’t score 7) 25. Jenna rolls a die. Event A is “rolling a number greater than 4.” Event B is “rolling a number less than 4.” Jenna claims that P(event A) + P(event B) = 1. Explain to Jenna why this is not true.
Independent Practice 1. The probability of an event occurring is 0.6. Explain why the probability of the event not occurring cannot be 0.2. 2. P(A occurring) and P(A not occurring) are the same. Find P(A). 3. P(A occurring) is twice P(A not occurring). Find P(A). Cynthia picks one card from a standard pack of 52 cards. She makes a note of the suit, then replaces the card and picks another one. The tree diagram below shows the possible outcomes. Card 1
Card 2 Outcomes
Now try these: Lesson 6.2.4 additional questions — p459
DD DC DH DS CD CC CH CS HD HC HH HS SD SC SH SS
D, C, H, and S represent diamonds, clubs, hearts, and spades. Calculate P(A) and P(not A), where event A is Cynthia picking the following: 4. At least one heart 5. Two diamonds 6. At least one red card 7. One heart and one club 8. Two cards the same suit
Round Up The rule P(A) + P(not A) = 1 is very useful and important. If you need to find a complicated-looking P(not A), the first thing you should think is “Is it easier to find 1 – P(A)?” The answer is quite often yes. 332
Section 6.2 — Theoretical Probability
Lesson
6.2.5
Venn Dia g rams Diag
California Standard:
It’s often tricky to figure out in your head how different events and outcomes are related. A Venn diagram is a way to show how different events are related, and they can make probabilities easier to visualize.
Sta tistics ta Anal ysis Statistics tistics,, Da Data Analysis ysis,, and Pr oba bility 3.1 Proba obability Repr esent all possib le present possible or compound outcomes ffor events in an or ganiz ed w ay org anized wa (e.g., tables, grids, tree diagrams) and express the theoretical probability of each outcome.
A Venn Dia g ram Is a Way to R epr esent Ev ents Diag Re present Events One outcome will often match more than one event. You can show situations where one or more outcomes match more than one event using a Venn diagram.
What it means for you: You’ll learn about Venn diagrams, which are useful in helping you to understand how different events relate to each other.
All the possible outcomes are inside the rectangle.
Event A
Event B
The circles represent events. All the outcomes that match an event are inside that event’s circle.
Key words: • Venn Diagram • outcome • event
The area where two circles overlap contains all the outcomes that match both events.
The next example should make the usefulness of a Venn diagram clearer. Example
1
The following are two events for rolling a die once: Event A: Rolling an even number Event B: Rolling a number less than 4 Use a Venn diagram to show how many outcomes match both events. Solution
The rectangle represents all possible outcomes. This means rolling 1, 2, 3, 4, 5, or 6. The blue circle represents event A, rolling an even number.
The red circle represents event B, rolling a number less than 4. The outcome “rolling a 2” is in both circles. The circles have to overlap, so that 2 can be in both at the same time. There is 1 outcome that matches both event A and event B. There is also 1 outcome (5) that matches neither event A nor event B. Section 6.2 — Theoretical Probability
333
Guided Practice Andres picks a card from a standard pack. Event A is “picking a spade.” Event B is “picking an ace.” Check it out: Many Venn Diagrams don’t show all the actual outcomes on the diagram. They often just show the circles representing the events.
1
In which section of this Venn diagram do the following outcomes belong? 1. Ace of clubs 2. King of hearts 3. Ace of spades 4. Three of spades
A B 2 3 4
5. Sketch a Venn diagram showing the events below if an integer from 1 to 25 is picked at random. Place the integers 1 to 25 in the correct areas of the Venn diagram. Event A: the number picked is a multiple of 4 Event B: the number picked is a multiple of 6
The Cir c les on a Venn Dia g ram Don ways Ov er la p Circ Diag Don’’ t Al Alw Over erla lap Venn diagrams can show some other situations: The circles don’t overlap at all if no outcomes match both event A and event B. Example
B
A
2
Draw a Venn diagram showing the following events for rolling one die: Event A: Rolling an even number Event B: Rolling an odd number Solution
The outcomes 2, 4, and 6 match event A. The outcomes 1, 3, and 5 match event B.
B
A
No outcomes match both events, so the circles don’t overlap.
In this diagram, all the outcomes matching event B also match event A. Some outcomes match A, but not B. Example
A
B
3
Draw a Venn diagram showing the following events for rolling one die: Event A: Rolling an odd number
Event B: Rolling less than 6
Solution A
B
The outcomes 1, 3, and 5 match event A. The outcomes 1, 2, 3, 4, and 5 match event B. All the outcomes matching event A also match event B. The circle representing event A is completely inside the one for event B.
334
Section 6.2 — Theoretical Probability
Guided Practice Use the Venn diagrams below to answer Exercises 6–11 1.
2.
A B
3.
A
B
4.
A B
A
B
Which diagram could show each of the following pairs of events for picking a number at random from 1 to 100? 6. Event A: odd Event B: even 7. Event A: less than 50 Event B: even 8. Event A: less than 50 Event B: less than 20 9. Event A: greater than 39 Event B: less than 74 10. Event A: greater than 86 Event B: less than 17 11. Event A: a multiple of 6 Event B: a multiple of 3
Independent Practice This Venn diagram shows two events when a number from 1 through 20 is chosen at random. 5 11 A B Use it to answer Exercises 1–6. Find how many outcomes match 1. event A 2. event B 3. both event A and event B 4. at least one of events A and B
Now try these: Lesson 6.2.5 additional questions — p460
14 20 6 18 9 19 4 8 15 16 12 1 10 2 3 7 13
17
5. Which of the following could be event A: A. multiple of 4 B. number less than 16
C. even number
6. Which of the following could be event B: A. multiple of 3 B. number less than 18
C. odd number
Sketch Venn diagrams for the following pairs of events. Place the integers from 1 to 12 in the correct areas of the Venn Diagram. 7. Event A: choosing an odd number Event B: choosing a multiple of 4 8.
Event C: choosing a number less than 6 Event D: choosing a prime number
9.
Event E: choosing a number greater than 4 Event F: choosing a multiple of 5
Round Up Venn diagrams often don’t give enough information on their own to figure out probabilities, but they can still be useful. In the next Lesson, you’ll see that when you combine events, a Venn diagram can help you to understand the situation better. Section 6.2 — Theoretical Probability
335
Lesson
6.2.6
Combining Ev ents Events
California Standards:
The words “and” and “or” are simple little words people use all the time, but in probability they’re very important. In this Lesson, you’ll see how they describe different probabilities when two events are combined.
Sta tistics ta Anal ysis Statistics tistics,, Da Data Analysis ysis,, and Pr oba bility 3.4 Proba obability
What it means for you: You’ll work out the probability of combinations of events.
Some probabilities combine two events. P(A and B) means “the probability that event A and event B both happen.” Example
1
This spinner is spun once.
10
Event A is spinning an even number Event B is spinning a number less than 7. Find P(event A and event B) Solution
Possible outcomes
1
Event A and Event B
3
2
Event A Event B
2
1
2 2
4
5
4 3
4
6
7
6 5
4
So P(A and B) = P(even number and less than 7) =
You can use a Venn diagram to help you understand what’s going on in Example 1.
7
1 4
3 = 30% 10
Exercises 1–4 use the spinner from Example 1. For one spin of the spinner, find P(event A and event B) for the following pairs of events: 1.
Event A: Spinning an odd number Event B: Spinning a number greater than 7
2.
Event A: Spinning a prime number Event B: Spinning a number less than 4
3.
Event A: Spinning a number greater than 2 Event B: Spinning a multiple of 3
4.
Event A: Spinning an odd number Event B: Spinning an even number
9 3
5
Event A is the green circle, event B is the purple circle. The outcomes that match both events are in the area where the circles cross.
336
10
Guided Practice
Check it out:
10 6
10
6
• • • • •
2
9
6
There are 10 possible outcomes. There are 3 outcomes that match both events A and B.
8
8 8
Key words: probability event outcome and or
1 2
3 4
Under stand tha Understand thatt the pr oba bility of either of tw o proba obability two disjoint e vents occur ring is ev occurring the sum of the tw o two indi vidual pr oba bilities and individual proba obabilities that the probability of one event following another, in independent trials, is the product of the two probabilities.
“And” Means “Both Ev ents Ha ppen” Events Happen”
8 9
Repr esent pr oba bilities as present proba obabilities opor tions tios,, pr propor oportions tions,, ra tios decimals betw een 0 and 1, between and per centa ges betw een 0 percenta centag between and 100 and vverify erify tha thatt the pr oba bilities computed ar e proba obabilities are r easona ble w tha easonab le; kno know thatt if P is the pr oba bility of an proba obability event, 1– oba bility 1–P is the pr proba obability of an e vent not occur ring ev occurring ring..
5 6 7
Sta tistics ta Anal ysis Statistics tistics,, Da Data Analysis ysis,, and Pr oba bility 3.3 Proba obability
Section 6.2 — Theoretical Probability
“Or” Means ““At At Least One Ev ent Ha ppens” Event Happens” Check it out: P(A or B) means the probability that either A happens, B happens, or they both happen.
P(A or B) means “the probability that at least one of event A and event B happens.” Example
2
This spinner is spun once.
10
3 4
8 9
Event A is spinning an even number Event B is spinning a number less than 7. Find P(event A or event B)
1 2
5 6 7
Solution
Possible outcomes Event A
Check it out: In this example, we are looking for the outcomes that are in either of the two circles in the Venn Diagram.
8 7
1
2
10 6
4
1
5
9 3
2
3
2
4
5
4
6 6
Event B
1
2
3
4
5
6
Event A or Event B
1
2
3
4
5
6
7
8
9
10
8
10
8
10
There are:
10 possible outcomes altogether, 5 favorable outcomes for event A, 6 favorable outcomes for event B. But you can’t add 5 to 6 to find the number of favorable outcomes for A or B. This is because 3 outcomes match both events, and you don’t count these twice. In fact there are 8 favorable outcomes — 2 match event A only, 3 match event B only, and 3 match both events. 8 So P(even number or less than 7) = = 80% 10
Guided Practice 5. Using the pairs of events from Exercises 1–4 for the spinner shown above, find P(event A or event B).
Venn Dia g rams Can Sho w You Ho w to Find P(A or B) Diag Show How You might find it helpful to use Venn diagrams to picture what happens when you combine outcomes. To find P(A), you count all the outcomes in circle A.
A
B
A
B
To find P(A and B), you count all the outcomes that are in both circles at the same time.
Section 6.2 — Theoretical Probability
337
Check it out: If the events have no outcomes in common, there is no overlap.
To find P(A or B) you count any outcomes that are in either circle, but you only count the outcomes in the overlap once. So
B
P(A or B) = P(A) + P(B) – P(A and B)
=
B
A
A
+
–
because P(A) + P(B) counts the overlap twice — once in A and once in B. So P(A and B) = 0 and P(A or B) = P(A) + P(B)
Check it out: You can use the rule above to show that the formula you met in the last Lesson is true: “P(A) + P(not A) = 1.” (i) P(A OR not A) = 1 (since an event either happens or it doesn’t). (ii) P(A AND not A) = 0 (since an event can’t both happen and not happen). Substitute these numbers into P(A or B) = P(A) + P(B) – P(A and B) using “not A” as event B. You get: 1 = P(A) + P(not A) – 0. Or 1 = P(A) + P(not A). The Venn diagram would look like this:
A
Guided Practice Aisha takes a standard pack of 52 playing cards. She picks one card at random. Calculate the following probabilities. 6. P(red) 7. P(5) 8. P(red and 5) 9. P(red or 5) 10. P(black) 11. P(face card) 12. P(black and a face card) 13. P(black or a face card) 14. P(hearts) 15. P(clubs) 16. P(hearts and clubs) 17. P(hearts or clubs) 18. For two events, A and B, P(A) = 25%, P(B) = 30%, and P(A or B) = 50%. Are there any outcomes that match both events? Explain your answer.
Independent Practice Bernardo picks one of the cards shown here at random. Calculate the following probabilities. Write your answers as decimals.
not A
Now try these: Lesson 6.2.6 additional questions — p460
1
2
3
4
5
6
11
12
13
14
15
16 17
1. P(odd) 2. P(even) 3. P(red) 4. P(blue) 6. P(odd and blue) 8. P(red and yellow) 10. P(yellow and even) 12. P(red and a multiple of 4) 14. P(yellow or less than 10)
7
8
9
10
18 19
20
5. P(yellow) 7. P(odd or blue) 9. P(red or yellow) 11. P(yellow or even) 13. P(blue and prime)
15. Sally spins the spinner shown in Examples 1 and 2. She calculates P(even or less than 5) = 90%. What mistake has Sally made? What is P(even or less than 5)?
Round Up The tricky thing to remember from this Lesson is what “or” means in probability. “A or B” doesn’t mean “either A or B” — it means “either A or B or both.” If you forget about that “or both” bit, you’ll wind up getting questions wrong. 338
Section 6.2 — Theoretical Probability
Section 6.3 introduction — an exploration into:
Pic k a Car d Pick Card You’re going to conduct two probability experiments with cards. The experiments are very similar, but there’s one big difference between them. As you do the Exploration, think about that difference, and the effects it has on your two sets of results. Before you do your two experiments, you need to create a table to record your results. Copy the table on the right. You need the Ace-King of Now you need some playing cards. any suit. You will need one complete suit of 13 playing cards. Shuffle them, and put them in a pile (with the cards face down).
A
A
Experiment 1 1) Choose one card from the pile. You can choose any card you like. 2) Record the result in your table. 3) Now put that card to one side. Do not put it back in the pile. 4) Repeat steps 1–3 until you have picked 13 times. There will be one less card to choose from each time. For your final pick, there will only be one card left.
Pick # 1 2 3 4 5 6 7 8 9 10 11 12 13
Experiment 1 Experiment 2 Card Card
Experiment 2 Use the same set of 13 cards. Shuffle them, and place them face down in a pile. 1) Choose one card from the pile. You can choose any card you like. 2) Record the result in your table. 3) Put the card back in the pile and reshuffle the cards. 4) Repeat steps 1–3 until you have picked 13 times. There will always be 13 cards to choose from.
Exercises 1. Did you pick the same card twice during either experiment? Explain why/why not. 2. Now think about the probabilities of choosing cards in Experiment 1. a. What was the probability you would choose the Ace on your 1st pick? b. If you had chosen the Ace on your 1st pick, what was the probability you would choose the Ace on your 2nd pick? And on the 3rd pick? On the 8th pick? On the 13th pick? 3. Now think about the probabilities of choosing cards in Experiment 2. a. What was the probability you would choose the Ace on your 1st pick? b. If you had chosen the Ace on your 1st pick, what was the probability you would choose the Ace on your 2nd pick? And on the 3rd pick? On the 8th pick? On the 13th pick?
Round Up This Exploration was all about how events can be dependent or independent — it’s a really important idea in probability. You’re going to learn much more about it in this Section. Section 6.3 Explor a tion — Pick a Card 339 Explora
Lesson
6.3.1 California Standard: Statistics, Data Analysis and Probability 3.5 Understand the difference between independent and dependent events.
Section 6.3
Independent and Dependent Events In Section 6.2, you came across the idea that you can figure out the probability that an event will happen. This Lesson is all about how different events can affect each other — how one event happening can affect the probability of another event happening.
What it means for you:
One Event Happening Can Affect Later Probabilities
You’ll learn the difference in math between events that don’t affect what happens later, and those that do.
A really important idea in probability is whether events are independent or dependent. Imagine Luis and Gabrielle both belong to a gym...
Key words: • • • •
independent events dependent events probability event
(i) Each morning when he wakes up, Luis decides completely at random whether he’ll go to the gym that day or not. Every day, the probability that Luis will go to the gym is 0.7. (ii) Gabrielle has a different system. If Gabrielle has been to the gym the previous day, she is less likely to want to go the next day too. If she went to the gym yesterday, the probability she will go to the gym today is 0.1. But if she didn’t go to the gym yesterday, then the probability she will go today is 0.8. Other events do not affect the probability that Luis will go to the gym, so this event is independent of others. However, the probability of the event “Gabrielle will go to the gym today” is dependent on the event “Gabrielle went to the gym yesterday.”
Check it out: In the gym example, Luis decided at random what to do. Events that happen at random are always independent.
Independent Events Two events are independent if one event happening doesn’t affect the probability of the other event happening. In other words, independent events have no influence on each other.
Example
1
Which of the following pairs of events are independent, and which are dependent? (i) “Throwing a head on a coin,” and “throwing a 6 on a die.” (ii) “Throwing a head on a coin on the first spin,” and “throwing a head on a coin on the second spin.” Solution
(i) The result of the coin toss doesn’t affect the result of throwing the die, so these events are independent. (ii) The result of the first coin toss doesn’t affect the result of the second toss, so these events are also independent.
340
Section 6.3 — Dependent and Independent Events
Guided Practice Check it out: For events to be independent, the result of the first should have no effect at all on the result of the second.
1. Mr. Garcia has an apple, a banana, and a peach. He randomly picks one piece of fruit and eats it. Then he picks another and eats it. Are the events “eats the apple first” and “eats the apple second” independent? Explain your answer. 2. Every day after school Dominic goes running, cycling, or swimming. He chooses randomly which activity to do. Are the events “Dominic runs on Monday” and “Dominic runs on Tuesday” independent? Explain your answer. 3. The basketball squad need to choose a captain and a co-captain. They pick a captain from the team. From the remaining players, they then select a co-captain. Are the events “Letisha becomes captain” and “Letisha becomes co-captain” independent? Explain your answer.
Coins and Spinners Give Independent Events Events based on tossing coins and spinning spinners are always independent. The result of one coin toss, or one spin on a spinner, doesn’t affect what happens on later tosses or spins. Example Check it out: The example talks about a fair coin. This means it’s just as likely to land on heads as tails. It’s not biased in any way.
2
You toss a fair coin three times. On your first toss, it lands on heads. What is the probability of the coin landing on heads with: (i) your second toss? (ii) your third toss? Solution
Coin tosses are always independent. This means that the result of one coin toss doesn’t affect the probability of any of the later results. Whenever you toss a fair coin, heads and tails are both equally likely. So for a fair coin, P(heads) = 0.5 and P(tails) = 0.5. (i) This means that for your second toss: P(heads) = 0.5. (ii) This also means that for your third toss: P(heads) = 0.5.
Example Check it out: Here, the events “spinner lands on blue” and “spinner lands on green” are not equally likely. However, they are independent — the result of one spin does not affect the probability of getting blue or green on later spins.
3
You spin the spinner shown on the right two times. The probability of the spinner landing on blue is 0.35. The first two spins land on blue. What is the probability that the third spin also lands on blue? Solution
The spins of a spinner are independent. So it doesn’t matter what the first two results were, the probability of it landing on blue with the third spin must be 0.35. Section 6.3 — Dependent and Independent Events
341
Guided Practice Say whether Events A and B in Exercises 4–7 are independent. 4. Kalisa tosses a coin twice. Event A: She tosses a head with her first throw. Event B: She tosses a tail with her second throw. 5. Jordan rolls a number cube twice. Event A: He throws a 2 on the first throw. Event B: He throws a number less than 3 on the second throw. 6. Madison spins the spinner shown two times and gets purple on both spins. He then spins the spinner two further times. Event A: He gets purple with his third spin. Event B: He gets purple with his fourth spin. 7. Ethan has two bags, each with 5 numbered chips in it. He picks one chip from each bag. Event A: He picks the chip numbered 2 from the first bag. Event B: He picks the chip numbered 2 from the second bag.
Independent Practice Explain your answer to each of Exercises 1–3.
Now try these: Lesson 6.3.1 additional questions — p460
1. Maria likes both fish and carrots, but she never eats them in the same meal. Are the events “eats fish for her evening meal” and “eats carrots with her evening meal” independent? 2. You spin a spinner numbered 1–10, then toss a coin, and then roll a number cube. Are the events “spin a 7,” “throw a tail,” and “throw a 4” independent? 3. There are 23 instruments. The 23 class members take it in turn to pick an instrument each to play. Are the events “the first student picks a drum” and “the second student picks a flute” independent? Say whether Events A and B in Exercises 4–5 are independent. 4. You toss a coin 10 times. Event A: the 9th toss is a head. Event B: the 10th toss is a head. 5. Ted has five different-colored pens. Each day, he picks one to take to school, but he never takes the same pen more than once in a week. Event A: He takes the red pen on Monday. Event B: He takes the blue pen on Tuesday.
Round Up If one event has no effect at all on the probability of another event, then they are independent of each other. It doesn’t mean that the probabilities have to be equal — remember the spinner in Example 3 — they just have to be unaffected by what’s gone before. There’s more about this next Lesson. 342
Section 6.3 — Dependent and Independent Events
Lesson
6.3.2 California Standard: Statistics, Data Analysis and Probability 3.5 Understand the difference between independent and dependent events.
What it means for you: You’ll learn how to recognize when events are affecting each other by looking at probabilities.
Key words: • • • • •
dependent event independent event outcome probability replacement
Don’t forget: You saw in Section 6.2 that the formula for calculating the probability of an event happening is: Favorable outcomes Probability = Possible outcomes
If you need a reminder, you can see more about this in Lesson 6.2.2.
Events and Probabilities In the last Lesson, you saw that some events affect what happens later, while others don’t. This Lesson talks about dependent and independent events in a bit more detail.
Probabilities of Independent Events Don’t Change To decide if events are independent, you can look at probabilities. If one event’s probability is affected by another event, they’re dependent. Example
1
There are 2 red marbles and 1 blue marble in a bag. You pick a marble, put it to one side, and then pick another. Show that the events “picking a red marble with the first pick” and “picking a red marble with the second pick” are dependent. Solution
Suppose you pick a red marble with your first pick. Then there is one red marble and one blue marble 1 left in the bag. So P(pick red with second pick) = . 2 Now suppose you don’t pick a red marble with your first pick. This means there are two red marbles left in the bag. So P(pick red with second pick) = 1. Whether the first event happens or not affects the probability of the second event. So the two events are dependent. This next example is identical, except the first marble is put back. Example
2
There are 2 red marbles and 1 blue marble in a bag. You pick a marble, put it back, and then pick another. Show that the events “picking a red marble with the first pick” and “picking a red marble with the second pick” are independent. Solution
Suppose you pick a red marble with your first pick, then put it back. There will still be two red marbles and one blue 2 marble in the bag. So P(pick red with second pick) = . 3 Now suppose you pick the blue marble with your first pick, then put it back. There will still be two red marbles and one
2 . 3 Whether the first event happens or not does not affect the probability of the second event. So the two events are independent.
blue marble in the bag. So P(pick red with second pick) =
Section 6.3 — Dependent and Independent Events
343
Guided Practice In Exercises 1–3, calculate the probability of Event B twice: (i) when Event A has happened, and (ii) when Event A did not happen. Use your answers to decide whether Events A and B are dependent. 1. Calvin has a bag with six counters in it, numbered 1–6. He pulls out a counter and puts it to one side. Then he pulls out another counter. Event A: He pulls out the counter numbered 1 on his first pick. Event B: He pulls out the counter numbered 1 on his second pick. 2. Madison spins the spinner shown on the right 2 times. Event A: She spins purple with her first spin. Event B: She spins green with her second spin. 3. Jordan rolls a number cube twice. Event A: He gets a 4 with his first roll. Event B: He gets an odd number with his second roll.
Selection with Replacement Gives Independent Events
Don’t forget: Examples 1 and 2 also show how replacing (or not replacing) the first item affects probabilities for the second pick.
A lot of the probability situations you’ll meet in this grade are connected with selecting various objects at random. Once you’ve made your first pick, there are two things you can do before you make your second pick: (i) you can put the first object back, or (ii) you can make your second pick without putting the first pick back. Putting the first pick back means your two selections are independent. Example
3
Kalisa picks a card from a standard deck, and then puts it back. She then picks another card. Show that the events “picking a red card with her first pick,” and “picking a black card with her second pick,” are independent. Solution
Since Kalisa has replaced her first card, it doesn’t matter what she got with her first pick. Either way, the deck will contain exactly the same cards for her second pick. Since there are 52 cards in total, and 26 of them are black, the probability that she gets a black card with her second pick is: 26 1 = P(black) = 52 2
26
26
If the first card is not put back, then the deck has changed for the second pick. This means that the probabilities for the second pick will change. So not replacing your first pick will always lead to dependent events. 344
Section 6.3 — Dependent and Independent Events
Example
4
Kalisa picks a card from a standard deck, and then puts it to one side. She then picks another card. Check it out: If the first selection is not replaced, this affects what you can choose from for your second pick. This means probabilities for your second pick are affected by what happens with your first pick, which makes events dependent.
Show that the events “picking a red card with her first pick,” and “picking a black card with her second pick,” are dependent. Solution
Suppose Kalisa picks a red card with her first pick. For her second pick, there will now be 51 cards 26 — 26 black and 25 red. So P(black) = . 51 But if she picks a black card with her first pick, then for her second pick, there will still be 51 cards
26
25
25
26
25 . 51 As these probabilities are different, the two events must be dependent.
— but this time, 25 black and 26 red. So P(black) =
Guided Practice Don’t forget: Vowels are A, E, I, O, and U. All other letters are consonants.
4. Kadema has 26 cards in a bag, each marked with a different letter of the alphabet. He picks one letter, puts it to one side, then draws out a second letter. Show that the events “picks a vowel with first pick” and “picks a consonant with second pick” are dependent. In Exercises 5–6, say whether the selections are independent. 5. Melisenda picks a four-digit combination for her school locker. She chooses each digit at random. 6. Corran also picks a four-digit combination for his locker. He decides not to use each digit more than once.
Independent Practice 1. Necie has a bag with six counters in it, numbered from 1 to 6. He pulls out a counter, keeps it and then pulls out another. By calculating probabilities, show that the events “picks odd number with first pick” and “picks even number with second pick” are dependent. Now try these: Lesson 6.3.2 additional questions — p461
In Exercises 2–3, say whether the selections are dependent. 2. Each month, the class picks a different student to be a hall monitor. 3. Jessica is holding three long straws and one short straw. The ends are hidden so that nobody can see which is the shortest. One at a time, Jessica’s four friends draw a straw until Jessica has no straws left. 4. When you pick items at random from a selection, why do they have to be put back in for the events to be independent?
Round Up To decide whether events are dependent or independent, you have to look at probabilities. But... there’s sometimes a shortcut — selecting without replacement always leads to dependent events. Section 6.3 — Dependent and Independent Events
345
Lesson
6.3.3
Calculating Probabilities of Independent Events
California Standard: Statistics, Data Analysis, and Probability 3.4 Understand that the probability of either of two disjoint events occurring is the sum of the two individual probabilities and that the probability of one event following another, in independent trials, is the product of the two probabilities.
What it means for you: You’ll see that there’s a formula you can use to figure out the probability that two events will both happen.
You’ve already seen in Section 6.2 how to figure out the probability that an event will happen. And you saw in the last Lesson how you can use those probabilities to show that two events are independent. In this Lesson, you’ll see a formula to work out the probability that two independent events happen in a row.
You Can Show Outcomes for Two Actions in a Table Think about rolling two dice. What’s the probability of rolling two numbers higher than 3? Example
3
1 5
1
You have a red die and a blue die. Draw a table showing all the possible outcomes of rolling these two dice. Shade the outcomes that show rolling two numbers higher than 3. What is the probability of rolling higher than 3 on both dice? Solution
• • • •
First draw your table. Blue die
Key words: independent event probability outcome
2
Don’t forget:
1 2 3 4 5 6
The purple boxes show outcomes greater than 3 on both rolls.
(1,1)(1,2) (1,3)(1,4) (1,5) (1,6) (2,1)(2,2) (2,3)(2,4) (2,5) (2,6) (3,1)(3,2) (3,3)(3,4) (3,5) (3,6) (4,1)(4,2) (4,3)(4,4) (4,5) (4,6) (5,1)(5,2) (5,3)(5,4) (5,5) (5,6) (6,1)(6,2) (6,3)(6,4) (6,5) (6,6)
}
Bottom 3 rows show a number greater than 3 on blue die.
}
Example 1 uses the same method as Section 6.2.
Red die 1 2 3 4 5 6
3 columns on the right show a number greater than 3 on red die.
The probability of rolling greater than 3 on both dice is: P(red die > 3 and blue die > 3) =
# favorable outcomes 9 1 = = # possible outcomes 36 4
Guided Practice 1. How many ways can you score over 3 using a single die? 2. How many ways can you get two scores over 3 using two dice? 3. What is the probability of scoring over 3 using a single die? 4. What is the probability of getting two scores over 3 using two dice? 5. When you roll two dice, are the events “roll over 3 on die 1” and “roll over 3 on die 2” independent? Justify your answer. 6. What do you notice about your answers to Exercises 1 and 2? And what do you notice about your answers to Exercises 3 and 4? 346
Section 6.3 — Dependent and Independent Events
Multiply Probabilities If Events Are Independent There’s actually a quicker way to work out the probability in Example 1. It means you don’t have to draw a table.
The probability of getting more than 3 on the blue die is
Don’t forget: 1 P(get over 3 on a die) = . 2
blue die
red die 1 2 3 4 5 6 1 2 3 4 5 6
1 . 2
blue die
To begin with, just imagine the table showing all the possible results for red die throwing the two dice. 1 2 3 4 5
6
1 2 3 4 5 6
The probability that any of these rolls will be 1 followed by more than 3 on the red die is also 2 .
To find the fraction of the possible outcomes that show over 3 on both dice, you’ve found
1 2
of
1 . 2
over 3 on both dice must be
This means that the probability of getting 1 1 1 × = 2 2 4
— you’ve multiplied probabilities.
In fact, for any two independent events...
P(A and B) = P(A) × P(B) Don’t forget: These events are independent. Look back to Example 1 of Lesson 6.3.1 if you don’t remember what this means.
Example
2
What is the probability of: (i) Throwing a head on a coin, and then throwing a 6 on a die? (ii) Throwing a head on a coin on the first toss, and then throwing another head on the coin on the second toss? Solution
1 1 , and P(6 on a die) = . 2 6 1 1 1 So P(head and a 6) = × = 2 6 12 1 1 1 1 (ii) P(head on a coin) = , so P(a head on two tosses) = × = 2 2 2 4 (i) P(head on a coin) =
Sometimes, you have to use the formula the “wrong way around.” Example
3
The probability of a spinner landing on red is 0.25. The probability of spinning red and then green is 0.0625. What is the probability of spinning green? Solution
Use a variable (x, say) for the probability of the spinner landing on green. This means P(green) = x, and P(red) = 0.25. These events are independent, so P(red, then green) = 0.25x. But you also know that P(red, then green) = 0.0625. Put these together and you find that 0.25x = 0.0625. So x = 0.0625 ÷ 0.25 = 0.25 Divide both sides by 0.25 to find x. So the probability of spinning green is 0.25. Section 6.3 — Dependent and Independent Events
347
Guided Practice Exercises 7–14 are about the bag of marbles on the right. You draw a marble from the bag shown, note its color, and replace it. Then you draw another one. 7. Will the result of the first pick affect the probability of picking a particular color with the second pick? Explain your answer. 8. Are the two draws independent or dependent? In Exercises 9–14, find the probabilities of the given events. 9. P(pick green) 10. P(pick blue) 11. P(pick red) 12. P(pick green, then red) 13. P(red, then green) 14. P(blue, then red, then green) Check it out: In Exercise 15, you know P(A and B), and you have to find P(B).
15. Pepe picks a hat and scarf to wear at random from his drawer. The probability he picks a red hat is 0.4. The probability he picks a red hat and a blue scarf is 0.1. Find the probability he picks a blue scarf.
Independent Practice In Exercises 1–6, find the probability of rolling the numbers shown on two number cubes. 1. P(6 and 6) 2. P(1 then 6) 3. P(even number, then odd number) 4. P(odd number, then less than 3) 5. P(less than 6, then less than 2) 6. P(more than 2, then less than 4) 7. Noah picks a red card from a standard deck. He keeps it and picks 1 1 1 another, which is also red. He says the probability of this is 2 × 2 = 4 . Is he correct? Explain your answer. 8. Ceri draws a marble from one bag, and a shape from another bag. The probability of picking a green marble and then a square is 5
Now try these: Lesson 6.3.3 additional questions — p461
5 . 24
If the probability of picking a green marble is 6 , what is the probability of picking a square? 9. Dominique has a bag of nine marbles, of which two are red. She draws a blue marble, puts it back, and then draws a red marble. 8 The probability of this combination of events was 81 . Calculate the number of blue marbles in the bag.
Round Up If two events are independent, then you can find the probability that they both happen by multiplying probabilities together. This can save you lots of time, because it means you don’t have to draw tables or tree diagrams. 348
Section 6.3 — Dependent and Independent Events
Section 6.4 introduction — an exploration into:
Shooting Bask ets Baskets You can use the past to predict the future — that much is almost common sense. But doing it accurately is where you need math. That doesn’t mean you can say for sure exactly what’s going to happen in the future, but it does mean you don’t have to make a wild guess. For this Exploration you need to work in teams. Your team is going to try to score “baskets” by throwing balls of paper into the trashcan from: (i) 2 meters, and (ii) 5 meters. First, the team will take a total of 20 shots from each distance. Before starting to throw, everyone has to make a prediction of how many shots the team will make. Record your predictions in a table like this one.
So if there are 4 people in your team, each person takes 5 shots shots.
Distance from trash can 2 meters 5 meters
Prediction for 20 shots
Actual result for 20 shots
The team should now take 20 shots from each distance. Record the results in your table.
Exercises 1. Calculate: a. the fraction of baskets that you predicted the team would make, Divide the number of baskets predicted/made by 20 20. b. the fraction of baskets that the team actually made. 2. How close was your prediction?
Subtract the percent made from the percent predicted predicted.
The team is now going to take 30 extra shots from: (i) 2 meters, and (ii) 5 meters. Before throwing, each person must again predict Prediction for Actual result how many baskets the team will make. 30 shots for 30 shots Use your data from the first 20 shots at each distance to help. Extend your table to record your predictions and the results. The team should now take 30 shots from each distance. Record the results in your table.
Exercises 3. Explain how you used the data from the first 20 shots to make predictions for 30 shots. 4. Subtract the fraction made from the fraction predicted to see how good the predictions were. 5. How many shots would you predict the team would make if it took 100 shots? You can use the data from your tables. Explain how you made your estimate this time.
Round Up Did the data from the first 20 shots help you to make a better prediction of what would happen on the next 30? Even if you didn’t predict exactly what would happen on the 30 shots, making the second prediction was hopefully easier than the first (which was probably just a guess). Section 6.4 Explor a tion — Shooting Baskets 349 Explora
Lesson
6.4.1
Relative Frequency
California Standard:
In Section 6.2, you saw how to work out the probability that an event will happen by looking at how likely different possible outcomes are. Relative frequency is another kind of probability measure — but it’s based on looking at outcomes from the past.
Statistics, Data Analysis and Probability 3.2 Use data to estimate the probability of future events (e.g., batting averages or number of accidents per mile driven).
What it means for you: You’ll see how to use the results of experiments that you’ve already done to predict the results of future experiments.
Relative Frequency Shows Past Results If you repeat an action such as tossing a coin or spinning a spinner a number of times, you can look at how often one particular outcome occurs. This is called the relative frequency of the outcome. You calculate relative frequency by dividing the number of times you saw one outcome by the number of times that you did the experiment. Relative Frequency:
Key words: • • • •
Relative Frequency =
relative frequency experimental probability outcome trial
Example Check it out: The word “trial” means an action. So tossing a coin once is one trial, tossing a coin twice is two trials, and so on. The word “experiment” can also be used.
number of times outcome occurred number of trials
1
You toss a coin 100 times. It lands heads 47 times, and tails 53 times. Find the relative frequency of the coin landing on tails. Solution
number of times outcome occurred number of trials 53 = 100
Relative Frequency =
Check it out: Like a probability, relative frequency can be a fraction, a decimal, or a percentage. So you could write the answer to Example 1 as 53%.
53 100
Example
2
You roll a number cube 500 times. You get the following results. Score
1
2
3
4
5
6
Frequency
89
82
85
77
88
79
, 0.53, or
Calculate the relative frequency of each score to 3 decimal places. Solution
Check it out: If you work out the relative frequencies of all the possible results, then they should add up to 1 — just like probabilities.
350
To calculate a score’s relative frequency, divide the frequency by the total number of rolls. 89 For example, the relative frequency of 1 is = 0.178 500 Doing the same for the other scores gives the following: Score
1
2
3
4
5
6
Relative Frequency
0.178
0.164
0.170
0.154
0.176
0.158
Section 6.4 — Experimental Probability
Guided Practice Find the relative frequencies described in Exercises 1–3. 1. You toss a coin 56 times. Heads comes up 23 times. Find the relative frequencies of heads and tails. 2. A 3-section spinner was spun 99 times. Blue came up 37 times, red 58 times, and green the remainder. Find the relative frequency of each color. 3. For a probability experiment, Tim picked a card from a standard deck, recorded the suit, and then replaced the card. He then repeated this until he had picked a total of 200 times. His results are shown in the table. Find the relative frequency of each suit.
Hearts
58
Diamonds
43
Clubs
51
Spades
48
Relative Frequency Can Be Used as a Probability Check it out: A theoretical probability is one that you work out by comparing how likely it is that each possible outcome of a trial will happen. You can say that, since a coin has two sides, the theoretical probability of a tail is 0.5. An experimental probability is one that you calculate by working out how often an outcome has occurred in past experiments — its relative frequency. You can say that if you flip a coin 100 times and get 52 tails, the experimental probability of a tail is 0.52.
Relative frequency can be used to estimate probability. When you do this, you’re using the past to predict the future. This kind of probability is called an experimental probability. Example
3
The spinner on the right was spun 1000 times. The results are shown in the table. Use these results to estimate the probability of the next spin landing on: (i) red, (ii) green. Solution
Red
514
Green
2 21
Relative frequency can be used as a probability. Blue 265 (i) Use the relative frequency of red. So the experimental probability of red is 514 ÷ 1000 = 0.514. (ii) Use the relative frequency of green. So the experimental probability of green is 221 ÷ 1000 = 0.221. Relative frequency can be used as a probability in all sorts of situations. Example
4
In a medical study of 1000 people suffering from a particular condition, 630 patients saw an improvement after taking a new drug. Based on these results, estimate the probability that another patient with the same condition will see an improvement after taking the new drug. Solution
Use relative frequency as a measure of probability. Relative frequency =
Number of people improved 630 = = 0.63 Number of people in study 1000
P(another patient improving) = 0.63 Section 6.4 — Experimental Probability
351
Guided Practice 4. A school is looking at the data concerning where its pupils live. The school has 1121 pupils altogether, and 267 of them come from Town A. Based on this, what is the probability that the next student to enroll at the school will live in Town A? 5. A school wants to start a new sports club. It takes a poll of the students to see which sport is most popular. The numbers of votes so far for each sport are shown in the table on the right. One more student is still to vote. Use the results to find the probability this student will vote for each sport.
Spor t
Votes
Tae kwon do 219 Lacrosse
11 3
Surfing
37 5
6. The Green Club asks members how many aluminium cans their families had bought and how many of these they had recycled, during the past month. They found that 247 cans had been bought, of which 229 had been recycled, the rest thrown away. Estimate the probability that the next can bought by a member will be thrown away.
Independent Practice 1. A spinner is spun 100 times, and lands on red 27 times. Calculate the relative frequency of landing on red. 2. You plan to toss a coin 25 times. What should you do to find the relative frequency of heads? Don’t forget: The frequency of an outcome is just how many times it happens. So if you see 5 red cars in an hour, you can say that the frequency of red cars in your survey is 5.
Now try these: Lesson 6.4.1 additional questions — p461
3. Peppa spins a triangular spinner 20 times. It lands on red 7 times and lands 5 times on blue. If Peppa spins the spinner again, estimate the probability that it will land on its third side – green. 4. Pepe does a survey of the color of cars that pass his house in one hour. His data is shown in the table on the right. Calculate the relative frequency of each car color.
Car Color
Frequency
Red
23
Blue
7
Silver
16
5. A factory randomly tests its goods for faults. The whole day’s output is tested if the probability of getting a faulty item is greater than 0.02. On day A, 21 items out of 948 randomly tested were faulty. On day B, 14 out of 999 tested were faulty. Does the whole day’s output for either of these days need to be tested again? If yes, then which day(s)? Color
Frequency Relative Frequency
Green
12%
Orange
60%
Purple
28%
6. A 3-section spinner was spun 25 times, giving the results shown in the table on the left. Fill in the frequency of each color.
Round Up You can use relative frequency as a measure of how likely something is to happen. And you can use it for events where you couldn’t work out a probability in other ways. It’s a way of using your experience of the past to predict an outcome in the future. You’ll see more about all this in the next Lesson. 352
Section 6.4 — Experimental Probability
Lesson
6.4.2
Making Predictions
California Standard:
Experimental probability is all about using the relative frequency of an outcome to work out the probability that the same outcome will happen again. Like all probability, it’s about making predictions.
Statistics, Data Analysis and Probability 3.2 Use data to estimate the probability of future events (e.g., batting averages or number of accidents per mile driven).
What it means for you: You’ll see how to use the results of experiments that you’ve already done to predict the results of future experiments.
Key words: • • • •
experimental probability theoretical probability relative frequency estimate
Check it out: If a die is “biased,” it means it doesn’t behave like a die should — for example, maybe it comes down on “1” much more often than you’d expect. In the same way, a coin that landed on heads 80 times out of 100 is probably biased — you’d expect about 50 heads, and 80 is a lot more than this. A coin/die that isn’t biased is usually described as either “fair” or “unbiased.”
Don’t forget: This is the same as with theoretical probability — multiply the number of trials by an event’s probability to predict the number of times that event will happen. Experimental probability is useful in areas like business and sports — where theoretical probability is tricky to calculate and lots of old data is available.
Create Your Own Data to Estimate Probability You can use experimental probability to predict the outcome of an experiment when you can’t use theoretical probability. For example, using experimental probabilities might be the only way you can make predictions about a biased die. Example
1
Pepe has an odd-shaped die, which he thinks will be biased. Describe how he could find the probability of scoring a 3 using this die. Solution
3
1 2
Pepe would need to undertake some trials. He should roll the die, say, 100 times, and count the number of times he scored 3. He can then work out the relative frequency. For example, if he scores 3 on 40 occasions, he would 40 2 = say that the probability of scoring 3 in the future would be: 100 5
Guided Practice 1. Sandi rolled a biased die 150 times and found that it landed on the number 4 a total of 40 times. What is the experimental probability that the coin will land on 4 with the next throw? 2. Juan stuck some modeling clay to one side of a coin and tossed it 20 times. He found that it landed on heads 15 times. Find the experimental probability that the coin will land on heads next time.
Use Experimental Probabilities with Real-Life Events You can use relative frequency to estimate the number of times something will happen in the future. Example
2
At a factory, the relative frequency of tires failing a safety test is 0.01. Estimate how many tires will fail the test in a batch of 4000. Solution
The experimental probability that a tire will fail is 0.01. Multiply the number of tires by this probability to estimate the number that will fail: 4000 × 0.01 = 40 tires Section 6.4 — Experimental Probability
353
Don’t forget: To work out a batting average, divide the number of hits by the number of times at bat.
Baseball batting averages are examples of relative frequency, or experimental probability. Example
3
Barry has had 15 hits in 50 times at bat. What is his batting average? If he has the same batting average for his next 30 at bats, how many hits will he get? Solution
Batting average = Hits ÷ Number of at bats = 15 ÷ 50 = .300 To find the number of hits in Barry’s next 30 at bats, multiply 30 by his batting average: 30 × .300 = 9 hits
Guided Practice 3. On the basketball court, Kobe made 17 shots out of 20. Use relative frequency to estimate how many baskets he will make in 300 shots. 4. In a survey of 50 students, 23 said that they rode the bus to school. If these 50 students are representative of the whole school, how many students out of 1350 would you estimate will ride the bus to school? 5. In the past, School A’s football team has beaten School B’s team in 40% of their games. Based on these results, how many of next season’s 5 games would you estimate School A will win?
The More Data You Have, the Better When you work out experimental probability, the more trials that you do, the more accurate your estimate is likely to be. A probability based on 1000 trials is likely to be more accurate than one based on 10. Example
Don’t forget: To work out the experimental probability of rolling a 6, divide the number of favorable outcomes by the total number of possible outcomes. See Lesson 6.2.2 for more information.
4
Alan rolls a fair number-cube 10 times, and records the scores. He then rolls it a further 90 times and records these scores too. Calculate the experimental probability of Score 1 2 3 4 each score based on: In 10 rolls 2 0 1 3 (i) 10 rolls, (ii) 100 rolls. In 100 rolls 19 17 15 14
5
6
3
1
17 18
Which set of probabilities is likely to give more accurate predictions? Solution
You can use a table. 1
2
3
4
5
6
Relative Frequency after 10 rolls
Score
2÷10 = 0.2
0÷10 = 0
1÷10 = 0.1
3÷10= 0.3
3÷10 = 0.3
1÷10 = 0.1
Relative Frequency after 100 rolls
19÷100 = 0.19
17÷100 = 0.17
15÷100 = 0.15
14÷100 = 0.14
17÷100 = 0.17
18÷100 = 0.18
The probabilities based on 100 rolls are likely to give better predictions. 354
Section 6.4 — Experimental Probability
Example
5
Score 1 2 3 4 5 6 Alan then rolls his number In 1000 rolls 171 164 166 170 161 168 cube a further 900 times. He now has a total of 1000 results. How do his results for 10 rolls, 100 rolls, and 1000 rolls compare with the theoretical probability for each score?
Solution
Find the experimental probabilities for 1000 rolls in exactly the same way as for Example 4. Score
Check it out: As you get more and more data, your experimental probabilities should get closer and closer to the theoretical probabilities.
Relative Frequency after 1000 rolls
1
2
3
4
5
6
0.171
0.164
0.166
0.17
0.161
0.168
The theoretical probability of each score is 1 ÷ 6 = 0.1666... = 0.166 . The greater the number of rolls, the closer most of the experimental probabilities are to the theoretical probability.
Guided Practice Sarah has a fair five-color spinner, as shown. 6. What is the theoretical probability of it landing on each color? 7. She spins it 12 times and gets the results in the table. Colour Frequency
Don’t forget: You can use experimental probability to predict the outcome of any event. But it’s most useful when you’re looking at real-life events whose theoretical probabilities are impossible to calculate.
Now try these: Lesson 6.4.2 additional questions — p462
Red
Blue
Yellow
Purple
Green
1
1
3
2
5
Find the experimental probability for each color, using these results. 8. How would you expect her experimental probabilities to change if she spun the spinner 1000 times?
Independent Practice In Exercises 1–4, use the relative frequency given to estimate the number of times that event A will occur in 20 trials. 1. P(A) = 0.2 2. P(A) = 30% 1 3. P(A) = 5 4. P(A) = 0 Manny has a batting average of .350. Use this to find: 5. the expected number of hits if he batted 100 times. 6. the expected number of hits if he batted 40 times. 7. the expected number of hits if he batted 220 times. 8. Mrs. Hill cycles to work. She notes what color the traffic lights are when she reaches them. She finds they are red on two-fifths of the times she reaches them. Based on these figures, how many times in 20 trips are the lights likely to be red when she arrives?
Round Up You can use experimental probability to predict the outcome of an event based upon data you’ve collected previously. This is all about using information from the past to predict the future. Section 6.4 — Experimental Probability
355
Chapter 6 Investigation
A Game of Chance The producers of a new TV game show, “Wheel of Dollars,” need a wheel for contestants to spin. The wheel will be divided up into equally-sized sections, and each section marked with a prize. Part 1: The producers need the wheel to be designed so that the llars $$$ heel of Do W $ $ $ probabilities of winning the amounts shown below are: $100 • P(win $100) = 0.35 • P(win $200) = 0.3 • P(win $300) = 0.2 • P(win $500) = 0.1 • P(win $1000) = x (There are no other possible outcomes.) 00 $1
2
$
$100
0 0
$500
Design a wheel for use on the game show.
Remember... each section on the wheel must be the same size.
Things to think about: Clue: write P(win $1000) as a fraction. • What is the probability of winning $1000? • What is the minimum number of sections your wheel will need? Part 2: There will be 26 programs in the game show’s first season. In each show, the wheel is spun a total of 15 times. How much should the producers expect to pay out in prizes? Things to think about: • How many times would you expect each event to occur? Extension To increase the excitement, one “$100” section is made into a “Bankrupt” section. If a contestant spins “Bankrupt,” they leave, and lose all the prizes they had won to that point. If they don’t spin “Bankrupt,” they can either spin again, or keep their prizes and leave. • What is the probability of landing on the “Bankrupt” section on your wheel? • What is the probability that a contestant will land on “Bankrupt” in 2 spins? In 3 spins? Open-ended Extensions 1) The producers want to make the show even more exciting for the viewers. • How could you arrange the prizes on your wheel to make it as exciting as possible? • How could you change the rules of the game to make the show more exciting? 2) Select a simple card game, game show, board game, or computer game that is based on chance. Describe the game, the outcomes, and the number of ways to obtain each outcome. Calculate the theoretical or experimental probability of each outcome. Does the game try to make it appear that there is a good chance of winning? How?
Round Up Lots of games are based on chance. That’s where probability can be very important. You might not be able to say for certain what will happen, but you can say what’s likely. pter 6 In vestig a tion — A Game of Chance Chapter Inv estiga 356 Cha
Chapter 7 Geometry
Section 7.1
Exploration — Circle Ratios ...................................... 358 Circles ....................................................................... 359
Section 7.2
Exploration — Folding Angles ................................... 369 Angles ....................................................................... 370
Section 7.3
Exploration — Sorting Shapes .................................. 388 Two-Dimensional Figures .......................................... 389
Section 7.4
Exploration — Building Cylinders .............................. 405 Three-Dimensional Figures ....................................... 406
Section 7.5
Generalizing Results ................................................. 419
Chapter Investigation — Reclining Chairs ........................................ 427
357
Section 7.1 introduction — an exploration into:
Cir c le R a tios Circ Ra Circles are everywhere. For example, wheels are circular. So are CDs. So are some mugs, trashcans, tables, plates, pies... Even the Earth is circular when you look at it from space. It’s no wonder you have to learn about them. But there’s a lot you can work out for yourself. First, there are two technical words about circles you need to know: • The distance all the way around the outside of a circle is called the circumference. • The distance through the center from one side of a circle to the other is called the diameter. Now find 5 circular objects of various sizes.
circumference
diameter
For example, a trashcan, a cup, a plate, a CD... anything anything.
You need to measure their circumference and diameter as accurately as possible. 1) Measure the diameter of each of your circles using a meter stick or a ruler. 2) Use some string to find the circumference of each circle. • Loop the string around the the object. • Mark the length needed to go all the way around. • Now measure this length, and record the result.
Object Mug Plate
Diameter (cm) Circumference (cm) 9.2 28.9 24.3 76.6
3) Record your measurements in a table like this one. Example For each of your circles, find the ratio of the circumference to the diameter. Solution
Divide the bigger
Add an extra column to your table. number by the smaller smaller. Divide each circle’s circumference by its diameter. Record the result as a decimal, to the nearest hundredth. Use a calculator.
Ratio: circumference ÷ diameter 3.14 3.15
Exercises 1. Is there a pattern to the ratios in the third column of your table? 2. Compare your results with a partner. Are they similar? Or different? What about the results of the rest of the class? Are they similar or different? 3. Use your results to complete this sentence: For any circle, the ratio of the circumference to the diameter ___________________. 4. A regulation basketball hoop has a diameter of 18 inches. Find its circumference.
Hint... Use your result from Exercise 3.
Round Up The link between a circle’s circumference and its diameter is one of the most important results in math. And if this Exploration went well, then you’ve just worked it out for yourself. There’s a lot more information about circles in this Section of this book. a tion — Circle Ratios Explora 358 Section 7.1 Explor
Lesson
Section 7.1
7.1.1
Parts of a Circle
California Standards:
Circles are all around you. They’re an important type of shape because they have some special features that other shapes just don’t have. You’re going to learn about these special features in this Lesson.
Algebra and Functions 3.1 Use variables in expressions describing geometric quantities (e.g., 1 P = 2w + 2l, A = bh, C = pd 2 — the formulas for the perimeter of a rectangle, the area of a triangle, and the circumference of a circle, respectively). Algebra and Functions 3.2 Express in symbolic form simple relationships arising from geometry.
All Circles Have Certain Features in Common The parts of a circle have special names. The center of a circle is the point in the middle that is an equal distance from all points of the circle. The red dot in this picture is at the center.
What it means for you: You’ll learn about the parts of a circle, including how some of them relate to each other.
The radius of a circle is the distance from the center to the edge of the circle. It is shown by a straight line from anywhere on the edge to the center of the circle.
Key words: • • • •
The blue line in this picture represents the radius of the circle.
circle center radius diameter
The diameter of a circle is the greatest distance from one edge of the circle to the other. It is shown by a straight line that passes through the center. The green line in this picture represents the diameter of the circle.
Example
Check it out:
1
Which letters show: • a radius of this circle? • a diameter?
The plural of radius is radii.
Solution
Lines A and C are both radii of the circle. Line B is a diameter of the circle. Notice that half of line B, from either edge to the center, would also be a radius.
Section 7.1 — Circles
359
Guided Practice 1. On which of these pictures does the blue line show the radius? Why do the others not?
A
B
C
D
2. On which of these pictures does the red line show the diameter? Why do the others not?
A
B
C
D
3. Mrs. Batalles is buying a circular tablecloth for her kitchen table. The tabletop is circular with a diameter of 60 in. She wants the tablecloth to hang down over the table about another 3 in. all around. Which of these tablecloths should she buy? Explain your answer. a. 66 in. diameter b. 180 in. diameter c. 63 in. diameter d. 33 in. diameter 4. This figure shows concentric circles. Use the diagram to complete this definition: Concentric circles are circles that share the same _____.
The Diameter of a Circle Is Double Its Radius The diameter of a circle is twice its radius. Example
2
A circle has a radius of 8 inches. What is its diameter? Solution
The diameter is twice the radius. So the diameter of the circle is 8 × 2 = 16 inches
In math, the letter r is often used to represent the radius. And d is used for the diameter. So you can write “the radius is 5 cm” as “r = 5 cm,” and “the diameter is 10 cm” as “d = 10 cm.” Don’t forget: The letters r and d are variables in this formula. That means they’re standing in for numbers you don’t know yet.
360
Section 7.1 — Circles
“The diameter is 2 times the radius” can be written as the formula
d = 2r
Example
3
A circle has a diameter of 22 feet. What is its radius? Solution
You can use the formula d = 2r here. d = 2r
Write out the formula
22 = 2r
Substitute d = 22
r = 22 ÷ 2
Divide by 2
r = 11 feet
Guided Practice 5. A circle has diameter 40 inches. What is its radius? 6. A circle has radius 7 cm. What is its diameter? 7. A circle has radius 3 feet. What is its diameter? 8. A circle has diameter 25 inches. What is its radius? In Exercises 9–14, find the missing measure: 9. r = 30 cm, d = ? 10. d = 30 in., r = ? 11. r = 4.6 m, d = ?
12. d =
13. d = 6.7 cm, r = ?
14. r =
2 3 2 3
yd, r = ? ft, d = ?
Independent Practice 1. A circle has radius 17 cm. What is its diameter? Now try these: Lesson 7.1.1 additional questions — p462
2. A circle has diameter 9 yards. What is its radius? In Exercises 3–6, find the missing measure: 3. r = 49 ft, d = ? 4. r = 11.5 m, d = ? 5. d = 63 in., r = ?
6. d =
4 9
yd, r = ?
7. Mr. Rodriguez has a circular clock on his wall. The minute hand reaches exactly from the center of the face to the edge, and is 6 inches long. What is the diameter of the clock face? 8. Russell is standing at the edge of a circular lake. Looking straight across the center of the lake, he sees his friend Salvador on the other side. They are 35 yards apart. What is the radius of the lake?
Round Up The radius and diameter are really important, because we can use them to work out things like the area and circumference of a circle. Make sure you know what each one means, because you will see them again soon. Section 7.1 — Circles
361
Lesson
7.1.2
Circumference and p
California Standards:
In this Lesson, you’ll learn about the distance around the outside of a circle, which is called the circumference. There is a clever way to work out the circumference if you know the radius or diameter of the circle, and it involves a special number.
Algebra and Functions 3.2 Express in symbolic form simple relationships arising from geometry. Measurement and Geometry 1.1 Understand the concept of a constant such as p ; know the formulas for the circumference and area of a circle. Measurement and Geometry 1.2
The Circumference Is Another Important Measurement The distance around the edge of a circle is called the circumference.
Circu
m
Imagine that a circle is like a loop of string. If you cut the string and laid it out flat, you could measure the length of the string.
ence fer
Algebra and Functions 3.1 Use variables in expressions describing geometric quantities (e.g., 1 P = 2w + 2l, A = bh, C = p d 2 — the formulas for the perimeter of a rectangle, the area of a triangle, and the circumference of a circle, respectively).
If you could cut a circle and lay it out in the same way, the length you measured would be the circumference.
Know common estimates of p (3.14; 22/7) and use these values to estimate and calculate the circumference and the area of circles; compare with actual measurements.
Norman is walking around the edge of a large, circular rug. By the time he is one-third of the way around it, he has walked 8 feet. What is the circumference of the rug?
What it means for you:
Solution
You’ll learn about a special number which you can use in a formula to find the circumference of a circle.
The circumference is the distance all the way around the rug.
Example
1
Norman has walked one-third of the way around, so the circumference is 3 times the distance he has walked. 3 × 8 = 24
Key words: • • • • •
circumference diameter radius pi (p) estimate
The circumference of the rug is 24 feet.
Guided Practice 1. Jack is running around a circular track. When he is halfway around, he has run 150 yards. What is the circumference of the track? 2. Judy’s scarf is 60 cm long. The scarf will go around a telephone pole exactly 3 times. What is the circumference of the pole? 3. Jorge is riding his bicycle. When he travels 5 m, his front wheel goes around 5 times. What is the circumference of the wheel?
362
Section 7.1 — Circles
Circumference Is Related to Diameter You’ve seen that the diameter is always twice the radius. Diameter and circumference are related too — the circumference of a circle is always just over three times its diameter. The diameter of this circle is 1. Its circumference is about 3.14. 1
If the diameter is doubled, the circumference also doubles.
3.14 2
6.28
In fact, circumference ÷ diameter for any circle is always 3.1415926535897... This number is very important in math. The full version of it is a never-ending decimal. There’s a special symbol mathematicians use instead of writing out the full number. The symbol is a Greek letter called pi (pronounced “pie”). It looks like this:
p
p is kind of tricky — it’s a symbol, so it looks like a variable. This is one time where a symbol doesn’t stand for an unknown. It’s just the name of a special number.
You Can Use p in a Formula to Find Circumference Using C for circumference and d for diameter, you can use p to make a formula for the circumference of a circle: Check it out: These formulas are the same because d = 2r, so p × d = p × 2r =2×p×r
C = pd This formula is often written using r for radius, like this:
C = 2p 2pr
Section 7.1 — Circles
363
Example
2
A circle has a diameter of 12 inches. What is its exact circumference? Solution
You can use the formula C = p d to solve this question. C
= p × 12 p in. = 12p
You can leave p in an exact answer, because it is a number.
Guided Practice Don’t forget: To get an exact answer, you’ll need to leave p in your answer.
In Exercises 4–9, find the exact circumference of a circle with the given diameter or radius. 4. d = 20 in. 5. d = 13 ft. 6. d = 3.5 m 7. r = 5 ft. 8. r = 26 cm 9. r = 6.3 cm
You Can Often Use a Rounded Value of p If an exercise asks you to give the exact circumference, you should always leave p in the answer. Numbers with p in them are not very easy to work with, so it is more usual to estimate the circumference by using a rounded value of p. If the radius or diameter is given as a decimal, you should use 3.14 or 3.142 as the value of p. Check it out: If you need to use an estimate of p in a question, you will usually be told what value you should use.
If the question uses fractions, you should use
22 . 7
You should normally use a rounded value of p in real-life situations.
Example
3
Alicia has a circular swimming pool. The diameter of the pool is 30 ft. What is the circumference of Alicia’s pool? Solution
Using the formula C = p d, and estimating p as 3.14, you can solve the question: C = 3.14 × 30 = 94.2 The circumference of Alicia’s pool is 94.2 ft.
364
Section 7.1 — Circles
Example
4
T.J. is cutting a circle of paper to use on a poster for a math project. 1
The circle has a radius of 3 4 in. What is its circumference? Solution
You can use the formula C = 2p 2pr, and estimate p as 22 7 22 7
C =2× =2× Check it out: Be careful when you are answering circumference questions. Make sure that you check whether the question gives the radius or the diameter.
=
572 28
= 20
=
1
22 : 7
× 34 ×
13 4
143 7
3 7
So the circumference of the circle is 20
3 7
in.
Guided Practice In Exercises 10–21, calculate the circumference of a circle with the given diameter or radius. Use p = 3.14 for Exercises 10–15. 10. d = 15 cm 11. r = 11 yd 12. d = 42 in. 13. r = 6 m 14. d = 9.2 ft 15. r = 13.5 m 22 7
Use p = 16. r =
1 3
for Exercises 16–21.
yd
19. r = 12 3 m 7
17. d =
5 8
m
20. d = 9 1 ft 9
18. r =
4 11
21. d = 1
in.
5 17
ft
Independent Practice Now try these: Lesson 7.1.2 additional questions — p462
In Exercises 1–9, use a suitable estimate of p to find the circumference of a circle with the given diameter or radius: 1. r =
5 6
ft
2. r = 2.75 cm
3. d = 6.3 m
4. d = 13 in.
5. r =
7 11
m
6. d = 0.75 yd
7. r = 3.3 ft
8. d =
4 3
m
9. r = 8.1 in.
10. Complete the sentence by filling in the blanks. p is the ratio of a circle's c _ _ _ _ _ _ _ _ _ _ _ _ to its d _ _ _ _ _ _ _. 11. A good estimate of the height of an elephant from its foot to its shoulder is twice the circumference of its (roughly circular) foot. A zoologist in Africa finds an elephant footprint with a 12-inch diameter. Estimate how tall the elephant that left the footprint is.
Round Up p is a really useful and important number. It might look a little strange, but remember, it is just a fixed number. You will see p again many times in math, including in the next Lesson of this book. Section 7.1 — Circles
365
Lesson
7.1.3
Area of a Circle
California Standards:
You’ve already worked out the circumference of a circle using p. You might need to find the area of a circle too.
Measurement and Geometry 1.1 Understand the concept of a constant such as p; know the formulas for the circumference and area of a circle. Measurement and Geometry 1.2 Know common estimates of p (3.14; 22/7) and use these values to estimate and calculate the circumference and the area of circles; compare with actual measurements.
What it means for you: You’ll learn a formula that will help you find the area of a circle. You’ll use the formula with exact and estimated values of p.
There are a couple of ways of working out the area of a circle. You can either estimate it, or you can work it out using a formula that uses the radius and a value of p. That’s what this Lesson is about.
The Area of a Circle Is Measured in Square Units The area of a circle refers to the number of square units inside the circle. Example
1
Estimate the area of a circle with a radius of 2. Solution
Draw the circle on grid paper and count the squares inside the circle. There are 4 squares completely inside the circle. There are 8 squares almost completely inside the circle.
Key words: • • • • •
area radius diameter pi (p) estimate
There are 4 squares that are about one-quarter inside the circle. Estimated area = 4 + 8 + (4 ×
1 ) 4
= 13.
So the area is about 13 square units.
Don’t forget: One square unit is the amount of space taken up by a square with a side length of one unit.
Guided Practice In Exercises 1–2, find the area of each circle by counting squares. 1. 2.
In Exercises 3–6, find the area of each circle by drawing the circle and counting squares. 3. r = 6 4. r = 4 5. r = 3.5 6. d = 10 366
Section 7.1 — Circles
You Can Find the Area of a Circle Using a Formula
Don’t forget:
Counting squares is okay if the circle is small, but not if the circle is large. There is a formula you can use to find the area of a circle. The formula for the area of a circle is:
A = p r2 where r is the radius of the circle. For more about parts of a circle, see Lesson 7.1.1
Example
2
Find the area of a circle with a diameter of
2 3
mile.
Solution
First find the radius. r =
2 3 1 3
1 2 1 × = 2 3 3
Since the radius is a fraction, use a fraction estimate for p. A = pr2 ⎛ 1 ⎞⎟ ⎜⎜ ⎟ ⎜⎝ 3 ⎟⎠
2
=
22 7
=
22 1 × 7 9
Simplify exponents first.
=
22 63
Then multiply out.
Substitute
1 3
for r and substitute
The area of the circle is approximately
22 63
22 7
for p .
square miles.
Don’t forget: If you use an estimated version of p (such as 3.14 ), then the answer is only or 22 7 an estimate too. If an exercise asks for an exact area, you just need to leave p in the answer.
Guided Practice In Exercises 7–15, find the estimated area of each circle with the given radius or diameter. Use p = 3.14. 7. r = 11 in. 8. r = 2.1 cm 9. d = 14 yd 10. d = 20 m 11. r = 6.4 m 12. d = 42.2 cm 13. d = 16.1 yd 14. d = 40 m 15. d = 86.4 m In Exercises 16–24, find the exact area of each circle with the given radius or diameter. 16. r = 8 in. 17. r = 7.1 cm 18. d = 44 yd 19. d = 19 m 20. r = 9 m 21. d = 21.4 cm 22. d = 12.5 yd 23. d = 16.4 m 24. d = 99.6 m
Section 7.1 — Circles
367
Use a Rounded Value of p in Real-Life Situations When you’re applying the area formula in real-life situations, you usually need to use a rounded value of p. Example
3
A weed-control company charges $1.25 a square foot to spray for weeds. If a circular flower garden has a radius of 7 ft, how much will the company charge to spray it for weeds? Solution
A = pr2 = 3.14 × r2 = 3.14 × 72 = 3.14 × 49 = 153.86
First substitute p for a rounded value. Substitute 7 for r. Simplify all exponents.
The approximate area is 153.86 ft2. Now work out the cost: Cost = (Area in square feet) × (Cost per square foot) = 153.86 × 1.25 = 192.325. Rounded to the nearest cent, the company will charge $192.33.
Guided Practice 25. A water sprinkler can be set to rotate in a full circle. If the radius of the spray is 5 ft, how many square feet of grass will the sprinkler cover in one full rotation? 26. A dessert plate has a radius of 3 in. A dinner plate has a radius of 5 in. How many more square inches of area does a dinner plate have?
Independent Practice Now try these: Lesson 7.1.3 additional questions — p463
Check it out: A semicircle is half a circle:
In Exercises 1–5, find the estimated area of each circle. Use p = 3.14. 1. r = 8 in. 2. r = 3.2 cm 3. d = 12 yd 4. d = 18 m 5. r = 5.8 m 6. Paint is purchased in 1-quart cans. A quart of paint covers between 110 and 125 square feet. If you need to paint four circles that are 6 feet in diameter, how many cans of paint should you buy? 7. If you order two 12-in.-diameter pizzas, is this the same amount of pizza as one 24-in.-diameter pizza? Explain your answer. 8. Write a formula that could be used to find the area of a semicircle.
Round Up This is another example of how useful p is. Using the formula makes calculating the area of a circle a whole lot easier than if you tried to count squares on graph paper. Later in this Chapter, you’ll use the circle area formula as a first step in calculating the volume of a cylinder. 368
Section 7.1 — Circles
Section 7.2 introduction — an exploration into:
Folding Ang les Angles This Exploration is all about angles and triangles. The first part of the Exploration involves making all the angles and triangles you’re going to need by folding a sheet of paper. Do this as carefully as you can or the second part of the Exploration will become much more difficult. For this exploration you’ll need a square sheet of paper. You need to fold your sheet of paper by carefully following the directions below. 1) Fold the paper in half lengthwise. Unfold the paper.
3) Fold the left edge of the paper down, so that it meets the other folded edge. Then unfold your paper.
2) Fold the bottom-right corner of the square to the center line — with the fold going through the bottom-left corner.
4) Repeat steps 2 and 3, but this time fold the bottom-left corner to the middle, then the right edge over. Now unfold your paper, and make pencil lines along each fold. Use a ruler ruler.
Example obtuse angle
Identify one of each of the following types of angle: • right (= 90°), • acute (less than 90°), • obtuse (greater than 90°). Solution
acute angle right angle
See the diagram.
Exercises 1. What is an equilateral triangle? Identify any equilateral triangles formed by the folds. 2. Identify as many types of triangles and other shapes formed by the folds as you can. 3. Identify the measures of as many angles formed by the folds as you can. Say whether each angle is acute, obtuse, or right. Justify your answers.
Work them out. Don’t use a protractor.
Round Up You’ll see lots about angles and their properties in this Section. Once you’ve studied the Section, look back at this Exploration and see if you can identify any special pairs of angles. Section 7.2 Explor a tion — Folding Angles 369 Explora
Lesson
Section 7.2
7.2.1
Describing Angles
California Standards:
Whenever two straight lines meet at a point, there is an angle between them. This Lesson is about describing angles by naming the different types and by measuring them.
Measur ement and Measurement Geometr y 2.1 Geometry Identify ang les as vertical, angles adjacent, complementary, or ovide supplementary and pr pro descriptions of these terms. Measur ement and Measurement Geometr y 2.2 Geometry Use the properties of complementary and supplementary angles and the sum of the angles of a e pr ob lems triangle to solv solve prob oblems in volving an unkno wn inv unknown ang le angle le.
Ang les Ar e Measur ed in De g rees Angles Are Measured Deg You can measure how large the angle between two straight lines is. The angle is measured at the point where the lines meet. The units used to measure an angle are called degrees. The symbol ° means degrees. A right angle measures exactly 90° and looks like the corner of a square or rectangle.
What it means for you:
An angle measuring between 0° and 90° is called an acute angle.
You’ll learn about angles – some names for different types, how to measure them, and how to draw them.
An angle measuring between 90° and 180° is called an obtuse angle.
Key words: • • • • • • •
angle acute right angle obtuse protractor ray vertex
Because you know exactly what a 90° right angle looks like, you can use it to estimate the measure of other angles. Example
Don’t forget: An angle is an amount of a turn, for example,
45°
90° 180°
1
Estimate the sizes of angles A and B shown on the right. Are theses angles acute, obtuse, or right angles?
B A
Solution
Check it out: An angle that measures 180° would be a straight line.
The small squares on the picture show where there are right angles, which you know are 90°. You can use this to estimate angles A and B. It looks like angle A is about one-third as big as the right angle shown on the left. So 90 ÷ 3 = 30° might be a good estimate for angle A.
Check it out: Right angles are usually marked by a little square at the corner.
370
Section 7.2 — Angles
Angle B looks like it is about half the other right angle shown. So a good estimate of the size of angle B might be 90 ÷ 2 = 45°. Both angles are less than 90°, so they are both acute.
Guided Practice Use the diagram to answer Exercises 1–6: 1. Which of the angles shown are acute and which are obtuse?
1 5
2. Estimate the size of angle 1. 2
3. Estimate the size of angle 3.
3
4
4. Estimate the size of angle 4. 5. Esteban estimates that the size of angle 2 is about 45°. Is this a reasonable estimate? Give a reason for your answer. 6. Abigail estimates that the size of angle 5 is about 175°. Is this a reasonable estimate? Give a reason for your answer.
A Pr otr actor Can Be Used to Measur e Ang les Protr otractor Measure Angles Check it out:
40 0
14
90
100
90
80
110 70
12 0
60
13 0 50
30 15 0
30
10
170
0
180
0
170 10
180
20 160
160 20
2
Use a protractor to measure the size of this angle. Solution
Place the bottom line on the protractor directly over one of the rays of the angle. The point marking the center should be above the vertex of the angle.
60
40 0
14
13 0 50
30
30 15 0
12 0
60
0
160 20
20 160
110 70
15
180
0
170 10
10
100 80
170
90 90
40
0
80 100
0 14
50 0 13
0 12
70 110
180
The lines that form an angle can also be called the rays of the angle. The vertex is another name for the point where the rays meet.
0 12
80 100
0
Check it out:
50 0 13
70 110
15
Example
60
40
A protractor is a tool you can use to measure the size of an angle. The protractor has 180 small marks around its edge, each representing an angle of 1°.
0 14
There are usually two sets of numbers around the edge of a protractor. Make sure you go for the one that makes sense, by thinking about whether the angle is more or less than 90°.
You read the angle at the point where the second ray crosses the curved edge of the protractor. This angle measures 40°.
Section 7.2 — Angles
371
Guided Practice Use the diagram to answer Exercises 7–13. In Exercises 7–11, use a protractor to measure the angles: 7. Angle 1 8. Angle 2 9. Angle 3 10. Angle 4 11. Angle 5
1 6 5
Check it out: The total angle measure going all the way around one point is 360°.
7 2 4 3
12. Michael used a protractor to measure angle 6. His answer was 139°. What did he do wrong? What is the real size of angle 6? 13. Work out the measure of angle 7 from the values of angles 1–6, then check your answer with a protractor.
A Pr otr actor Can Also Be Used to Dr aw Ang les Protr otractor Dra Angles You can use a protractor for drawing angles of a given size. Example
3
Draw an angle that measures 54°. Solution 90 90
100 80
110 70
12 0 60
13 0 50
15
30
160 20
10
170
0
Follow the scale around the protractor until you get to 54°. Make a mark at the 54° point.
Finally, remove the protractor and join the mark up with the end of your original ray. 372
Section 7.2 — Angles
54°
180
0
170 10
180
0
30 15 0
80
100
20 160
70 110
40
14 0
60 0 12 50 0 13
0 14
Line it up with the bottom line on the protractor, with one end of the ray at the center mark of the protractor.
40
Start by drawing a straight line. This will be one of the rays that form the angle.
Guided Practice Use a protractor to draw the following angles: 14. 80° 15. 27° 16. 99° 17. 165° 18. 49° 19. 133°
Independent Practice Exercises 1–6 are about hands on a clock. For each time, say whether the angle made by the clock hands will be acute, obtuse, or right. 1. 3:00 2. 11:00 3. 1:15 4. 9:26 5. 7:30 6. 6:55 7. When cutting paper with a pair of scissors, is the angle between the blades of the scissors usually acute, obtuse, or right? Explain why. a.
8. Without measuring, match each of these angles with its measure. Choose from 15°, 80°, 90°, 105°, or 168°.
b.
c.
d.
e.
In Exercises 9–13, use the protractor to determine the measure of each angle. Say whether each angle is acute, right, or obtuse. 9.
10. 60
90
130 50
14 0 40
0 14 15 0 30
100
80
90
80 100
50 0 13
0 12
70 110
60
0
10 20
180 170 160
30
15 0
20
10 170
180
15 0 30 140 40 50
13
0
60
12
100
110
120
80
70
60
11 0 70
80
90 90
10 0
80 100 90
90
80
100
110 70
12 0
60
13 0 50
60
0
40
0 14
14
90
100
90
80
110 70
12 0
60
13 0 50
40
15 30
40
30 15
10
0
180
170
170 10
0
20 160
160 20
180
0
30 15 0
80 100
0 14
50 0 13
0 12
70 110
50 40 130 0 14 30 0 15
70 110
13.
20
0
10
16 0
0 12
60
Lesson 7.2.1 additional questions — p463
0
40 40 0
170
14
180
0
30 15 0
10 17 0
12.
14
0
0
40
0 1 80
20 160
50 0 13
Now try these:
10
170
180
170 10
30
30 15 0
0
160 20
20 160
160
40 14 0
180 0
90
13
0
0 17 10
11. 180
0
110 70
20
50
13 0 50
70
12 0
80
15
12 0
60
120
0 16
80 0 10 70 0 11
110
60
90
1 00
0
160 20
170 10
0 180
Round Up Angles are useful for describing the different shapes you’ll learn about later in this Chapter. You should be able to picture how big an angle is when you are told its size in degrees. Section 7.2 — Angles
373
Lesson
7.2.2
Pair s of Ang les airs Angles
California Standard:
In the last Lesson, you learned about different types of angles. Sometimes angles come in pairs. In this Lesson, you’ll meet three special types of angle pairs.
Measur ement and Measurement Geometr y 2.1 Geometry Identify ang les as vver er tical, angles ertical, adjacent, complementary, or ovide supplementary and pr pro descriptions of these ter ms terms ms..
What it means for you: You’re going to learn about some special types of angles.
Key words: • • • • • •
angle adjacent angles linear pair vertical angles ray vertex
Adjacent Ang les Shar e One R ay and Don er la p Angles Share Ra Don’’ t Ov Over erla lap A pair of angles are said to be adjacent if they are found next to each other on either side of one ray. No parts of the angles overlap each other. The two angles shown here are adjacent. The ray marked in red is a ray of both angles, but no parts of the angles themselves are the same. Example
1
In this diagram: • Is angle 1 adjacent to angle 2? • Is angle 1 adjacent to angle 3? Solution
2. 1. 3.
The middle ray forms a side of both angle 1 and angle 2. They have one ray in common and don’t overlap, so angles 1 and 2 are adjacent. Angles 1 and 3 share one ray, on the left-hand side of the diagram. But they aren’t adjacent, because they overlap each other. Don’t forget: The rays are the sides of the angle. The vertex is the point where the rays meet.
Guided Practice In Exercises 1–10, say whether the following pairs of angles are adjacent. If they are not, give a reason why. D 1. A and B 2. A and C A 3. C and E 4. C and F C G B 5. D and E 6. A and G E 7. B and E 8. B and G F 9. D and F 10. D and G
A Linear P air Is a Special P air of Adjacent Ang les Pair Pair Angles In a pair of adjacent angles, there are two “outside” rays that are part of only one angle each. When these two rays form a straight line, the pair of angles is called a linear pair. 374
Section 7.2 — Angles
In this picture, the blue ray is part of the angle on the left, and the green ray is part of the angle on the right.
Check it out: The measures of a linear pair of angles always add up to 180°.
Together these rays form a straight line, so the pair of angles is a linear pair. Example
2
Which two angles in this diagram are a linear pair? Solution
V U
Only two rays form a straight line. The two angles formed by one shared ray and two other rays that make the straight line are R and S.
T
R S
So angles R and S form a linear pair.
Guided Practice For Exercises 11–12, use the diagram in the previous Guided Practice. 11. Which one of the following pairs of angles is a linear pair: A and B A and D A and F 12. Give another pair of angles on the diagram that are a linear pair.
Ver tical Ang les Shar e the Same Ver te x ertical Angles Share erte tex
Check it out: Vertical angles are equal, because... 4
1
2
3
... angles 1 and 2 are a linear pair ... angles 1 and 4 are also a linear pair. The measures of a linear pair of angles always add up to 180°, so angles 2 and 4 must have the same measure.
Vertical angles are formed when two straight lines cross. The rays of one angle are opposite the rays of the other angle. A pair of vertical angles share the same vertex and have the same size. In this diagram: 1
• •
4
Angles 1 and 3 are vertical angles. Angles 2 and 4 are also vertical angles. Example
2
3
3
On the diagram below, identify: • the vertical angles
•
the linear pairs
Find the measures of angles 7 and 8. Solution
Angles 5 and 7 are vertical angles. Angles 6 and 8 are also vertical angles.
5 33°
6 147° 7 8
The linear pairs are: Angles 5 and 6, angles 5 and 8, angles 6 and 7, and angles 7 and 8. Vertical angles always have the same measure, so angle 7 measures 33° and angle 8 measures 147°. Section 7.2 — Angles
375
Guided Practice Use the diagram to answer Exercises 13–15. 13. Identify the vertical angles in the diagram. J
I
Angle H measures 28° and angle I measures 152°. 14. Find the size of angle J. 15. What is the size of angle K?
H
K
Use the diagram below to answer Exercises 16–19. 16. Identify the linear pairs and vertical angles in the diagram. Angle S measures 55°. 17. Work out the size of angle Q. 18. Find the size of angle R. 19. Find the size of angle T.
Independent Practice In Exercises 1–4, say whether the statements are true or false. If false, explain why. 1. A linear pair of angles always forms a straight line. 2. Two adjacent angles can never have equal measures. 3. The angles forming a linear pair are never adjacent. 4. Two adjacent angles can also be vertical angles.
In Exercises 5–7, use the numbered angles on this stained-glass window design to identify each of the following: Now try these: Lesson 7.2.2 additional questions — p464
5. a pair of vertical angles 6. a linear pair of angles 7. a pair of adjacent angles that are not a linear pair
A B D C
4
5 1 2 3
8. Gordon and Ana are looking at this diagram. Gordon claims it is a sketch of vertical angles. Ana says it is a sketch of adjacent angles. Explain why both students are correct.
Round Up You should make sure you know these three types of angle pairs pretty well. Adjacent angles and linear pairs will be particularly important in the rest of this Section. 376
Section 7.2 — Angles
Lesson
7.2.3
Supplementar y Ang les Supplementary Angles
California Standards:
You don’t always have to measure with a protractor to find the size of an angle. There are times when you can figure out the size of one of a pair of angles if you know the size of the other.
Measur ement and Measurement Geometr y 2.1 Geometry Identify ang les as vertical, angles adjacent, complementary, or supplementar y and pr ovide supplementary pro descriptions of these ter ms terms ms.. Measur ement and Measurement Geometr y 2.2 Geometry Use the pr oper ties of proper operties complementary and supplementar y ang les and supplementary angles the sum of the angles of a e pr ob lems triangle to solv solve prob oblems in volving an unkno wn inv unknown ang le angle le..
Supplementar y Ang les Ad d Up to 180° Supplementary Angles Add Any two angles with measures that add up to 180° are called supplementary angles. Example
1
Angle K measures 65°. Angle M measures 115°. Are angles K and M supplementary? K
M
What it means for you:
Solution
You’ll meet another special type of angle pair, and see how this can be useful for solving problems.
You can add the sizes of two angles together to see if they are supplementary. 65° + 115° = 180° So angles K and M are supplementary.
Key words: • • • •
angle supplementary angles supplement linear pair
When two angles are supplementary, we say that one is the supplement of the other. In Example 1, angle K is the supplement of angle M. Don’t forget: For more about equations, see Section 2.2.
Example
2
Angle S measures 108°. Find the measure of its supplement. Solution
This type of question can be solved using an equation. 108 + n = 180
Let n = the measur e of the supplement to S measure
n = 180 – 108
Subtr act 108 fr om both sides Subtract from
n = 72
Solv e to find n Solve
The supplement of angle S measures 72°.
Section 7.2 — Angles
377
Guided Practice 1. Match the angles below into three supplementary pairs. b
e
20° 160°
a
c
105°
f 45°
d
75°
135°
In Exercises 2–6, find the size of the supplement of an angle with the given measure: 2. 80° 3. 1° 4. 65° 5. 124° 6. 16° 7. 32° 8. Use a protractor to measure the angles on this diagram. Which two angles are a supplementary pair?
Linear P air s Ar e Al ways Supplementar y Pair airs Are Alw Supplementary
Check it out:
Any angle with two rays that make a straight line measures 180°.
The fact that an angle on a straight line measures 180° is very useful and important. Make sure you remember it.
180°
The angles in a linear pair are on a straight line, so added together they must measure 180°. So the angles in a linear pair are always supplementary to each other.
Don’t forget: For more about linear pairs, see Lesson 7.2.2.
Example
3
Angle A measures 30°. Find the size of angle B. Solution
A and B are a linear pair of angles, so together they add up to 180°. You can now use an equation to find the measure of angle B. 30 + n = 180
Let n = the measur e of ang le B measure angle
n = 180 – 30 n = 150 Angle B measures 150°.
378
Section 7.2 — Angles
Guided Practice Find the measure of the missing angles in Exercises 9–11. 9.
10.
11.
Ther e Ar e Two Ways to Dr aw Supplementar y Ang les here Are Dra Supplementary Angles If you’re asked to draw a supplementary angle, there are two ways to do it. Example
4
Angle F measures 140°. Draw an angle that is supplementary to angle F. Solution
Method 1: Work out the angle you need using the fact that supplementary angles add up to 180°. 180 – 140 = 40, so the angle you need to draw is 40°. You can draw this with a protractor. Method 2: You can use a straightedge to extend one of the rays of angle F. This makes a linear pair. You could create either one of the linear pairs shown.
Don’t forget: You’ll need to use a protractor to copy these angles. First you’ll need to measure how big each one is. Then you’ll need to draw an angle that size in your book. See Section 7.2.1 if you’ve forgotten how to do this. Or you could trace the angles instead.
Guided Practice In Exercises 12–15, copy the angle shown, then draw its supplement by adding only one more ray to the sketch. 12.
13.
14.
15.
Section 7.2 — Angles
379
You Can Wor k Out Mor e Than Two Ang les on a Line ork More Angles Using what you’ve learned about supplementary angles and linear pairs, you can solve more complex problems. Example
5
Find the measure of angle Q.
S 80° Q
Solution
34° R
You can break this question into two easier parts. First, imagine combining angles Q and R to make a new angle, T. Angles T and S are a linear pair so their measures add up to 180°. So the measure of angle T is 180° – 80° = 100°. But you know that angle R + angle Q = angle T. So
34 + n = 100 n = 100 – 34 = 66
Let n = the measur e of ang le Q measure angle
The measure of angle Q is 66°.
Guided Practice In Exercises 16–18, calculate the missing angle on each sketch. 16.
17.
18. 49°
50°
45°
77°
33°
135°
Independent Practice In Exercises 1–3, use the diagram shown to calculate the measure of: 1. Angle A 2. Angle B 3. Angle C 4. Can two acute angles be supplementary? Explain your answer. Now try these:
5. Can two obtuse angles be supplementary? Explain your answer.
Lesson 7.2.3 additional questions — p464
6. Can two right angles be supplementary? Explain your answer. 7. Two angles are supplementary. One angle is three times as large as the other. Find the measure of each angle. 8. Two angles are supplementary. One angle measures 30° more than the other angle. Find the measure of each angle.
Round Up Supplementary angles are really useful because you can often use them to find the exact measure of an unknown angle. This is better than using a protractor, because it’s easy to make mistakes with a protractor and be a degree or two out. 380
Section 7.2 — Angles
Lesson
7.2.4
The Triang le Sum riangle
California Standard:
So far you’ve mostly been thinking about angles around a single point. But when you have a drawing of a shape, there’s an angle made at each corner. In this Lesson, you’ll be learning one of the most useful things you’ll ever learn about a triangle.
Measur ement and Measurement Geometr y 2.2 Geometry Use the pr oper ties of proper operties complementary and supplementary angles and the sum of the angles of a triang le to solv e pr ob lems triangle solve prob oblems in volving an unkno wn inv unknown ang le angle le..
Ang les in a Triang le Ad d Up to 180° Angles riangle Add A triangle is made up of three angles. These angles always add up to 180°.
What it means for you: You’ll learn some useful rules about the angles in a triangle.
11 + 33 + 136 = 180
80°
11°
33° 136°
Key words: • angle • triangle sum • linear pair
Check it out: Cut a triangle out of paper.
Tear off the three angles
50° 50° 50 + 50 + 80 = 180
25° 65° 25 + 65 + 90 = 180
This fact is called the triangle sum. If you have a triangle where one angle is unknown, you can use the triangle sum to work out the missing angle. Example
1 L
Find the measure of the angle at M.
28°
You can place them along a straight line
Solution
This shows that the angles’ measures add up to 180º.
You could try to find the measure using a protractor. But if the diagram isn’t accurate, a protractor is useless.
Check it out:
However, you can work out the exact measure, because you know that all the angles in a triangle add up to 180°.
There are different ways to refer to an angle. In Example 1, you could say the angle is “the angle at point M.” Or you could call it “the angle LMN” (written –LMN), since if you point to points L, M, and N in that order, you trace out the angle at M.
N
46° M
So you can write an equation to solve this problem. x + 28 + 46 = 180
Let x = the measur e of ang le M measure angle
x + 74 = 180 x = 180 – 74 x = 106 So angle M measures 106°.
Check it out: “The triangle with corners L, M, and N” is often written DLMN.
Section 7.2 — Angles
381
Guided Practice In Exercises 1–8, find the missing angle on each sketch. 1.
2.
4.
3.
5. 6.
7.
8.
Sometimes You Can Find Two Missing Ang les Angles Sometimes you’ll only know the measure of one angle inside a triangle. But there might be other information you can use to work out the size of the other two angles. Example
2
Find the values of the angle measures x° and y°.
x°
Solution
Don’t forget: The angles in a linear pair always sum to 180°.
The first step is to find y. The angle of y° makes a linear pair with the 130° angle: y + 130 = 180 y = 180 – 130 y = 50 Now you can use the triangle sum to find x. x + 60 + 50 = 180 x + 110 = 180 x = 180 – 110 x = 70 So x° = 70° and y° = 50°.
382
Section 7.2 — Angles
60°
y° 130°
Guided Practice In Exercises 9–12, find the two missing angles in the triangles shown. 9. 10. 64°
44°
70°
125°
11.
95°
12. 47° 129° 151°
Independent Practice In Exercises 1–4, find the missing angles. 1. 2. 28°
47°
62°
111° 70°
3.
4.
126°
37°
5. The measures of angles A and B have a sum of 62°. Find the measure of angle C.
116°
C
A
B
6. Which three of the following could be the measures of three angles of a triangle: 155° 25° 10° 20° 15° 100° Now try these:
Exercises 7–8 are about Andrea, who is using her protractor to draw a triangle. She decides to use angles measuring 20°, 85°, and 12°.
Lesson 7.2.4 additional questions — p464
7. Explain to Andrea why the measures she has chosen will not work. 8. If she draws a triangle with angles of 20° and 85°, what must the measure of the third angle be? 9. Gil measured the angles of a triangle and recorded them as 40°, 140°, and 100°. Looking at his notebook, he realized that he used the wrong scale on his protractor when measuring one of the angles. Which angle was wrong and how can you tell?
Round Up This is just one of many math rules to do with triangles that you’ll come across — but it’s one of the most important. You’ll need to use the triangle sum in many areas of math in the future. Section 7.2 — Angles
383
Lesson
7.2.5
Complementar y Ang les Complementary Angles
California Standard:
A couple of Lessons ago, you learned all about supplementary angles, which add up to 180°. Complementary angles are very similar, but they add up to 90°.
Measur ement and Measurement Geometr y 2.2 Geometry Use the pr oper ties of proper operties complementar y and complementary les and supplementary ang angles the sum of the angles of a e pr ob lems solve prob oblems triangle to solv in volving an unkno wn inv unknown ang le angle le..
What it means for you: You’ll find out about another special type of angle pair.
Complementar y Ang les Ad d Up to 90° Complementary Angles Add Any two angles with measures that add up to 90° are called complementary angles. When two angles are complementary, we can say that each angle is the complement of the other. Example
1
Key words: • • • •
angle complementary angle complement right angle
Which two of these three angles are complementary? 45°
35°
Solution
Complementary angles add up to 90°. 35 + 45 = 80
Not complementar y complementary
45 + 55 = 100
Not complementar y complementary
35 + 55 = 90
Complementar y Complementary
55°
So the angles 35° and 55° are complementary.
Example
2
Angle A is the complement of angle B. Angle B measures 65°. Find the size of angle A. Solution
You know that complementary angles add up to 90°. So you can solve this question using an equation. n + 65 = 90 n = 90 – 65 n = 25 So angle A measures 25°.
384
Section 7.2 — Angles
Let n = the measur e of ang le A measure angle
Guided Practice In Exercises 1–8, find the complement of the given angle. 1. 60° 2. 11° 3. 27° 4. 37° 5. 84° 6. 59° 7. 41° 8. 76° 9. Make three complementary pairs from the angles below.
c
a b
26°
56° d
64°
74°
e
f 16°
34°
Adjacent Complementar y Ang les Mak e a Right Ang le Complementary Angles Make Angle You know that complementary angles have a sum of 90°, and you also know that an angle that measures 90° is a right angle. So if you put a pair of complementary angles together to make an adjacent pair, the two angles together form a right angle.
This can help you to identify complementary angles that are adjacent. Example
3
Identify an angle in the diagram that is complementary to angle 3. Don’t forget: A little square at an angle means it is a right angle, measuring 90°.
Solution
1
2
3 4
There is a right angle identified on the diagram, so you can look for two adjacent angles that form the right angle. Angles 3 and 4 share a common ray and together they form the right angle shown. Angle 4 is the complement of angle 3.
Guided Practice In Exercises 10–13, find the missing angle. 10. 11. 12.
13. 65°
71°
27°
46°
Section 7.2 — Angles
385
Ther e Ar e Two Ways to Dr aw Complementar y Ang les here Are Dra Complementary Angles Example
4
Angle C measures 15°. Draw the complement of angle C. Don’t forget: These are very similar to the methods you saw in Section 7.2.3 for drawing supplementary angles.
Solution
Method 1: You can work out that 90 – 15 = 75, then use a protractor to draw a 75° angle. Method 2: Use a protractor to draw angles of 15° and 90° with the same vertex. You will automatically create an adjacent complement to the 15° angle.
Guided Practice In Exercises 14–19, use a protractor to draw the given angle. Then draw its complement by adding only one ray to the sketch. 14. 58° 15. 7° 16. 42° 17. 76° 18. 19° 19. 36°
Triang les Can Contain Complementar y Ang les riangles Complementary Angles A right triangle always has one right angle (90°). Using the triangle sum, you know that the three angles in a triangle have measures that add up to 180°. So if one of the angles has a measure of 90°, then the other two angles must add up to 180° – 90° = 90°. So if one angle in a triangle is a right angle, the other two angles are complementary angles.
386
Section 7.2 — Angles
Example
Check it out: You could work out the size of the third angle using the triangle sum. The third angle must be 180° – 54° – 67° = 59°. But the question asks you to use complementary angles, so that’s what you have to do.
5
A triangle has two angles that measure 54° and 67°. Use complementary angles to determine whether the third angle of the triangle is a right angle. Solution
If the third angle is a right angle, the measures of the other two angles will have a sum of 90°. 54° + 67° = 121° So the third angle is not a right angle.
Guided Practice Exercises 20–25 give the measures of two angles of a triangle. For each pair, determine whether the third angle is a right angle. 20. 28°, 78° 21. 65°, 35° 22. 13°, 77° 23. 39°, 21° 24. 41°, 49° 25. 69°, 41°
Independent Practice
Now try these: Lesson 7.2.5 additional questions — p465
In Exercises 1–6, find the complement of the given angle. 1. 11° 2. 26° 3. 88° 4. 75° 5. 52° 6. 43° Use the diagram shown to answer Exercises 7–11. Angle 3 and angle 4 are complementary. 3 Find the measure of each angle below. 2 30° 7. Angle 6 8. Angle 4 9. Angle 2 10. Angle 5 6 1 5 11. Angle 1
4
12. Two angles are complementary. The measure of one angle is two times the measure of the other. Find the measure of each angle. 13. Two angles are complementary. The measure of one angle is 50° less than the measure of the other. Find the measure of each angle. 14. Angle R is complementary to angle S. Angle S is complementary to angle T. Is angle R complementary to angle T? Explain your answer.
Round Up All of this stuff about complementary angles should look quite familiar. Most of it’s based on what you’ve already learned in this Section. If there’s anything in this Lesson that you’ve seen before but have forgotten about, go back and take another look at the Lesson it came from to remind yourself. Section 7.2 — Angles
387
Section 7.3 introduction — an exploration into:
Sor ting Sha pes Sorting Shapes In this Exploration, you’ll see that there are many different ways to sort shapes into groups. You’ll get to use your imagination to come up with your own ways to sort them. Your teacher will give you a set of 32 shapes. You’re going to sort them into two groups. Example Sort all 32 shapes into any two groups. What do the shapes in each group have in common? Solution
The shapes on the left have a right angle. The shapes on the right do not have a right angle. Look at your shapes and put the shapes into any two groups — use your imagination. The only thing to remember is that all the shapes in a group must have something in common.
Exercises 1. How did you sort the shapes? What do the shapes in each group have in common? Sort the shapes again, this time sort the shapes by the number of sides.
Exercises 2. What is the name for each group of shapes? Now take the group of shapes with three sides, and sort these shapes into two groups.
Exercises 3. Describe how you sorted the three-sided polygons. 4. Draw a three-sided shape that has two equal sides and a right angle. What is the name of this shape? Now take the group of shapes with four sides, and sort these shapes into two groups.
Exercises 5. Describe how you sorted the four-sided polygons. 6. Draw a shape that is both a rectangle and a rhombus. What is the name of this shape?
A rhombus is a 4-sided shape whose sides are all the same length. Like this, for example.
Round Up There are different ways you can sort shapes — you’ve used a few during this Exploration. Most often in math, you’ll see that polygons are sorted and named using their sides or angles. a tion — Sorting Shapes Explora 388 Section 7.3 Explor
Lesson
7.3.1
Section 7.3
Classifying Triang les b y riangles by Angles
California Standard: Measur ement and Measurement Geometr y 2.3 Geometry Dr aw quadrilaterals and Dra triang les fr om gi ven triangles from giv inf or ma tion a bout them infor orma mation about (e.g., a quadrilateral having equal sides but no right angles, a right isosceles triangle).
What it means for you: You’ll learn one way of naming triangles and how to draw a triangle described in that way.
There are different types of triangles. There are a couple of ways to name the different types, depending on whether you’re looking at the angles or the sides. You’ll learn about the two sets of names over two Lessons. In this Lesson, you’ll learn how to name triangles by looking at their angles.
You Can Identify Triang les b y Their Ang les riangles by Angles Depending on the measures of their angles, you can sort triangles into three types: An acute triangle has three acute angles. These triangles are acute triangles.
Key words: • • • • •
triangle acute angle obtuse angle right angle classify
An obtuse triangle has an obtuse angle. The other two angles are acute. These triangles are obtuse triangles.
Don’t forget: Right angles measure 90°. Acute angles measure less than 90°. Obtuse angles measure more than 90°.
A right triangle has a right angle. The other two angles are acute. These triangles are right triangles.
Example
1
Classify each of these triangles by its angles: Check it out:
1.
A
B
3.
2.
“Classify...” means “name the type of...”
4.
G
D I C
H E
J
K
L
F
Solution
Triangle 1 is an obtuse triangle, because angle A is obtuse. Triangle 2 is an acute triangle, because all three angles are acute. Triangle 3 is an acute triangle, because all three angles are acute. Triangle 4 is a right triangle, because angle J is a right angle. Section 7.3 — Two-Dimensional Figures
389
Example
2
Below are the measures of the angles of three triangles. Determine whether each triangle is acute, obtuse, or right. E. 20°, 74°, 86°
F. 12°, 16°, 152°
G. 90°, 40°, 50°
Solution
Triangle E is acute, because all three angles are less than 90°. Triangle F is obtuse, because one angle is more than 90°. Triangle G is a right triangle, because one angle is 90°.
Guided Practice In Exercises 1–6, classify each of these triangles as acute, obtuse, or right. 1.
2.
3.
4.
5.
6.
In Exercises 7–12, the three measures of a triangle are given. Identify whether each triangle is acute, obtuse, or right. 7. 10°, 20°, 150° 8. 18°, 90°, 72° 9. 80°, 80°, 20° 10. 50°, 70°, 60° 11. 34°, 60°, 86° 12. 42°, 42°, 96°
390
Section 7.3 — Two-Dimensional Figures
You Can Dr aw Triang les Using a Pr otr actor Dra riangles Protr otractor Example
3
Use a protractor to draw an acute triangle with angle measures of 60°, 45°, and 75°. Don’t forget: Look back at Lesson 7.2.1 for a reminder of how to use a protractor to draw angles.
Solution
You can start by using your protractor to draw any of the three angles. You can make the sides any length you want — the question only gives you information about the angles.
45°
Now you need to draw the second angle. Use the free end of one of the rays you have already drawn as the vertex for the new angle.
You can now use a straightedge to extend the two rays that are not joined until they meet. You should erase any parts of the rays that extend outside the triangle. Check it out:
The third angle will automatically be the right measure to complete the triangle. In this example, the third angle is 75° since
It’s a useful check to measure the third angle. If that is wrong, then one of the others will also be wrong.
180° – 60° – 45° = 75°.
Example
4
Use a protractor to draw a right triangle with a 25° angle. Solution
You need to draw a right triangle, so you know one of the angles is 90°. You can start by drawing the right angle. Then draw a 25° angle at the end of one of the rays. Check it out: You can use the triangle sum to work out the size of the third angle as 180° – 90° – 25° = 65°
You can now join up the sides of the triangle.
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Guided Practice In Exercises 13–20, use a protractor to draw the triangles with the following angle measures: 13. 50°, 60°, 70° 14. 35°, 55°, 90° 15. 25°, 50°, 105° 16. 44°, 62°, 74° 17. 17°, 68°, 95° 18. 39°, 51° 19. A right triangle with one angle of 77° 20. 122°, 29°
Independent Practice In Exercises 1–8, identify each triangle shown as acute, obtuse, or right. 1.
2.
4.
6.
3.
5.
7.
8. a. b.
Now try these: Lesson 7.3.1 additional questions — p465
For each statement in Exercises 9–11, determine what type(s) of triangle(s) could be being described. 9. A triangle that has exactly one right angle. 10. A triangle that has no more than two acute angles. 11. A triangle that has at least two acute angles. In Exercises 12–13, use your protractor to draw a triangle that satisfies the condition. Then state the type of triangle you’ve drawn, and the measure of the missing angle. 12. Two of the three angles each measure 80°. 13. Two of the three angles each measure 25°.
Round Up This is just one of the ways to sort triangles into different types. The angles of a triangle only tell you part of the story. To know exactly what the triangle looks like, you also need to know about the lengths of the sides. The next Lesson will show you how side lengths are used to name triangles. 392
Section 7.3 — Two-Dimensional Figures
Lesson
7.3.2
Classifying Triang les b y riangles by Side Lengths
California Standard: Measur ement and Measurement Geometr y 2.3 Geometry Dr aw quadrilaterals and Dra triang les fr om gi ven triangles from giv inf or ma tion a bout them infor orma mation about (e .g ., a quadrilateral having (e.g .g., equal sides but no right angles, a right isosceles triangle).
What it means for you: You’ll learn another way of naming triangles and how to draw a triangle described in that way.
You’ll have seen the names equilateral, isosceles, and scalene triangles before in earlier grades. In this Lesson, you’re going to get a reminder of what each of those names means, and you’ll learn how to draw these three types of triangles.
You Can Identify Triang les b y Their Side Lengths riangles by A triangle with all three sides the same length is called an equilateral triangle. A triangle with two sides the same length is called an isosceles triangle. A triangle with no sides the same length is called a scalene triangle.
Key words: • • • • •
triangle equilateral isosceles scalene arc
Check it out: Sides with equal numbers of tick marks are all the same length. For example, all sides with 1 tick mark are the same length, all sides with two tick marks are the same length, and so on.
You can classify a triangle as equilateral, isosceles, or scalene by looking at a sketch. To make this easier, some triangles have little lines called tick marks on their sides. Sides of the same length have matching tick marks.
Example
1
Classify each of these triangles by the length of its sides:
Check it out: The number of sides of equal length in a triangle is the same as the number of angles of equal measure. So all three angles in an equilateral triangle are equal, an isosceles triangle always has two angles the same, and all three angles in a scalene triangle are different.
Solution
Triangle 1 is an equilateral triangle, because all three sides are the same length. Triangle 2 is a scalene triangle, because none of the sides are the same length. Triangle 3 is a scalene triangle, because none of the sides are the same length. Triangle 4 is an isosceles triangle, because two sides are the same length.
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393
You can also classify a triangle from a list of the lengths of its sides. Example
2
Below are the measures of the side lengths of three triangles. Determine whether each triangle is equilateral, isosceles, or scalene. A. 5 in., 8 in., 5 in.
B. 9.2 cm, 9.2 cm, 9.2 cm
C. 10 ft, 2 ft, 9 ft
Solution
Triangle A is isosceles, since two sides measure the same. Triangle B is equilateral, as all three sides are the same length. Triangle C is scalene, because the sides all have different measures.
Guided Practice In Exercises 1–6, classify each of these triangles as equilateral, isosceles, or scalene. 1. 2. 3.
4.
5.
6.
In Exercises 7–12, the lengths of the sides of triangles are given. Identify whether each triangle is equilateral, isosceles, or scalene. 7. 6 cm, 6.5 cm, 6.1 cm 8. 3 ft, 3 ft, 4 ft 9. 10 in., 20 in., 28 in. 10. 5 yd, 5 yd, 5 yd 11. 0.4 m, 0.4 m, 0.4 m 12. 8 in., 3 in., 6 in. In Exercises 13–16, identify whether the highlighted triangles are equilateral, isosceles, or scalene. 13. 14.
A
15.
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Section 7.3 — Two-Dimensional Figures
16.
You Can Dr aw Triang les Using a R uler and Compass Dra riangles Ruler In the last Lesson, you used a protractor to draw triangles with certain angle properties. To draw triangles based on the measures of their side lengths, you’ll need different tools. Example Check it out: If you use a compass to draw a curve that is not a full circle, the curve is called an arc.
3
Use a compass and a ruler to draw a scalene triangle with side lengths 7 cm, 3 cm, and 9 cm. Solution
You could try to draw the triangle using just a ruler. After some trial and error, you could probably get it right. By using a compass and a ruler together, you can get the drawing right the first time. Step 1: Use your ruler to measure and draw a line 7 cm long. Step 2: Use the ruler to measure 3 cm between the compass tip and the pencil tip. Step 3: Place the compass tip on one end of the 7 cm line. Use the compass to make an arc.
Step 4: Open the compass to 9 cm. Place the compass tip on the opposite end of the 7 cm line. Make another arc. Continue the arc until it crosses the first arc you made. Step 5: Use the straight edge of your ruler to connect the end points of the 7 cm line to the point where the arcs cross. You now have a scalene triangle with side lengths 7 cm, 3 cm, and 9 cm.
Guided Practice In Exercises 17–22, draw a triangle with the side lengths described. 17. 4 cm, 5 cm, 6 cm 18. 3 in., 3 in., 3 in. 19. 5 cm, 7 cm, 8 cm 20. 5 cm, 5 cm, 8 cm 21. 2 in., 5 in., 5 in. 22. 4 in., 7 in., 5 in. Section 7.3 — Two-Dimensional Figures
395
Ther e is a Method ffor or Dr awing Eac h Type of Triang le here Dra Each riangle If you are not given specific side lengths, you can still draw an example of each type of triangle. There are slightly different methods for drawing equilateral, isosceles, or scalene triangles. Example
4
Use a compass and ruler to draw an equilateral triangle. Solution
Check it out: You can draw a right isosceles triangle using a method similar to that shown in Example 5. The first step is to draw a right angle. 1) Draw a line. 2) Put the compass tip on one end of the line, and draw two arcs — one either side of the line. 3) Put the compass tip on the other end of the line, and again, draw two arcs — one either side of the line.
Step 1: Draw a straight line. Open your compass and place the compass tip on one end of the line and the pencil tip on the other end. Step 2: With the compass tip still on the end of the line, draw an arc. Step 3: Keep the compass opening the same and place the compass tip on the other end of the line. Draw another arc. Make the arcs long enough so that they cross. Step 4: Use the straight edge of your ruler to connect the ends of the line to the point where the arcs cross.
Step 1
Example
Step 2
Step 3
Step 4
5
Use a compass and ruler to draw an isosceles triangle. Solution 4) Join the crossing points of the arcs to form a right angle. 5) Now measure points on the two lines, equal lengths away from your right angle. Joining these gives you a right isosceles triangle.
×
Step 1: Draw a line. Open your compass to any distance more than half the length of the line, but not the same length as it. Step 2: Place the compass tip on one end of the line and draw an arc. Step 3: Keep the compass opening the same and repeat Step 2 from the other end of the line. Make the arcs long enough so that they cross. Step 4: Use the straight edge of your ruler to connect the ends of the straight line to the point where the arcs cross.
×
Step 1
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Section 7.3 — Two-Dimensional Figures
Step 2
Step 3
Step 4
Example
6
Use a compass and ruler to draw a scalene triangle. Solution
Step 1: Draw a line. Open your compass to any distance that is not equal to the length of the original line. Step 2: Place the compass tip on one end of the line and draw an arc. Step 3: Place the compass tip on the other end of the line. Change the compass opening and draw a new arc. The new compass opening can be any length other than the length of the original line, so long as the second arc crosses the first one. Step 4. Use the straight edge of your ruler to connect the ends of the line to the point where the arcs cross.
Step 1
Step 2
Step 3
Step 4
Guided Practice 23. Use a compass and a ruler to draw an equilateral, an isosceles, and a scalene triangle, each with a base measuring 4 cm. 24. Leila was using her compass to construct an isosceles triangle, but she was not able to make a triangle. Her work is shown here. Explain what happened and what she should do instead.
Independent Practice Now try these: Lesson 7.3.2 additional questions — p465
In Exercises 1–4, draw a triangle with the side lengths described. Determine whether each triangle is equilateral, isosceles, or scalene. 1. 6 cm, 6 cm, 6 cm 2. 8 in., 5 in., 6 in. 3. 3 in., 4 in., 5 in. 4. 5 cm, 5 cm, 7 cm 5. Use your compass to construct several equilateral triangles. Measure each angle of your triangles. Look for a pattern in the measures of the angles. 6. Is it possible for a triangle to be an obtuse equilateral triangle? Explain using your answer to Exercise 5.
Round Up This way of naming triangles is quite common, so you should get to know the names equilateral, isosceles, and scalene really well. Practice the methods for drawing the three types of triangles. Section 7.3 — Two-Dimensional Figures
397
Lesson
7.3.3
Types of Quadrila ter als Quadrilater terals
California Standard:
In the last couple of Lessons, you’ve been looking at triangles. Now you’re going to move on to four-sided shapes. These are also called quadrilaterals.
Measur ement and Measurement Geometr y 2.3 Geometry Dr aw quadrila ter als and Dra quadrilater terals om gi ven triangles fr from giv inf or ma tion a bout them infor orma mation about (e.g., a quadrilateral having equal sides but no right angles, a right isosceles triangle).
A Quadrila ter al Has F our Sides Quadrilater teral Four Polygons are named by the number of angles or number of sides they have. You know that 3-sided figures with 3 angles are called triangles.
What it means for you: You’ll learn the names and special properties of some four-sided shapes, also known as quadrilaterals.
Key words: • • • • • • •
quadrilateral rectangle square parallelogram rhombus trapezoid isosceles
When a polygon has 4 sides and 4 angles, it is called a quadrilateral. Rectangles are a type of quadrilateral. In a rectangle, the opposite sides are the same length. All the angles in a rectangle measure 90°. A square is a special type of rectangle where all the sides are the same length. So squares are quadrilaterals too.
Don’t forget:
A Quadrila ter al’ s Ang le-Measur es Sum to 360° Quadrilater teral’ al’s Angle-Measur le-Measures
Matching tick marks show sides of the same length.
The angles inside a rectangle have measures that add up to 4 × 90° = 360°. In fact, the angles inside any quadrilateral add up to 360°. Example
1
Find the measure of the missing angle in this quadrilateral. Solution
You know the measures of three of the angles. You also know that the sum of the four angles’ measures is 360°. So you can write an equation to find the missing angle. 50° + 60° + 100° + n = 360°
Let n = measur e of missing ang le measure angle
210° + n = 360° n = 360° – 210°
Rear e the equa tion earrrang ange equation
n = 150°
Solv e to find n Solve
The missing angle measures 150°.
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Section 7.3 — Two-Dimensional Figures
Guided Practice In Exercises 1–4, find the missing angle measure in each quadrilateral. 1.
2.
3.
4.
AP ar allelo g ram Has Two P air s of P ar allel Sides Par arallelo allelog Pair airs Par arallel
Don’t forget:
A parallelogram is any quadrilateral with two pairs of parallel sides. Parallel sides are often marked with matching arrows.
Parallel lines are lines that always stay the same distance apart.
Opposite sides of a rectangle are parallel. So a rectangle is a special type of parallelogram.
A parallelogram with all four sides the same length is called a rhombus. Check it out:
So a square is a special type of rhombus.
The plural of rhombus is rhombi.
In a parallelogram or rhombus, the measures of the angles at either end of any side add up to 180°. You can see this if you place two identical parallelograms next to each other. Don’t forget: For a reminder about linear pairs of angles, see Lesson 7.2.2.
Where two angles meet, they make a linear pair of angles, so their measures add up to 180°.
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Guided Practice In Exercises 5–8, find the measure of the missing angle. 5. 6. 25° ?
?
7.
8.
84°
?
57°
?
138°
A Tra pe zoid Has Exactl y One P air of P ar allel Sides pez Exactly Pair Par arallel A trapezoid is a quadrilateral with exactly one pair of parallel sides. The figure below is a trapezoid. The parallel sides of a trapezoid are called the bases. Check it out: The bases of a trapezoid must be different lengths. If they were the same length, you’d have a parallelogram.
Check it out: An isosceles trapezoid is like an isosceles triangle that’s had the top bit “chopped off” parallel to the base.
The bases of a trapezoid are never the same length. If the nonparallel sides of a trapezoid are the same length, then the trapezoid is called an isosceles trapezoid. The angles at the ends of a base of an isosceles trapezoid have the same measure. The measures of the angles at the ends of a nonparallel side of an isosceles trapezoid add up to 180°. You can see this by putting two isosceles trapezoids together. When the angles at the ends of the nonparallel sides are lined up next to each other, they make a linear pair.
Guided Practice In Exercises 9–14, say which of these shapes are trapezoids. If they are not, give a reason. 9. 10. 11. 12.
13.
14.
15. Which one of the trapezoids above is an isosceles trapezoid? 400
Section 7.3 — Two-Dimensional Figures
In Exercises 16–18, find the missing angle measures in each trapezoid. 16.
17.
18.
Independent Practice In Exercises 1–6, complete the sentences by filling in the blank with the correct word. 1. A rectangle is a parallelogram with four _____ angles. 2. A rhombus with four right angles is called a ______. 3. A rectangle with four equal-length sides is called a ______. 4. A quadrilateral is a _____ -sided polygon. 5. Isosceles trapezoids have at least __________ sides of equal length. 6. A trapezoid has exactly ____ pair of sides that are parallel. In Exercises 7–9, find the measure of the missing angle. 7. 8. 9.
Now try these: Lesson 7.3.3 additional questions — p466
In Exercises 10–14, say whether the statements are true or false. If false, explain why. 10. A trapezoid is a quadrilateral. 11. A polygon can be both a trapezoid and a parallelogram. 12. In an isosceles trapezoid, the parallel sides can be the same length. 13. A parallelogram cannot have any right angles. 14. A quadrilateral must be either a trapezoid or a parallelogram. 15. If all the angles in a quadrilateral are equal, how much does each angle measure? 16. One side of a parallelogram measures 16 cm. What do you know about the measure of the opposite side? 17. Use the clues below to determine the polygon being described. I have at least 1 pair of parallel sides. I do not have any right angles. My sides are all the same length. What am I?
Round Up The names of the special types of quadrilaterals in this Lesson should be familiar from lower grades. Try to remember the special properties of each type of quadrilateral — they will be useful when you learn how to draw them in the next Lesson. Section 7.3 — Two-Dimensional Figures
401
Lesson
7.3.4
Dr awing Quadrila ter als Dra Quadrilater terals
California Standard:
In the last Lesson, you met some of the most important of the special types of quadrilaterals. In this Lesson, you’ll learn how to draw them.
Measur ement and Measurement Geometr y 2.3 Geometry Dr aw quadrila ter als and Dra quadrilater terals om gi ven triangles fr from giv inf or ma tion a bout them infor orma mation about (e .g ., a quadrila ter al ha ving (e.g .g., quadrilater teral having equal sides b ut no right but ang les angles les, a right isosceles triangle).
What it means for you: You’ll find out how to draw the special quadrilaterals you learned about in the last Lesson.
Use a Pr otr actor and R uler to Dr aw R ectang les Protr otractor Ruler Dra Rectang ectangles You can draw rectangles and squares using a protractor and ruler. Example
1
Draw a rectangle measuring 3 cm by 5 cm. Solution
Start by using a protractor to make a right angle.
Key words: • • • • • • •
quadrilateral rectangle square parallelogram rhombus trapezoid isosceles
Then use a ruler to extend the rays to 3 cm and 5 cm.
3 cm
5 cm
Using your protractor again, make new right angles at the free ends of the sides you have drawn.
Check it out: The diagonals from one corner of a rectangle to the other are always the same length. So you can check that you have drawn a true rectangle by measuring the diagonals to make sure they are both the same.
Now extend the new rays until they join up and complete the rectangle.
Guided Practice In Exercises 1–6, draw a rectangle with the measurements described. 1. 2 in. by 5 in. 2. 3 cm by 4 cm 3. 1 cm by 5 cm 4. 4 in. by 4 in. 5. 6 in. by 1 in. 6. 3.5 cm by 8.5 cm
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Use a Pr otr actor and R uler to Dr aw P ar allelo g rams Protr otractor Ruler Dra Par arallelo allelog Example
2
Draw a parallelogram with side lengths 3.6 cm and 4.8 cm. Solution
First, draw one side of the parallelogram. Then draw the second side starting at one end of the first. The angle between the sides can be any size you want. Use a protractor to measure the angle between the sides. Don’t forget:
Now make an angle that is supplementary to the one you’ve measured. In this case, that will measure 180° – 68° = 112°.
Supplementary angles add up to 180°. See Lesson 7.2.3 for more.
Draw the supplementary angle at the free end of either of the sides you have already drawn. 3.6 cm 68°
112°
Then extend the new ray so that it is the same length as the side it is opposite to.
Check it out: You can use exactly the same method to draw a rhombus (a quadrilateral with 4 sides of the same length).
Finally, join up the two remaining free ends. The final side will automatically be parallel to the side it is opposite, and the same length.
Guided Practice Check it out: If you are given the angle measures instead of side lengths, you’ll need to start by using a protractor to draw an angle. Then extend the sides to whatever length you like before drawing the next angle.
In Exercises 7–10, draw a parallelogram with the given measurements. 7. Side lengths 3 in. and 5 in. 8. Side lengths 5.0 cm and 6.0 cm. 9. Angles measuring 77° and 103°. 10. One side of length 4 in. and at least one angle of 65°.
Ther e Ar e Two Ways to Dr aw Isosceles Tra pe zoids here Are Dra pez Example
3
Use a protractor and ruler to draw an isosceles trapezoid. Solution
First draw a line for one of the bases. Use your protractor to create two angles of equal measure from the endpoints of the base. Make the sides the same length, then connect the fourth side. This way the bases will be parallel.
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Example
4
Use a compass and straight edge to draw an isosceles trapezoid. Solution
Begin by constructing an isosceles triangle. Don’t forget: For a reminder of how to construct an isosceles triangle, see Lesson 7.3.2.
Don’t forget:
Measure and mark a point on each of the two equal length sides. Both points should be the same distance from the base of the triangle. Join the points to make a line parallel to the base. Erase the corner opposite the base of the triangle. The remaining figure is an isosceles trapezoid.
You could use the compass to mark points the same distance from the corners.
Guided Practice In Exercises 11–14, draw an isosceles trapezoid that has the given measurements. 11. One base of length 4 cm. 12. Two angles measuring 50°. 13. One base of length 3 in. and two sides of length 2 in. 14. One base of length 4 in. and two angles measuring 102°.
Now try these: Lesson 7.3.4 additional questions — p466
Independent Practice In Exercises 1–8, draw the shapes described. 1. A square with side length 1 in. 2. A rectangle measuring 6.8 cm by 3.9 cm. 3. A parallelogram in which one angle measures 140°. 4. A rhombus in which one of the angles measures 130°. 5. A rhombus with side length 4.4 cm. 6. A parallelogram with one angle measure of 56° and one side length of 5 in. 7. An isosceles trapezoid with two angles measuring 72° and two sides of length 2 in. 8. An isosceles trapezoid with one base of length 4.3 cm and two sides of length 2.6 cm.
Round Up The best way to get the hang of drawing these shapes is to practice doing it. It is important that you understand flat shapes before you move on to 3-D shapes in the next Section of this Chapter. 404
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Section 7.4 introduction — an exploration into:
Building Cylinder s Cylinders This Exploration involves some light engineering — you’re going to build a cylinder. But it can’t be just any cylinder — it has to be a cylinder with a particular volume. And if that’s too easy, you’ve then got to try and do it using the minimum amount of materials. Just like real engineering.
height
You are going to make some cylinders out of paper. • Roll up a rectangular piece of paper, and tape the edges to form the walls. • Make the bases by carefully drawing round your paper roll onto a sheet of paper. Then cut out these circles, and attach them to the cylinder. diameter Your aim is to build a cylinder with a volume between 460 cm3 and 480 cm3. Example For each cylinder you build, record the relationship between the diameter, height, and volume. Solution
diameter (cm) height (cm) volume (cm3) 5 15 295 10 15 1178 10 5 393
The height and diameter are marked on the diagram above. Use a table to record these for each cylinder you make. To find the volume, fill your cylinder with rice. Then empty the rice into a measuring jug and record the volume of rice.
Exercises 1. Calculate the base area of each of your cylinders. Describe the relationship between the base area, height, and volume.
Use the formulas d “ r = ” and “A = pr2.” 2
Now try to make a cylinder with a volume of 460-480 cm3, but using as little paper as possible. Experiment, by making some different-sized cylinders. Then find their volume and surface area. Add an extra column to your table to record all your results. Example Find the surface area of a cylinder with diameter 8 cm and height 9 cm. radius (r) = 4 cm
Solution
The surface area is the total area of the one rectangular piece of paper, plus the two circular pieces. Work them out like this: So the total surface area is 226 + 50.3 + 50.3 = 326.6 cm2
9 cm
Area = pr2
= 50.3 cm2
Area of rectangular piece = cylinder circumference × height = (p × d) × h 2 = (3.14 × 8) × 9 = 226.0 cm
Exercises 2. What do you think is the least possible surface area of a cylinder with volume 460-480 cm3?
Round Up That was a very difficult task, but it’s a good review of circles. And it did get you working with 3D shapes. That’s good, because there are many more 3D shapes in this Section of the book. Section 7.4 Explor a tion — Building Cylinders 405 Explora
Lesson
Section 7.4
7.4.1
Thr ee-Dimensional Figur es hree-Dimensional Figures
California Standard:
The 3D shapes you’re going to see in this Lesson should be familiar. You’ll have seen them before in math lessons in earlier grades, and also all around you in real life.
Measur ement and Measurement Geometr y 1.3 Geometry Know and use the formulas for the volume of triangular prisms and ccylinder ylinder s (area ylinders of base × height); compare these formulas and explain the similarity betw een them between and the formula for the ectangular volume of a rrectangular solid solid.
What it means for you: You’ll see some examples of 3D figures, with their names. You’ll learn about faces, edges, and vertices of 3D figures.
A Prism Is a Special Type of 3D Figur e Figure A prism is a type of 3D shape that looks like a polygon that’s been stretched out along a straight line. A prism is exactly the same size and shape all the way through. If you cut through a prism in line with one end, the slice is always the same shape and size and the same way around. Example
Polygon Stretched along this line
1
Which of these figures is a prism? Explain why the others are not. Key words: • • • • • • • •
rectangular prism cube triangular prism cylinder face edge vertex polygon
Don’t forget: A polygon is a flat shape with straight sides. Polygons include triangles, rectangles, parallelograms, and trapezoids — plus many other shapes.
A.
B.
C.
D.
Solution
The ends of Figure A have a curved side, so they are not polygons. So Figure A isn’t a prism. Figure B isn’t a prism because the triangles at the front and back of the figure are different sizes. The figure will be the same shape all the way through, but not the same size. Figure C isn’t a prism. If you cut through it as shown here, the slice is the same shape as the end, but not the same way up, so it’s not exactly the same. Figure D is a prism. It’s the same shape, the same size, and the same way around all the way through.
Guided Practice In Exercises 1–6, identify whether or not the shapes are prisms. If they are not, give reasons why. 1.
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Section 7.4 — Three-Dimensional Figures
2.
3.
4.
5.
6.
Some Prisms Ha ve Special Names Hav The sides of a 3D figure are called faces. The polygons at the ends of a prism are called bases. Check it out: Another name for a rectangular prism is a cuboid.
A 3D figure in which all the faces are rectangles is called a rectangular prism.
A cube is a special type of rectangular prism where all the faces are squares. A triangular prism is a prism in which the bases are triangles. The other faces in the prism are rectangles. This shape isn’t actually a prism, but it’s very similar. A cylinder is a three-dimensional figure with two circular bases that are parallel. But a cylinder is not actually a prism because circles are not polygons.
Guided Practice In Exercises 7–12, name the 3D figures shown. 7.
11.
8.
9.
10.
12.
Section 7.4 — Three-Dimensional Figures
407
3D Figur es Ha ve F aces es tices Figures Hav Faces aces,, Edg Edges es,, and Ver ertices The line where two faces of a figure meet is called an edge. A point where edges meet is called a vertex. Check it out: A vertex is the mathematical name for what in everyday language we just call a corner point.
Example
2
Name this figure and determine how many faces, edges, and vertices it has. Solution
This is a rectangular prism, since all the faces are rectangles. Check it out: The plural of vertex is vertices.
When you count the faces, edges, and vertices, you need to remember to count the ones you can’t see on the picture, but you know are there. In the picture you can see 3 faces. Each one is opposite a matching face that you can’t see. So there are 6 faces in total. There are 4 edges of length 3.4 cm, 4 of length 5 cm, and 4 of 8 cm.
That makes a total of 12 edges. Finally, the prism has 8 vertices. You can see 7 of them on the original picture. The eighth is on the other side. Example
3
Name this figure and determine how many faces, edges, and vertices it has. Solution
This is a triangular prism. There are 2 parallel triangles and all the other faces are rectangles. It has 5 faces. It has 2 faces that are triangles, and 3 that are rectangles joining the bases together. Each triangle has 3 edges. There are another 3 edges where the rectangular faces meet. So there are 9 edges in total. There are 6 vertices. You can count the 3 vertices on the front triangle and 3 on the back triangle.
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Section 7.4 — Three-Dimensional Figures
Guided Practice In Exercises 13–16, say how many vertices each figure has. 13.
15.
14.
16.
17. Charlie is making a cube-shaped box from square panels of wood. How many square panels does she need?
Independent Practice In Exercises 1–3, say whether each statement is true or false. If any are false, explain why. 1. The faces of a triangular prism are triangles. 2. A cube is not a rectangular prism. 3. A cylinder is not a prism. In Exercises 4–8, use the word face(s) or base(s). Choose the best word to complete each sentence. 4. Prisms are named by their ______. All the other ______ are rectangles. 5. In a cylinder, the ______ are circles. 6. In a triangular prism, there are three rectangular ______. 7. A ______ is one of the ______. 8. In rectangular prisms, all the ______ are rectangles. 9. Identify which of the figures are rectangular prisms. a. b. c. d. e.
Now try these:
In Exercises 10–15, name the 3D figures shown. 10. 11. 12.
Lesson 7.4.1 additional questions — p466
13.
14.
15.
Round Up You will have seen these special types of 3D figures before in earlier grades. You should make sure you understand all the facts you have seen in this Lesson. They’ll be useful in the next few Lessons as you learn about the volume of 3D figures. Section 7.4 — Three-Dimensional Figures
409
Lesson
7.4.2
Volumes of R ectangular Rectangular Prisms
California Standard: Measur ement and Measurement Geometr y 1.3 Geometry Kno w and use the formulas Know for the volume of triangular prisms and cylinders (area of base × height); compare these formulas and explain the similarity between them or mula ffor or the and the ffor orm volume of a rrectangular ectangular solid solid.
What it means for you: You’ll learn a formula that you can use to work out how much space there is inside a rectangular prism.
It’s really useful to be able to work out the volume of different solid figures. The easiest type of figure to work out the volume of is a rectangular prism. That’s what you’ll learn about in this Lesson.
Volume Is the Amount of Space Inside a Figur e Figure The amount of space inside a 3D figure is called the volume of the figure. Volume is measured in cubic units. One cubic unit is the volume of a unit cube — a cube with a side length of 1 unit.
1
volume rectangular prism cube base area height
1
When you work out the volume of a 3D figure, you are working out how many unit cubes would fit inside it.
Key words: • • • • • •
1
2 cm 1 cm 2 cm 1 cm
1 cm
1 cm
If you put two 1 cm cubes together, you make a rectangular prism with side lengths 1 cm, 1 cm, and 2 cm. The volume of the prism is the total of the two cubes together, 2 cm3.
1 cm 2 cm
1 cm
Guided Practice Check it out: A unit cube could have side lengths of 1 cm, 1 m, 1 yd... it all depends on the units you’re using. For bigger volumes, you need bigger units — you might measure volumes using a 1-meter unit cube.
The rectangular prisms shown in Exercises 1–6 are made up of 1 cm cubes. Find the volume of each prism. 1.
2.
3.
4.
5.
6.
In Exercises 7-10, choose a reasonable estimate of each volume from the three choices. 7. Volume of a bathroom 13 cm3 9 m3 90 m3 8. Volume of a bar of soap 8 in3 80 in3 0.8 in3 3 3 9. Volume of a bathtub 0.75 yd 7.5 yd 75 yd3 10. Volume of a medicine cabinet 10 in3 900 in3 1 in3 410
Section 7.4 — Three-Dimensional Figures
You Can Calcula te Volume Using a F or mula Calculate For orm For larger prisms, it isn’t possible to find the volume by counting cubes. It is more convenient to use a formula to work out the volume.
Check it out:
This is the formula you need to find the volume of a rectangular prism.
V = Bh
Area =4×3 2 = 12 units
B represents the area of one of the prism’s bases.
4 layers
Think of it like this... The area of the base (B) is the number of unit cubes in one layer (here, 4 × 3 = 12). The volume V (the total number of unit cubes) is the number of cubes in one layer multiplied by the number of layers.
h stands for the height of the prism. In a rectangular prism, you can pick any pair of parallel faces as the bases. The height is the distance between the bases. Example
1
Find the volume of this rectangular prism. Solution
You need to start by choosing a rectangle as the base.
8 ft
For example, you might choose the 2 ft by 5 ft rectangle as the base, making the height 8 ft. Don’t forget:
5 ft 2 ft
You can now find the volume using the formula V = Bh
The area of a rectangle is given by the formula A = lw where l and w are the length and width of the rectangle.
V = Bh
l
V = (l × w) × h
Replace B with rrectang ectang le ar ea ffor or mula ectangle area orm
V = (2 × 5) × h
Substitute in length and width of base
V = 10 × h
w
V = 10 × 8
Substitute in height of the prism
V = 80 So the volume of the prism is 80 cu. ft, or 80 ft3. Check it out: 1 ft3 or 1 cu. ft is 1 cubic foot — the volume of a unit cube whose sides are 1 foot long.
Guided Practice In Exercises 11–16, use the formula V = Bh to find the volume of each prism. 11.
12.
5m
3m
13.
2m
4 in.
8 ft
4 in.
12 ft
12 ft
3 in.
14.
15.
16. 8m
7 cm
6m
11 cm 9 cm
8 in.
11 in.
17 in.
2.5 m
Section 7.4 — Three-Dimensional Figures
411
You Can Find the Volume Without a Pictur e Picture You don’t have to have a drawing of a prism to find its volume. You can use the formula V = Bh to find the volume of a rectangular prism when only the lengths of the sides are given. Example
2
Find the volume of a rectangular prism with length of 3.4 cm, width of 5.6 cm, and height of 3 cm. Solution
Check it out: The formula for the volume of a rectangular prism is often written V=l×w×h (or V = lwh) — where l, w, and h are the length, width, and height.
Use the formula V = Bh. V = (l × w) × h
Replace B with rrectang ectang le ar ea ffor or mula ectangle area orm
V = (3.4 × 5.6) × 3
Substitute vvalues alues of l,, w,, and h
V = 19.04 × 3 V = 57.12 The volume of the prism is 57.12 cm3.
Guided Practice Exercises 17–22 each give the length, width, and height of a rectangular prism. Calculate the volume of each prism. 17. 4 cm, 9 cm, 1 cm 18. 3 ft, 7 ft, 10 ft 19. 8 in., 8 in., 11 in. 20. 5.5 cm, 7 cm, 9.5 cm 21. 1.2 m, 5.2 m, 6.9 m 22. 3.25 in., 4.5 in., 6 in.
Independent Practice In Exercises 1–3, find the volume of each rectangular prism. 6 in. 9 in. 1. 2. 3. 5 in.
Now try these: Lesson 7.4.2 additional questions — p467
4 cm
10 ft 13 ft
7 ft
5.5 cm
5.5 cm
4. The area of the base in a rectangular prism is 35 cm2 and the height is 2 cm. What is the volume of the prism? 5. A cube has an edge length of 2.5 ft. What is its volume? 6. A box measures 18 cm by 25 cm by 6 cm. What is the volume of the box? 7. The area of the base of a rectangular prism is 40 m2. Its volume is 120 m3. What is the height of the prism? 8. A cube has a volume of 1728 ft3. Find its length, width, and height.
Round Up The formula V = Bh is one you should learn, because you don’t just use it with rectangular prisms. As you’ll see in the next couple of Lessons, it’s handy for finding the volumes of other 3D figures too. 412
Section 7.4 — Three-Dimensional Figures
Lesson
Volumes of Triangular Prisms and Cylinder s Cylinders
7.4.3
California Standard: Measur ement and Measurement Geometr y 1.3 Geometry Kno w and use the ffor or mulas Know orm for the vvolume olume of triangular prisms and ccylinder ylinder s (ar ea ylinders (area of base × height) height); compare these formulas and explain the similarity between them and the formula for the volume of a rectangular solid.
The formula you used last Lesson, V = Bh, doesn’t just allow you to find the volume of a rectangular prism. In this Lesson, you’ll be using it to find the volumes of triangular prisms and cylinders.
What it means for you:
You can use the same formula to find the volume of a triangular prism.
You’ll use the formula V = Bh to work out the volumes of triangular prisms and cylinders.
Key words: • • • •
V = Bh Can Be Used with Triangular Prisms The formula you used to find the volume of a rectangular prism is V = Bh.
Example
1
Find the volume of this triangular prism. The area of its base is 45 in2. 11 in.
Solution
volume triangular prism cylinder base
You can find the volume using the formula V = Bh V = Bh
Don’t forget: In the formula V = Bh: • B is the area of one of the bases of the prism • h is the height of the prism, or the distance between the two bases.
V = 45 × h
Replace B with ar ea of the base area
V = 45 × 11
Height of prism is distance betw een bases between
V = 495 So the volume of the prism is 495 in3.
Sometimes you’ll need to work out B first.
Check it out:
Example Area = (4 × 3) ÷ 2 2 = 6 units
2
Find the volume of this triangular prism. 4 layers
The formula is the same as the one from the previous Lesson, because, again, the volume is the number of unit cubes in one layer multiplied by the number of layers.
Check it out: The formulas for area of a triangle and volume of a prism both have a lowercase h in them. One is the height of the triangle and one is the height of the prism. Don’t get them mixed up.
Solution
6 cm
You need to find the area of the triangular base first. A=
1 bh 2
A=
1 2
5 cm
7 cm
× 5 × 6 = 15 cm2
You can now find the volume using the formula V = Bh V = Bh V = 15 × 7 V = 105 So the volume of the prism is 105 cm3. Section 7.4 — Three-Dimensional Figures
413
Guided Practice Find the volume of the triangular prisms shown in Exercises 1–6. 1. 2. 3. 4m Base area = 2 6m
9 in.
Base area 2 = 14 in
10 cm 6 cm
4 cm
4.
5.
17 in.
6. 3 in.
9.5 in.
7 cm
9 ft 8 cm
10.5 ft 13 ft
11 cm
You Can Find the Volume of a Cylinder Using V = Bh The formula V = Bh even works for finding the volume of a cylinder. Example
3
Find the volume of this cylinder. The area of the base is 50.24 in2.
4 in.
Solution
You can find the volume using the formula V = Bh V = Bh V = 50.24 × h
Replace B with ar ea of the base area
V = 50.24 × 4
Height of prism is distance betw een bases between
V = 200.96 So the volume of the prism is 200.96 in 3. Don’t forget: Evaluate exponents before doing any multiplication.
Again, you might need to work out B first. Example
4
Find the volume of this cylinder. Use 3.14 for p. Don’t forget: See Lesson 7.1.3 for how to find the area of a circle.
You need to find the area of the circular base first. B = pr2 B = 3.14 × 32
Check it out: The formula for the volume of a cylinder is often written V = pr2h
Put in vvalues alues ffor or p and r
B = 28.26 ft2 You can now find the volume using V = Bh V = Bh V = 28.26 × 6 = 169.56 So the volume of the cylinder is 169.56 ft3.
414
6 ft
Solution
Section 7.4 — Three-Dimensional Figures
3 ft
Guided Practice In Exercises 7–12, find the volume of each cylinder. Use 3.14 for p. 7.
8.
2 cm
9.
10 m
3 cm
3 in.
14 m
4.5 in.
Don’t forget: You need the radius of a circle to find its area. If the question gives the diameter instead, then you need to divide that by 2 to get the radius.
10.
11.
12.
6.6 in.
11.4 m 4.2 ft
15 ft
8.1 in.
9.9 m
Independent Practice In Exercises 1–6, find the volumes of the triangular prisms and cylinders shown. Use 3.14 for p. 10 m 1. 2. 3. 7 m 6 cm 12 in.
12 cm
12 in.
4.
13 m 15 in.
5.
6.
14.1 cm
4.4 in. 5.5 in.
8.1 cm 5.7 ft 7.9 ft 7.3 ft
Now try these: Lesson 7.4.3 additional questions — p467
7. The volume of a triangular prism is 248 in3. The height of the prism is 8 in. What is the area of the base? 8. A swimming pool company makes a 15 ft diameter circular pool that is available in 3 ft, 3.5 ft, and 4 ft depths. Find the approximate number of cubic feet of water needed to fill each pool. 9. A candy company has decided to change the shape of its box. The old box was a rectangular prism with l = 12 cm, w = 6 cm, and h = 2 cm. Instead they will make a triangular prism in which the base of the triangle is 6 cm, and the height of the triangle is 12 cm. The height of the prism is 2 cm. How does this new design change the volume of the box?
Round Up This Lesson shows how useful the formula V = Bh can be. If you learn the formula, and how to use it, you can work out the volume of any prism so long as you can find the area of the base. Section 7.4 — Three-Dimensional Figures
415
Lesson
7.4.4
Volumes of Compound Solids
California Standard: Measur ement and Measurement Geometr y 1.3 Geometry Kno w and use the ffor or mulas Know orm for the vvolume olume of triangular prisms and ccylinder ylinder s (area ylinders of base × height); compare these formulas and explain the similarity between them and the formula for the volume of a rectangular solid.
What it means for you: You’ll use the formula V = Bh to figure out the volumes of more complicated 3D figures.
Key words: • • • • • • •
volume rectangular prism triangular prism cylinder base area height
Now you know how to find the volume of rectangular prisms, triangular prisms, and cylinders. Sometimes you’ll want to find the volume of solid shapes that look complicated, but can be broken down into smaller, simpler shapes.
Sometimes 3D Figur es Need to Be Br ok en Do wn Figures Brok oken Down Example
1 2 in.
Find the volume of this figure. 7 in.
6 in.
Solution
The figure is made up of two rectangular 4 in. prisms. All you need to do is figure out the volume of each prism, then add the two together.
6 in.
The volume of the first prism is V = Bh V = (l × w) × h V=6×4×6 V = 144 in 3
6 in. 4 in. 6 in.
The volume of the second prism is
6 in.
2 in.
V = Bh V = (l × w) × h V=6×7×2 V = 84 in 3
7 in.
So the total volume of the figure is 144 in 3 + 84 in 3 = 228 in 3
Check it out: This is the 3D equivalent of the material in Lesson 2.3.3.
Guided Practice In Exercises 1–5, find the volume of each figure. Use 3.14 for p. 12 cm 1. 2. 14 m 1.3 m
8 cm 30 cm
3.
4.
4 ft 8 ft
6.2 m
7.5 m
24 cm
6 cm
5. 3 cm
4.5 in. 4.5 in.
6 cm 3 cm
10 ft
416
Section 7.4 — Three-Dimensional Figures
11 in.
20 ft
11 in.
Some 3D Figur es Ha ve Holes in Them Figures Hav Example
2
Find the volume of this figure. Use 3.14 for p.
10 ft
Solution 5 ft
The figure is a cylinder with a cylinder-shaped hole through the middle. You first need to find 5 ft the volume that the cylinder would have without the hole. Then you need to find the volume of the hole and take that away. The volume of the full cylinder is 10 ft
V = Bh V = pr 2 × h V = 3.14 × 102 × 5 V = 1570 ft3
5 ft
The volume of the hole is V = Bh V = pr 2 × h V = 3.14 × 52 × 5 V = 392.5 ft3
5 ft 5 ft
So the total volume of the figure is 1570 ft 3 – 392.5 ft 3 = 1177.5 ft 3
Guided Practice In Exercises 6–9, find the volume of each figure. Use 3.14 for p. 11 in. 6. 7. 4.5 in. 8. 7m
11 in.
7 in.
4m 4m
4 in.
9 in.
3m
7m
8 in.
9. 5 cm
4 cm 2 cm
5 cm
10 cm 4 cm 3 cm 25 cm
Section 7.4 — Three-Dimensional Figures
417
Independent Practice In Exercises 1–6, find the volume of each figure. Use 3.14 for p. 1.
2.
9 ft
3.
11 in.
23 ft 8 ft
7m
3 in.
13 in.
4m 6m
5 in.
4.
12 cm
30 ft
11 ft
7m
8 in.
8 cm
5.
6. 18 in.
22 cm
7 in.
2 cm
6 in.
12 cm
4 cm
9 in.
15 cm
3 cm
10 cm
15 in.
3 cm
4.5 cm
Don’t forget: If there isn’t a picture in the question, it usually helps to draw one for yourself.
Don’t forget: For questions involving real-life situations, you can take the value of p to be 3.14.
7. A cylinder-shaped bottle of aspirin is packaged inside a box that is a square-based prism. The bottle has radius 2 cm and height 6.6 cm. The box has length 4.5 cm, width 4.5 cm, and height 7 cm. How much empty space is left inside the box once the bottle has been put in it? 8. Mount Prism is a mountain that is shaped like a triangular prism. The size of the mountain is shown in the diagram. A tunnel is built straight through the mountain from one triangular side to the other. The tunnel is circular, with a diameter of 100 ft. What is the approximate remaining volume of the mountain once the tunnel has been dug out?
1400 ft
900 ft 1200 ft
Now try these: Lesson 7.4.4 additional questions — p468
9. This building has a volume of 51.12 m3. What is the length of the longer side?
1.7 m 2.7 m 3.2 m
10. This figure has a volume of 360 in3. How long is the distance marked x?
x
8.5 in. 5 in. 10 in.
6 in.
Round Up If you need to find the volume of a tricky-looking 3D figure, you should always try to split it up into simpler shapes. Remember to look out for any cases where it’s easier to take a shape away rather than add them together. 418
Section 7.4 — Three-Dimensional Figures
Section 7.5
Lesson
7.5.1
Generalizing Results
California Standards:
There are lots of rules and patterns in math. In this Lesson, you’ll learn to think about patterns of numbers in a way that will help you to figure out rules of your own.
Mathematical Reasoning 1.2 Formulate and justify mathematical conjectures based on a general description of the mathematical question or problem posed. Mathematical Reasoning 3.1 Evaluate the reasonableness of the solution in the context of the original situation. Mathematical Reasoning 3.2 Note the method of deriving the solution and demonstrate a conceptual understanding of the derivation by solving similar problems. Mathematical Reasoning 3.3 Develop generalizations of the results obtained and the strategies used and apply them in new problem situations.
To Find Patterns You Need to Collect Information Often in math, you can collect data and look for patterns. This will help you make predictions and develop rules. To look for a pattern, it helps to record your data in an organized way. Example
1
Make a table of the number of diagonals in polygons of 3 to 7 sides. Solution
Number of sides
Polygon
Triangle
0
What it means for you: You’ll see how to use information you have collected to find a rule and make predictions.
3 0
0
Quadrilateral 1
The number at each vertex shows how many new diagonals start there.
4
1+1+0+0=2
5
2+2+1+0+0=5
6
3+3+2+1+0+0=9
7
4 + 4 + 3 + 2 + 1 + 0 + 0 = 14
0
rule pattern prediction generalization formula
0
Pentagon 2 1 2
Check it out:
0
0
The examples in this Lesson are all about diagonals of polygons. L
0+0+0=0
1
Key words: • • • • •
Number of diagonals
M
Hexagon
3 2
3
0
1 0
Heptagon P
4
4 3
N
A diagonal of a polygon is a line joining two vertices that aren’t already joined by the sides of the polygon. A rectangle has two diagonals.
0 2 0
1
Section 7.5 — Generalizing Results
419
Guided Practice The numbers represented by this pattern of dots are called square numbers. Exercises 1–2 are about the pattern. 1. Copy and complete the table below. Place in the pattern
Number of dots
1st
1
2nd
4
3rd 4th
2. Draw the 5th and 6th figures in the pattern. Add the number of dots in the 5th and 6th figures to the table.
You Can Make Rules and Predictions When you’ve collected your information, you can look for patterns in the numbers. If you can figure out how the pattern will continue, then you can make a rule. Example
2
Find the number of diagonals in an octagon (8-sided polygon). Solution
You need to look for a pattern in the sums used to find the number of diagonals in Example 1. Check it out: The sketch for an 8-sided polygon is pretty confusing to look at and complicated to draw. For polygons with large numbers of sides, it would be almost impossible to draw a sketch. That’s why it would be useful to find a rule, so you could work out the number of diagonals without having to draw a sketch.
For each shape, the number of diagonals that can be drawn from the first vertex is 3 less than the number of sides. This number is repeated once, then decreased by one until 0 is reached on the last two vertices. So an 8-sided polygon will have 5 + 5 + 4 + 3 + 2 + 1 + 0 + 0 = 20 diagonals. You can check this with a sketch.
5
5
4 0 3 0 2 1
420
Section 7.5 — Generalizing Results
Example
3
Find the number of diagonals in a 10-sided polygon. Solution
You can find the answer using the same pattern as in Example 2: 7+7+6+5+4+3+2+1+0+0 There is a way to add the numbers up quickly — you can match the numbers into pairs that add up to 7, the first number in the sum.
7+7+6+5+4+3+2+1+0+0 4+3=7 5+2=7 6+1=7 7+0=7 7+0=7
Check it out: With an odd number of vertices, you can still pair up the numbers in the sum. There will be one number left unpaired. The unpaired number is always half the total of each pair of numbers. When you add this half-pair to the others, the number of pairs is still half the number of vertices. Try this for yourself, using the heptagon from Example 1.
Each number in the sum represents one vertex. There are 10 vertices, so there are 10 ÷ 2 = 5 pairs that each add up to 7. So there are 5 × 7 = 35 diagonals in a 10-sided polygon.
Example
4
Find a rule for working out the number of diagonals in a polygon with n sides. Solution
Don’t forget: The letter n is a variable — the letter is just standing in for a number you don’t know yet.
You can say the same things for a polygon with n sides as you said for a polygon with 10 sides in Example 3. Number of vertices
Number of diagonals from f irst vertex
Number of pairs of vertices
10
10 – 3
10 ÷ 2
n
n–3
n÷2
For any n-sided polygon, we can generalize that the number of diagonals is always (n – 3) × (n ÷ 2)
Section 7.5 — Generalizing Results
421
Guided Practice Exercises 3–5 use the table you made in Exercises 1–2. 3. Study the pattern of the numbers of dots. Predict how many dots would be in the 10th figure. 4. How could you predict the number of dots in the 100th figure? 5. Which of the following could be used as a general rule for finding the number of dots in the nth figure of the pattern? (n – 1)(n – 1) 2n n+3 n2
You Need to Test Rules to Make Sure They Work When you figure out a new rule, you need to test it to see if it is true. Example
5
Test the rule for working out the number of diagonals in any polygon. Solution
The rule from Example 4 is that for any n-sided polygon, you can generalize that the number of diagonals is always (n – 3) × (n ÷ 2). Don’t forget: You could check that the rule works for all the polygons you’ve looked at so far — just to make sure.
You can test the generalization on polygons for which you already know the results. The table from Example 1 says a hexagon has 9 diagonals. If the rule works, it will give the answer 9 when n = 6. (n – 3) × (n ÷ 2) = (6 – 3) × (6 ÷ 2) =3×3
Write out the expression Replace n with 6 Simplify
=9 The rule gives the right answer of 9 diagonals.
Guided Practice Exercises 6–7 use the rule you found in Exercises 3–5. 6. Test the rule you chose in Exercise 5 by checking that it gives the same answers as in the table from Exercises 1–2. 7. Use the rule to predict how many dots would be in the 7th figure in the pattern. 8. Draw the 7th figure in the pattern and count the dots to see if your answer to Exercise 7 was right. 422
Section 7.5 — Generalizing Results
Independent Practice The measures of the angles in a triangle always add up to 180°. You can use this fact to make a generalization about the sum of the measures of the angles in any polygon. 1. In the chart below, diagonals are drawn from one vertex of each polygon to divide it into triangles. Copy and complete the chart.
Polygon
Number of sides
Number of triangles
Sum of measures of the angles
3
1
1 × 180° = 180°
4
2
2 × 180° = 360°
5
?
3 × 180° = ?
?
?
? × 180° = ?
Triangle
Quadrilateral
Pentagon
Hexagon
2. Continue your chart for a heptagon (7-sided polygon) and an octagon (8-sided polygon). Remember to draw diagonals from one vertex only to break the polygon into nonoverlapping triangles. 3. Without drawing it, predict the number of triangles and the sum of the measures of the angles in a 10-sided polygon. 4. Study your chart. What is the relationship between the number of sides and number of triangles in a polygon? Now try these: Lesson 7.5.1 additional questions — p468
5. For an 85-sided polygon, which of the following could be used to find the sum of the measures of its angles? 83 × 180° 84 × 180° 85 × 180° 86 × 180° 87 × 180° 6. Describe in words a rule that could be used for finding the sum of the measures of the angles in any polygon. 7. Write a formula for finding the sum of the angle measures in an nsided polygon. Use S to represent the sum of the angle measures. It may help to first complete each column of your table for an n-sided polygon.
Round Up That might seem like a lot to take in, but the Lesson boils down to three basic steps for finding math rules. First, collect and organize your data. Next, look for a pattern and use it to find a rule. Finally, test the rule to make sure it works. Section 7.5 — Generalizing Results
423
Lesson
7.5.2
Proving Generalizations
California Standards:
When you make a generalization, you might not be able to tell if it’s always true. It isn’t true if you can find even one example where the rule doesn’t work. If it is true, there’s always a mathematical reason why.
Mathematical Reasoning 1.2 Formulate and justify mathematical conjectures based on a general description of the mathematical question or problem posed. Mathematical Reasoning 3.2 Note the method of deriving the solution and demonstrate a conceptual understanding of the derivation by solving similar problems. Mathematical Reasoning 3.3 Develop generalizations of the results obtained and the strategies used and apply them in new problem situations.
What it means for you: You’ll find out about using examples to prove when generalizations are not true, and about using math to prove when they are.
Key words: • • • •
generalization prove example test
Generalizations Can Sometimes Be Proved Wrong You might find some information that leads you to make a generalization. But sometimes you’ll then be able to find an example that proves the generalization isn’t true. Example
1
Tara measured the angles of this trapezoid. She then made a generalization that no two angles in a trapezoid ever have the same measure. Give an example of a trapezoid that shows that this generalization is incorrect.
110°
100°
80°
70°
What was wrong with Tara's method of making a generalization? Solution
The trapezoid shown here is one example that proves that Tara’s generalization is wrong. You could show any isosceles or right trapezoid.
110°
110°
70° 70° Tara based her generalization on only one example. If she’d looked at more trapezoids, she might have found one with two angles of the same measure. She didn’t test if she could draw a trapezoid with more than one angle of equal measure.
Guided Practice Don’t forget: A trapezoid is a quadrilateral with only one pair of parallel sides.
In Exercises 1–5, use an example to prove that each generalization is not true. 1. In a right triangle, the two angles that are not right angles always have the same measure. 2. A trapezoid can never contain a right angle. 3. If two sides of a triangle are the same length, the third side is always a different length. 4. Rectangles with different side lengths always have different areas. 5. The two sides of equal length in an isosceles triangle are each always longer than the other side.
424
Section 7.5 — Generalizing Results
You Can Prove a Generalization Using Math Example
2
Is it possible to draw a triangle with two right angles? Solution
If you can draw a triangle with two right angles, then the answer is yes. This attempt has failed because the shape doesn’t close. This attempt has failed because there’s only one right angle. This attempt has failed because there are four sides. Check it out: If you’re using examples, you only need one to prove that something isn’t true. But proving that something is true using examples can be tricky.
It looks like you can make the following generalization: “it’s impossible for a triangle to contain two right angles.” But there’s no way to tell for sure if that’s true by drawing examples — there might be a way to do it that you just haven’t thought of. The way to show a generalization is true is to start with some known facts. Example
3
Use the triangle sum to show that a triangle can’t have two right angles. Solution
A right angle measures 90°. Don’t forget: The triangle sum tells us that the measures of the angles in a triangle add up to 180°.
If a triangle has two right angles, the sum of their measures is 90° + 90° = 180°. For the sum of all three angles to be 180°, the third angle would have to measure 0°. But a triangle can’t contain an angle of 0°, so a triangle can’t have two right angles.
Guided Practice In Exercises 6–9, try to draw a quadrilateral with the following characteristics. 6. Exactly 1 right angle 7. Exactly 2 right angles 8. Exactly 3 right angles 9. Exactly 4 right angles 10. Use math to prove the generalization: “It is not possible to draw a four-sided shape that contains exactly three right angles.”
Section 7.5 — Generalizing Results
425
Independent Practice There are two diagonals that can be drawn in every rectangle. A few examples are shown below. Don’t forget: Matching tick marks show line segments of equal length.
From looking at the examples above, say whether the following generalizations about the diagonals of all rectangles are true. 1. The two diagonals are always the same length. 2. The two diagonals cut each other into equal-length segments. 3. The diagonals intersect at a 90° angle.
Don’t forget: “Rhombi” is the plural of “rhombus.” See Lesson 7.3.3 for more information.
In Exercises 4–7, draw several examples of the given type of quadrilateral. Determine which of the generalizations from Exercise 1–3 are true for all quadrilaterals of that type. 4. Squares 5. Rhombi 6. Parallelograms 7. Trapezoids Use the parallelograms shown below to answer Exercises 8–13. A
D
45°
135°
135° C
45°
B
A
60° D
120°
120°
A 106°
B 60°
C
D
74°
B
A
B
D
C
74°
106°
C
8. Compare the measures of angles A and C in each parallelogram. What generalization can you make about the opposite angles A and C in the parallelograms above? 9. Is this generalization true for opposite angles B and D in each parallelogram above? Now try these:
10. Is it possible to draw a parallelogram where these generalizations are not true? If so, draw it. If not, explain why not.
Lesson 7.5.2 additional questions — p469
11. For each parallelogram, find the sum of angles A and B. 12. What generalization can you make about the neighboring angles A and B in the parallelograms above? 13. There are other pairs of neighboring angles in each of the parallelograms. Is your generalization in Exercise 12 true for the pairs B and C, C and D, and D and A as well?
Round Up Sometimes you can prove a generalization using examples, but usually you’ll need to use the math behind the generalization to show that it is always true. That doesn’t mean that examples aren’t important — they’re often the best way to prove when a generalization isn’t true. 426
Section 7.5 — Generalizing Results
Chapter 7 Investigation
Rec lining Chair s eclining Chairs Reclining lounge chairs at hotel pools can usually be set to recline at various angles. The support bar at the back of the seat is moved into designated slots — like in the picture below. The nearer the slot to the base of the seat, the more upright the back will be. When viewed from the side, the support bar, the back of the seat, and the slotted bar form a triangle, as shown. • The support bar is 10 inches long. • It is permanently attached to the back of the chair at a length of 8 inches from the hinge of the seat. • The support bar can be moved into slots at 6, 8, 10, 12, or 14 inches from the hinge of the seat to create different reclining positions. It can also be laid flat.
8 in.
x°
Part 1: How is the reclining angle x° related to angle 1?
10 in.
2 1
3 6 in.
10 in. 14 in. 8 in. 12 in.
How is angle 1 affected when the support brace is moved into different slots? Part 2: Investigate the different types of triangles that can be created by the support bar, the back of the seat, and the slotted bar. Use a compass and ruler to draw the triangles. Extension The reclining angle (x°) has a special relationship to the other angles in the “support triangle.” • Describe any patterns you notice between x and the measures of angles 1, 2, and 3. Try to make a generalization about any relationships you see. • Test your generalization for other triangles. Is this relationship true of all triangles? If not, explain why not. If so, generalize your findings. Open-ended Extension Create a design for a lounge chair different than the model given. • What reclining angles would your chair have? • Could you use the same basic design? If so, what length would the chair back and support bar be? Why? If not, explain why not, and design an alternative reclining mechanism.
Round Up This is just the kind of thing some engineers have to do. They’re given a specification — a description of something that someone needs, giving information about what it must do. The engineer must then come up with a design that will do all of the things necessary. Cha pter 7 In vestig a tion — Reclining Chairs 427 Chapter Inv estiga
Additional Questions Lesson 1.1.1 — Comparing Integers In Exercises 1–3, say which of the integers is greater. 1. –4, 4 2. 5, –6
3. –2, –4
In Exercises 4–6, list all the numbers from –3 through 4 that are: 4. whole numbers 5. integers
6. natural numbers
The change in the position of a football after Timmy ran it was +2 yards. The last time Jamal ran the football the change of position was –4 yards. –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 8 7. Circle these two integers on a number line Day A 6 8. Use the correct symbol, “<”, “>”, or “=”, to compare these two integers. When the water level in a hotel pool is below the 0 marker, Andrew needs to add more water. When it is at or above 0 he does not. The water level during two different days is shown on the water meter to the right. 9. Write all the integers between the two levels of water shown. 10. Describe the range of numbers shown on the water meter at which Andrew does not need to add water as natural numbers, whole numbers, or integers.
4 2 0 –2 –4
Day B
–6
Lesson 1.1.2 — Adding and Subtracting Integers In Exercises 1–6, work out the result of each calculation. 1. 8 + 4 2. 6 + (–5) 4. –3 – 2 5. –14 – (–8)
3. –12 + (–3) 6. –9 – (–6)
7. How is subtracting a negative number using the number line different from adding a negative? $14 $12 $10 $8
Mo
nda y Tue sday Wed nesd ay Thu rsda y Frid ay
$6
The graph shows the price of a stock over four days. Use this for Exercises 8–10. 8. Write the integer that represents the change in the price of the stock between Monday and Tuesday. 9. Write the integer that represents the change in the price of the stock between Tuesday and Wednesday. 10. The stock price changed by –$3 overall from Monday to Friday that week. Write the price of the stock on Friday.
Golf Scores Janet played four rounds of golf in a tournament. Her scores for the first three rounds are shown in the table. Use this for Exercises 11–13. Player Rd 1 Rd 2 Rd 3 Rd 4 11. What was Janet’s overall score after round 2? –3 –1 2 Janet 12. Janet’s overall score for all four rounds was –7. –4 1 –1 Melissa What was Janet’s score in round 4? 13. Janet’s overall score after four rounds was 2 lower than Melissa’s. Write Melissa’s score for round 4.
Lesson 1.2.1 — Multiplying with Integers In Exercises 1–3, work out the result of each calculation. 1. –6 × 8 2. –3 × (–5)
3. –12 × (–3) × (–2)
4. Explain why these two expressions have the same value: 6 × 3 and 3 × 6 A cafeteria used five cans of soup every day, for eight days. To work out the change in the number of soup cans in stock, a worker wrote this expression: –5 × 8. 5. Write the value of this expression. 6. The number of cans of soup left at the end of the eight days was 164. How many cans of soup did the cafeteria have at the beginning of the eight days? 428
Additional Questions
Lesson 1.2.2 — Dividing with Integers In Exercises 1–3, work out the result of each calculation. 1. 72 ÷ (–9) 2. –24 ÷ 3
3. –90 ÷ (–5)
Mary did the division, 54 ÷ (–2) and got the answer –26. 4. Describe how Mary could check her result without repeating the division. 5. Say whether –26 is the correct answer to the division. If it isn’t, find the correct answer. 6. Write an expression with a quotient of –12. A worker at a laundromat needed to do 15 basketfuls of laundry for a hotel client. Each basket had 10 pounds of laundry in it. The washers could hold 25 pounds of laundry each. 7. Write the total number of pounds of laundry the worker needed to do. 8. Write the least number of washes the worker could do to complete the order. Malcolm is building a fence. He plans to use 18 fence posts, and to put a topper on every third post. 9. How many toppers will Malcolm need? 1
10. The fence posts are 3 feet high without the toppers. The toppers are 3 the height of the original fence posts. What will be the height of a fence post with a topper added?
Lesson 1.2.3 — Integers in Real Life In Exercises 1–4, work out the result of each calculation. 1. 1.5 miles multiplied by four. 2. 100 people divided into four equal groups 3. 150 feet, dropping by 40 feet 4. $20 plus $134 In Exercises 5–6, use the number line below. –15 –12 –9 –6 –3
0
3
6
9
12 15 18
°C
5. Write an integer to show the temperature change if a thermometer reading goes from –1 °C to –8 °C. 6. Write two integers from the number line with a difference of 21 °C. 7. A deliveryman delivered 20 packages each hour for three hours. After this time he still had 70 packages left to deliver. If he carried on delivering packages at the same rate, how long would it take him in total to deliver all the packages?
Lesson 1.3.1 — Decimals In Exercises 1–3, write the digit in the hundredths place. 1. 152.685 2. 565.642
3. 33.896
The changes in a gymnastic team’s average score after each of four events are shown in the table to the right. 4. The change in the average score after the balance beam was an increase of 3 tenths. Represent this change using a decimal. 5. The change in the average score after the floor exercise was a drop of 5 tenths. Represent this change using a decimal. Package weight
Mailing cost
< 0.5 lb
$2.00
0.5 to 0.999 lb
$4.00
1.0 to 1.499 lb
$6.00
1.5 to 1.999 lb
$8.00
Event
Change in average score
Vault
+0.4
Uneven Bars
+0.1
Balance Beam Floor
Suzanne worked in a mail room. She found the weight of several packages and charged the customers based on that weight as shown in the table to the left. 6. Suzanne weighed a package that was between 0.6 and 0.7 pounds. Find the mailing cost. 7. Suzanne then weighed another package. The cost of mailing the first package (in Exercise 6) and the second package together was $12.00. Write a possible weight for the second package. Additional Questions
429
Lesson 1.3.2 — Ordering Decimals In Exercises 1–2, place the numbers in order of increasing value. 1. –2.67, –0.67, –1.275, 0, –5 2. 0.1, –0.03, –3.2, –3.26, 0.15 3. Explain how to add zeros to compare the decimal numbers 4.52 and 4.5. After a rainstorm a class measured three puddles on the school playground. The width of the first puddle was 21.4 cm. 4. The width of the second puddle was less than the width of the first, but greater than 20 cm. Write a decimal number that could represent the width of the second puddle in centimeters. 5. The third puddle was more than two times wider than the second puddle. Write a decimal number that could represent the width of the third puddle in centimeters.
Lesson 1.4.1 — Rounding Numbers In Exercises 1–3, write the numbers –1001.55, –479.133, and 49.806, to the nearest: 1. one 2. hundred 3. hundredth 4. A scientist showed that the average raindrop measures 0.001655 meters across. What is this to the nearest thousandth of a meter? 5. When Marsha rounded three numbers to the nearest hundredth, she got a result of 0.53 all three times. Two of the numbers were greater than 0.53 and one was less. Write three numbers that could have been the numbers that Marsha rounded. Country
Area (km2)
Russia
8,502,290
Brazil
5,326,884
Canada
2,456,996
United States
2,262,039
China
1,629,600
The table to the left shows the forested area of various countries in the world. Use the table to answer Exercises 6–8. 6. Write the forested area in Russia to the nearest hundred thousand square kilometers. 7. Write the forested area in Canada to the nearest ten thousand square kilometers. 8. When the forested area of the United States was rounded, it came to 2,000,000 km2. What place value was the forested area rounded to?
Lesson 1.4.2 — Using Rounded Numbers In Exercises 1–3, use estimation to find approximate answers 1. 518 + 287 2. 16,994 – 5527
3. 3889 × 689
4. Use the sum 847 + 145 to explain how the place value to which a number is rounded affects the accuracy of the estimation. The table to the right lists the average snowfall in five places in the US. Use this information in Exercises 5–7. 5. List the average snowfalls from the chart, when rounded to the nearest 10. 6. Estimate the difference between the average snows in Valdez, Alaska and Marquette, Michigan. 7. Estimate the total number of inches of snow the people in Yakutat, Alaska should expect during a 9 year period.
Place
Ave. inches snow/year
Valdez, Alaska
326
Mount Washington, NH
260
Blue Canyon, California
240
Yakutat, Alaska
195
Marquette, Michigan
141
8. The greatest TV audience for one program was 114,970,000 people. Write the greatest and least possible audience figures, given that this number is rounded to the nearest 10,000 people.
430
Additional Questions
Pro duc tB
In Exercises 1–3 use the bar graph of sales during September to answer the questions. 1. Estimate the number of times greater the sales for Product A were, than for Product B. 2. Product B had sales of between 200,000 and 250,000. Estimate the sales for Product A 1 3. In the month of October, the sales for each product rose by 4 of the amount they sold in September. Estimate the sales for each product in October.
Pro duc tA
Lesson 1.4.3 — Estimation
The lengths of coastlines in several states are shown on the graph. Alaska has a coastline of 6640 miles. Use these facts to answer Exercises 4–6. 4. Write an estimate for the length of the Californian coastline. Explain your estimate. 5. Write an estimate of the coastline length in Texas. Explain your answer. 6. Write an estimate to describe the difference between the coastline lengths in Texas and Florida. Explain your estimate.
Alaska Florida California Hawaii Louisiana Texas
Lesson 1.4.4 — Using Estimation In Exercises 1–3, say whether a precise answer or an estimate is appropriate. 1. Coach Henderson is measuring the distance jumped by 10 students during a long-jump competition. 2. Pauline is deciding how much paint to buy to paint the outside of her house. 3. Mrs. Jones is making a cake and is measuring the amount of flour she needs. A boat traveling down a river was located at the point marked X after 42.5 hours of travel.
Start
42.5 hours
X
Finish
4. Estimate the total amount of time it will take the boat to travel from start to finish. Explain your estimate. 5. The distance from start to finish was 360 miles. Estimate the distance the boat had left to travel. 6. Say whether or not the boat would be able to travel 500 miles in 100 hours. Explain your answer.
Lesson 2.1.1 — Variables 1. Gina has scored 47 more points this basketball season than Isabella. What letter would be a sensible variable to use to represent the number of points scored by Isabella. In Exercises 2–7, show the expressions on a number line. 2. a – 2 3. r – (–4) 5. s ÷ 5 6. w × 3
4. y + (–6) 7. b ÷ (–2)
In Exercises 8–11, determine whether or not it is necessary to use a variable to describe the problem. 8. Emilio has twice as many spelling awards as Jake. Determine the number of spelling awards that Jake has. 9. Mrs. Gooden planted 3 times the number of daisies that Mr. Kearns planted. Mr Kearns planted 12 daisies. Determine the number of daisies Mrs. Gooden planted. 10. A baker sold 36 more doughnuts than sweet rolls during one day. She wants to know how many doughnuts she sold. 11. Luke pitched 12 baseball games this season. Jose pitched one-third of the number of baseball games that Luke pitched. Their coach wants to know how many baseball games Jose pitched this season. 12. Write a problem in which the h in h ÷ 4 is an unknown quantity in a situation. 13. Write a situation in which the unknown is the number of movies Andre saw last summer. Additional Questions
431
Lesson 2.1.2 — Expressions In Exercises 1–5, write an expression for each situation. 1. A number, p, is increased by 12. 2. A number, y, is multiplied by 7. 3. A number, h, is decreased by 36. 4. A number, w, is divided by 2. 5. A florist divided 64 roses equally between v vases. How many roses were in each vase? Evaluate Exercises 6–8, if the value of a is 9. 6. (a × 2) + 13 7. 55 – (a ÷ 3)
8. (9 ÷ 3) + a
In Exercises 9–11, describe each expression in your own words. 9. ( j × 9) + 5 10. 23 + (d ÷ 2)
11. 15 – (3 × e) + 1
12. The cost of bananas was $0.32p, where p is the number of pounds of bananas sold. How much would 4 pounds of bananas cost? 13. Write an expression in which d is multiplied by 5, and then the product is decreased by 12. 14. Evaluate your expression from Exercise 13 for d = 8.
Lesson 2.1.3 — Multi-Variable Expressions In Exercises 1–3, evaluate the expressions for x = 1 and y = 2. 1. x + 2y 2. (6 – y) + (x × 4)
3. 5 × (y ÷ x) + 3
In Exercises 4–6, write an expression for each situation. 4. c is increased by the product of y and r 5. The ratio of j to k is decreased by the sum of w and z 6. 6m is decreased by the product of g, h, and i 7. There were red, green, and yellow footballs in a P.E. equipment box. There were three times as many red footballs as the difference between the number of green and yellow footballs. Write an expression for the number of red footballs. Exercises 8–9 use the diagram to the right, which shows an amusement park ride that travels up and down. 8. The ride travels straight up to a maximum height of 150 feet off of the ground. From there it drops x feet, and goes up another y feet, then goes down the remaining z feet to the ground. Write an expression to show the total distance, in feet, a person on the ride would travel. 9. Find the total distance, in feet, a person on the ride would travel when x = 70, y = 45, and z = 115.
Lesson 2.1.4 — Order of Operations 1. Rewrite w5 as a repeated multiplication of w. 2. Rewrite the expression b × b × b × b × b × b × b × b in base and exponent form. 3. What does PEMDAS stand for? 4. Explain the order of operations in the expression 15 + (23 ÷ 3 × 4 – 1). 5. Ms. Vaughn is giving 15 questions to every student in each of her three math groups. There are 8 students in one math group, 7 in a second math group, and s students in the third math group. Write an expression that can be used to calculate the total number of questions Ms. Vaughn will have given out. 6. Given that there are 9 students in Ms. Vaughn's third math group, determine the value of your expression from Exercise 5. 432
Additional Questions
Lesson 2.2.1 — Equations 1. What is the main difference between an equation and an expression? 2. Use the guess and check method to solve the equation x – 12 = 10. Give 2 examples of a guess and check, then show the correct value of x. In Exercises 3–5, write down the value of n that satisfies each equation. 3. 100 = 9n + 19 4. n ÷ 6 = 2 5. 5n – 45 = 90 In Exercises 6–9, write an equation that represents each given situation. 6. Aaron is 6 inches taller than Mario. 7. Hailey is 9 years younger than Cassandra. 8. Gavin ate half as many hot dogs as Lin. 9. Trina needs twice as much fabric as Bea. 10. A hairstylist makes $25 per haircut and $80 per color. Write an expression for the total amount of money the hairstylist makes for h haircuts and c colors. 11. On Saturday, the hairstylist gave 7 haircuts and 4 colors. Write an equation for the total amount of money, m, that the hairstylist made that day. Solve your equation for m.
Lesson 2.2.2 — Manipulating Equations 1. When you manipulate equations, what is the only way to keep both sides of the equation balanced? 2. The diagram to the right shows a balanced equation. If you add 5 triangles to the left side of the equation, what has to be done to the right side?
8+7=9+6 =
In Exercises 3–5, fill in the missing parts. 3. d – c = 34 4. 6y = 24 d – c + ? = 44 6y ÷ ? = 12
5. t – l = 64 ? = 64 – 64
In Exercises 6–8, explain what step was done incorrectly. 6. a × b = 130 7. q = 50 a = 130 + b 5q = 10
8. m – 9 = n m + 2 = n – 11
9. Ike saw that Jenna had 32 ¢ more than he did. Write an equation to represent this situation. 10. Using your answer from Exercise 23, if Jenna has 96 ¢, how many cents does Ike have?
Lesson 2.2.3 — Solving + and – Equations Tomas has solved the equation x + 12 = 20 and found that x = 8. 1. Explain how he can check that this answer is correct. 2. Is Tomas's solution to the equation correct? Show your work. Solve the equations in Exercises 3–11 using manipulation. Remember to check your answers. 3. y + 5 = 13 4. 2.2 + y = 5.3 5. –15 = y + 12 6. 75 = y – 25 7. y + 20 = –20 8. y + –6 = –31 9. –17 = y – 9 10. –7 = –7 + y 11. y – 3.7 = 9.8 In Exercises 12–15, write an equation for each situation, then solve it. Use n to represent the variable. 12. Three plus a number is 9. 13. A number decreased by 10 is equal to 91. 14. When five is added to a number the result is 52. 15. A number increased by 0.2 equals 1.8 16. Kelsey's plant has grown 3 inches since last Tuesday and is now 10 inches tall. Write an equation to represent this situation. Use h to represent the height of Kelsey's plant last Tuesday. Then solve your equation to find h. Additional Questions
433
Lesson 2.2.4 — Solving × and ÷ Equations Write down the solution to the multiplication equations in Exercises 1–3. 1. 15a = 90 2. j × 3 = 108 3. 20·q = –480 Write down the solution to the division equations in Exercises 4–6. 4. 12 ÷ h = 4 5. 32 ÷ m = 2
6. g ÷ 5 = –9
7. Ryan made a mistake in solving the equation to the right. What mistake did Ryan make in his work? 8. Solve Ryan's equation correctly.
3t = 27 t = 81
In Exercises 9–12, write an equation for each situation, then solve it. Use y to represent the variable. 9. A number multiplied by seven is equal to 21. 10. A number divided by –5 equals twenty. 11. The product of a number and –3 is eighteen. 12. Four divided by a number equals 16. 13. The Bobcats football team scored 18 points this year from 2-point safeties. Write and solve a multiplication equation that can be used to determine the number of safeties, s, that the Bobcats made this year.
Lesson 2.2.5 — Graphing Equations 1. What is a different way to write the equation b ÷ 5 = 40? 2. What are different ways to write the equation m × 6 = 22? 3. How does the equation x + 3 = 12 explain the location of x on a number line? 4. How does the equation y – 2 = 7 explain the location of y on a number line? In Exercises 5–6, look at the diagram and write down the equation that goes with each. 5.
y
6.
3 4 5 6 7 8 9 10
y 0
3+7
5
10
15
5×3
In Exercises 7–11, identify the type of equation and solve it. 7. k – 40 = 92 8. 19j = –38 10. a ÷ –7 = 11 11. n × 25 = 200
9. w + 14 = 65
12. Coach Peters had 80 uniforms that are to be shared equally among x players. However, he lost 23 of the uniforms leaving each player with u uniforms. If there are 19 players, calculate how many uniforms each player receives by writing and solving a suitable equation.
Lesson 2.3.1 — Expressions About Length 1. Write down an expression that represents the length of this object.
Diagram not to scale
3m
2m
4m
?m
2. The equation (x + y) + z = x + (y + z) is an example of what property of addition? 3. Prove the equation using x = 7, y = 4 and z = 9. In Exercises 4–6, write down a shape whose perimeter could be represented by the expression 4. 5 + 6 + x 5. 3 + 3 + 3 + 3 + x 6. 9 + 12 + 4 + x 7. Malik is cutting balloon strings. The strings that came with the balloons are too long. Each string is i inches long and needs to be exactly 12 inches. Write an expression that could be used to represent the length of the string that has been cut off. 8. Miss Ortiz is arranging desks in her classroom. Each desk is rectangular in shape and has a length of 26 inches and a width of 16 inches. She arranged 6 desks so the widths of the desks were side-byside. Find the perimeter of the row of 6 desks Miss Ortiz arranged.
434
Additional Questions
Lesson 2.3.2 — Expressions About Areas In Exercises 1–6, write an expression for the area of each shape. 7 in 5 cm 1. 2. x in
y cm
4.
k cm j cm
5.
y in
3. 6.
y
7 ft
14 in
z
y ft
In Exercises 7–10, write an expression that represents the area of each rectangle described. 7. Base = 3 cm and Height = h cm 8. Base = d inches and Height = 12 inches 9. Base = m feet and Height = w feet 10. Base = 17 inches and Height = 11 inches 11. The equation (xy)z = x(yz) is an example of what property of multiplication? Show an example using x = 7, y = 4, and z = 9. In Exercises 12–13, find the length of the unknown side when the area of each figure is 48. 13. 12. ? cm ? yd 8 cm
12 yd
14. Aubrey wants to frame a picture that is 8 inches long by 10 inches wide. He wants to use a frame that is 16 inches long by 20 inches wide. Determine the area of the picture and the frame. Find the amount of space inside the frame that will not be occupied by the picture.
Lesson 2.3.3 — Finding Complex Areas 1. Dora and Ben divided the same complex shape differently to determine the area. Their drawings are shown to the right. Explain how Dora and Ben get the same answer for the area when they each divided the shape up differently. In Exercises 2–10, find the area, or an expression for the area, of each figure. 2.
3.
4 cm 8 cm
6.
8 yd
5 in
4 in
b
4 in a
4 in
6 yd
10 yd
19 in c
7.
4 in 4 in
10 yd
5 in
5 in
6 units
5.
Ben’s shape
5 in
4.
2 units 2 units
8 cm
12 cm
Dora’s shape
6 yd
a
6 yd
8.
9.
x
10.
h k
x
k
y
k
a b c
k
c
In Exercises 11–13, use subtraction to determine the area of each figure. 11.
x y
z z z
12.
8 cm
12 cm
13. 2 ft
5 ft 5 ft
5 ft
8 cm 4 cm
12 ft12 ft
14. A gazebo is being built in a park. The floor of the gazebo is shaped like a 12 ft regular hexagon, as pictured to the right. Determine the area of the floor of the gazebo. Explain, in words, how you determined your answer.
12 ft
Additional Questions
435
Lesson 2.3.4 — The Distributive Property 1. The equation a(b + c) = ab + bc is an example of what mathematical property? Verify this property when a = 4, b = 9 and c = 2. 2. Verify the distributive property by finding the area of the figure to the right, in two different ways. In Exercises 3–5, draw a picture to represent each expression. 3. 3 × (4 + 5) 4. 8 × (6 + 7)
a b
c
5. 2 × (9 + y)
Use the distributive property in Exercises 6–8 to determine the value of each expression. 6. 3 × (12 – 9) 7. –1 × (13 + 27) 8. –5 × (–6 – 9) Remove the parentheses from each expression in Exercises 9–11. 9. k × (g – d) 10. (5 – c) × 2
11. (m – v) × x
Lesson 2.3.5 — Using the Distributive Property 1. Explain how the distributive property can help with mental math. Show how you would use the distributive property to solve Exercises 2–7. 2. 7 × 15 3. 4 × 17 4. 18 × 6 5. 17 × 8 6. 14 × 9 7. 23 × 5 8. Deshaun had one dollar, made up of the following coins: 3 quarters, 2 dimes, and 1 nickel. The total number of coins is given by 3q + 2d + n, where q is the number of quarters, d is the number of dimes, and n is the number of nickels. Write an expression for the number of coins Deshaun would have if he had 5 dollars made up of the same combination of coins given above. Rewrite Exercises 9–11 using the distributive property. 9. g × (h + 19) 10. 11 × (4 – w)
11. a × (b + d + f)
12. A sheet of stickers contains dogs, cats, birds, and fish. The total number of stickers on each sheet is given by 3d + 5c + 8b + 4f, where d is the number of dog stickers, c is the number of cat stickers, b is the number of bird stickers, and f is the number of fish stickers. Write two expressions for the total number of stickers on 4 of these sticker sheets. Evaluate your expression for d = 2, c = 6, b = 3, and f = 5.
Lesson 2.3.6 — Squares and Cubes In Exercises 1–4, write an expression for the total area of each figure. 1. A square with sides of length m 2. Three identical squares with sides of length c 3. Two identical rectangles of side lengths b and h 4. Six identical squares with sides of length a In Exercises 5–7, use the distributive property to determine the surface area of a cube whose edges have the given length. 5. Edge length = 4 in 6. Edge length = 7 cm 7. Edge length = 11 yd 8. Using the formula, determine the volume of a cube with edges 10 feet long. In Exercises 9–11, the volume of a cube is given. Determine the length of the edges of each cube. 9. 8 m3 10. 27 yd 3 11. 343 cm3 12. Morgan is designing a water container and he wants it to be big enough for y cm 512 ml water (512 ml water = 512 cm3 water). He decides to make it in the shape of a cube. What is the suface area of the container he designs?
y cm y cm
436
Additional Questions
Lesson 2.3.7 — Expressions and Angles 1. The angles in a triangle always add up to what degree measure? In Exercises 2–3, determine the measure of Angle C inside each triangle. 2. Angle A = 45°, Angle B = 90°, Angle C = ? 3. Angle A = 30°, Angle B = 70°, Angle C = ? Use the diagrams in Exercises 4–5 to write equations involving b. Solve them, given that c = 40. 5. 4. b°
b° c°
25°
39°
c°
In Exercises 6–8, find the value of x in each diagram. 6. x° 7. 2x°
8. 53° x°
41°
x°
In Exercises 9–11, draw a picture to go with the given description of the angles. 9. The measure of angle MNP is 90°. The measure of angle MNR is 90 – y. 10. The measure of angle GHJ is x°. The measure of angle GHK is 45 + x. 11. The measure of angle ABC is 180°. The measure of angle ABD is 90°.
Lesson 2.4.1 — Analyzing Problems In Exercises 1–3, explain what is happening in each pattern. 1.
2.
3.
In Exercises 4–9, draw the next shape in the pattern.
?
?
?
9.
P
In Exercises 10–15, write down the next number in the pattern. 10. 9, 12, 15, 18, 21, 24, … 11. 64, 32, 16, 8, 4, … 13. 91, 82, 73, 64, 55, 46, … 14. 123, 234, 345, 456, 567, …
P
P
7. P
8.
?
5.
P
6.
?
P
4.
?
12. 5, 15, 25, 35, 45, 55, … 15. 96, 84, 72, 60, 48, 36, …
16. Four years ago, Garrison was 6 inches shorter than he is now. At that time, he was 8 inches shorter than his brother. Garrison is now 64 inches tall. How tall was Garrison’s brother four years ago?
Lesson 2.4.2 — Important Information In Exercises 1–4 write down whether you have too much or not enough information to solve each problem. If too much, write down the information you don’t need, and solve the problem. If not enough, write down the information you need. 1. Lisa’s picture frame has a length of 6 inches and a width of 4 inches. The picture Lisa wants to put in the frame has an area of 30 inches. What is the area, in square inches, of Lisa’s picture frame? 2. Each serving in a can of tuna fish contains 60 calories. What is the total number of calories that a can of tuna fish contains? 3. Xavier used a flashlight that provides 12 hours of light. If the flashlight were left on for 12 hours, what time would it be when the flashlight no longer provided light? 4. Melanie and Tara have known each other for 12 years. They were both 3 years-old when they met. They each have a birthday on March 18. How old are Melanie and Tara now? Additional Questions
437
Lesson 2.4.3 — Breaking Up a Problem
2 cm
30
1. Find the last digit of 2 by making a table for small exponents. 9 cm
2. The diagram on the right shows a postage stamp on a postcard. What area of the postcard is NOT covered by the postage stamp?
Hi guys, Wish you were here. We’re having a great time! Tim ‘n’ Amie
2.4 cm Hugo First Rickty Cottage Scare Street Frightsville 91481
16 cm
Determine the area of each of the shapes in Exercises 3–5. All measurements are given in inches. 11
4
3. 2
12
Orangesb 99 ¢/l
4.
11
2
Bananas 33 ¢/lb Apples 1.29 $/lb
5.
11
7
16
14
6. When Jeremy saw the signs on the fruit market store window, he went inside to get some fruit. He bought 3 lb of oranges, 4 lb of apples, and 2 lb of bananas. He had only planned on buying 2 lb of oranges, 2 lb of apples, and 1 lb of bananas. How much additional money did Jeremy end up spending on his fruit?
Lesson 3.1.1 — Understanding Fractions In Exercises 1–3, write down a fraction that is equivalent to the one that is given. 1. 6 2. 1 3. 7 18 3 56 4. Where will a negative fraction with a numerator smaller than the denominator be located on a number line? In Exercises 5–7, mark each fraction on a number line 5. − 3 6. − 1 7. − 2 7 3 5 In Exercises 8–10, write the fraction that is represented on each number line. 8.
9. –1
0
10. 0
1
–1
0
11. Five pairs of Rhonda’s 15 pairs of shoes are brown. Express this as a fraction. What fraction of Rhonda’s shoes are NOT brown?
Lesson 3.1.2 — Improper Fractions In Exercises 1–3, write down the improper fraction and the mixed number each drawing represents. 1. 2. 3. 4. Which part of the improper fraction is always the same as in the mixed number? 5. Which 2 operations are used when converting a mixed number to an improper fraction? In Exercises 6–8, write each mixed number as an improper fraction. 2 6. 8 7. 8. 5 1 8 3 9 3 9 9. Which operation is used to convert an improper fraction to a mixed number? In Exercises 10–12, convert the improper fractions into mixed numbers. 10. 20 11. 11 12. 23 7 6 3 3 3 13. Ramon and his friend ate 2 square cakes. Make a drawing to show 2 cakes. Write a mixed number 3 that means the same as 2 . 438
Additional Questions
Lesson 3.1.3 — More on Fractions 1. When putting
9 8
on a number line, what should you do first?
In Exercises 2–4, convert the improper fractions to mixed numbers. 18 80 3. 4. 36 2. − − 8 7 9 In Exercises 5–7, place the mixed numbers on a number line. 5 2 1 5. 6. 7. 2 −9 −1 7 3 2 3 1 8. Which one of these mixed numbers has the greater value: −6 4 , − 6 4 ? Explain your reasoning. 9. Which one of
−
14 11 ,− 10 10
has the lesser value? Explain your reasoning.
10. Show these fractions on the same number line.
−
1
19 2 , −3 5 5
3
11. On Friday, Jake’s thermometer read −1 3 degrees Celsius. On Saturday, Jake’s thermometer read −1 4 degrees Celsius. On which day was the temperature greater? Explain your reasoning.
Lesson 3.1.4 — Fractions and Decimals Convert each of Exercises 1–3 into decimal numbers. 3 1 1. 2. 3. 13 2 8 12 15 8 4. Where do a fraction and its equivalent decimal lie on a number line? In Exercises 5–7, write each fraction as a decimal, without using a calculator. 5. 3 6. 1 7. 3 5 8 4 8. What does it mean when a decimal is said to be a terminating decimal? Tristan has four drill bits. The drill bit sizes are shown on the diagram using fractions. 9. Convert each of the drill bit sizes in the diagram to decimals. 10. Tristan needs to drill a hole that is a minimum of 27 inches across. 64 3 in. 1 in. 15 in. 7 in. Which of the drill bits could he use? 32 16 8 2
Lesson 3.2.1 — Multiplying Fractions by Integers 1. Explain how to multiply a fraction by an integer. 2. In the problem 7 ×
2 3
which numbers get multiplied?
In Exercises 3–5, evaluate each expression. 3 2 3. 4. 3× 11 × 4 3 6. Write down the symbol that goes in the [space] in this equation:
5. 6 ×9 8
4 4 × 16 [space] 16 × 5 5
In Exercises 7–12, evaluate each expression. Give your answer as an integer or mixed number. 3 1 7. 8. 9. 2 –2 × –9 × × –4 4 2 6 1 10. 11. 1 12. 3 7 × –3 5 ×5 –8 × 1 3 4 7 13. Mr. Harris needs to cut pieces of yarn for all of the students in his art class. Each of his 25 students 3 will receive 6 4 inches of yarn. Write an expression that can be used to determine the total number of inches of yarn that Mr. Harris will cut. Evaluate your expression. Write your answer as an integer or mixed number AND an equivalent decimal. Additional Questions
439
Lesson 3.2.2 — More on Multiplying Fractions by Integers Evaluate the expressions in Exercises 1–3. 1 1 2. 3. 1. 2 –2 × 4 10 × –5 3 ×5 3 4 3 4. Three friends are equally sharing 37 feet of fishing line. How many feet will each friend receive? 1
5. A rectangular table with a perimeter of 22 feet is placed against a wall. 4 of the table’s perimeter is touching the wall. What is the total length of the sides of the table that are NOT touching the wall. Find the error that was made in working Exercises 6–8. 6. 2×15 = 15 7. − 4 ×3 = − 12 4
7
8 7 =1 8
8.
7
5 =1 7 1
1 12 3 ×8 = ×8 4 4 96 = 4 = 24
9. Avril’s bottle of sunscreen contained 475 milliliters of sunscreen. 5 of the bottle has already been used. Determine the number of milliliters of sunscreen that have already been used. Then, determine the number of milliliters of sunscreen that are left in the bottle.
Lesson 3.2.3 — Multiplying Fractions by Fractions 1. Write down the formula for multiplying fractions using variables a, b, c, d. Evaluate the expressions in Exercises 2–7. 2. 1 8 3. 29 1 × × 4 9 10 3 2 5. 9 6. 2 7 ×1 × 16 5 7 2 3 8. Jenna usually runs for 4 hours. Today she only ran for Jenna run today?
3 2 1 × 5 6 7. 1 3 × 2 8 11 that time. What fraction of an hour did
4.
1 2
In Exercises 9–11, find the error that was made when the fractions were multiplied. 3 9. 1 5 5 10. 1 3 11. 4 2 8 × = × = 1 × = 3 8 11 1 5 3 10 4 8 32 1 12. Becca played 4 of the first 2 of her basketball game. What fraction of the game did Becca play? 5
13. DeShaun’s mom worked 8 6 hours on Friday. On Saturday, she worked How many hours did DeShaun’s mom work on Saturday?
1 3
that many hours.
Lesson 3.3.1 — Dividing by Fractions Use multiplication to solve each of Exercises 1–5. 2 2 1 1. 1 2. 1 3. 1 3 ÷ ÷ ÷ 4 3 20 5 3 2 7 5 3 5 4. 5. ÷ 2 ÷2 9 6 4 8 6. Fill in the four blank spaces in this statement. “If two numbers which are reciprocals of one another 4 5 are __________, the answer will always be ___. For example, 5 ___ 4 = ___” 7. What is another name for a reciprocal? In Exercises 8–9 write the following expressions: 1 8. m × as a quotient. n
1 2
9.
d e
as a product.
10. Bret has pound of turkey to use for sandwiches. This is shared between 7 equally sized turkey sandwiches. Write an expression to show this situation and evaluate your expression to give the number of pounds of turkey in each sandwich. 440
Additional Questions
Lesson 3.3.2 — Solving Problems by Dividing Fractions 1. What are the 2 types of division problems? 2.
3 4
of a cup of sugar is shared between 3 people. How much sugar does each person get? 1
3. An 18 3 mile drive is divided into 2 equal-length sections. How long is each section? 4. 50 1 oz of juice is divided equally between 4 glasses. How much juice is put into each glass? 2 1 5. A 14,000 gallon pool is being drained. The pool drains 2 2 gallons of water per minute. How many minutes before the pool is empty? 6. Troy drove from his house to his grandmother’s house. After driving 157 miles, he was there. How many miles is it from Troy’s house to his grandmother’s house?
3 4
of the way
5
7. A gardener collected 120 9 lb of grass clippings. He took the clippings to a compost site in 3 equally-sized truckloads. What was the weight of grass in each truckload? 8. A tunnel is 14 3 feet high. How many sections of 18 in. does this height divide into? 4
Lesson 3.4.1 — Making Equivalent Fractions Find the simplest form of each fraction in Exercises 1–3 by canceling. 1. 8 2. 5 3. 9 12 35 15 4. Equivalent fractions look different but have the same ____________ . 5. Describe what is special about a fraction that’s equal to 1. Give an example. In Exercises 6–8, write down 2 fractions that are equivalent to each given fraction. 6. 3 7. 5 8. 9 7 8 12 18 9. Anastasia got 20 of her spelling words correct on her spelling test. What is a simpler form of the fraction of words Anastasia spelled correctly? Write down the decimal equivalent to this fraction. 1
10. Parker drew a pattern, shown here. Shade 3 of Parker’s diagram. Write down 3 fractions that are equivalent to the part you shaded.
Lesson 3.4.2 — Finding the Simplest Form In Exercises 1–3, find the simplest form of each fraction. 1. 22 2. 5 3. 21 88 13 105 4. Three-sixteenths of the crayons in Kaia’s crayon box are some shade of orange. Write the fraction that means three-sixteenths. Explain whether this is the simplest form of your fraction. In Exercises 5–7, find the greatest common divisor of each pair of numbers. 5. 45, 30 6. 35, 49 7. 200, 26 8. Dividing the numerator and denominator of a fraction by their greatest common divisor is the same as doing what? 9. Write the fraction that means the same as 0.16. What is the simplest form of this fraction? 10. Kobi put 50 staples into his stapler when it was empty. He used 15 of the staples for his homework. What fraction of the staples in the stapler did Kobi use for his homework? Write your answer in its simplest form.
Additional Questions
441
Lesson 3.4.3 — Fraction Sums Find the sum or difference in Exercises 1–6. If possible, simplify your answer. 1 1. 9 2. 5 2 3. 30 29 10 + + − + 12 12 8 8 33 33 33 8 1 2 3 4. 7 5. 2 2 2 6. 7 + − + + − + 14 14 14 9 9 9 16 16 16 Three friends shared an 8-slice pizza. Holly ate 1 slice, Malik ate 3 slices, and Darren ate 2 slices. In Exercises 7–11 write down the simplest form of the fraction of pizza that: 7. each friend ate. 8. Holly and Malik ate. 9. Holly and Darren ate. 10. Malik and Darren ate. 11. all 3 friends ate. 12. Using the fraction of pizza that all 3 friends ate, find the fraction of pizza that was left. Write expressions to represent the following situations, then evaluate your expressions. 3 13. Luis had a full glass of water. He drank 4 of the water in his glass. 14. Luis then took the water in his glass and added another 1 glass to it. 4 15. Finally, Luis drank all of the water in his glass.
Lesson 3.4.4 — Fractions with Different Denominators 1. What is a common denominator? Give an example in your answer. In Exercises 2–7, find a common multiple of each pair of numbers. 2. 3, 5 3. 1, 9 5. 13, 12 6. 5, 18
4. 9, 5 7. 12, 7
In Exercises 8–10, work out the sums and differences. Write your answers in their simplest form. 8. 3 2 9. 9 1 10. 2 1 + − + 7 3 12 4 13 2 11. Explain which of these two fractions has a greater value: 3
13 9 , 7 5
1
12. Khalid has grown 1 4 inches since last year. Jason has grown 2 3 inches since last year. What is the combined growth, in inches, of the two boys since last year? Show your work, and explain each step. 7
13. In Last Thursday’s game, Sherron made 8 of her free throw attempts. Today in practice, she made of her free throw attempts. Explain which is greater, showing what you know about common denominators and comparing fractions.
Lesson 3.4.5 — Least Common Multiples In Exercises 1–6, find the least common multiple (LCM) of the two numbers. 1. 8, 12 2. 4, 12 3. 4, 6 4. 22, 20 5. 10, 17 6. 64, 28 7. What does GCD stand for? Explain how you use it to determine an LCM of two numbers. 8. When adding two fractions, with different denominators, what do you have to do first? Determine the sums in Exercises 9–11. Write your answers in their simplest form. 10. 4 3 11. 5 3 9. 2 3 + + + 5 4 17 8 49 2 21
3
12. Mrs. Greene tried to calculate the sum 10 + 5 , giving her answer in the simplest form. 21 3 42 12 58 + = + = 10 5 20 20 20 What errors did Mrs. Greene make in working this problem? Correct her errors.
442
Additional Questions
2 3
Lesson 3.4.6 — Mixed Numbers and Word Questions 1. What do you do if the fraction parts of the mixed numbers you’re adding have different denominators? 2. When fractions have the same denominators, how can you compare them? In Exercises 3–5, determine the sum or difference. 1 2 5 4. 3. 5 +8 12 − 7 2 3 9
5.
1 3 19 + 12 2 5
In Exercises 6–8, use this diagram of a podium. Give all your answers as simplified fractions. 6. Kyle took 2nd place in a race. How far did he have to 3 climb to get onto his position on the podium? 6 8 in 7. Julio came 1st. How far did he have to climb? 4 52 in 8. Lin came 3rd. What was the difference in height 8 43 in between Lin’s and Julio’s podium positions.
2
1
3
Lesson 3.5.1 — Fractions and Percents 1. Explain how to use equivalent fractions to convert
3 25
into a percent.
In Exercises 2–7, write each fraction as a percent. 2. 19 3. 14 4. 50 50 35 2500 5. 363 6. 13 7. 3 500 40 14 In Exercises 8–10, write each percent as a fraction in its simplest form 8. 40% 9. 65% 10. 22.5% 11. Archie is watching birds for a project. What is this as a percentage?
64 144
of these birds are Northern Mockingbirds.
12. A group of people are kayaking down a river. 37.5% of the group capsized. Write the fraction of the group that capsized in its simplest form. 13. A soccer team has 11 players on the field. One of the players is sent off. Write the fraction and the percentage of the original team that remains on the soccer field.
Lesson 3.5.2 — Percents and Decimals 1. Explain how to write any percent as a decimal. In Exercises 2–7, the values are shown as a percent, a decimal, or a fraction. Write each value in the two forms which are not shown. Give all fractions in their simplest form. 2. 37% 3. 0.64 4. 0.05 8 5. 6. 99.6% 7. 33.8% 42 In Exercises 8–10, order each set of numbers from least to greatest. 8. 2 , 0.637, 65.2% 9. 0.102, 1 , 10 10. 3 , 0.006, 5%, 4 3 8 85 55 70 11. A football team has 46 active players. Only 11 of them are permitted on the field at any one time. What percentage of the players in this team must remain off the pitch? 12. Tommy does a survey of his class on the colour of pens they use. 18 use black pens, 9 use blue pens, 4 use green pens and 1 uses a red pen. What percent of the class use blue pens? Additional Questions
443
Lesson 3.5.3 — Percents of Numbers In Exercises 1–3, find the following percents: 1. 33% of 55 2. 12% of 60 4. 69% of 69 5. 13% of 89
3. 95% of 108 6. 15% of 564
7. It is thought that about 42% of people in California speak languages other than English at home. How many people from a class of 31 would you expect to speak a non-English language at home? 8. Monica takes a 500 ml bottle of water with her on her morning run. When she returns she has 24% of the water remaining. How many ml of water are left in the bottle? 9. Monica’s friend joins her on her run the next day, and returns with 142 ml water in her bottle. This is 25% of the amount she started out with. How much did she have to begin with? 10. Yoshiko did a survey and found that 56% of people said their favorite color was red, and 24% said that their favorite color was blue. If 18 people opted for blue, how many chose red?
Lesson 3.5.4 — Circle Graphs and Percents Exercises 1–5 use the following circle graphs. Graph A
Basketball: 20%
Graph B
Y
Graph C
Fall: 28.4% Summer: 12.5%
Football: 43.1%
Z Baseball
X
Winter: 44%
Spring
1. What are the missing percents in circle graphs A and C? 2. Graph A refers to a group of 65 people who were asked which sport they preferred, from a choice between football, basketball and baseball. How many people said basketball? 3. Graph B gives the results of an environmental traffic survey, detailing the number of people traveling in each car. A total of 492 cars were observed and the following results obtained: 1 person only: 246 cars, 2 people: 153 cars, 3+ people: 93 cars. Which section of the graph represents each group, and what percentage of the total do they each make up? 4. Graph C shows the annual sales made by a shop in each of the four seasons. If the sales made in summer came to a total of $45,843, how much money was taken during the winter season? 5. The following year, the shop took an annual total of $500,000, but the percentage of the takings in each season remained the same. How much more was made in the fall season of this year compared to the previous one?
Lesson 3.5.5 — Percent Increase 1. If you multiplied a number by 1.09, what percentage increase would you be calculating? In Exercises 2–4, calculate the total after the increase: 2. Increase 64 by 25% 3. Increase 140 by 8%
4. Increase 87 by 65%
5. Explain the steps you would take to calculate a 151% increase to a number. In Exercises 6–7, calculate the total after the increase: 6. Increase 35 to 134% of its original value 7. Increase 65 to 245% of its original value. 8. Vincent’s bill at a restaurant is $45, he leaves a 16% tip. How much did he pay in total? 9. DeMorris’s taxi fare came to $13.40, but he rounded up to $15 to include the tip. What percentage tip did DeMorris add to the fare? 10. Sharon pays $65.27 for a meal, including a 22% tip. How much did her meal cost before the tip? 444
Additional Questions
Lesson 3.5.6 — Percent Decrease In Exercises 1–3, calculate the total after the increase: 1. Decrease 64 by 25% 2. Decrease 140 by 8%
3. Decrease 87 by 65%
4. If you multiplied a number by 0.8, what percentage decrease would you be calculating? In Exercises 5–7, work out the percentage decrease from the first number to the second. 5. 178 decreased to 78 6. 18 decreased to 2 7. 579 decreased to 96.5 8. Alison works at a kayak shop and can buy kayaks for trade price. Trade price is 57% of the retail price. If she buys a kayak that should retail at $1049, how much will she actually pay? 9. Three t-shirts are in a sale. Their original price, and the discount are shown in the following table. Write the t-shirts in ascending order of price following the discount.
T-Shirt #
1
2
3
Price
$16
$21
$ 28
Discount
5%
25%
45%
10. Arnold buys a laptop for $599 after negotiating a discount of 26%. How much would it have cost before the discount?
Lesson 3.5.7 — Simple Interest Enoc borrows $800. He makes a $945 payment after 36 months which pays off his loan. 1. What was the Principal? 2. How much interest did he have to pay? 3. How many years did it take Enoc to repay his loan? 4. The formula for simple interest is I=Prt. What do each of the variables mean, and what are the most common units of I, P and t? Use the formula for simple interest from Exercise 4 to answer Exercises 5–8. 5. Simone borrowed $8500. She paid back the interest 18 months later. If the rate she paid interest at was 11.5% per year, how much interest did she have to pay? 6. Elvera puts $1655 into a savings account which pays 8% interest. 48 months later she checks her balance. How much should she have? 7. Benny has $1651 in his bank account. Three years ago he paid in $1300. What is the rate of interest on his account? 8. Juana pays off her loan with a single payment of $998.75. $148.75 of this is interest. If the interest rate was 7%, how long has she had this loan for?
Lesson 4.1.1 — Ratios Chloe has a bag of marbles. The bag contains 7 green marbles, 2 blue marbles, 5 red marbles, 1 orange marble and 4 purple marbles. Write down the ratios of the following colors of marbles: 1. Green to orange 2. Red to purple 3. Orange to blue 4. Purple to green 5. Blue to red In Exercises 6–8, write the ratio 2 other ways than the way it is given. 5 6. 6 to 2 7. 2:1 8. 7 9. There are 4 volleyball players and 3 soccer players in Tim's class. What is the ratio of volleyball players to soccer players? Write your answer three different ways. 10. The ratio of squirrels to acorns is 3:1. Express the ratio of acorns to squirrels in three different ways. Felix wrote the following ratio to compare the length of 2 pencils: 8 inches: 6 inches 11. What mistake did Felix make with his ratio? 12. Write the correct ratio of longer to shorter pencil length in two different ways. Additional Questions
445
Lesson 4.1.2 — Equivalent Ratios In Exercises 1–6, write down an equivalent ratio that is simpler than the given ratio. 10 18 1. 5:15 2. 3. 100 6 4. 90 to 15 5. 13:39 6. 16 to 180 Carlito bought 4 red apples, 8 yellow apples, and 2 green apples at the store. 7. What is the ratio of yellow apples to green apples? Write your answer in its simplest form. 8. What is the ratio of red apples to yellow apples? Write your answer in its simplest form. 9. What is the ratio of red apples to green apples? Abe has 20 CDs. There are 10 pop CDs, 2 rock CDs, 3 rap CDs, and the rest are country CDs. Write the following ratios in their simplest form: 10. pop CDs to rap CDs 11. rock CDs to pop CDs 12. rap CDs to country CDs 13. country CDs to pop CDs 14. Find a number x, so that the ratio 2:7 is equivalent to x :35. 15. Find a number y, so that the ratio 3:13 is equivalent to y :39.
Lesson 4.1.3 — Proportions In Exercises 1–4, write down whether the ratios can be written as proportions. Explain your answer. 1. 4:3 and 8:3 2. 2:5 and 4:10 3. 1:4 and 2:6 4. 6:1 and 18:3 Shawna is tiling the area around a sink, using 6 tiles like the one shown on the right. Write a proportion involving each of the following: 5. The numbers of red squares and purple squares in 1 tile and 6 tiles. 6. The numbers of green squares and pink squares in 1 tile and 6 tiles. 7. The numbers of blue squares and orange squares in 1 tile and 6 tiles. A fruit cup was made using 6 parts strawberries and 5 parts raspberries. 8. Write a proportion involving the number of strawberries and raspberries needed in 1 fruit cup and 8 fruit cups. 9. If s is the number of strawberries needed to mix with 20 raspberries, solve for s. 10. If r is the number of raspberries needed to mix with 36 strawberries, solve for r.
Lesson 4.1.4 — Proportions and Cross-Multiplication 1. Explain what it means to cross multiply. In Exercises 2–4, use cross-multiplication to check if the ratios are equivalent. 2. 2 and 14 3. 3:2 and 21:12 4. 6 to 5 and 42 to 35 35 5 In Exercises 5–7, determine the value of x in each proportion. 42 6 5. 7 = 14 6. x = 36 7. = x 7 2 x 5 10 8. Paul is packing for a trip to a basketball tournament. For each day of the tournament, he needs 2 pairs of socks. Is he correct to pack 8 pairs of socks for a 3-day tournament? Explain your answer. Include a proportion. 9. A baker has 20 cups of flour and 12 cups of sugar to use for muffins. One batch of muffins contains 1 2 cups of flour and 2 cup of sugar. How many batches of muffins can the baker make? Will any flour or sugar be left over? Explain your reasoning. 446
Additional Questions
Lesson 4.2.1 — Similarity Exercises 1–2 each show a pair of shapes. Explain whether the shapes in each pair are similar. Give reasons for your answers. 2.4 m 5 in. 4.2 m 1. 2. 45°
135°
3 in.
5 in.
3 in. 135°
60°
45°
5 in.
120°
3 in.
1.8 m
2.6 m
3.4 m
3 in. 120°
1.8 m
60°
1.8 m
5 in.
3.6 m
3. Explain what ratios have to do with shapes being similar. The triangles below are all similar to each other. Use the diagrams to answer Exercises 4–7. 3 in.
1.5 in.
3 in.
2 in.
A
B
4 in.
4.5 in.
1 in. C
0.75 in.
6 in.
4 in.
D
2 in.
E 1 in. 0.5 in.
6 in. 8 in.
Find the ratios of the length of corresponding sides between each of the following pairs of triangles: 4. A and C 5. D and B 6. E and A 7. B and C In Exercises 8–9, find the missing lengths in each pair of shapes by determining the ratios of the lengths of corresponding sides. 3 ft 8. 1.4 cm 9. 1.4 cm 1 cm
1 cm
5 ft
2 cm
2 cm
5 ft
2 cm 4.5 ft
6 ft
4 cm
Lesson 4.2.2 — Proportions in Geometry C 9 in. A
H 7 in.
y
27 in.
B F
30 in.
Use the diagram on the left to answer Exercises 1–4. 1. Write a proportion using the ratios AC:FH and BC:GH. x 2. Solve your proportion to find x. 3. Write a proportion using the ratios AC:FH and AB:FG. G 4. Solve your proportion to find y.
5. If you know the ratio of the lengths of 2 sides in one shape, what can you say about the ratio of the corresponding sides in any similar shape? X
6. The two figures shown on the right are similar. Use proportions to find the missing lengths a, b and c.
R
9 cm
6 cm 5 cm
Q 2 cm
b
W a
T
c
S
Z
6 cm
Y
7. Carlita has two photos on her wall. One photo measures 5 cm by 8 cm, and the other measures 6 cm by 9 cm. Are the shapes of the two photos similar? Use a proportion in your explanation. 8. Marcus has cut a figure with sides that measure 11 inches, 15 inches, and 17 inches in length. He wants to cut a second figure that is similar to his first figure. The length of the longest side of the new figure is 51 inches. Write and solve a proportion to find the lengths of the other two sides Marcus should cut.
Additional Questions
447
Lesson 4.2.3 — Scale Drawings The map below represents the location of some buildings in a small town. The scale is 0.25 inches = 0.4 miles, and the grid squares on the drawing measure 0.25 in. by 0.25 in. In Exercises 1–6, calculate each of the following distances in miles. 1. From the library to the school. Key G = Grocery store L = Library 2. From the park to the library. P = Park S = School 3. From the movie theatre to the park. M = Movie theater 4. From the grocery store to the movie theatre. 0.25 in. = 0.4 miles 5. From the park to the grocery store. 0.25 in. 6. From the school to the grocery store. Donovin has a drawing of a pencil made with a scale of 1 mm : 0.025 inches. 7. The length of the pencil in the drawing is 240 millimeters. What is the length of the real pencil? 8. The real pencil is 0.25 inches wide. How wide is the pencil in the drawing? 9. The real pencil's eraser is 0.4 inches long. How long is the eraser in the drawing? A builder made a scale drawing of a retail store using a scale, in inches, of 2:y. The height of the store in the drawing is 3 inches. 10. The height of the actual store is 12 feet. Find the value of y. 11. The width of the actual store is 50 feet. Find the width of the store in the drawing.
Lesson 4.3.1 — Customary and Metric Units The length of a painter's ladder is 8 yards. 1. What is the length of the ladder in feet?
2. What is the length of the ladder in inches?
3. When you convert to a smaller unit, the number of units gets ____________ . 4. When you convert to a larger unit, the number of units gets _____________ . Convert the values in Exercises 5–9. 5. 12 miles to yards 6. 4 feet to inches 8. 8800 yards to miles 9. 2 miles to inches
7. 144 inches to feet 10. 4500 inches to yards
The height of a tree in Erin’s garden is 2.25 meters. Find the height of the tree in each of the following units: 11. decimeters 12. centimeters 13. millimeters 14. hectometers
Lesson 4.3.2 — Conversions and Proportions 1. What is the ratio of inches to miles? Explain your answer. 2. Convert 342.25 mm to meters. Use a proportion to determine your answer. Keyshawn ran a 1600 meter race. Use proportions to convert the race distance to each of the following: 3. dekameters 4. hectometers 5. kilometers 6. What is the ratio of inches to yards? 7. Convert 9.37 yards into inches using your ratio from Exercise 7. A professional baseball player hit a home run ball 527 feet. Use proportions to calculate the following: 8. the number of inches the ball was hit 9. the number of yards the ball was hit 10. the number of miles the ball was hit
448
Additional Questions
Lesson 4.3.3 — Converting Between Unit Systems A flight from Minnesota to Mexico took 6.25 hours. Convert the length of the flight into each of the following units. Use a proportion for each conversion. 1. minutes 2. seconds 3. days The largest anaconda in the city zoo is 28 feet long. 4. Use a proportion to convert 28 ft to inches. 5. Use a proportion to convert 28 ft to yards. 6. Use the conversion factor 2.54 to convert 28 ft to centimeters. 7. Use a proportion to convert 28 ft to meters. Darla and Rex converted 1 foot into meters. Their work is shown below. Darla's Conversion feet to inches: 1 foot = 12 inches inches to cm: 12 × 2.54 = 30.48 cm cm to m: 30.48 ÷ 100 = 0.3048 m
Rex's Conversion 1 yard = 0.91 meters 1 foot =
1 3
yard
0.91 ÷ 3 = 0.3033 m
8. Explain how Darla did her conversion. Is her conversion correct? 9. Explain how Rex did his conversion. Is his conversion correct? 10. Why are Darla's and Rex's answers close but not equal?
Lesson 4.3.4 — Other Conversions In Exercises 1–3, say which of the temperatures given is higher. 1. 50 °F or 8 °C 2. 5 °F or –10 °C
3. 170 °F or 79 °C
The temperature at 7 a.m. was 12 °C. By 10 a.m. the temperature had increased by 21 °F. 4. What was the temperature at 10 a.m. in °F? 5. What was the temperature at 10 a.m. in °C? Scientists often measure temperature in units called Kelvins (K), instead of °F or °C. If K is a temperature in Kelvin, then: K = C + 273.15 Convert the following: 6. 40 °C to K 7. –116 °C to K 8. 86 °F to K 9. 200.15 K to °C 10. 453.2 K to °C 11. 68 K to °F
Lesson 4.4.1 — Rates In Exercises 1–4, use the given quantities to work out two unit rates. 1. 30 players for 15 basketballs 2. $2.40 for 12 cookies 3. 12 mouse toys for 3 cats 4. 288 dominoes shared between 24 containers 5. Two chefs each bought salmon for their restaurant. Chef Ana got 12 pounds of salmon for $83.88. Chef Todd got 15 pounds of salmon for $95.85. Which Chef got a better price per pound of salmon? Explain your answer. 6. A van owner and a SUV owner were comparing the efficiency of their vehicles. The van drove 108 miles on 9 gallons of gas. The SUV drove 240 miles on 16 gallons of gas. Which vehicle drove more miles per gallon of gas? Show your work and explain your answer. A small gym has 40 45-pound weights, 75 35-pound weights, and 90 25-pound weights that weightlifters can use. There are 5 racks that each hold an equal number of each type of weight. 7. How many of each type of weight are there per rack? 8. What is the total weight in pounds held by each rack? Additional Questions
449
Lesson 4.4.2 — Using Rates 1. Dora made 9 necklaces using 36 beads per necklace. How many beads did Dora use in total? 2. Joey's mom has 6 strings of outdoor lights with 250 lights per string. Find the total number of lights. 3. Judy ran 10.5 miles at a speed of 6 miles/hour. How long, in minutes, was Judy running for? 4. Aaron spent $47.76 on mulch for his yard. If the mulch costs $1.99/yd3, how many yd3 of mulch did Aaron buy? Lyron and Gabriela are each making a car journey. 5. Lyron’s car travels 140 miles in 240 minutes. What is his average speed in miles per hour? 6. Gabriela’s average speed is 40% faster than Lyron’s. What is Gabriela’s average speed? 7. How long would it take Gabriela to travel 180 miles at this speed? 8. How far would Gabriela travel in 90 minutes at this speed?
Lesson 4.4.3 — Finding Speed 1. Victoria rides her bike for half an hour. If she travels a total of 7 miles, what is her average speed? 2. Andres walks a distance of 10 miles in 3 hours. What is his average speed? 3. An airplane cruises at a constant speed for a distance of 1200 miles. What is the speed of the airplane if it takes 2.25 hours to travel this distance? An Indy car once recorded a speed of 239 miles/hour. 4. What is this speed in miles/minute? 5. What is this speed in miles/second? 6. What is this speed in kilometers/hour? (Use the conversion factor 1 mile = 1.61 km) 7. What is this speed in meters/second? 8. A dragonfly flies at a speed of 20 miles per hour. A hornet flies at a speed of 21.5 kilometers per hour. A bee flies at a speed of 4 meters per second. Convert the speeds of the bee and dragonfly to km/hour. Which of these insects flies at the fastest speed? (Use the conversion factor 1 mile = 1.61 km)
Lesson 4.4.4 — Finding Time and Distance 1. How can you find the distance an object travels in a certain amount of time if you know its speed? Malik walks at an average speed of 0.6 meters/second. Jessica walks at an average speed of 3.5 miles/hour. 2. If Malik walks for 9000 seconds, how many meters will he travel? 3. If Jessica walks for 2.5 hours, how many miles will she travel? 4. Which person walked for a longer period of time? Explain your reasoning. 5. Which person walked a greater distance? Explain your reasoning. 6. The data below shows the speed that two toy robots moved at, and the time they traveled for. Explain how you could compare the two distances that could be calculated from the data. You don’t need to work out the actual distances traveled. 56 cm/second for 5 minutes 40 ft/hour for 2 hours Suri set a treadmill to move at a speed of 6.5 miles/hour. She ran on the treadmill at that rate for 45 minutes and 30 seconds. 7. What is the distance Suri ran in miles? 8. Calculate Suri’s speed in meters/second? 9. What is the distance that Suri covered in kilometers? 450
Additional Questions
Lesson 4.4.5 — Average Rates David went on a bike ride. He cycled at an average speed of 29 miles/hour for the first hour. During the next 2 hours, he biked at an average speed of 21 miles/hour. 1. How many miles did David cycle during the first hour? 2. How many miles did he ride during the next two hours? 3. What is the total distance David rode? 4. What was his average speed for the three hours? Carmen ran at a rate of 5.5 miles/hour for 0.75 hours. During the next 0.5 hours, she ran at a rate of 6.5 miles/hour. 5. How far did Carmen run in the first 0.75 hours? 6. How far did Carmen run in the next 0.5 hours? 7. What was the total distance that Carmen ran? 8. What was the total length of time she ran for? 9. What was Carmen's average speed in miles/hour? 10. What was Carmen's average speed in meters/second?
Lesson 5.1.1 — Median and Mode In Exercises 1–2, find the median of each data set. 1. {12, 10, 15, 14, 17, 11, 20}
2. {25, 53, 12, 46, 31, 45, 72, 11}
In Exercises 3–4, find the mode of each data set. 3. {27, 21, 27, 23, 29, 25, 27, 21}
4. {2, 15, 11, 2, 16, 5, 2, 2, 12}
5. Can a data set have 2 modes? Explain your answer. 6. Can a data set have no mode? Explain your answer. Item Soup Salad Sandwich Hot dog Tacos
Price $1.25 $2.00 $1.75 $1.50 $1.50
Use the lunch menu shown on the left for Exercises 7–10. 7. What is the median price on the lunch menu? 8. What is the mode of the prices on the menu? 9. Two other items, "Hamburger and Fries" and "Grilled Cheese and Fries" will be added to the sign tomorrow, each with a price of $2.25. What will be the new median price on the menu? 10. What will be the new mode price on the sign?
Lesson 5.1.2 — Mean and Range In Exercises 1–3, calculate the mean of the data in each set. 1. {6, 1, 3, 2, 9, 14, 11, 20} 2. {14, 14, 11, 9, 4, 7, 13}
3. {60, 12, 15, 57}
4. In his first 5 games, Amare averaged 10.8 rebounds/game. He had the rebounds listed in the data set {9, 12, 7, 15}, but he was missing one of the rebound numbers. Use the mean to determine Amare's missing rebound number. 5. The mean of 5 numbers is 1. Explain why the set of data cannot be {0, 2, 4, 3, 3}. 6. Explain why range is not considered a measure of central tendency. Below is a list of the prices of eight shirts on a clothing store rack: $24.99, $16.99, $15.99, $18.99, $9.99, $8.99, $15.99, $19.99 7. What is the range of shirt prices on the rack? 8. What is the mean shirt price on the rack? 9. What is the median shirt price on the rack? 10. What is the mode shirt price on the rack? 11. Write a data set that meets the following criteria: Mode = 10 Mean = 10.6 Range = 20 Greatest value is 22 5 pieces of data Mode occurs twice in data set
Median = 10
Additional Questions
451
Lesson 5.1.3 — Extreme Values 1. What is an outlier? 2. What can outliers do to the mean of a data set? The data below shows the number of days 7 neighborhood houses were up for sale before they sold. {7, 4, 9, 6, 92, 5, 10} 3. Identify the outlier. 4. Calculate the mean with the outlier. 5. Calculate the mean without the outlier. 6. Explain which mean would be better to use as a typical value. The number of pairs of glasses prescribed by an eye doctor each day for 6 days is shown below. {20, 6, 4, 3, 4, 5} 7. Find the mean. 8. Explain why the mean of this data set is not a good choice as a typical value. 9. Find the mode. 10. Find the median. 11. What is the most representative measure of central tendency for this data set?
Lesson 5.1.4 — Comparing Data Sets Give a summary of each set of data in Exercises 1–3. 1. {2, 18, 15, 8, 15, 20} 2. {14, 8, 8, 8, 4, 15}
3. {6, 17, 1, 17, 3, 4}
The statements in Exercises 4–5 are about the three data sets in Exercise 1–3. Say whether you agree with each statement, and explain your answers. 4. The spread of the data is greatest in Exercise 3. 5. The data set in Exercise 1 has the highest typical value. Two gymnasts' scores for their 4 events at a gymnastics competition are listed below. Katie {8.60, 9.75, 9.15, 9.90} Jazmin {9.85, 9.00, 8.55, 9.20} 6. Find the mean, median, mode, and range of scores for each gymnast. 7. Use your answer to Exercise 5 to compare the sets of scores for the two gymnasts. 8. What does it mean if your data set has a small range? 9. What could it mean if your data set has a very large range?
Lesson 5.2.1 — Including Additional Data: Mode, Median and Range 1. Is there a value that could be added to the data set {5, 3, 1, 2, 2, 3, 4, 7} to change the mode? Explain your answer. 2. How could you add as many values as you want to a data set without changing the range? The data set below contains the number of brothers and sisters of each student in Terrell's class. {0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 4, 5, 6, 6, 8} 3. What is the mode of this data set? 4. How would the mode be affected if another student with 1 brother were added to the data set? 5. What is the median of this data set? 6. How would the median change if another student with 2 sisters and 1 brother was added? 7. What is the range of this data set? 8. What value could you add to the data set to change the range to 13? Explain your answer. 9. Terrell says that there is no value you could add to the data set to change the range to 5. Is he correct? Explain your answer. 452
Additional Questions
Lesson 5.2.2 — Including Additional Data: Mean 1. What kind of effect does adding values near the original mean have on the mean? 2. What kind of effect can adding an outlier to the data set have on the mean? The grade point averages of the starting players on Lily's basketball team are: {3.25, 3.25, 3.5, 3.75, 4.0} The grade point averages of the 7 reserve players on the team are: {3.0, 3.4, 3.5, 3.67, 3.8, 3.9, 4.0} In Exercises 3–5, find the mean grade point average of the following: 3. The 5 starting players. 4. The 7 reserve players. 5. All 12 players the team? 6. A new reserve player joins the team with a 3.15 grade point average. How does this change the mean grade point average of the reserve players? 7. How does this change the mean grade point average of all players? 8. Was the change you found in Exercise 7 large or small? Explain why adding a value of 3.15 affected the mean in this way. 9. Jordan has been told that the mean of a data set is 15 and the mean of a second data set is 17. He wants to find the mean of the two data sets combined. He decides the mean of the combined set must be (15 + 17) ÷ 2 = 32 ÷ 2 = 16. Is Jordan’s method correct? Explain your answer.
Lesson 5.3.1 — Analyzing Graphs X X X X X X X X
X X X X
X X X X X X X
X X X X X
Use the data display on the left to answer Exercises 1–3. 1. What type of data display is this? 2. What is the mode of the data? 3. What is the range of the data?
11 12 13 14 15 16 17
Use the bar graphs below to answer Exercises 4–8. 4. Which data set has the greatest range? 5. Which data set has a mode of 16? 6. Which data set has more than one mode? 7. Which two data sets have the same range? 8. Which data set has the smallest mode? A6
B 6
4
4
2
2
0
11 12 13 14 15 16 17 18 19 20
0
C6
D 6
4
4
2
2
0
11 12 13 14 15 16 17 18 19 20
0
11 12 13 14 15 16 17 18 19 20
11 12 13 14 15 16 17 18 19 20
Miss Allaire's science class went on a nature walk to collect leaves. The number of leaves each student in the class collected is displayed below. 9. Who collected the most leaves? How many did he/she collect? Student 10. Who collected the least number of leaves? Amir How many did he/she collect? Cecilia 11. Which student collected exactly 7 leaves? Chad 12. What was the total number of leaves collected? Mark 13. What is the mode number of leaves collected? Tawana 14. What is the range in number of leaves collected?
Number of leaves collected
Key:
= 2 leaves
Additional Questions
453
Lesson 5.3.2 — Finding the Mean and Median from Graphs Use the line plot on the right for Exercises 1–2. 1. Find the median of this data set. 2. Determine the mean of this data set. Graph A 11
11
12
12
13
13
14
14
15
15
16
16
17
17
18
X X 5
Graph B
X X X X X X X X X 6
7
8
X X
9 10 11 12 13 14
Use the the bar graphs on the left to answer Exercises 3–6. 3. Which graph has the greatest mode? 4. Which graph has the greatest range? 5. Which graph has the greatest median? 6. Which graph has the greatest mean?
18 0
1
2
3
4
5
0
1
2
3
4
5
Number of students
Hayley collected data from her classmates as to the number of people each student had in their family. She used the data to make the bar graph below. 6 7. What is the mean number of people in the students' families? 5 8. What is the median number of people in the students' families? 4 3 9. What is the range in number of people in the students' families? 2 10. Find the new mean and median if the value 15 is added to 1 Hayley's data set. 0 1
2
3 4 5 6 7 8 9 10 11 Number of people in family
Lesson 5.3.3 — Other Types of Graph Mrs. Martinez asked the girls and boys in her class to choose their favorite from four colors. She made the circle graphs below to display the girls' data and the boys' data. Yellow 1. Which color was most popular among the girls? 2. Which color was most popular among the boys? Yellow 3. Which color was least popular among the girls? Green 4. Which color was least popular among the boys? Green Red Red 5. Which group of students liked yellow more? Blue Blue 6. Which group of students liked green more? 7. Mrs. Martinez asked the class how many girls and boys voted for red. Kirk said 11. Boys Girls Explain the problem with Kirk's answer.
Height (in inches)
60
Abby's mom made the data display on the left to show Abby’s height between the ages of 5 and 12. 8. Identify this type of data display. 9. What was Abby's height at age 5? 10. How old was Abby when her height was exactly 55 in.? 11. How much did Abby grow between the ages 7 of 11? 12. Between which two ages did Abby grow the most?
50 40 30 20 10 0
5
6
7
8
9
10
11
12
Age (in years)
13. Explain the difference between a single bar graph and a double bar graph. 454
Additional Questions
Lesson 5.4.1 — Using Samples 3000 girls were surveyed for a study to determine the number of hours of television teens watch per day. 1. Explain why this study is biased. 2. How could this sample be changed to be more representative of the population? Identify the population and the sample in Exercises 3–5. 3. A uniform company wants to know what style of jerseys college basketball players prefer. The company sent jersey samples to 500 of their college clients to find out which the players liked best. 4. A pool manager needs to determine the amount of chlorine in the adult pool versus the amount of chlorine in the children's pool. A small bottle of water is filled from each of the pools. 5. A veterinary office wants to know whether to add dog grooming services to the list of services they offer. A receptionist calls 50 dog owners who use the vet office to ask if they'd be interested. 6. Explain why a sample of 10 people out of a city of 100,000 won't give very useful results. 7. A meeting was held on Tuesday at 10:00 a.m. to discuss a new school policy on the use of cell phones. People at the meeting were surveyed as to whether they thought cell phones should be allowed on school grounds. Based on these results, a school policy was made banning the use of cell phones on school grounds. Give 3 reasons why this sample may have been biased.
Lesson 5.4.2 — Convenience, Random and Systematic Sampling In Exercises 1–3, identify the type of sampling being used, and give an advantage and a disadvantage of using that type of sampling in the situation described. 1. A mall manager is surveying mall shoppers as to what kind of store they would like to see open in the vacant store area. She interviewed 50 shoppers who walked by her mall office. 2. A TV news station asks random people coming out of a movie theatre grand opening what they think about the theatre. 3. Every 3rd person coming to a bank drive thru between 1–2 p.m. was surveyed as to whether bank hours should be extended. A cell phone store wants to know if customers are pleased with the service plans that are available to them through their company. 4. Give an example of convenience sampling that could be used in this situation. 5. Give an example of random sampling that could be used in this situation. 6. Give an example of systematic sampling that could be used in this situation.
Lesson 5.4.3 — Samples and Accuracy 1. If a sample is representative, what can you say about how it compares to the population? 2. If a sample accurately represents a population, what can you say about the sampling error? The mass of 450 different laboratory mice was determined and recorded. The data below shows the mass in grams of 4 different samples of mice in the laboratory. Sample A: {4.5, 3, 3.2, 5, 2.9} Sample B: {6.1, 4.4, 3, 2.8, 4.6} Sample C: {6.5, 3.1, 4, 3.7, 5} Sample D: {5.9, 3.1, 2.6, 3.8, 4.2} 3. Find the mean of each of the 4 samples. 4. Find the median of each of the 4 samples. The mean mass of the population of mice is 5.75 grams. 5. Find the sampling error for Sample A. 6. Find the sampling error for Sample B. 7. Find the sampling error for Sample C. 8. Find the sampling error for Sample D. 9. The sampling error for the median between Sample A and the population is 1.25 grams . What is the median of the population? 10. Based on your median in Exercise 9, find the sampling error for the median for Samples B, C, and D. Additional Questions
455
Lesson 5.4.4 — Questionnaire Surveys A company wants to collect data on what types of junk foods people in Texas buy at the grocery store. The store surveys shoppers as they are leaving the store, asking about their junk food purchases. 1. What is the population? 2. What is the sample? 3. What are the benefits of collecting data this way? 4. What are the problems with collecting data this way? 5. What is a biased question? 6. When might a biased question be asked? A student council is trying to gather information about whether a school should change its school colors. 7. Write an unbiased question that could be used to gather opinions in the survey. 8. Write a biased question in favor of changing the school colors. 9. Write a biased question against changing the school colors. 200 sixth graders were asked, "Isn't it unfair that teachers aren't graded on their teaching?" As a result a claim was made that 100% of students think that they should get an opportunity to evaluate their teachers' performance. 10. Is this a valid claim? Explain your reasoning. 11. Write an unbiased question that could have been used to gather more valid opinions.
Lesson 5.5.1 — Evaluating Claims Ms. Clark made the line plot on the right to show her students' scores on their last spelling test. She showed the line plot to the class and they discussed their results. X In Exercises 1–6, explain whether the data from the line plot supports each claim. X 1. Martin said, "Most of us scored an 18 on our last test." X X X 2. Trey said, "The mode score on the test was 14 and 20." X X X X X X X X X X 3. Liz said, "At least this time the range of scores was only 6." X X X X X X X 4. Attiana said, "The median score was 17 on this test." 5. Write a different claim that is supported by the data on the line plot. 14 15 16 17 18 19 20 6. Write a different claim that is not supported by the data on the line plot. Spelling test scores A teacher made the chart below to show the first 3 test scores of 3 different students in her class. 7. Write a claim about Angelo's test Student Test 1 Test 2 Test 3 scores that can be supported by the data. 8. Write a claim about Angelo's test 89 95 92 Angelo scores that cannot be supported by the data. 9. Write a claim about Brianna's test 94 81 92 Brianna scores that can be supported by the data. 10. Write a claim about Brianna's test 85 89 94 Corinne scores that cannot be supported by the data. 11. Write a claim about Corinne's test scores that can be supported by the data. 12. Write a claim about Corinne's test scores that cannot be supported by the data. 13. Write a claim about all three students' combined scores that can be supported by the data. 14. Write a claim about all three students' combined scores that cannot be supported by the data.
456
Additional Questions
Lesson 5.5.2 — Evaluating Displays The graphs below show the number of students in grades 6 and 7 who play sports. Graph B Number of students
Number of students
Graph A 200 150 100 50 0 Grade 6
190 185 180 175 0
Grade 7
Grade 6
Grade 7
1. Which graph makes it easier to find the exact number of students who play sports? Explain why. 2. What are the exact numbers of students that play sports in grades 6 and 7? Winnie is looking at one of the graphs. She decides that the number of grade 7 students who play sports is double the number of grade 6 students who play sports. 3. Which graph is Winnie looking at? Explain your answer. 4. Is Winnie correct? If not, explain what mistake she has made. 5. Explain how changing the spacing on the horizontal axis of a line graph can change the appearance of the graph.
Number of students
Mrs. Beckman made the graph on the right to display the test scores on last chapter's math test. 6. Give an advantage of grouping the test scores the way 8 Mrs. Beckman did on her graph, instead of plotting each score separately. 6 7. Explain how the grouping of this data could be 4 misleading. 8. Can you use this data display to determine the range of 2 test scores? Explain your answer. 0 9. Can you use this data display to find the mode test score? 60 70 1–80 1–90 –100 50– 8 7 61– 91 Explain your answer. Scores
Lesson 6.1.1 — Listing Possible Outcomes 1. What is the difference between an outcome and an event? 2. Can one outcome match more than one event? Explain your answer using an example. For Exercises 3–5, sketch a spinner with 8 sections. 3. Add squares to the spinner sections so that 3 possible outcomes match the event "spin a square". 4. Add triangles to the spinner sections so that 2 possible outcomes match the event "spin a triangle". 5. Add another shape to the remaining sections. Write an event that matches these outcomes. Mr. Sanchez is making 2 spinners for his students to use to determine the subject and type of project each student must complete. He wants the subject spinner to have 3 events: spinning history, geography, or culture. 6. Sketch a spinner that matches the description of the subject spinner. 7. How many outcomes match the event "does not spin history"? Mr. Sanchez wants the type of project spinner to have 4 events, spinning diorama, poem, art picture, or report. 8. Sketch a spinner that matches the description of the type of project spinner. 9. How many outcomes match the event "does not spin report"? Each student will spin the subject spinner and the type of project spinner. 10. List all the outcomes of spinning both spinners that match the event "does not spin geography". Additional Questions
457
Lesson 6.1.2 — Tree Diagrams 1. Malcolm is using a tree diagram to find how many outcomes there are for spinning a spinner and rolling a die. Explain whether or not it matters which action he puts first in his tree diagram. Elvin is decorating for the school dance. He can choose from the lists of decorations on the right.
Streamer Colors Red
Table Top Decorations
Hanging Decorations Stars
Yellow
Rockets
Moons
Blue
Airplanes
Planets Clouds
2. Make a tree diagram that shows the number of combinations using one color of streamer, one type of hanging decoration and one type of table top decoration. 3. How many outcomes do not include Rockets as a table top decoration? 4. Elvin doesn't have enough clouds to hang over the last 10 tables at the dance. How many outcomes do not include clouds hanging over the table? 5. Elvin doesn’t have enough blue streamers for the last four tables. How many outcomes do not include clouds or blue streamers?
Lesson 6.1.3 — Tables and Grids Oliver is choosing from three different hats, and three different types of nose to put on his snowman. The types of hat are: top hat, cowboy hat, beret. The types of nose are: carrot, button, quarter. 1. Make a table to show all the combinations of one hat and one nose Oliver can make. 2. How many combinations match the event "with a button nose"? 3. List the combinations that match the event "with a cowboy hat". 4. Oliver could not find a quarter. How many possible combinations of hats and noses are there now? Spinner A Spinner B Exercises 5–9 are about the spinners shown on the right. Kayla spins Spinner A once, then spins Spinner B once, 13 3 8 2 and then multiplies the two numbers together. 5 11 5. Make a table to show all the combinations of products 6 4 9 7 Kayla can make with one spin on each spinner. 6. How many combinations match the event "product is an odd number"? 7. How many combinations match the event "product is an even number"? 8. Make a list of all the combinations that match the event "product greater than 60". 9. Make a list of all the combinations that match the event "product greater than 30 but less than 40".
Lesson 6.2.1 — Probability Exercises 1–4 are about the set of cards shown below. The cards are turned over and shuffled, and then one card is picked. Find the probability of each of the following. Write your answer as a decimal. 1. Picking a card with an I 2. Picking a card with an F N I C A L I F R A 3. Picking a card with an A 4. Picking a card with a J O 5. List three ways the probability of an event occurring can be written. 6. Write two fractions that represent the same probability as 30%. Tyrese wants to make a spinner with the following probabilities: • 40% chance of spinning the number 1. • 10% chance of spinning the number 8. • 20% chance of spinning the number 4. • 30% chance of spinning the number 2. 7. Draw a spinner that fits this description. 8. Tyrese is going to spin the spinner once. Find the probability that he spins a number greater than 9. 9. Find the probability that he spins a number greater than 3. 458
Additional Questions
Lesson 6.2.2 — Expressing Probability In Exercises 1–4 write each answer as a fraction in its simplest form. Maggie has a box of 120 envelopes. It contains 40 red envelopes, 30 green envelopes and 50 yellow envelopes. She is going to pick one envelope out. Find: 1. P(Maggie picks a green envelope) 2. P(Maggie picks a red envelope) 3. P(Maggie picks a yellow envelope) 4. P(Maggie picks an envelope that is not yellow) Tracie is playing a game where she mixes up 36 cards and picks one without looking. Each card has a picture of a shape on it. There are 16 circles, 6 squares, 9 triangles, and 5 stars. Write each of the following probabilities as a fraction in its simplest form: 5. P(Tracie picks a circle) 6. P(Tracie picks a square) 7. P(Tracie does not pick a square) Manuel owns a pet store. He sells four types of bird feed. He currently has 144 bags of safflower seed, 24 bags of sunflower seed, 36 bags of cracked corn and 36 bags of Spanish peanuts. Manuel picks a bag of bird feed at random from his stock. Write each of the following probabilities as a percent: 8. P(picking sunflower) 9. P(picking safflower) 10. P(picking cracked corn)
Lesson 6.2.3 — Counting Outcomes Sarah spins each of the spinners shown below and multiplies the two numbers she spins together. 1. Draw a table to show the possible outcomes. 1 2 4 1 8 3 Find the following probabilities, if x is the product of the two numbers spun. 7 4 2. P(x = 32) 3. P(x is less than 10) 3 2 6 5 4. P(x is 20 or greater) 5. P(x is even) When the probability of an event occuring is written as a fraction, what is represented by: 6. the numerator? 7. the denominator? Eldrick is making a pattern out of tiles. For each of three spots he randomly chooses either a green, white or yellow tile. He can use the same color more than once. 8. Draw a tree diagram to show the possible combinations of tiles Eldrick can choose. Use your diagram from Exercise 8 to find the following probabilities. Give each answer as a decimal to the nearest thousandth. 9. P (3 green tiles) 10. P (3 tiles of the same color) 11. P (exactly 2 tiles with the same color) 12. P (the 2nd and 3rd tiles being the same color)
Lesson 6.2.4 — Probability of an Event Not Happening Find P(A) and P(not A) where event A is picking each of the following at random from the cards shown on the right. Write your answers as fractions. 1. a black card 2. a card with a circle 3. a striped card 4. The probability of event B happening is 20%. What is the sum of P (B) and P (not B)?
Shorts
Lionel has 5 T-shirts and 4 pairs of shorts in his drawer. This table shows the combinations of T-shirts and shorts he can choose. Orange Purple Find P(C) and P(not C) where event C is Lionel choosing each of Blue OB PB the following from the drawer. Give your answers as percents. White OW PW 5. Striped shirt 6. Dotted shirt and red shorts Grey OG PG Lionel added a pair of tan shorts to the drawer. OR PR Red 7. What is the new total number of combinations? 8. Explain how your answer to Exercise 6 would change with the tan shorts included. 9. Find the probability that the tan shorts would not be pulled out of the drawer.
T-Shirts Striped Dotted Yellow SB
DB
YB
SW
DW
YW
SG
DG
YG
SR
DR
YR
Additional Questions
459
Lesson 6.2.5 — Venn Diagrams A number spinner is spun twice and the results multiplied. In which section of the Venn Diagram does each of the following outcomes belong? 1. 4 × 4 2. 9 × 2 3. 2 × 6 4. 9 × 9 5. 9 × 1 6. 4 × 2
A
B
C
Greater than 12
D Even
7. Nicole placed an outcome outside of the circles on a Venn Diagram. What does this tell you about the outcome? 8. One circle on a Venn Diagram was placed entirely within another circle. What does this mean? The Venn Diagram on the right shows three events for picking a number from 1 to 200 at random. Event A is greater than 100. Event B is a multiple of 25. Event C is an even number. 9. Copy the diagram and place the numbers 1, 2, 25, 50, 100, 150, 175 and 200 in the A correct sections of the diagram. 10. Name an outcome that would only match event A. 11. Name an outcome that would be placed in the overlapping area between circles A C and C, but not in circle B.
B
Lesson 6.2.6 — Combining Events Exercises 1–7 are about the set of 24 colored and numbered balls shown below. One ball is picked at random from the set. 1 2 3 4 5 6 7 8 Calculate each probability and write your answers as percents. 1. P(green) 2. P(green or yellow) 9 10 11 12 13 14 15 16 3. P(orange and greater than 6) 4. P(red or less than 21) 17 18 19 20 21 22 23 24
Event A is “picking a number greater than 10 and less than 20,” Event B is “picking a purple ball,” and Event C is “picking an even number.” Find the following probabilities: 5. P(Event A and C) 6. P(Event B or C) 7. P(Event A and B and C) 8. Explain how P(A and B) is different from P(A or B). 9. Explain why you can't always just add P(A) to P(B) to finding P(A and B) 10. P(A) is 30% and P(B) is 20% with no overlap between events. What is P(neither A nor B)?
Lesson 6.3.1 — Independent and Dependent Events Sierra and Noah are each choosing a number between 1 and 100. Say whether each of the following pairs of events for picking the numbers are dependent or independent. Explain your answers. 1. Event A: Sierra thinks of a number, but doesn’t tell Noah what it is. Event B: Noah thinks of a number, but doesn’t tell Sierra what it is. 2. Event A: Sierra thinks of a number and tells Noah what she picked. Event B: Noah thinks of a different number. 3. Event A: Sierra randomly picks one of 100 cards numbered 1–100 and keeps it. Event B: Noah picks one of the remaining 99 cards. 4. Event A: Sierra randomly picks one of 100 cards numbered 1–100 and keeps it. Event B: Noah randomly picks one of 100 balls numbered 1–100 out of a bag and keeps it. 5. Event A: Sierra randomly picks one of 100 balls numbered 1–100 out of a bag, then replaces it. Event B: Noah randomly picks one of 100 balls numbered 1–100 out of the same bag. 6. A number cube is rolled, and a number of red counters equal to the number rolled are put into a box with 10 blue counters. Are the events “picking a red counter from the box” and “rolling 6 on the number cube” independent or dependent? Explain your answer. 460
Additional Questions
Lesson 6.3.2 — Events and Probabilities Charles has a box containing the 8 cards shown on the right. He picks one card from the box, and then picks a second card without replacing the first. Find the following, writing your answers as decimals to 3 decimal places: 1. The probability that the first card picked is red 2. The probability that the first card picked is blue 3. The probability that the second card picked is blue, if the first was red 4. The probability that the second card picked is blue, if the first was blue 5. The probability that the first card picked is a circle 6. The probability that the first card picked is a square 7. The probability that the second card picked is a square, if the first was a circle 8. The probability that the second card picked is a square, if the first was a square 9. What would the answers to Exercises 3 and 4 be if Charles put the first card back in the box before he picked the second card? 10. Complete the following sentence: The events “picking a red card on the first pick” and “picking a blue card on the second pick” are ___________ if the first card is replaced and _________ if the first card is not replaced.
Lesson 6.3.3 — Calculating Probabilities of Independent Events In Exercises 1–3, calculate P(A and B). A and B are independent events. 1. P(A) = 0.15; P(B) = 0.5 2. P(A) = 1; P(B) = 0 3. P(A) = 0.99; P(B) = 0.75 4. The probability that it will rain today is 20%. The probability that it will rain tomorrow is 30%. What is the probability that it will rain both today and tomorrow? Each Monday a cooking teacher puts a star under one of six placemats. Six students, three boys and three girls, come in and sit randomly in front of the placemats. Calculate the probabilities of the following events: 5. Amy, a student in the class, will sit at the place with the star the first and second weeks. 6. A boy will sit at the place with the star the first and second week. 7. Amy will sit at the place with the star the first week and a boy will sit at it the second week. 8. Someone other than Amy will sit at the placemat the first week and a boy will sit at it the 2nd week.
Lesson 6.4.1 — Relative Frequency Spinner A has the outcomes X and Y. Spinner B has outcomes 1, 2 and 3. The two spinners were spun 12 times. The results are recorded on the table below. Spinner A Spinner B
Y 2
X 3
X 1
Y 2
X 1
Y Y 1 3
X 2
Y 2
Y 1
Y 3
X 2
Calculate the relative frequency of the following events. Write your answers as fractions. 1. Spinner A landing on Y 2. Spinner B landing on 3. 3. Spinner B landing on 1 or 2. 4. Spinner B not landing on 1 5. Spinner A landing on Y and spinner B landing on 2 6. Explain the difference between theoretical probability and experimental probability. The colors of 120 gumballs that came out of a gumball machine were Number of Relative Color gumballs frequency recorded. The table on the right shows the number and relative frequency of the four colors of gumball. Write the relative frequencies of the Red 26 7 following as fractions in their simplest form: Green 24 7. Blue gumballs 8. Red gumballs 31 Blue 9. How many green gumballs were there? 7 Yellow 30 10. How many yellow gumballs were there? Additional Questions
461
Lesson 6.4.2 — Making Predictions In Exercise 1–3, use the relative frequencies to predict how many times event A will happen in 50 trials. 1. P(A) = 0.8 2. P(A) = 4% 3. P(A) = 34% If the relative frequency of Event A is 0.15, predict how many the event will happen in: 4. 380 trials 5. 640 trials 6. 2000 trials 7. Describe the difference between biased and unbiased experiments. One card from the set shown on the left was picked at random then replaced.
A B C D E F A
B
4
2
C 5
D
E
6
3
F 5
The results of doing this 25 times are shown in the table on the left. 8. Find the experimental probability of each card being pulled.
9. How would you expect the experimental probabilities you calculated in Exercise 8 to be affected if the number of trials was increased to 1000?
Lesson 7.1.1 — Parts of a Circle In Exercises 1–4, choose the correct definition from the table below to match each word. 1. 2. 3. 4.
Diameter Circle Radius Center
A. The set of all points in a plane that are all the same distance from the center B. The distance from the center to the edge of a circle C. The point at the middle of a circle that is an equal distance from all points on the circle D. The distance from one edge of a circle to the other in a straight line through the center
5. The radius of a circle is y inches. Write an expression that could be used to determine the diameter of the circle. Harry is placing equally-sized water bottles into a box as shown in the diagram below. 6. How long is the shortest side of the box? Harry wants to know how many bottles will fit into a box. In Exercises 7–9, find how many water bottles would fit into a box with the following lengths: 7. 84 centimeters 8. 30 centimeters
3.5 cm
9. 68 centimeters
Lesson 7.1.2 — Circumference and p d = 6.1 mm
4
1
5
2
9
6
3
+ -
Vol
8
0
r = 4.1 mm
3. The circumference of the red button is 12.25 mm. What is its diameter?
7
+ -
Ch
Exercises 1–3 are about the remote control shown in the picture. Use 3.14 as p. Find the circumference of: 1. Button number 1 2. The volume button.
4. Kyle says that the value of p changes in every equation. Is he correct? Explain your answer. 5 . The diameter of a large circle was 10 times as long as the diameter of a smaller circle. How many times greater is the circumference of the large circle? A cylindrical container is used to send items back and forth to a drive thru bank teller. 6. The radius of the container’s circular lid is 4.3 inches. Find the circumference of the lid. The container passes through a tube. The diameter of the tube is 2 inches greater than the diameter of the container. 7. Calculate the circumference of a circle with the same diameter as the tube.
462
Additional Questions
Lesson 7.1.3 — Area of a Circle Use 3.14 for p in Exercises 1–7.
9.2 in.
6.1 in.
The diagram on the right is from a poster showing a car wheel. 1. Find the approximate area of the smaller circle. 2. Find the approximate area of the larger circle. 3. What is the approximate area of the part of the diagram between the two circles? Marlene has a rectangular piece of cardboard measuring 16 inches by 8 inches. She wants to cut out 2 circles each with a diameter of 7 inches from the cardboard. 4. Calculate the approximate area of each circle. 5. Find the approximate area of the cardboard that is left after the circles have been cut out. Marcie is making a circular spinner like the one shown in the diagram. The spinner has a radius of 6.25 centimeters. 6. Find the approximate area of the spinner. Round your answer to the nearest tenth. 7. Find the approximate area of the spinner that will be colored yellow.
Lesson 7.2.1 — Describing Angles 1. Copy and complete the following sentence: All right angles have measures _______ than acute angles but _______ than obtuse angles.
le
ap
M Cedar
Elm
Oak
The map on the right shows some of the road in Jamal's hometown. 2. Which two pairs of roads appear to meet at right angles on the map? Measure the angles between them to check this. 3. Which pair of roads appear to meet at an acute angle on the map? Measure the angle that they create. 4. Which two pairs of roads appear to meet at obtuse angles on the map? Measure the angles that they create. 5. Birch Road is under construction and will divide the angle created by Maple and Cedar exactly in half. What will be the measure of the angle created by Birch and Maple road?
In Exercise 6–9, choose the correct definition from the table below to match each word. 6. Angle A. A part of a straight line that has one end point and goes on forever in one direction 7. Protractor B. A point where the two sides of an angle meet 8. Ray C. A figure made when two rays or two line segments meet at the same end point 9. Vertex D. An instrument used to measure angles in degrees
Additional Questions
463
Lesson 7.2.2 — Pairs of Angles Use the diagram on the left to answer Exercises 1–9. Say whether the following pairs of angles are adjacent, linear, vertical, or none of these. 1. Angle 1 and angle 2 2. Angle 1 and angle 7 3. Angle 2 and angle 5 4. Angle 2 and angle 4 5. Angle 6 and angle 7 6. Angle 3 and angle 6
1. 7.
2. 3. 4.
6. 5.
7. Angle 1 measures 121°. What is the measure of angle 4? If angle 3 measures 87°, find the measures of: 8. Angle 6 9. Angle 5
In Exercises 10–12 use the diagram on the right. Steve used chalk to draw the two lines shown on the sidewalk. 11. 8. 10. Name 2 angles that appear to be vertical angles in the sketch. 10. 9. 11. Name 2 angles that appear to be linear angles on the sketch. 12. Jake then added another line to the sketch that divided angle 8 exactly in half. If angle 10 measures 142°, what is the measure of half of angle 8? Explain your answer.
Lesson 7.2.3 — Supplementary Angles 1. Explain how to determine if two angles are supplementary. 2. Marsha said that all linear pairs must be supplementary. Is this true? Explain your answer. 3. Can two angles be supplementary if one is acute and one is right? Explain your answer. Kendra and Alejandro are cutting up a circular pie. The pie is cut exactly in half in a straight line, and they are each given exactly half of the pie to cut into slices. 4. What is the angle measure of each half of the pie? 5. From her half, Kendra cuts one slice of pie with an angle measure of 72º, and one with an angle measure of 45°. What is the measure of the angle made by Kendra’s remaining piece of pie? 6. From the second half of the pie, Alejandro cuts one slice with an angle of 68º, then cuts the remaining piece exactly in half. What is the measure of the angle made by each of the two slices Alejandro makes with his second cut?
Lesson 7.2.4 — The Triangle Sum The angle measures of three triangles were recorded on the table shown on the right. Find: 1. x 2. y 3. z
Triangle A B C
Angle 1 Angle 2 50° y° 6°
Angle 3
60° 31° z°
x° 19° 108°
4. Explain how knowing the measure of a supplementary angle to an angle in a triangle can help determine the measure of the other angles in a triangle. In this diagram, the measures of angles 1 and 2 are equal. The measure of angle 3 is 35°, and angle 5 measures 109°. Find the measure of: 5. Angle 1 6. Angle 2 7. Angle 6 8. Angle 7
464
Additional Questions
5. 6.
1. 3. 2.
4. 7.
Lesson 7.2.5 — Complementary Angles 1. Copy the following sentence and fill in the missing words: __________ angles add up to 90º but __________ angles add up to 180º. 2. If one of the angles in a triangle is a right angle, what does that tell you about the other two angles? 3. Will found that two angles in a triangle are complementary. What is the measure of the third angle? In this diagram, angle 2 measures 12º less than angle 1. 4. Find the measure of angle 2. 5. Find the measure of angle 3.
3.
1. 2.
The top view of a box used to hold chocolates is shown in the diagram on the right. The triangle is an isosceles triangle. 6. Find the measure of angle 5. 7. Find the measure of angle 6. The company that makes the box decides to change the measures of angles 5 and 6 so that they are still complementary, but not equal. The new measure of angle 5 will be 52.5º. 5. 8. Find the measure of angle 6 in the new triangle.
6.
4.
Lesson 7.3.1 — Classifying Triangles by Angles In Exercises 1–4, say whether each sentence is true or false, and explain your answer. 1. In an obtuse triangle, two of the angles must be obtuse. 2. In a right triangle, only one angle can be a right angle. 3. An acute triangle must have 3 angles that are less than 90º. 4. All the angles in an obtuse triangle have greater measures than all the angles in an acute triangle. A design for a stained glass window is shown in the diagram. 5. Find the measure of the missing angle in triangle A. 6. Is triangle A angle right, obtuse or acute? 7. Name two triangles in the picture that appear to be obtuse. 8. If the largest angle in triangle G was 3° larger, it would be a right angle. What is the measure of this angle? 9. The other angles in triangle G are the same size. Find their measures. 10. Is triangle G right, obtuse or acute?
B.
C.
D.
A.
E. 47°
F.
G.
H.
Lesson 7.3.2 — Classifying Triangles by Side Lengths This table has been used to record the side lengths of seven triangles. Are the following equilateral, isosceles or scalene? 1. Triangle A 2. Triangle B 3. Triangle C 4. Triangle D 5. The missing lengths from the last three rows of the table are 2 cm, 3 cm and 6 cm. Match each of these missing length to the correct triangle.
Triangle A B C D E F G
Side 1 9 mm 5.86 m 9.2 ft 1.3 in 4.5 cm 6 cm 2 cm
Side 2 10 mm 12 m 9.2 ft 3.5 in 9 cm 2 cm
Side 3 9 mm 5.9 m 9.2 ft 1.3 in 5.5 cm
Type
Scalene Isosceles Equilateral
6. Linda says that a right triangle could not be an isosceles triangle. Is this true? Explain your answer. 7. Luis says that if you flip an equilateral, isosceles or scalene triangle over to get a mirror image, the new triangle will be the same type of triangle as the original. Is this true? Explain your answer.
Additional Questions
465
Lesson 7.3.3 — Types of Quadrilaterals Quadrilateral Angle 1 Angle 2 Angle 3 Angle 4 a° 50° 55° 226° 1. 115° b° 76° 2. 110° 82° 82° 98° c° 3. e° d° 74° 106° 4. g° f° 35° 35° 5.
This table shows the angle measures of five quadrilaterals. Use it to answer Exercises 1–6. 1. Find a 2. Find b 3. Find c 4. One of quadrilaterals 1, 2 and 3 is an isosceles trapezoid. Which one is it? Explain you answer.
5. Quadrilateral 4 is an isosceles trapezoid. If d > e, find d and e. 6. Quadrilateral 5 is a parallelogram. Find f and g. In Exercises 7–9, use the words below to complete each sentence. Use each word only once. Square Rhombus Trapezoid Parallelogram Rectangle 7. A _________ has 4 sides of equal length, but does not have to have 4 equal angles. 8. A ________ is a _________ with 4 sides the same length. 9. A square is not a _______, but is a ________.
Lesson 7.3.4 — Drawing Quadrilaterals Andrew started drawing a parallelogram as shown on the right. The two sides he has drawn are the correct length. 1. What is the next step Andrew should take to draw the parallelogram? 2. Find the measure of the next angle Andrew will need to draw.
122°
Reggie is making a greeting card for his mom in the shape of an isosceles trapezoid. 3. One angle on the trapezoid must be 130°. Write the measures of the other 3 angles in the trapezoid. 4. The bottom base of the trapezoidal card will be 2 times as long as the top base. Suggest 2 possible measures for the bases. 5. Draw the trapezoid described. Label the angles and the base lengths on the trapezoid you drew. 6. Describe how to draw a rhombus when only 1 side length and 1 angle is known.
Lesson 7.4.1 — Three Dimensional Figures In Exercise 1–4 say whether each sentence is true or false, and explain your answer. 1. The number of edges on a rectangular prism is always 4 more than the number of vertices. 2. A cube and a triangular prism each have 6 faces. 3. A cylinder has 2 circular faces. 4. A triangular prism has 3 fewer edges than a rectangular prism. Exercises 5–10 are about the prisms shown below. A. B.
C.
For each of the prisms shown, find the number of: 5. faces 6. edges 7. vertices If the base of a prism is a polygon with n vertices, find an expression for: 8. The number of faces in the prism 9. The number of edges in the prism 10. The number of vertices in the prism 466
Additional Questions
Lesson 7.4.2 — Volume of Rectangular Prisms In Exercises 1–4 choose a reasonable estimate of the volume of each object. Choose from: 100 cm3, 1000 cm3, 10,000 cm3, or 100,000 cm3 1. Volume of a toybox 2. Volume of a shoe box 3. Volume of a slice of bread 4. Volume of a small loaf of bread 5. List the dimensions of a cube that has a volume of 1 cubic yard. 6. Describe how the volume of a cube changes when 3 cubes of the same size are added to it. A construction worker built a wall with dimensions as shown in the picture. 7. Find the volume of the wall. 8. Find the volume of the wall if the thickness was increased from 0.25 meters to 0.5 meters. Josie had these 2 suitcases available for a trip. 9. Find the volume of Suitcase A. 10. Find the volume of Suitcase B. 11. Find the difference in these volumes.
3m 0.25m
8.75 m
Case A
4 in
Case B
18 in
10 in
12 in
Josie took the suitcase with the bigger volume on her trip. She only filled it
3 4
3 in
20 in
of the way full.
12. Find the volume of empty space left in the suitcase.
Lesson 7.4.3 — Volume of Triangular Prisms and Cylinders In Exercises 1–4, calculate the volume of each figure. Use 3.14 for p. 6.2 m 1. 2. 3.
4. 4.32 m
4.5 m
16 cm
9.8 m
9.93 miles
20 cm 8 cm
5.8 m
3 miles
5. Explain how the volume of a rectangular prism would change if the prism was divided in half. 6. Explain how to find the height of a triangular prism if you know the volume of the prism and the area of its base. 2.8 cm
In Exercises 7–9, use 3.14 for p. The lid on water bottle is pictured in the diagram. 7. Find the volume of the lid. 8. Find the volume of the bottle. 9. Find the difference between these volumes.
12.8 cm
3 cm
19 cm
10. The volume of a triangular prism is 3.5 cm3 and the area of the base is 7.0 cm2. What is the height of the prism? 11. A cylinder has a volume of 366 in3. Find the height of the cylinder if its base has a radius of 4 in. Use 3.14 for p. Additional Questions
467
Lesson 7.4.4 — Volume of Compound Solids Each of the figures below has a hole cut out of them in the shape of a cylinder. Determine the volume of each figure after the hole was cut out. Use 3.14 for p. 1. 2. 3. 20 cm 6 cm
2.5 in
6 in
7.5 m 3m
6 in
2m 6 in
11 cm
28 cm
3.5 in
6 in
4. Describe how the volume of a 3D figure would change if a cube was cut out of the middle of it. In Exercises 5–8, use the diagram on the left. Boxes A, B and C all extend all the way through the cylinder. Allison is practicing pitching. She first tried to pitch into box A as shown in the figure. 5. Find the volume of box A.
9 ft 15 ft
A B 6 ft
2 ft
C
Next, she pitched into box B. The volume of box B is volume of box A. 6. Find the volume of box B.
2 3
of the
Then, Allison pitched into box C. The volume of box B is twice as much as the volume of box C. 7. Find the volume of box C. 8. Find the volume of the cylinder that was left after the 3 boxes were cut out of it. Use 3.14 for p.
Lesson 7.5.1 — Generalizing Results 1. What does the letter n in the expression n + 6 represent? 2. Explain why it is important to test a generalization. This table shows information about the faces, vertices and edges of prisms. 3. 4. 5. 6.
Determine the value of p. Determine the value of q. Determine the value of r. Determine the value of s.
3D Figure
Number of Number of faces vertices
Number of faces Number of edges and vertices
Triangular Prism
5
6
11
9
Rectangular Prism
6
8
14
12
Pentagonal Prism
7
10
17
15
Hexagonal Prism
8
12
20
18
Heptagonal Prism
9
14
p
q
Octagonal Prism
10
16
r
s
7. Write how you could predict the number of edges on a prism if the number of faces was 100 and the number of vertices was 196. 8. Write an equation to find the number of edges on any prism. Use v for the number of vertices, f for the number of faces and e for the number of edges.
468
Additional Questions
Lesson 7.5.2 — Proving Generalizations In Exercises 1–4 use the diagram below, showing four equilateral triangles. 8.27 in 7 cm
3.3 mm
7 cm 8.27 in
8.27 in
11.2 miles
3.3 mm
11.2 miles
3.3 mm 11.2 miles
7 cm Area = 21.2 cm
2
2
Area = 29.6 in
Area = 4.7 mm
2
2
Area = 54.3 miles
Using the equilateral triangles shown above, decide whether the statements in Exercises 1–4 appear to be true or false. If any are false, give an example that proves this. 1. All equilateral triangles have 3 sides the same length. 2. If a line is drawn from any vertex of an equilateral triangle to the midpoint of the side opposite it, 2 right angles are created at the midpoint. 3. The length of any side on an equilateral triangle is always equal to half the perimeter of the triangle. 4. The area of an equilateral triangle is always half as much as the perimeter of the triangle. 5. The area of an equilateral triangle is always more than 3 times the side length of the triangle. 6. Draw several right triangles and use your sketches to decide which of the statements about equilateral triangles from Exercises 1–5 are true for right triangles, and which are false. 7. Andrew said that a total of 2 examples are needed to prove a statement is true. Is Andrew correct? Explain your answer.
Additional Questions
469
Appendixes Glossary
................................................... 471
Formula Sheet ................................................... 473 Index
470
................................................... 475
Glossary Symbols < > £ ≥
is less than is greater than is less than or equal to is greater than or equal to
π N W Z
is not equal to the natural numbers the whole numbers the integers
A adjacent ang les two angles that share a common side and a angles vertex ang le the amount of turn between two straight lines meeting at angle a point; there are 360° in a full turn ar c part of a circle's circumference; can be drawn with a arc compass associa ti ve pr oper ties (of ad dition and m ultiplica tion) associati tiv proper operties addition multiplica ultiplication) for any a, b, c: a + (b + c) = (a + b) + c a(bc) = (ab)c
B base in the expression bx, the base is b bias unfair influence on a result
C centr al tendenc y the value of "typical" items in a data set central tendency er ence the distance around the outside of a circle cumf circumf cumfer erence cir common ffactor actor a number or expression that is a factor of two or more other numbers or expressions common m ultiple a multiple of two or more different integers multiple v oper ties (of ad comm uta ti dition and m ultiplica tion) for proper operties addition commuta utati tive pr multiplica ultiplication) any a, b: a + b = b + a and ab = ba complementar y ang les two angles whose measures sum to 90° complementary angles con ver sion ffactor actor the ratio of one unit to another; used for conv ersion converting between units cube a three-dimensional figure with six identical square faces customar y units the system of units that includes: inches, feet, customary yards, miles, ounces, and pounds c ylinder a three-dimensional figure with two circular bases and a constant cross-section
D da ta set a collection of information, often numbers data decimal a number including a decimal point; digits to the right of the decimal point show parts of a whole number denomina tor the bottom expression of a fraction denominator diameter a straight line from one side of a circle to the other, passing through the center ty (of m ultiplica tion o ver ad dition) distrib uti ve pr oper multiplica ultiplication ov addition) distributi utiv proper operty for any a, b, c: a(b + c) = ab + ac
E edg e on a three-dimensional figure, an edge is where two faces edge meet
equa tion a mathematical statement showing that two quantities equation are equal equila ter al triang le a triangle whose sides are all the same equilater teral triangle length equi v alent fr actions fractions are equivalent if they have the equiv fractions same value estima te an inexact judgement about the size of a quantity; an estimate "educated guess" evalua te find the value of an expression by substituting actual aluate values for variables event a description that matches one or more possible outcomes exponent in the expression bx, the exponent is x expr ession a collection of numbers, variables, and symbols xpression that represent a quantity
F face a flat surface of a three-dimensional figure factor a number or expression that can be multiplied to get another number or expression — for example, 2 is a factor of 6, because 2 × 3 = 6 fr equenc y the number of times that something happens frequenc equency
G g rea test common ffactor actor (GCF) the largest expression that is a eatest common factor of two or more other expressions; all other common factors will also be factors of the GCF g rouping symbols symbols that show the order in which mathematical operations should be carried out — such as parentheses and brackets
I impr oper fr action a fraction whose numerator is greater than improper fraction its denominator — for example
7 4
inde pendent e vents events whose probabilities are not independent ev affected by whether the other happens or not inte ger s the numbers 0, ±1, ±2, ±3,...; integ ers the set of all integers is denoted Z inter est extra money you pay back when you borrow money, or interest that you receive when you invest money in ver ses a number’s additive inverse is the number that can be inv erses added to it to give 0; a number’s multiplicative inverse is the number that it can be multiplied by to give 1 isosceles triang le a triangle with two sides of equal length triangle
L least common m ultiple (L CM) the smallest integer that has multiple (LCM) two or more other integers as factors
M manipula te to change an expression or equation manipulate
Glossar y 471 Glossary
Glossary (continued) mean a measure of central tendency; the sum of a set of values, divided by the number of values in the set median the middle value when a set of values is put in order metric the system of units that includes: centimeters, meters, kilometers, grams, kilograms, and liters mix ed n umber a number containing a whole number part and a mixed number fraction part mode the value that occurs most frequently in a set
N na tur al n umber s the set of numbers 1, 2, 3,...; natur tural number umbers the set of all natural numbers is denoted N numer a tor the top expression of a fraction umera numeric e xpr ession a number or an expression containing only expr xpression numbers and operations (and therefore no variables)
O origin on a number line, the origin is at zero outcome a possible result of an experiment
P par allelo g ram a four-sided shape with two pairs of parallel parallelo allelog sides per cent value followed by the % sign; corresponds to the percent numerator of a fraction with 100 as the denominator perimeter the sum of the side lengths of a polygon popula tion the full group that samples can be taken from population wer an expression of the form bx, made up of a base (b) and po pow an exponent (x) prime ffactoriza actoriza tion a factorization of a number where each actorization factor is a prime number, for example 12 = 2 × 2 × 3 umber a natural number that can only be divided by prime n number itself and 1, with exactly 2 factors prism a three-dimensional figure with two identical bases and a constant cross-section pr oba bility a number which shows how likely an event is to proba obability happen, written as a number from 0 to 1, a percent, or a fraction pr oduct the result of multiplying numbers or expressions product together pr opor tion an equation showing that two ratios are equivalent propor oportion
Q quadrila ter al a two-dimensional figure with four straight sides quadrilater teral quotient the result of dividing two numbers or expressions
R radius the distance between a point on a circle and the center of the circle rang e the difference between the lowest and highest values in a ange data set ra te a kind of ratio with units ra tio the amount of one thing compared with the amount of another thing recipr ocal the multiplicative inverse of an expression eciprocal
rela ti ve fr equenc y the number of times an outcome occurred, elati tiv frequenc equency divided by the number of trials rhomb us a two-dimensional figure with four equal-length sides rhombus in two parallel pairs r ounding replacing one number with another number that’s easier to work with; used to give an approximation of a solution
S sample a small part of the population chosen to represent the whole population sampling in convenience sampling, each person or object is chosen because they are easy to access; in random sampling, each person or object in the population has an equal chance of being selected; in systematic sampling, people or objects are chosen by a set pattern; self-selected sampling relies on people choosing to give information sampling er errror a measure of how accurately a sample represents the population; a sampling error for the mean is the difference between the mean of the sample and the mean of the population scalene triangle a triangle with three unequal sides sign of a n umber whether a number is positive or negative number similar two figures are similar if all of their corresponding sides are in proportion and all of their angles are equal simplify to reduce an expression to the least number of terms, or to reduce a fraction to its lowest terms solv e to manipulate an equation to find out the value of a solve variable sum the result of adding numbers or expressions together supplementar y ang les two angles whose measures sum to supplementary angles 180° sur vey information collected by contacting people by phone, surv mail, or in person and asking them to provide some data
T ter ms the parts that are added or subtracted to form an terms expression tr a pe zoid a four-sided shape with exactly one pair of parallel tra pez sides
V varia ble a letter that is used to represent an unknown number ariab Venn dia g ram a diagram showing how different items in sets diag are related, making probabilities easier to visualize ver te x the point on an angle where the two rays meet on a erte tex two-dimensional figure; the point where three or more faces meet on a three-dimensional figure ver tical ang les angles that are opposite each other when two ertical angles lines cross volume a measure of the amount of space inside a three-dimensional figure
W whole n umber s the set of numbers 0, 1, 2, 3,...; number umbers the set of all whole numbers is denoted W
472 Glossar y Glossary
Formula Sheet Order of Operations Perform operations in the following order: 1. Anything in g r ouping symbols — working from the innermost grouping symbols to the outermost. 2. Exponents Exponents. visions tions and di divisions visions, working from left to right. 3. Multiplica Multiplications ditions and subtr actions subtractions actions, again from left to right. 4. Ad Additions
Fractions Adding and subtracting fractions with the same denominator::
a c a+c + = b b b
a c a−c − = b b b
Adding and subtracting fractions with different denominators:
a c ad + bc + = b d bd
a c ad − bc − = b d bd
Multiplying fractions: a ⋅
Dividing fractions:
c ac = d d
and
a c a d ad ÷ = ⋅ = b d b c bc
a c ac ⋅ = b d bd Reciprocals:
d c is the called reciprocal of c d
Rules for Multiplying
Rules for Dividing
positive × positive = positive positive × negative = negative negative × positive = negative negative × negative = positive
positive ÷ positive = positive positive ÷ negative = negative negative ÷ positive = negative negative ÷ negative = positive
Axioms of the Real Number System For any real numbers a, b, and c, the following properties hold: Pr oper ty Name Proper operty Commutative Property: Associative Property: Distributive Property of Multiplication over Addition:
Ad dition Addition a+b=b+a (a + b) + c = a + (b + c)
Multiplica tion Multiplication a×b=b×a (ab)c = a(bc)
a(b + c) = ab + ac and (b + c)a = ba + ca
Applications Formulas In vestments Inv The return (I) earned in t years when p is invested at an interest rate of r (expressed as a fraction or decimal): Speed
speed =
distance time
distance = speed × time
time =
I = prt
distance speed
For m ulas 473 orm
Formula Sheet (continued) Units Lengths in Customar y Units Customary 1 foot (ft) 1 yard (yd) 1 mile (mi)
= 12 inches (in.) = 3 feet (ft) = 1760 yards (yd)
Customar y to Metric Con ver sions Customary Conv ersions 1 inch (in.) 1 foot (ft) 1 yard (yd) 1 mile (mi)
= 2.54 cm = 30.48 cm = 0.91 m = 1.61 km
Lengths in Metric Units 1 dekameter (dam) = 10 meters (m) 1 hectometer (hm) = 100 meters 1 kilometer (km) = 1000 meters 1 decimeter (dm) = 0.1 meters 1 centimeter (cm) = 0.01 meters 1 millimeter (mm) = 0.001 meters Metric to Customar y Con ver sions Customary Conv ersions 1 cm 1 cm 1m 1 km
Area
Volume
Area of a rectangle:
A = bh
Area of a triangle:
A=
1 bh 2
where b stands for the length of the base and h stands for the height.
The ratio “a to b” can also be written:
where B represents the area of one of the prism’s bases, and h stands for the height of the prism.
Volume of a cube:
where s represents the length of the side of the faces.
Circles a or a : b b
Probability number of favorable outcomes total number of outcomes P(not A) = 1 – P(A) P(A or B) = P(A) + P(B) – P(A and B) P(A and B) = P(A) × P(B), for independent events
474 For m ulas orm
A = 6s 2 V = s3
Surface area of a cube:
Diameter: Circumference: Area:
P(A) =
F = 1.8C + 32 C = (F – 32) ÷ 1.8
Cubes
V = Bh
Ratios
Con ver ting Betw een Conv erting Between Temper a tur es in empera tures Fahr enheit and Celsius ahrenheit
= 0.39 inches (in.) = 0.033 feet (ft) = 1.09 yards (yd) = 0.62 miles (mi)
d = 2r C = pd A = pr2
di
am
et
radius, r
er,
d
Statistics For any numerical data set:
sum of values total number of values Median = middle value if the values are ordered Mode = value that occurs most often Range = maximum value – minimum value Mean =
Index Symbols
B
p 363, 364, 367, 368
balanced equations 63 bar graphs 275, 307 double 284 bases (of prisms) 407 bias and survey questions 297-299 and samples 288-291
A accuracy in predictions 354 in sampling 293-295 acute angles 370, 389 triangles 389 addition 18, 55-57 associative property 76 commutative property 76 of fractions 152, 153, 156, 161, 164 of integers 6, 7 of mixed numbers 163 of negative numbers 6 of variables 45 to solve equations 67 additional data 268-270, 272-274 adjacent angles 374, 385 angle sums quadrilateral 398 triangle 95, 381 angles 94, 370-372, 374, 375, 377-382, 384-391 acute 370 adjacent 374, 385 complementary 384-386 in similar shapes 209 in triangles 95, 381 linear pairs 374, 375, 378, 399, 400 missing 95, 382 obtuse 370 right 370 supplementary 377-380 vertical 375 annual interest rate 190 area 79, 366, 367 expressions for 78 of circles 366, 367 of complex shapes 82-84, 104 of rectangles 79 of squares 90 of triangles 80 associative property of addition 76 of multiplication 80 average 259, 272 rates 250, 251 see also mean, median, mode average speed 245, 246
C calculators 235 canceling fractions 146, 147 central tendency measures of 256 circle graphs 177-179, 282, 306 circles 359-369 area 366, 367 circumference 362, 363 diameter 359, 360, 363 radius 359, 360, 367 circumference 362-364 claims 300-304 and data displays 304-306 invalid 289, 298 common denominators 156, 157, 164 common factors or divisors 146, 147, 150, 200 common multiples 155, 156, 160 least 160, 161 common percents 172 commutative property of addition 76 of multiplication 80 comparison of data sets 265-267, 284 of decimals 26, 27, 123 of integers 4, 123 of fractions 111, 112, 157 compasses 395-397 complementary angles 384-386 complex shapes area 82-84 compound areas 82-84, 104 solids 416, 417 constants 53 convenience sampling 290 conversions 222-224, 226-235 conversion factors 222, 223, 229-231, 233 proportional quantities 226-228 temperatures 234
corresponding sides/angles 209-212 cross-multiplication 228 of fractions 158 with proportions 205-207, 227 cubes 91, 92, 407 volume 92 cubic units 410 cuboids 407 customary units 222, 223 cylinders 407, 414
D data displays 275-277, 279, 280, 282-284, 304-307 data sets 256, 265, 268, 272 combining 270, 273 comparing 265-267 decimals 23-27, 121, 169, 171, 172 comparing 27 plotting on a number line 24, 25 repeating 122 terminating 122 degrees 370 denominators 111, 125, 133, 136, 156, 157, 164, 168, 200, 204, 205 common 156, 157, 164 dependent events 340, 343-345 diameter 359, 360, 363 discounts 186, 187 distributive property 85, 86, 89 and mental math 88 division 10-12, 14-17, 19, 55-57 as division of multiplication 142 of negative numbers 16, 17 of fractions 136-138, 140-142 of mixed numbers 137, 138 of variables 46 to solve equations 69 divisors common 146, 147, 150, 200 greatest common 150, 170, 200 double bar graphs 284
E edges 408 equations 60, 77, 81, 95 graphing 73, 74 manipulating 63, 64 one-step 72 solving 66, 69, 72 by manipulation 66, 67, 69, 70 guess and check 61 equilateral triangles 393, 396
Inde x Index
475
equivalent fractions 110, 145, 168 ratios 199-202 estimation 35, 36, 38 front-end 39 evaluating claims 300-303 displays 304-307 expressions 49, 53 events 311-313, 321, 322, 324, 330, 331, 333, 334, 336, 337 independent/dependent 340, 341, 343-347 exponents 56 expressions 48, 51-53 and angles 94, 95 and areas 79, 81 and lengths 75 evaluating 49, 53 simplifying 76 extreme values 262-264
F
hundredths 171
I improper fractions 114-116, 118-120 converting to mixed numbers 116, 118-120 independent events 340, 341, 343-347 information in questions 100, 101 integers 4, 5, 125 addition 6, 7 division 14-17 multiplication 10-13 subtraction 7, 8 interest 189-191 inverses multiplicative 136, 207 isosceles trapezoids 400 triangles 393, 396
L
faces 91, 407 factors 10 common 146, 147, 150, 200 for conversions 222, 223, 229-231, 233 fractions 110-113, 121, 125, 146, 147, 155, 168-170, 172 addition 152, 153, 164 and ratios 197, 198 canceling 146, 147 converting to decimals 121, 122 converting to percents 168, 169 cross-multiplication 228 division 136-138, 140-142 equivalent 110, 145, 156 improper 114-116, 118, 119, 122 multiplication 125, 126, 129-131, 133-135 ordering 111-113 frequency 275 relative 350, 351 tables 275 front-end estimation 39
G generalizations 419-422, 424, 425 graphs 275-277, 279, 280, 282-284, 304-307 bar graphs 275, 307 double 284 circle graphs 177-179, 282, 306 conversion graphs 233 line graphs 283, 284 line plots 276, 279 pictographs 276 greatest common divisors (GCD) 150, 170, 200 grids (of outcomes) 317, 318, 346, 347
476
H
Inde x Index
leading questions 298 least common multiples 160, 161 lengths expressions for 75, 76 line graphs 283, 284 line plots 276, 279 linear pairs of angles 374, 375, 378, 379, 399, 400 loans 189
multiples common 155, 156 least common 160, 161 multiplication 10-12, 19, 55-57 associative property 80 commutative property 80 of fractions 125, 126, 129-131, 133-135 of integers 10-13 of mixed numbers 127 of negative numbers 11-13 to solve equations 70 ways to write 50 with variables 46 multiplicative inverses 136, 207
N natural numbers 4, 5 negative numbers 3, 6-8, 11, 12, 16 fractions 112, 113, 126 numbers types of 4, 5 number lines 3, 4, 10, 11, 24, 25, 73, 74, 111, 112, 118, 119, 125, 131 and addition 6 and decimals 24-26 and division 15 and fractions 111-113, 118, 119 and multiplication 11, 125 and subtraction 7, 8 numerators 111, 125, 133, 136, 204, 205
O M magic squares 101, 102 manipulating equations 63, 64, 66, 67, 69, 70 maps 219 mean 259, 260, 263, 265-267, 272-274, 280, 293 and extra data 272-274 and graphs 280 and missing values 260 measures of central tendency 256 median 256, 257, 264-269, 279, 294 and extra data 269 and graphs 279 mental math 88 metric units 223, 224 mixed numbers 114-116, 118-120, 163 addition 163, 164 converting to improper fractions 115 division 137, 138 multiplication 127 modal value — see mode mode 257, 258, 264-268, 277 and extra data 268 and graphs 277
obtuse angles 370 obtuse triangles 389 one-step equations 72 order of operations 55-57, 235 ordering numbers 3, 4, 173 decimals 26 fractions 111-113 integers 3, 4 outcomes 311-315, 317, 318, 324, 325, 327, 333, 334 and tables/grids 317, 318, 346, 347 and tree diagrams 314, 315 favorable 324, 327 outliers 262-264
P parallel 399, 400 parallelograms 399, 403 parentheses 55, 56, 76, 80, 85, 86, 235 and calculators 235 and distributive property 85, 86 partitioning 140 patterns 97, 98, 104, 419, 420 PEMDAS 56, 57
percent decrease 185-187 percent increase 181-183 percents 168-175 and circle graphs 177-179 converting to decimals 171 converting to fractions 170 decrease 185-187 discounts 186, 187 increase 181-183 tips 182, 183 perimeter 76 pictographs 276 place values 30, 32 populations 287-290, 293, 297 versus samples 287 positive numbers 3 prime factorization 147, 150, 161 principal 189, 190 prisms 406, 407 bases 407 edges 408 faces 407 rectangular 407 triangular 407, 413 vertices (vertex) 408 probability 311, 321, 322, 324, 325, 327, 328, 330, 331, 340, 343, 346, 351 A and B 336, 347 A or B 337, 338 dependent events 340, 341, 343-345 experimental 351, 353, 354 independent events 340, 341, 343-347 of an event not happening 330, 331 of complement 330, 331 theoretical 322, 351, 355 Venn diagrams 333, 334, 336-338 problem-solving strategies 104, 105, 165 product 10 see also multiplication proportional quantities 233 proportions 202-207, 209, 213, 226-228, 282, 231 and circle graphs 282 and converting units 226-233 and scale drawings 218, 219 cross-multiplication 205-207, 227 solving 204, 206 protractors 371, 372, 391, 402
Q quadrilaterals 398-400, 402-404 angle sum 398 parallelograms 399, 403 rectangles 90, 398, 402 squares 90, 398, 402 trapezoids 400, 403, 404 questions leading 297-299 survey 296-299
questionnaires 296-299 quotient 14 see also division
R radius 359, 360, 367 random sampling 291 range 260, 261, 265-268, 270, 277 and extra data 270 and graphs 277 rates 237-242, 244, 247, 248, 250, 251 average 250, 251 and units 237, 238 speed 244-248 ratios 196-202, 226 and probability 324 and scale 217-219 and similar shapes 210-215 equivalent 199-201 simplifying 199, 200 writing 197 rays 374, 375 reciprocals 136, 239 and division 137 rectangles 90, 398, 402 rectangular prisms 407, 410–412 relative frequency 350, 351 repeated subtraction 141 repeating decimals 122 replacement 344, 345 representative samples 288, 290, 291, 296 rhombi (rhombus) 399 right angles 370, 389 right triangles 389 rounding 29, 30, 32 and accuracy 32, 33 decimals 30 to check work 33, 106
S sample size 295 samples 287-295, 297 accuracy 293-295 bias 288-291 convenience 290 random 291 sampling errors 293-295 self-selecting 288, 296 systematic 292 versus populations 287 sampling errors 293-295 savings 189-191 scale drawings 217-219 and maps 219 and units 217, 218 scalene triangles 393, 397
selection with replacement 343, 344 without replacement 343, 345 self-selected sampling 296 sequences 97, 98 similar shapes 209-210 and proportions 210-215 simple interest 190 simplifying expressions 76 fractions 146, 147, 149, 150 ratios 199, 200 solving equations by manipulation 66, 67, 69, 70 guess and check 61 speed 237, 244-248, 250, 251 and units 247 average 245, 246 spinners 341 squares 90, 398, 402 area 90 subtraction 18, 55-57 of fractions 152 of integers 7, 8 of negative numbers 7, 8 of variables 45 to solve equations 66 supplementary angles 377-380 surface area of cubes 91 surveys bias 297, 298 questionnaires 296-299 systematic sampling 292
T tables of outcomes 317, 318, 346, 347 tally charts 275 temperature conversions 234 tenths 23 terminating decimals 122 terms 53 thousandths 23 tips 182, 183 trapezoids 400, 403, 404 isosceles 400, 403, 404 tree diagrams 314, 315, 328 triangle sum 381, 382 triangles 381, 389, 393 acute 389 area 80 equilateral 393 isosceles 393, 396 obtuse 389 right 386, 389 scalene 393 angle sum 95, 381 triangular prisms 407, 413
Inde x Index
477
U unit cubes 92, 410 units 92, 106 and rates 237-239 and scale 217, 218 converting 222-224, 226, 227, 229-235 customary 222, 223 metric 223, 224
V variables 44-46, 57, 60, 61, 75, 76, 83, 200, 203 Venn diagrams 333, 334, 337, 338 vertical angles 375 vertices (vertex) 408 volume 92, 410-414, 416 of complex shapes 416, 417 of cubes 92 of cylinders 414 of prisms 413
W whole numbers 4, 5 word problems 18-20, 48, 60, 99, 165, 203
Z zero 3, 4
478
Inde x Index
CGP
Education
California
Mathematics Course Two
Student Textbook California Standards-Driven Program
California
Mathematics Course Two
Student Textbook California Standards-Driven Program
i
Contents This Textbook provides comprehensive coverage of all the California Grade 7 Standards. The Textbook is divided into eight Chapters. Each of the Chapters is broken down into small, manageable Lessons and each Lesson covers a specific Standard or part of a Standard.
California Standard
Chapter 1 — The Basics of Algebra
AF 1.1 AF 1.2 AF 1.3 AF 1.4
Section 1.1 — Variables and Expressions Exploration Variable Tiles ........................................................................................................................ Lesson 1.1.1 Variables and Expressions ................................................................................................... Lesson 1.1.2 Simplifying Expressions ........................................................................................................ Lesson 1.1.3 The Order of Operations ...................................................................................................... Lesson 1.1.4 The Identity and Inverse Properties ...................................................................................... Lesson 1.1.5 The Associative and Commutative Properties ......................................................................
2 3 7 10 13 17
AF 1.1 AF 1.4 AF 4.1 MR 1.1 MR 2.1
Section 1.2 — Equations Exploration Solving Equations ................................................................................................................. Lesson 1.2.1 Writing Expressions .............................................................................................................. Lesson 1.2.2 Equations .............................................................................................................................. Lesson 1.2.3 Solving One-Step Equations ................................................................................................. Lesson 1.2.4 Solving Two-Step Equations ................................................................................................. Lesson 1.2.5 More Two-Step Equations .................................................................................................... Lesson 1.2.6 Applications of Equations ..................................................................................................... Lesson 1.2.7 Understanding Problems ......................................................................................................
20 21 24 28 32 35 38 41
AF 1.1 AF 1.4 AF 1.5
Section 1.3 — Inequalities Lesson 1.3.1 Inequalities ............................................................................................................................ 44 Lesson 1.3.2 Writing Inequalities ............................................................................................................... 47 Lesson 1.3.3 Two-Step Inequalities ........................................................................................................... 50 Chapter 1 — Investigation Which Phone Deal is Best? .......................................................................................................................... 53
Chapter 2 — Rational and Irrational Numbers
ii
NS 1.3 NS 1.4 NS 1.5
Section 2.1 — Rational Numbers Lesson 2.1.1 Rational Numbers ................................................................................................................. 55 Lesson 2.1.2 Converting Terminating Decimals to Fractions ..................................................................... 59 Lesson 2.1.3 Converting Repeating Decimals to Fractions ........................................................................ 62
NS 2.5 AF 1.1
Section 2.2 — Absolute Value Lesson 2.2.1 Absolute Value ...................................................................................................................... 65 Lesson 2.2.2 Using Absolute Value ............................................................................................................ 68
NS 1.1 NS 1.2 NS 2.2
Section 2.3 — Operations on Rational Numbers Lesson 2.3.1 Adding and Subtracting Integers and Decimals .................................................................... Lesson 2.3.2 Multiplying and Dividing Integers .......................................................................................... Lesson 2.3.3 Multiplying Fractions ............................................................................................................. Lesson 2.3.4 Dividing Fractions ................................................................................................................. Lesson 2.3.5 Common Denominators ........................................................................................................ Lesson 2.3.6 Adding and Subtracting Fractions ......................................................................................... Lesson 2.3.7 Adding and Subtracting Mixed Numbers ..............................................................................
71 75 78 81 84 87 90
California Standard
Chapter 2 Continued Section 2.4 — More Operations on Rational Numbers Lesson 2.4.1 Further Operations with Fractions ........................................................................................ Lesson 2.4.2 Multiplying and Dividing Decimals ........................................................................................ Lesson 2.4.3 Operations with Fractions and Decimals .............................................................................. Lesson 2.4.4 Problems Involving Fractions and Decimals .........................................................................
93 96 99 102
NS 1.1 NS 1.2 AF 2.1
Section 2.5 — Basic Powers Exploration Basic Powers ........................................................................................................................ Lesson 2.5.1 Powers of Integers ................................................................................................................ Lesson 2.5.2 Powers of Rational Numbers ................................................................................................ Lesson 2.5.3 Uses of Powers .................................................................................................................... Lesson 2.5.4 More on the Order of Operations .........................................................................................
105 106 109 112 116
NS 1.4 NS 2.4 MR 2.7
Section 2.6 — Irrational Numbers and Square Roots Exploration The Side of a Square ........................................................................................................... Lesson 2.6.1 Perfect Squares and Their Roots .......................................................................................... Lesson 2.6.2 Irrational Numbers ................................................................................................................ Lesson 2.6.3 Estimating Irrational Roots ...................................................................................................
119 120 123 126
NS 1.2 NS 2.2 MR 2.2
Chapter 2 — Investigation Designing a Deck .......................................................................................................................................... 130
Chapter 3 — Two-Dimensional Figures
MG 2.1 MG 2.2
Section 3.1 — Perimeter, Circumference, and Area Exploration Area and Perimeter Patterns ................................................................................................ Lesson 3.1.1 Polygons and Perimeter ........................................................................................................ Lesson 3.1.2 Areas of Polygons ................................................................................................................ Lesson 3.1.3 Circles ................................................................................................................................... Lesson 3.1.4 Areas of Complex Shapes .................................................................................................... Lesson 3.1.5 More Complex Shapes .........................................................................................................
MG 3.2
Section 3.2 — The Coordinate Plane Exploration Coordinate 4-in-a-row ........................................................................................................... 149 Lesson 3.2.1 Plotting Points ....................................................................................................................... 150 Lesson 3.2.2 Drawing Shapes on the Coordinate Plane ............................................................................ 154
MG 3.2 MG 3.3
Section 3.3 — The Pythagorean Theorem Exploration Measuring Right Triangles .................................................................................................... Lesson 3.3.1 The Pythagorean Theorem ................................................................................................... Lesson 3.3.2 Using the Pythagorean Theorem .......................................................................................... Lesson 3.3.3 Applications of the Pythagorean Theorem ........................................................................... Lesson 3.3.4 Pythagorean Triples & the Converse of the Theorem ...........................................................
158 159 163 167 171
Section 3.4 — Comparing Figures Exploration Transforming Shapes ............................................................................................................ Lesson 3.4.1 Reflections ............................................................................................................................ Lesson 3.4.2 Translations ........................................................................................................................... Lesson 3.4.3 Scale Factor .......................................................................................................................... Lesson 3.4.4 Scale Drawings ..................................................................................................................... Lesson 3.4.5 Perimeter, Area, and Scale ................................................................................................... Lesson 3.4.6 Congruence and Similarity ...................................................................................................
174 175 178 182 185 189 192
MG 1.2 MG 2.0 MG 3.2 MG 3.4
132 133 136 139 142 145
iii
Contents California Standard
MG 3.1
AF 1.1 MG 3.3 MR 1.2 MR 2.2 MR 2.4 MR 3.3
Chapter 3 Continued Section 3.5 — Constructions Lesson 3.5.1 Constructing Circles ............................................................................................................. 196 Lesson 3.5.2 Constructing Perpendicular Bisectors ................................................................................... 199 Lesson 3.5.3 Perpendiculars, Altitudes, and Angle Bisectors .................................................................... 202 Section 3.6 — Conjectures and Generalizations Lesson 3.6.1 Geometrical Patterns and Conjectures ................................................................................. 206 Lesson 3.6.2 Expressions and Generalizations ......................................................................................... 210 Chapter 3 — Investigation Designing a House ........................................................................................................................................ 213
Chapter 4 — Linear Functions AF 1.1 AF 1.5 AF 3.3
Section 4.1 — Graphing Linear Equations Exploration Block Patterns ...................................................................................................................... Lesson 4.1.1 Graphing Equations .............................................................................................................. Lesson 4.1.2 Systems of Linear Equations ................................................................................................ Lesson 4.1.3 Slope .....................................................................................................................................
215 216 220 223
AF 3.4 AF 4.2 MG 1.3
Section 4.2 — Rates and Variation Exploration Pulse Rates .......................................................................................................................... Lesson 4.2.1 Ratios and Rates .................................................................................................................. Lesson 4.2.2 Graphing Ratios and Rates .................................................................................................. Lesson 4.2.3 Distance, Speed, and Time .................................................................................................. Lesson 4.2.4 Direct Variation .....................................................................................................................
227 228 231 235 238
AF 4.2 MG 1.1 MG 1.3
Section 4.3 — Units and Measures Lesson 4.3.1 Converting Measures ............................................................................................................ Lesson 4.3.2 Converting Between Unit Systems ....................................................................................... Lesson 4.3.3 Dimensional Analysis ............................................................................................................ Lesson 4.3.4 Converting Between Units of Speed ....................................................................................
241 244 248 251
AF 1.1 AF 4.1
Section 4.4 — More on Inequalities Lesson 4.4.1 Linear Inequalities ................................................................................................................. 254 Lesson 4.4.2 More on Linear Inequalities .................................................................................................. 258 Lesson 4.4.3 Solving Two-Step Inequalities ............................................................................................... 261 Chapter 4 — Investigation Choosing a Route ......................................................................................................................................... 264
Chapter 5 — Powers
iv
NS 2.1 NS 2.3 AF 2.1
Section 5.1 — Operations on Powers Lesson 5.1.1 Multiplying with Powers ......................................................................................................... 266 Lesson 5.1.2 Dividing with Powers ............................................................................................................. 269 Lesson 5.1.3 Fractions with Powers ........................................................................................................... 272
NS 1.1 NS 1.2 NS 2.1 AF 2.1
Section 5.2 — Negative Powers and Scientific Notation Lesson 5.2.1 Negative and Zero Exponents .............................................................................................. Lesson 5.2.2 Using Negative Exponents .................................................................................................... Lesson 5.2.3 Scientific Notation ................................................................................................................. Lesson 5.2.4 Comparing Numbers in Scientific Notation ...........................................................................
275 278 281 284
California Standard
Chapter 5 Continued
AF 1.4 AF 2.2
Section 5.3 — Monomials Exploration Monomials ............................................................................................................................ Lesson 5.3.1 Multiplying Monomials .......................................................................................................... Lesson 5.3.2 Dividing Monomials ............................................................................................................... Lesson 5.3.3 Powers of Monomials ........................................................................................................... Lesson 5.3.4 Square Roots of Monomials .................................................................................................
287 288 291 294 297
AF 3.1 MR 2.3 MR 2.5
Section 5.4 — Graphing Nonlinear Functions Exploration The Pendulum ....................................................................................................................... Lesson 5.4.1 Graphing y = nx2 .................................................................................................................... Lesson 5.4.2 More Graphs of y = nx2 ......................................................................................................... Lesson 5.4.3 Graphing y = nx3 ....................................................................................................................
301 302 306 309
Chapter 5 — Investigation The Solar System ......................................................................................................................................... 313
Chapter 6 — The Basics of Statistics
SDAP 1.1 SDAP 1.3 MR 2.6
SDAP 1.2 MR 2.3
Section 6.1 — Analyzing Data Exploration Reaction Rates ..................................................................................................................... Lesson 6.1.1 Median and Range ............................................................................................................... Lesson 6.1.2 Box-and-Whisker Plots ......................................................................................................... Lesson 6.1.3 More on Box-and-Whisker Plots ........................................................................................... Lesson 6.1.4 Stem-and-Leaf Plots ............................................................................................................. Lesson 6.1.5 Preparing Data to be Analyzed ............................................................................................. Lesson 6.1.6 Analyzing Data ......................................................................................................................
315 316 319 322 325 329 332
Section 6.2 — Scatterplots Exploration Age and Height ..................................................................................................................... Lesson 6.2.1 Making Scatterplots .............................................................................................................. Lesson 6.2.2 Shapes of Scatterplots ......................................................................................................... Lesson 6.2.3 Using Scatterplots ................................................................................................................
335 336 339 342
Chapter 6 — Investigation Cricket Chirps and Temperature ................................................................................................................... 345
Chapter 7 — Three-Dimensional Geometry MG 2.1 MG 2.2 MG 2.3 MG 3.5 MG 3.6 MR 1.3 MR 2.2 MR 3.6
Section 7.1 — Shapes, Surfaces, and Space Exploration Nets ...................................................................................................................................... Lesson 7.1.1 Three-Dimensional Figures .................................................................................................. Lesson 7.1.2 Nets ...................................................................................................................................... Lesson 7.1.3 Surface Areas of Cylinders and Prisms ................................................................................ Lesson 7.1.4 Surface Areas & Perimeters of Complex Shapes ................................................................. Lesson 7.1.5 Lines and Planes in Space ...................................................................................................
AF 3.2 MG 2.1 MR 2.2 MR 2.3 MR 3.2
Section 7.2 — Volume Exploration Build the Best Package ......................................................................................................... 366 Lesson 7.2.1 Volumes ................................................................................................................................ 367 Lesson 7.2.2 Graphing Volumes ................................................................................................................ 371
347 348 352 356 359 362
v
Contents California Standard MG 1.2 MG 2.1 MG 2.3 MG 2.4
Chapter 7 Continued Section 7.3 — Scale Factors Exploration Growing Cubes ..................................................................................................................... Lesson 7.3.1 Similar Solids ........................................................................................................................ Lesson 7.3.2 Surface Areas & Volumes of Similar Figures ....................................................................... Lesson 7.3.3 Changing Units .....................................................................................................................
374 375 379 383
Chapter 7 — Investigation Set Design .................................................................................................................................................... 387
Chapter 8 — Percents, Rounding, and Accuracy NS 1.3 NS 1.6
Section 8.1 — Percents Exploration Photo Enlargements ............................................................................................................. Lesson 8.1.1 Percents ................................................................................................................................ Lesson 8.1.2 Changing Fractions and Decimals to Percents ..................................................................... Lesson 8.1.3 Percent Increases and Decreases ........................................................................................
389 390 393 396
NS 1.3 NS 1.7
Section 8.2 — Using Percents Exploration What’s the Best Deal? .......................................................................................................... Lesson 8.2.1 Discounts and Markups ........................................................................................................ Lesson 8.2.2 Tips, Tax, and Commission ................................................................................................... Lesson 8.2.3 Profit ..................................................................................................................................... Lesson 8.2.4 Simple Interest ...................................................................................................................... Lesson 8.2.5 Compound Interest ...............................................................................................................
400 401 404 407 410 413
NS 1.3 MR 2.1 MR 2.3 MR 2.7 MR 2.8 MR 3.1
Section 8.3 — Rounding and Accuracy Exploration Estimating Length ................................................................................................................. Lesson 8.3.1 Rounding .............................................................................................................................. Lesson 8.3.2 Rounding Reasonably .......................................................................................................... Lesson 8.3.3 Exact and Approximate Answers .......................................................................................... Lesson 8.3.4 Reasonableness and Estimation ..........................................................................................
416 417 420 423 426
Chapter 8 — Investigation Nutrition Facts .............................................................................................................................................. 429
Additional Questions Additional Questions for Chapter 1 ............................................................................................................... Additional Questions for Chapter 2 ............................................................................................................... Additional Questions for Chapter 3 ............................................................................................................... Additional Questions for Chapter 4 ............................................................................................................... Additional Questions for Chapter 5 ............................................................................................................... Additional Questions for Chapter 6 ............................................................................................................... Additional Questions for Chapter 7 ............................................................................................................... Additional Questions for Chapter 8 ...............................................................................................................
430 435 443 452 457 462 465 470
Appendixes Glossary ........................................................................................................................................................ 475 Formula Sheet ............................................................................................................................................... 478 Index ............................................................................................................................................................. 480
vi
California Grade Seven Mathematics Standards The following table lists all the California Mathematics Content Standards for Grade 7 with cross references to where each Standard is covered in this Textbook. Each Lesson begins by quoting the relevant Standard in full, together with a clear and understandable objective. This will enable you to measure your progression against the California Grade 7 Standards as you work your way through the Program. California Standard
Number Sense
Chapter
1.0
Students know the properties of, and compute with, rational numbers expressed in a variety of forms:
2, 5, 8
1.1
Read, write, and compare rational numbers in scientific notation (positive and negative powers of 10), compare rational numbers in general.
2, 5
1.2
Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to whole-number powers.
2, 5
1.3
Convert fractions to decimals and percents and use these representations in estimations, computations, and applications.
2, 8
1.4
Differentiate between rational and irrational numbers.
2
1.5
Know that every rational number is either a terminating or a repeating decimal and be able to convert terminating decimals into reduced fractions.
2
1.6
Calculate the percentage of increases and decreases of a quantity.
8
1.7
Solve problems that involve discounts, markups, commissions, and profit and compute simple and compound interest.
8
2.0
Students use exponents, powers, and roots and use exponents in working with fractions:
2, 5
2.1
Understand negative whole-number exponents. Multiply and divide expressions involving exponents with a common base.
5
2.2
Add and subtract fractions by using factoring to find common denominators.
2
2.3
Multiply, divide, and simplify rational numbers by using exponent rules.
5
2.4
Use the inverse relationship between raising to a power and extracting the root of a perfect square integer; for an integer that is not square, determine without a calculator the two integers between which its square root lies and explain why.
2
2.5
Understand the meaning of the absolute value of a number; interpret the absolute value as the distance of the number from zero on a number line; and determine the absolute value of real numbers.
2
California Standard
Algebra and Functions
1.0
Students express quantitative relationships by using algebraic terminology, expressions, equations, inequalities, and graphs:
1, 2, 3, 4, 5
1.1
Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represents a verbal description (e.g., three less than a number, half as large as area A).
1, 2, 3, 4
1.2
Use the correct order of operations to evaluate algebraic expressions such as 3(2x + 5)2.
1
1.3
Simplify numerical expressions by applying properties of rational numbers (e.g., identity, inverse, distributive, associative, commutative) and justify the process used.
1
1.4
Use algebraic terminology (e.g., variable, equation, term, coefficient, inequality, expression, constant) correctly.
1, 5
1.5
Represent quantitative relationships graphically and interpret the meaning of a specific part of a graph in the situation represented by the graph.
1, 4
2.0
Students interpret and evaluate expressions involving integer powers and simple roots:
2, 5
2.1
Interpret positive whole-number powers as repeated multiplication and negative whole-number powers as repeated division or multiplication by the multiplicative inverse. Simplify and evaluate expressions that include exponents.
2, 5
2.2
Multiply and divide monomials; extend the process of taking powers and extracting roots to monomials when the latter results in a monomial with an integer exponent.
5
vii
California Grade Seven Mathematics Standards 3.0
Students graph and interpret linear and some nonlinear functions:
4, 5, 7
3.1
Graph functions of the form y = nx2 and y = nx3 and use in solving problems.
5
3.2
Plot the values from the volumes of three-dimensional shapes for various values of the edge lengths (e.g., cubes with varying edge lengths or a triangle prism with a fixed height and an equilateral triangle base of varying lengths).
7
3.3
Graph linear functions, noting that the vertical change (change in y-value) per unit of horizontal change (change in x-value) is always the same and know that the ratio (“rise over run”) is called the slope of a graph.
4
3.4
Plot the values of quantities whose ratios are always the same (e.g., cost to the number of an item, feet to inches, circumference to diameter of a circle). Fit a line to the plot and understand that the slope of the line equals the ratio of the quantities.
4
4.0
Students solve simple linear equations and inequalities over the rational numbers:
1, 4
4.1
Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results.
1, 4
4.2
Solve multistep problems involving rate, average speed, distance, and time or a direct variation.
4
California Standard
viii
Measurement and Geometry
1.0
Students choose appropriate units of measure and use ratios to convert within and between measurement systems to solve problems:
3, 4, 7
1.1
Compare weights, capacities, geometric measures, times, and temperatures within and between measurement systems (e.g., miles per hour and feet per second, cubic inches to cubic centimeters).
4
1.2
Construct and read drawings and models made to scale.
3, 7
1.3
Use measures expressed as rates (e.g., speed, density) and measures expressed as products (e.g., person-days) to solve problems; check the units of the solutions; and use dimensional analysis to check the reasonableness of the answer.
4
2.0
Students compute the perimeter, area, and volume of common geometric objects and use the results to find measures of less common objects. They know how perimeter, area, and volume are affected by changes of scale:
3, 7
2.1
Use formulas routinely for finding the perimeter and area of basic two-dimensional figures and the surface area and volume of basic three-dimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders.
3, 7
2.2
Estimate and compute the area of more complex or irregular two- and three-dimensional figures by breaking the figures down into more basic geometric objects.
3, 7
2.3
Compute the length of the perimeter, the surface area of the faces, and the volume of a threedimensional object built from rectangular solids. Understand that when the lengths of all dimensions are multiplied by a scale factor, the surface area is multiplied by the square of the scale factor and the volume is multiplied by the cube of the scale factor.
7
2.4
Relate the changes in measurement with a change of scale to the units used (e.g., square inches, cubic feet) and to conversions between units (1 square foot = 144 square inches or [1 ft.2] = [144 in.2]; 1 cubic inch is approximately 16.38 cubic centimeters or [1 in.3] = [16.38 cm3]).
7
3.0
Students know the Pythagorean theorem and deepen their understanding of plane and solid geometric shapes by constructing figures that meet given conditions and by identifying attributes of figures:
3, 7
3.1
Identify and construct basic elements of geometric figures (e.g., altitudes, midpoints, diagonals, angle bisectors, and perpendicular bisectors; central angles, radii, diameters, and chords of circles) by using a compass and straightedge.
3
3.2
Understand and use coordinate graphs to plot simple figures, determine lengths and areas related to them, and determine their image under translations and reflections.
3
California Grade Seven Mathematics Standards 3.3
Know and understand the Pythagorean theorem and its converse and use it to find the length of the missing side of a right triangle and the lengths of other line segments and, in some situations, empirically verify the Pythagorean theorem by direct measurement.
3
3.4
Demonstrate an understanding of conditions that indicate two geometrical figures are congruent and what congruence means about the relationships between the sides and angles of the two figures.
3
3.5
Construct two-dimensional patterns for three-dimensional models, such as cylinders, prisms, and cones.
7
3.6
Identify elements of three-dimensional geometric objects (e.g., diagonals of rectangular solids) and describe how two or more objects are related in space (e.g., skew lines, the possible ways three planes might intersect).
7
California Standard
Statistics, Data Analysis, and Probability
1.0
Students collect, organize, and represent data sets that have one or more variables and identify relationships among variables within a data set by hand and through the use of an electronic spreadsheet software program:
6
1.1
Know various forms of display for data sets, including a stem-and-leaf plot or box-and-whisker plot; use the forms to display a single set of data or to compare two sets of data.
6
1.2
Represent two numerical variables on a scatterplot and informally describe how the data points are distributed and any apparent relationship that exists between the two variables (e.g., between time spent on homework and grade level).
6
1.3
Understand the meaning of, and be able to compute, the minimum, the lower quartile, the median, the upper quartile, and the maximum of a data set.
6
California Standard
Mathematical Reasoning
1.0
Students make decisions about how to approach problems:
1.1
Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, identifying missing information, sequencing and prioritizing information, and observing patterns.
1.2
Formulate and justify mathematical conjectures based on a general description of the mathematical question or problem posed.
1.3
Determine when and how to break a problem into simpler parts.
2.0
Students use strategies, skills, and concepts in finding solutions:
2.1
Use estimation to verify the reasonableness of calculated results.
2.2
Apply strategies and results from simpler problems to more complex problems.
2.3
Estimate unknown quantities graphically and solve for them by using logical reasoning and arithmetic and algebraic techniques.
2.4
Make and test conjectures by using both inductive and deductive reasoning.
2.5
Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning.
2.6
Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language; support solutions with evidence in both verbal and symbolic work.
2.7
Indicate the relative advantages of exact and approximate solutions to problems and give answers to a specified degree of accuracy.
2.8
Make precise calculations and check the validity of the results from the context of the problem.
3.0
Students determine a solution is complete and move beyond a particular problem by generalizing to other situations:
3.1
Evaluate the reasonableness of the solution in the context of the original situation.
3.2
Note the method of deriving the solution and demonstrate a conceptual understanding of the derivation by solving similar problems.
3.3
Develop generalizations of the results obtained and the strategies used and apply them to new problem situations.
Illustrated throughout Program
ix
Published by CGP Education Editor: Sharon Keeley Core material written by: William D. Nolan and Raven Deerwater William is a Middle School Math teacher in New York State and is experienced in delivering math courses to diverse student groups. Raven is a freelance math textbook and test item writer and independent tax consultant, and has worked as a math teacher in Illinois and California. California General Advisor: Phyllis Zweig Chinn Phyllis is Director of the Redwood Area Mathematics Project and also Professor at Humboldt State University, Arcata. She is a former elementary, middle and high school math teacher. California English Language Learner Advisor: Judith A. McGinty Judith is a Middle School Math teacher in Oakland Unified School District, California. She has vast experience of teaching English Language Learners and holds a CLAD certificate (Cross-cultural Language and Academic Development). California Advanced Learner Advisor: William D. Nolan Additional Writers: Joiclyn Austin and Ingrid Brown-Scott Joiclyn is an Assistant Curriculum Specialist in Louisiana and has previously worked as a Math Team Leader in public schools in New Orleans. Ingrid is a College math professor in Maryland and has previously worked as a High School Math teacher. Additional Contributors: Paul R. Allrich and Darell Dean Ballard
Supporting Editors: Tim Burne Mary Falkner Tom Harte Sarah Hilton Kate Houghton Paul Jordin John Kitching
Simon Little Tim Major Andy Park Glenn Rogers Emma Stevens Claire Thompson Julie Wakeling
Proofreading: Judith Curran Buck, Ingrid Brown-Scott, and Amanda Jones Graphic Design: Caroline Batten, Russell Holden, Jane Ross and Ash Tyson
Mathematics Content Standards for California Public Schools reproduced by permission, California Department of Education, CDE Press, 1430 N Street, Suite 3207, Sacramento, CA 95814. ISBN 13: 978 1 60017 024 9 website: www.cgpeducation.com Printed by Elanders Hindson Ltd, UK and Johnson Printing, Boulder, CO Clipart sources: CorelDRAW and VECTOR. Text, design, layout, and illustrations © CGP, Inc. 2007 All rights reserved.
x
Chapter 1 The Basics of Algebra Section 1.1
Exploration — Variable Tiles ......................................... 2 Variables and Expressions ............................................ 3
Section 1.2
Exploration — Solving Equations ................................ 20 Equations .................................................................... 21
Section 1.3
Inequalities .................................................................. 44
Chapter Investigation — Which Phone Deal is Best? ......................... 53
1
Section 1.1 introduction — an exploration into:
Alg ebr a Tiles Alge bra You can write expressions using algebra tiles. This shows what an expression actually “looks like,” and how you can simplify it. You’ll be using two types of tile. Their areas represent their values — x-tile: 1
1-tile: 1
x
1
+
Here’s the sum 4 + 3 shown using algebra tiles: 4
= 3
+
=
7
The example below shows how you can simplify algebra expressions by rearranging the tiles. Example Show 3 + x + 1 + 2x using algebra tiles. Rearrange the tiles to simplify the expression. Solution
+
+
+
+
3 + x + 1 + 2x
=
3x + 4
Simplified expression
Exercises 1. Use algebra tiles to model the following addition problems: a. 5 + 2
b. 2 + 1
c. x + 2x
2. Write the algebra expressions that are modeled by these tiles. a.
b.
c.
3. Rearrange these tiles to make a simplified expression:
+
4. Write an algebraic expression to represent the tiles in Exercise 3. Rewrite this as a simplified expression. 5. Write expressions to represent the perimeter and area of these rectangles. a.
+
+
b.
Round Up When you’re simplifying expressions, you have to organize them so the bits that are the same are all grouped together. You can then combine these — so 2x + 3x = 5x, and 1 + 2 = 3. What you can’t do is combine different things together. For example, you could never add 4x and 7. 2
Section 1.1 Explor a tion — Algebra Tiles Explora
Lesson
Section 1.1
1.1.1
Varia bles and Expr essions ariab Expressions
California Standards:
When you write out a problem in math you often need to include unknown numbers. You have to use a letter or symbol to stand in for an unknown number until you figure out what it is — you did this before in earlier grades. That’s what a variable does. And that’s what this Lesson is about.
Alg ebr a and Functions 1.1 bra Alge Use vvaria aria bles and ariab appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represents a verbal description (e.g., 3 less than a number, half as large as area A). Alg ebr a and Functions 1.4 Alge bra Use alg ebr aic ter minolo gy alge braic terminolo minolog (e .g ., vvaria aria ble (e.g .g., ariab le,, equation, ter m, coef ficient term, coefficient ficient, inequality, expr ession, constant) xpression, cor y. corrr ectl ectly
What it means for you: You’ll learn how to use a letter or symbol to stand in for an unknown number, and how it can be exchanged for the real number when you know it.
Key words: • variable • expression • evaluate
A Varia ble R epr esents an Unkno wn Number ariab Re presents Unknown In algebra you’ll often have to work with numbers whose values you don’t know. When you write out math problems, you can use a letter or a symbol to stand in for the number. The letter or symbol is called a variable. Coefficient
2k + 4
Constant
Variable
The number that the variable is being multiplied by is called the coefficient — like the 2 above. Any number not joined to a variable is called a constant — like the 4 above. It’s called that because its value doesn’t change, even if the value of the variable changes. A term is a group of numbers and variables. One or more terms added together make an expression. For example, in the expression above, 2k is one term and 4 is another term. In the expression 3 + 4x – 5wyz, the terms are 3, 4x, and –5wyz.
An Expr ession is a Ma thema tical Phr ase Expression Mathema thematical Phrase Expressions are mathematical phrases that may contain numbers, operations, and variables. The operations act like a set of instructions that tell you what to do with the numbers and variables. For example, 2k + 4 tells you to double k, then add four to it.
Don’t forget: xy means “x multiplied by y.” It’s just the same as writing x × y or x • y.
There are two types of expressions — numeric and variable. • Numeric expressions have numbers in them, and often operations — but they don’t include any variables: Æ 5 + 13 Æ 2•5–6 Æ 8+7÷6 • Variable expressions have variables in them, and may also include numbers and operations: Æ 5h Æ 7x – 2 Æ 2k + 4 Section 1.1 — Variables and Expressions
3
Guided Practice Say whether the expressions in Exercises 1–8 are numeric or variable. 1. y
2. 4(x – y)
3. 10 – 7
4. 62
5. 8xy
6. 4(3 + 7)
7. 9(5 – y)
8. x2
Expr essions Can Be Described in Wor ds Expressions ords To show you understand an expression you need to be able to explain what it means in words. You can write a word expression to represent the numeric or variable expression. Example
1
Write the variable expression x + 5 as a word expression. Solution
In this question x is the variable. The rest of the expression tells you that five is being added on to the unknown number x. So the expression becomes: “x is increased by five” or “the sum of x and five” or “five more than x.” They all mean the same thing.
When you change a variable expression to a word expression you can say the same thing in several different ways. Check it out: These four groups of phrases are describing the four operations you’ll need: addition, subtraction, multiplication, and division.
+ Instead of “2 added to x” you could say “x increased by 2,” “2 more than x,” or “the sum of x and 2.”
– “2 subtracted from x” means the same as “2 less than x” or “x decreased by 2.”
× “x multiplied by 2” means the same as “the product of x and 2,” “x times 2,” or “twice x.”
÷ And you could say either “x divided by 3” or “one third x.” As long as it matches with the operation that you’re describing, you can use any of the phrases.
4
Section 1.1 — Variables and Expressions
Example
2
Write the variable expression 4(w – 3) as a word expression. Don’t forget:
Solution
You need to do the operations in the correct order. The PEMDAS rule helps you remember this — it tells you that you need to deal with the parentheses first. There’s more on the PEMDAS rule in Lesson 1.1.3.
In this expression w is the variable. The rest of the expression tells you that three is subtracted from w, and the result is multiplied by four. So the expression becomes: “four times the the result of subtracting three from w” or “three subtracted from w, multiplied by four.” They both mean the same thing.
Guided Practice Write a word expression for each of the variable expressions in Exercises 9–14. 9. z – 10 Check it out:
10. 6b
There is more than one possible word expression to describe each of these mathematical phrases. As long as it means the same thing, any variation you use is fine.
11. 12h + 4 12. 5(j + 6) 13. 6(4t) 14. 2c + 4d
When You Ev alua te a p Varia bles ffor or Number s Evalua aluate te,, You Sw Swa ariab Numbers When you’re given the actual numbers that the variables are standing in for you can substitute them into the expression. When you have substituted numbers for all the variables in the expression, you can work out its numerical value. This is called evaluating the expression. Example Don’t forget: Multiplication and division come before addition and subtraction in the order of operations. That’s why you do the multiplication first in this example.
3
Evaluate the expression 6f + 4 when f = 7. Solution
6f + 4 = 6 • 7 + 4 = 42 + 4 = 46
Substitute 7 in place of f Perf or m the m ultiplica tion fir st erfor orm multiplica ultiplication first Then do the addition
Section 1.1 — Variables and Expressions
5
Example
4
Evaluate the expression 11(p – q) when p = 7 and q = 3. Solution
11(p – q) = 11(7 – 3) = 11(4) = 44
Replace p and q with 7 and 3 Do the subtr action subtraction Do the m ultiplica tion to g et the ans wer multiplica ultiplication get answ
Guided Practice Evaluate the expressions in Exercises 15–22, given that j = 5 and t = 7. 16. t + j 15. 6t – 4j 17. 3(t – j)
18. 3t + 7j
19. jt
20. 2(j – t)
21. j + tj – 40
22. 3jt – j + 20
Independent Practice Write word expressions for the variable expressions in Exercises 1–3. 1. 6y + 10 2. 3(p – 6) 3. 18t Now try these: Lesson 1.1.1 additional questions — p430
4. Joe and Bonita went fishing. Joe caught j fish and Bonita caught j + 4 fish. Write a sentence that describes the amount of fish Bonita caught compared with Joe. Evaluate the expressions in Exercises 5–7, given that x = 3. 5. 4x 6. 1 – x 7. x2 8. An orchard uses the expression 0.5w to work out how much money, in dollars, it will make, where w = the number of apples sold. How much money does the orchard make if 250 apples are sold? 9. Given that a = –1, b = 3, and c = 2, evaluate the expression a2 + ab + c2 . 10. A car rental company uses the expression $30 + $0.25m to calculate the daily rental price. The variable m represents the number of miles driven. What is the price of a day’s rental if a car is driven 100 miles?
Round Up Variables are really useful — you can use them to stand in for any unknown numbers in an expression. When you know what the numbers are you can write them in — and then evaluate the expression. In Section 1.2 you’ll see how expressions are the building blocks of equations. 6
Section 1.1 — Variables and Expressions
Lesson
1.1.2
Simplifying Expr essions Expressions
California Standards:
When you’re evaluating an expression that’s made up of many terms, it helps to simplify it as much as possible first. The fewer terms you have to deal with, the less likely you are to make a mistake. This Lesson is about two ways that you can simplify expressions.
Alg ebr a and Functions 1.3 Alge bra umerical numerical Simplify n expr essions b y a ppl ying xpressions by appl pplying pr oper ties of rra a tional proper operties number s (e .g ., identity, umbers (e.g .g., uti ve, inverse, distrib distributi utiv associative, commutative) and justify the process used. Alg ebr a and Functions 1.4 Alge bra minolo gy alge braic terminolo minolog Use alg ebr aic ter (e .g ., vvaria aria ble (e.g .g., ariab le,, equation, ter m, coef ficient, inequality, term, coefficient, expr ession, constant) xpression, cor y. corrr ectl ectly
What it means for you: You’ll learn about some math techniques that will help you to simplify expressions.
Key words: • like terms • distributive property • simplify
Simplifying an Expr ession Mak es It Easier to Solv e Expression Makes Solve In math you’ll come across some very long expressions. The first step toward solving them is to simplify them. The first way to simplify an expression is to collect like terms. Like terms are terms that contain exactly the same variables. These are like terms because they contain no variables — they’re both constants.
h + 2 + 6h – 4
You need to bring like terms together to simplify the expression. When you do that, the plus or minus sign in front of a term stays with that term as it moves. h + 2 + 6h – 4 First swap the positions of the “+ 2” term and the “+ 6h” term.
Check it out: You can move the terms about because of the commutative and associative properties of addition. There’s more about these in Lesson 1.1.5.
h + 6h + 2 – 4
Example
The minus signs in the expression stay with the 5 and the y they were originally connected to. Remember — you can write subtractions as additions: x – 5 + 2y + 9 – y + 2x = x + (–5) + 2y + 9 + (–y) + 2x This often makes it easier to collect like terms.
Now all the terms containing the variable h are together and all the constants are together. Then simplify the grouped terms.
7h – 2
Check it out:
These are like terms because they both contain the same variable, h.
You can’t simplify this any more because there are no longer any like terms.
1
Simplify the expression x – 5 + 2y + 9 – y + 2x. Solution
x – 5 + 2y + 9 – y + 2x. = (x + 2x) + (2y – y) + (–5 + 9) = 3x + y + 4
Collect to gether the lik e ter ms tog like terms Simplify the par entheses parentheses
Guided Practice Simplify the expressions in Exercises 1–6 by collecting like terms. 1. a + 4 + 2a 2. 3r + 6 – 5r 3. c – 2 + 4 • c – 3 4. 4x + 5 – x + 4 – 2x 5. 7 – k + 2k + 3 – k 6. m + 4 – n + m – 2 – n Section 1.1 — Variables and Expressions
7
In an expression like 2(3m + 4) + m the parentheses stop you from collecting like terms. To simplify it any more you first need to remove the parentheses. You can use a property of math called the distributive property to do this.
Use the Distrib uti ve Pr oper ty to R emo ve P ar entheses Distributi utiv Proper operty Remo emov Par arentheses 5
Look at this rectangle. 2
Don’t forget: The formula for the area of a rectangle is: Area = Length × Width
4
You can find its total area in two different ways. • You could find the areas of both of the smaller rectangles and add them. Total area = (5 • 2) + (5 • 4) = 10 + 20 = 30 • Or you could find the total width of the whole rectangle by adding 2 and 4, and then multiply by the height. Total area = 5(2 + 4) = 5(6) = 30 Whichever way you work it out you get the same answer, because both expressions represent the same area. This is an example of the distributive property. So: To use the distributive property you multiply the number outside the parentheses by every term inside the parentheses.
5 • (4 + 2) = (5 • 4) + (5 • 2) = 20 + 10 = 30
Algebraically the property is written as: The Distributive Property a(b + c) = ab + ac
Check it out: No matter what numbers a, b, and c stand for, the rule will always be true.
The Distrib uti ve Pr oper ty and Varia ble Expr essions Distributi utiv Proper operty ariab Expressions Think about what would happen to the problem above if you didn’t know one of the lengths.
5
2
x
Find its total area using the two different methods again. • Adding the area of the two small rectangles: Total area = (5 • 2) + (5 • x) = 10 + 5x • Adding 2 and x to find the width and multiplying by the height: Total area = 5(2 + x) Again, you know that the two expressions have the same value because they represent the same area. So you can say that: 5(2 + x) = 10 + 5x You can use the distributive property to multiply out the parentheses in the first expression to get the second equivalent expression. 8
Section 1.1 — Variables and Expressions
Example
2
Write the expression 5(y – 3) without using parentheses. Solution
5(y – 3) Multipl y both y and –3 b y5 = (5 • y) + (5 • – 3) Multiply by Simplify the m ultiplica tions = 5y – 15 multiplica ultiplications You don’t know what number y is standing in for, so this is the most that you can simplify the expression.
Guided Practice Don’t forget: If you’re multiplying the contents of parentheses by a negative number, remember to include the negative sign when you multiply each term. For example: –3(2 – 1) = (–3 • 2) + (–3 • –1) = –6 + 3 = –3
Write the expressions in Exercises 7–14 in a new form using the distributive property. 7. 6(w + 5) 8. 4(7 – h) 9. –2(d + 2) 10. –3(f – 3) 11. –1(3 – x) 12. –2(–3 – m) 13. y(9 + t) 14. 3(3 + k) + 2(k – 0)
Independent Practice Simplify the expressions in Exercises 1–3 by collecting like terms. 1. 2p + 5 + 2p 2. 3 + 4x – 5 + x 3. 5h + 7 – h – 3 + 2h 4. Keon and Amy are collecting leaves for a project. On a walk Keon finds k leaves and Amy finds 6. Then Keon finds another 4k leaves, and Amy loses 2. Write an expression to describe how many leaves they ended up with, then simplify it fully. Using the distributive property, write the expressions in Exercises 5–7 in a new form. 5. 8(y + 1) 6. 5(w – 9) 7. –h(t – 4) Now try these: Lesson 1.1.2 additional questions — p430
8. Damian makes $7 an hour working at the mall. Last week Damian worked for 12 hours. This week he will work for x hours. a) Write an expression using parentheses to describe how much money he will have made altogether? b) Given that x = 8, evaluate your expression using the distributive property. Simplify the expressions in Exercises 9–12 as far as possible. 9. 2(m + 1) + 1(3 – m) 10. 3(b + 2 – b) 11. 2(h + 4) – 4(h – 1) 12. x(2 + y) + 3(x + y)
Round Up Collecting like terms and using the distributive property are both really useful ways to simplify an expression. Simplifying it will make it easier to evaluate — and that will make solving equations easier later in this Chapter. Section 1.1 — Variables and Expressions
9
Lesson
1.1.3
The Or der of Oper a tions Order Opera
California Standards:
When you have a calculation with more than one operation in it, you need to know what order to do the operations in.
Alg ebr a and Functions 1.2 Alge bra der of corrrect or order Use the cor oper a tions to e valua te opera ev aluate alg ebr aic e xpr essions suc h alge braic expr xpressions such as 3(2 3(2x + 5)2.
What it means for you:
For example, if you evaluate the expression 2 • 3 + 7 by doing “multiply 2 by 3 and add 7,” you’ll get a different answer from someone who does “add 7 to 3 and multiply the sum by 2.” So the order you use really matters.
You’ll learn about the special order to follow when you’re deciding which part of an expression to evaluate first.
There’s a set of rules to follow to make sure that everyone gets the same answer. It’s called the order of operations — and you’ve seen it before in grade 6.
The Or der of Oper a tions is a Set of R ules Order Opera Rules Key words: • parentheses • exponents • PEMDAS
An expression can contain lots of operations. When you evaluate it you need a set of rules to tell you what order to deal with the different bits in. Order of operations — the PEMDAS Rule ()[]{} Parentheses 2
7
First do any operations inside parentheses.
Check it out:
x y
Exponents
Then evaluate any exponents.
Another way to remember the order of operations is using the GEMA rule: G rouping — any symbol that groups things, like parentheses, fraction bars, or brackets. E xponents. M ultiplication and Division – done from left to right. A ddition and Subtraction – done from left to right. Use either PEMDAS or GEMA — whichever one you feel happier with.
× ÷
Multiplication or Division
Next follow any multiplication and division instructions from left to right.
+ –
Addition or Subtraction
Finally follow any addition and subtraction instructions from left to right.
Check it out: You’ll learn about exponents in Section 2.4.
When an expression contains multiplication and division, or addition and subtraction, do first whichever comes first as you read from left to right. 9 • 4 ÷ 3 Multiply first, then divide. 9 ÷ 4 • 3 Divide first, then multiply. 9 + 4 – 3 Add first, then subtract.
9 – 4 + 3 Subtract first, then add.
Following these rules means that there’s only one correct answer. Use the rules each time you do a calculation to make sure you get the right answer. Example
1
What is 8 ÷ 4 • 4 + 3? Solution
Follow the order of operations to decide which operation to do first. 8÷4•4+3 =2•4+3 =8+3 = 11 10
Section 1.1 — Variables and Expressions
Ther e ar e no par entheses or e xponents here are parentheses exponents You do the division first as it Do the di vision fir st division first comes before the multiplication, Then the m ultiplica tion multiplica ultiplication reading from left to right. Finall y do the ad dition to g et the ans wer Finally addition get answ
Guided Practice Evaluate the expressions in Exercises 1–6. 1. 3 – 2 + 6 – 1 2. 6 ÷ 2 + 1 4. 2 + 5 • 10 5. 40 – 10 ÷ 5 • 6
3. 4 + 3 – 2 + 7 6. 5 + 10 ÷ 10
Al ways Deal with P ar entheses Fir st Alw Par arentheses First When a calculation contains parentheses, you should deal with any operations inside them first. You still need to follow the order of operations when you’re dealing with the parts inside the parentheses. Example
2
What is 10 ÷ 2 • (10 + 2)? Don’t forget: Remember to show all your work step by step to make it clear what you’re doing.
Solution
The order of operations says that you should deal with the operations in the parentheses first — that’s the P in PEMDAS. 10 ÷ 2 • (10 + 2) You do the division = 10 ÷ 2 • 12 Do the ad dition in par entheses first here because it addition parentheses = 5 • 12 Then do the di vision division comes first reading from left to right. Finall y do the m ultiplica tion Finally multiplica ultiplication = 60
Guided Practice Evaluate the expressions in Exercises 7–14. 7. 10 – (4 + 3) 8. (18 ÷ 3) + (2 + 3 • 4) 9. 10 ÷ (7 – 5) 10. 41 – (4 + 2 – 3) 11. 10 • (2 + 4) – 3 12. (5 – 7) • (55 ÷ 11) 13. 6 • (8 ÷ 4) + 11 14. 32 + 2 • (16 ÷ 2)
PEMD AS A pplies to Alg ebr a Pr ob lems Too PEMDAS Alge bra Prob oblems The order of operations still applies when you have calculations in algebra that contain a mixture of numbers and variables. Example
3
Simplify the calculation k • (5 + 4) + 16 as far as possible. Solution
k • (5 + 4) + 16 = k • 9 + 16 = 9k + 16
Do the addition within parentheses Then the m ultiplica tion multiplica ultiplication
Section 1.1 — Variables and Expressions
11
Guided Practice Simplify the expressions in Exercises 15–20 as far as possible. 15. 5 + 7 • x 16. 2 + a • 4 – 1 17. 3 • (y – 2) 18. 10 ÷ (3 + 2) – r 19. 20 + (4 • 2) • t 20. p + 5 • (–2 + m)
Independent Practice
Now try these: Lesson 1.1.3 additional questions — p430
1. Alice and Emilio are evaluating the expression 5 + 6 • 4. Their work is shown below. Alice Emilio 5+6•4 5+6•4 = 11 • 4 = 5 + 24 = 44 = 29 Explain who has the right answer. The local muffler replacement shop charges $75 for parts and $25 per hour for labor. 2. Write an expression with parentheses to describe the cost, in dollars, of a replacement if the job takes 4 hours. 3. Use your expression to calculate what the cost of the job would be if it did take 4 hours. Evaluate the expressions in Exercises 4–7. 4. 2 + 32 ÷ 8 – 2 • 5 5. 4 + 7 • 3 6. 7 + 5 • (10 – 6 ÷ 3) 7. 3 • (5 – 3) + (27 ÷ 3) 8. Paul buys 5 books priced at $10 and 3 priced at $15. He also has a coupon for $7 off his purchase. Write an expression with parentheses to show the total cost, after using the coupon, and then simplify it to show how much he spent. 9. Insert parentheses into the expression 15 + 3 – 6 • 4 to make it equal to 48. Simplify the expressions in Exercises 10–12 as far as possible. 10. x – 7 • 2 11. y + x • (4 + 3) – y 12. 6 + (60 – x • 3)
Round Up If you evaluate an expression in a different order from everyone else, you won’t get the right answer. That’s why it’s so important to follow the order of operations. This will feature in almost all the math you do from now on, so you need to know it. Don’t worry though — just use the word PEMDAS or GEMA to help you remember it. 12
Section 1.1 — Variables and Expressions
Lesson
1.1.4
The Identity and In ver se Inv erse Pr oper ties Proper operties
California Standards: Alg ebr a and Functions 1.3 Alge bra umerical numerical Simplify n expr essions b y a ppl ying xpressions by appl pplying pr oper ties of rra a tional proper operties number s (e .g ., identity umbers (e.g .g., identity,, in ver se inv erse se, distributive, associative, commutative)) and justify the pr ocess process used.
When you’re simplifying and evaluating expressions you need to be able to justify your work. To justify it means to use known math properties to explain why each step of your calculation is valid. The math properties describe the ways that numbers and variables in expressions behave — you need to know their names so that you can say which one you’re using for each step.
What it means for you: You’ll learn how to use math properties to show why the steps of your work are reasonable.
The Identity Doesn e the Number Doesn’’ t Chang Change There are two identity properties — one property for addition and one property for multiplication:
Key words: • justify • identity • inverse • reciprocal • multiplicative • additive
The Additive Identity = 0 For any number, a, a + 0 = a. Adding 0 to a number doesn’t change it. For example: 5+0=5 x+0=x This is called the identity property of addition, and 0 is called the additive identity.
The Multiplicative Identity = 1 For any number, a, a • 1 = a. Don’t forget: Writing x is exactly the same as writing 1x or 1 • x.
Multiplying a number by 1 doesn’t change it. For example: 1•7=7 1•x=x This is called the identity property of multiplication, and 1 is called the multiplicative identity.
Guided Practice 1. What do you get by multiplying 6 by the multiplicative identity? 2. What do you get by adding the additive identity to 3y? Complete the expressions in Exercises 3–6. 3. x + __ = x 4. __ + 0 = h 5. k • __ = k
6. 1 • __ = t Section 1.1 — Variables and Expressions
13
The In ver se Chang es the Number to the Identity Inv erse Changes There are two inverse properties — one for addition and one for multiplication. Different numbers have different additive inverses and different multiplicative inverses.
The Ad diti ve In ver se Ad ds to Gi ve 0 Additi ditiv Inv erse Adds Giv The additive inverse is what you add to a number to get 0.
Don’t forget: Remember — adding a negative number can be rewritten as a subtraction: 2 + –2 = 2 – 2 = 0
The additive inverse of 2 is –2
2 + –2 = 0
The additive inverse of –3 is 3
–3 + 3 = 0
The additive inverse of
1 4
1 4
is – 1
4
+ –1 = 0 4
The Additive Inverse of a is –a. For any number, a, a + –a = 0.
Guided Practice Give the additive inverses of the numbers in Exercises 7–12. 8. 19 7. 6 9. –5 11.
1 7
10. –165 12.
–
2 3
The Multiplica ti ve In ver se Multiplies to Gi ve 1 Multiplicati tiv Inv erse Giv Don’t forget: A multiplicative inverse is sometimes called a reciprocal.
The multiplicative inverse is what you multiply a number by to get 1. So, a number’s multiplicative inverse is one divided by the number. The multiplicative inverse of 2 is 1 ÷ 2 =
1 . 2 1
The multiplicative inverse of 7 is 1 ÷ 7 = 7 . 1 7 •1 7 1 2 •1 2 To check, 2 • = = =1. = = 1 and 7 • = 7 7 7 2 2 2
14
Section 1.1 — Variables and Expressions
The Multiplicative Inverse of a is For any nonzero number, a, a •
1 a
1 . a
= 1.
Guided Practice Give the multiplicative inverses of the numbers in Exercises 13–16. 14. 10 13. 2 16. –5
15. –4
Fractions Ha ve Multiplica ti ve In ver ses Too Hav Multiplicati tiv Inv erses When you multiply two fractions together, you multiply their numerators and their denominators separately. 1 3 1• 3 3 = For example : • = 2 4 2•4 8 If you multiply a fraction by its multiplicative inverse, the product will be 1 — because that’s the definition of a multiplicative inverse. Don’t forget: Any fraction where the numerator is the same as the denominator is equal to 1. So, 12 9 4 1 = = = =1 12 9 4 1
For a fraction to equal 1, the numerator and denominator must be the same. So when a fraction is multiplied by its inverse, the product of the numerators must be the same as the product of the denominators. For example:
3 4 3 • 4 12 • = = =1 4 3 4 • 3 12
You can say that for any two non-zero numbers a and b, So, the multiplicative inverse of
a b
is
a b
•
b a
=
ab ab
= 1.
b . a
The multiplicative inverse, or reciprocal, of a fraction is just the fraction turned upside down. Example
1
Give the multiplicative inverse of
1 . 4
Solution
Don’t forget: Any number divided by one is equal to itself. So any whole number can be written as a fraction by putting it over 1. So 6 =
6 4 and 4 = . 1 1
4 1 , or 4, is the multiplicative inverse of . 1 4
To check your answer multiply it by
1 . 4
1 4 1• 4 4 = =1. • = 4 1 4 •1 4 1
So 4 is the multiplicative inverse of 4 .
Section 1.1 — Variables and Expressions
15
Guided Practice Give the multiplicative inverses of the fractions in Exercises 17–20. 17.
2 5
19.
–
18. 3 4
20.
1 10 1 – 2
You Can Use Ma th Pr oper ties to JJustify ustify Your Wor k Math Proper operties ork To justify your work you need to use known math properties to explain why each step of your calculation is valid. Check it out: To justify a step of your work you can write beside it which of the known math properties you have used.
Example
2
Simplify the expression 4(2 –
1 x). 4
Justify your work.
Solution 1
4(2 – 4 x) =4•2–4• = 8 – 1x =8–x
1 4
x
The distrib uti ve pr oper ty distributi utiv proper operty ver se pr oper ty of m ultiplica tion inv erse proper operty multiplica ultiplication The in The identity pr oper ty of m ultiplica tion proper operty multiplica ultiplication
Guided Practice Simplify the expressions in Exercises 21–24. Justify your work. 21. m • 1 + 6 1
23. 3 (9 + 3f)
22. d + 0 – d + 9 24. –5(2 – 4 •
1 a) 4
+ 10
Independent Practice Complete the expressions in Exercises 1–4. 1. 1 • __ = 5 2. __ + 0 = –2 3. 2.5 • __ = 2.5 4. –0.5 + __ = –0.5 Now try these: Lesson 1.1.4 additional questions — p431
Give the additive and multiplicative inverses of the numbers in Exercises 5–8. 6. –7 5. 6 7.
8. – 2
5 7
3
Simplify the expressions in Exercises 9–12. Justify your work. 1 9. a + a – a 10. • 4 • d 1 5
11. 5( + 2 – n)
4
1 2
12. 2( x + 4 + 0) – 1(5 – 5 + 7)
Round Up The identity property and the inverse property are two math properties you’ll need to use in justifying your work. Justifying your work is explaining how you know that each step is right. In the next Lesson you’ll cover two more properties that can be used in justifying your work. 16
Section 1.1 — Variables and Expressions
Lesson
1.1.5
The Associa ti ve and Associati tiv Comm uta ti ve Pr oper ties Commuta utati tiv Proper operties
California Standards: Alg ebr a and Functions 1.3 Alge bra umerical numerical Simplify n expr essions b y a ppl ying xpressions by appl pplying pr oper ties of rra a tional proper operties number s (e .g ., identity umbers (e.g .g., identity,, in ver se uti ve , inv erse se,, distrib distributi utiv associa ti ve, comm uta ti ve) associati tiv commuta utati tiv and justify the pr ocess process used used.
There are two more properties you need to know about to help simplify and evaluate expressions. They’re the associative and commutative properties. They allow you to be a little more flexible about the order you do calculations in. You used these properties already in earlier grades. It’s important to know their names and to practice using them to justify your work.
What it means for you: You’ll learn about some more math properties that will help you to justify your work.
The Associa ti ve Pr oper ties Associati tiv Proper operties If you change the way that you group numbers and variables in a multiplication or addition expression, you won’t change the answer.
Key words:
7 + (4 + 2) = (7 + 4) + 2 = 13 4 • (y • 5) = (4 • y) • 5 = 20y
• associative property • commutative property • justify
The numbers and variables don’t move — only the parentheses do.
These are the associative properties of addition and multiplication. In math language they are: The Associative Properties Addition: (a + b) + c = a + (b + c) Multiplication:
(ab)c = a(bc)
Sometimes changing the grouping in an expression using the associative property allows you to simplify it. Example
1
Simplify the expression (h + 12) + 13 using the associative property. Solution
(h + 12) + 13 ti ve pr oper ty of ad dition = h + (12 + 13) The associa associati tiv proper operty addition Do the ad dition = h + 25 addition Example
2
Simplify the expression 15(10y) using the associative property. Solution
15 • (10 • y) ti ve pr oper ty of m ultiplica tion = (15 • 10) • y The associa associati tiv proper operty multiplica ultiplication Do the m ultiplica tion = 150y multiplica ultiplication Section 1.1 — Variables and Expressions
17
Guided Practice Simplify the expressions in Exercises 1–6. Use the associative property. 1. 10 + (15 + k) 2. 5(7d) 3. (x + 5) + 7 4. –3(4f) 5. (y + 13) + (20 + m) 6. 0.5(3p)
The Comm uta ti ve Pr oper ties Commuta utati tiv Proper operties When you’re adding numbers together it doesn’t matter what order you add them in — the answer is always the same. For example: 10 + 14 = 24
and
14 + 10 = 24
Also, when you’re multiplying numbers it doesn’t matter what order you multiply them in — the answer is always the same. For example: 4 • x • 2 = 8x
and
2 • 4 • x = 8x
The numbers and variables move around, but the answer doesn’t change — these are the commutative properties. Algebraically they’re written as: The Commutative Properties
Check it out: You can’t use the associative or commutative properties on a subtraction or a division problem. But you could use the inverse property to change it into an addition or multiplication problem first. Subtraction is the same as adding a negative number, and division is the same as multiplying by a fraction. x – y = x + –y So 1 . y When they’re expressed in this way, you can use the associative and commutative properties. and
x÷y=x•
Example
Addition:
a+b=b+a
Multiplication:
ab = ba
3
Simplify the expression 18v + 9 + 2v + 4. Justify your work. Solution
18v + 9 + 2v + 4 = 18v + 2v + 9 + 4 = (18v + 2v) + (9 + 4) = 20v + 13 Example
The comm uta ti ve pr oper ty of ad dition commuta utati tiv proper operty addition The associa ti ve pr oper ty of ad dition associati tiv proper operty addition Do the ad ditions additions
4
Simplify the expression 4 • n • 9 using the commutative property. Solution
4•n•9 =4•9•n = (4 • 9) • n = 36n
The comm uta ti ve pr oper ty of m ultiplica tion commuta utati tiv proper operty multiplica ultiplication The associa ti ve pr oper ty of m ultiplica tion associati tiv proper operty multiplica ultiplication Do the m ultiplica tions multiplica ultiplications
Guided Practice Simplify the expressions in Exercises 7–12. 7. 7 + j + 15 8. 11 • x • 20 9. 3 + 4t + 7 + t 10. 3 • –y • 4 11. 5 + q + 3 + –q 12. –2 • f • –3 18
Section 1.1 — Variables and Expressions
You Can Use the Pr oper ties to JJustify ustify Your Wor k Proper operties ork Check it out: To justify your work means to use a math property to explain why each step of your calculation is valid.
Don’t forget:
You can use all the properties together to justify the work you do when solving a math problem. Example
5
Simplify the expression 3(x + 5 + 2x). Justify your work.
The math properties that you’ve seen in the last two Lessons are: • the distributive property and the • associative, • commutative, • identity, and • inverse properties of multiplication and addition.
Solution
And you’ve also seen how to collect like terms.
Simplify the expression (x • y) •
3(x + 5 + 2x) = 3x + 15 + 6x = 3x + 6x + 15 = 9x + 15 Example
The distrib uti ve pr oper ty distributi utiv proper operty The comm uta ti v e pr oper ty of ad dition commuta utati tiv proper operty addition Collect lik e ter ms like terms
6
Solution
(x • y) • = x • (y • =x•1 =x
1 y 1 ) y
1 y
. Justify your work.
The associa ti ve pr oper ty of m ultiplica tio n associati tiv proper operty multiplica ultiplicatio tion The m ultiplica ti ve in ver se pr oper ty multiplica ultiplicati tiv inv erse proper operty The identity pr oper ty of m ultiplica tion proper operty multiplica ultiplication
Guided Practice Simplify the expressions in Exercises 13–16. Justify your work. 1
13. 2( 2 h)
14. (c + –2) + 2
15. 5(1 + t + 3 + 2t)
16. p • 5 •
1 p
•2
Independent Practice
Now try these: Lesson 1.1.5 additional questions — p431
Simplify the expressions below using the associative properties. 1. 55 + (7 + z) 2. 6 • (10 • t) 3. –3(7k) 4. (–22 + q) + (q + 30) Simplify the expressions below. Use the commutative properties. 6. 9 • y • 4 5. 7 + f + 3 + f 8. –3 • c • 4 • –h 7. 2 + a + 18 + b Simplify the expressions in Exercises 9–12. Justify your work. 1 10. (r • 2 ) • 2 9. –5 + m + 5 11. 2(x + 5 + –x)
1
12. ( p • 3) • p • 2(4 + p – 4)
Round Up The associative and commutative properties are two more math properties. They’re all important tools to use when you’re simplifying expressions. By saying which property you are using in each step, you can justify your work. Section 1.1 — Variables and Expressions
19
Section 1.2 introduction — an exploration into:
Solving Equa tions Equations You can use algebra tiles to model algebra equations. You can also use them to model how you solve equations to find the value of x. The tiles on the right will be used to represent algebra equations:
= +1
= –1
=x
Here is the algebra equation 2x + 3 = –5 modeled using algebra tiles: This vertical line represents the equals sign.
To solve an equation, you need to get one green “x” tile by itself on one side of the equation, and only red and yellow tiles on the other. This way you’ll know the value of one green “x” tile. You’ll often need to use the idea of “zero pairs.” A yellow and a red tile together make zero. This is called a zero pair. 1 + (–1) = 0 Example Use algebra tiles to model and solve the equation x + 2 = –3. Solution
x + 2 = –3
x + 2 – 2 = –3 – 2
The answer is x = –5. The zero pairs are removed.
Add two red tiles to each side. zero pairs
Example Use algebra tiles to model and solve the equation 2x = 6. Solution
2x 6 = 2 2
2x = 6 Divide both sides into two equal groups (because there are two “x” tiles).
The answer is x = 3.
Remove one group from each side.
Exercises 1. Use algebra tiles to model these equations. a. x + 3 = 4 b. x – 5 = 3 c. 3x = 9
d. 4x = –8
2. Use algebra tiles to solve the equations above.
Round Up When you solve an equation, the aim is always to get the variable on its own, on one side of the equation. In this Exploration, the variable was the green tile “x.” To get it on its own, you have to do exactly the same to both sides of the equation — or else the equation won’t stay balanced. 20
Section 1.2 Explor a tion — Solving Equations Explora
Lesson
Section 1.2
1.2.1
Writing Expr essions Expressions
California Standards:
You can use both a word expression and a math expression to describe the same situation. In Section 1.1 you practiced changing numeric and variable expressions into word expressions. This Lesson is all about doing the reverse — changing word expressions into math expressions.
Alg ebr a and Functions 1.1 Alge bra ariab Use vvaria aria bles and a ppr opria te oper a tions to ppropria opriate opera write an e xpr ession expr xpression ession, an equation, an inequality, or a system of equations or epr esents a inequalities tha thatt rre presents verbal description (e.g., three less than a number, half as large as area A).
What it means for you: You’ll learn how to change word descriptions into math expressions.
Key words: • • • •
addition subtraction product quotient
Varia ble Expr essions Describe Wor d Expr essions ariab Expressions ord Expressions To write an expression to represent a word sentence you need to figure out what the word sentence actually means. Example
1
Write a variable expression to describe the sentence “A number, x, is increased by five.” Solution
The phrase “a number, x” is telling you that x is the variable being used. The words “increased by” are telling you that something is being added on to the variable x. In this case it is the number five. So the sentence “A number, x, is increased by five” translates as x + 5.
Don’t forget: There are lots of different phrases that describe the four operations. You could come across any of them, so you need to remember which of the operations each phrase refers to. For a reminder, see Lesson 1.1.1.
Example
2
Write a variable expression to describe the sentence “Nine is multiplied by a number, k.” Solution
The operation phrase being used is “multiplied by.” The rest of the sentence tells you that it is the number nine and the variable k that are being multiplied together. So “Nine is multiplied by a number, k” translates as 9 • k or 9k.
Guided Practice Write variable expressions to describe the phrases in Exercises 1–5. 1. Six more than a number, h. 2. Seven is decreased by a number, m. 3. A number, g, divided by 11. 4. The product of a number, w, and 10. 5. A number, k, divided into four equal parts.
Section 1.2 — Equations
21
You Need to Sor tant Inf or ma tion Sortt Out the Impor Important Infor orma mation You’ll often need to write a math expression as the first step toward solving a word problem. That might include choosing variables as well as working out what operations the words are describing. Example
3
Carla and Bob have been making buttons to sell at a fund-raiser. Carla made four more than Bob. Write an expression to describe how many buttons they made between them. Solution
First you need to work out what the expression you have to write must describe. In this case the expression must describe the total number of buttons made by Carla and Bob. See if there is an unknown number in the question: you don’t know how many buttons Bob made. You don’t know how many Carla made either, but you can say how many she made compared with Bob, so you only need one variable. Assign a letter or symbol to the unknown number: let b = the number of buttons Bob made. Don’t forget: To simplify this expression you can collect like terms, as you did in Lesson 1.1.2. If you collect the two b terms together you get 2b.
Then you need to identify any operation phrases: “more than” is an addition phrase. Carla made four more buttons than Bob, so she made b + 4 buttons. Which means that together they made b + 4 + b buttons. This expression can be simplified to 2b + 4.
Guided Practice Write variable expressions to describe the sentences in Exercises 6–9. Use x as the variable in each case, and say what it represents. 6. A rectangle has a length of 2 inches. What is its area? 7. Jenny has five fewer apples than Jamal. How many apples does Jenny have in total? 8. The student council is selling fruit juice at the prom for $0.75 a glass. How much money will they take? 9. A gym charges $10 per month membership plus $3 per visit. What is the cost of using the gym for a month?
Some Expr essions Describe Mor e than One Oper a tion Expressions More Opera You can also translate sentences with multiple operations in the same way. You just need to spot all the separate operations and work out what order to write them in.
22
Section 1.2 — Equations
Example
4
Write a variable expression to describe the phrase “ten decreased by the product of a number, y, and two.” Solution
In this question the phrase contains two different operation phrases, so you need to work out which operation is carried out first. The two operations here are “decreased by,” which is a subtraction phrase, and “product,” which is a multiplication phrase. You’re told to subtract the product from 10. So you need to work out the product first — this is the product of y and 2, which is 2y. Now you have to subtract this product, 2y, from 10. So the phrase “ten decreased by the product of a number, y, and 2.” translates as 10 – 2y.
Guided Practice Write variable expressions to describe the phrases in Exercises 10–14. Don’t forget: You might need to include parentheses in your expression to show which operation needs to be done first.
10. Five more than twice a number, q. 11. Sixteen divided by the sum of a number, m, and 7. 12. Twenty decreased by a quarter of a number, j. 13. The product of 7 and six less than a number, t. 14. The product of a number, k, and the sum of 5 and a number, x.
Independent Practice
Now try these: Lesson 1.2.1 additional questions — p431
Write variable expressions to describe the phrases in Exercises 1–5. 1. The product of six and a number, h. 2. A number, y, decreased by eleven. 3. A fifteenth of a number, p. 4. Nine more than twice a number, w. 5. Sixteen increased by the product of a number, k, and three. 6. A pen costs half as much as a ruler. Write an expression to describe how much the pen costs, using r as the cost of the ruler. 7. Peter has three fewer cards than Neva. Write an expression to describe how many cards they have together, using c as the number of cards Neva has.
Round Up Changing word expressions into algebra expressions is all about spotting the operation phrases and working out what order the operations need to be written in. Writing expressions is the first step toward writing equations — a skill that you’ll use when solving problems later in this Section.
Section 1.2 — Equations
23
Lesson
1.2.2
Equa tions Equations
California Standards:
In Lesson 1.1.2 and Lesson 1.2.1 you learned how to write expressions. Writing equations takes writing expressions one step further — equations are made up of two expressions joined by an equals sign.
Alg ebr a and Functions 1.4 Alge bra Use alg ebr aic ter minolo gy alge braic terminolo minolog e.g ., variable, equa tion (e .g., equation tion, term, coefficient, inequality, expr ession xpression ession, constant) cor y. corrr ectl ectly
What it means for you: You’ll learn what an equation is, and how it’s different from an expression.
An Equa tion Has an Equals Sign Equation An equation is made up of two expressions joined together by an equals sign. The equals sign is really important — it tells you that the expressions on each side of the equation have exactly the same value.
Key words:
5x + 8 = 10x – 12
• equation • expression • formula
Expression 1... ...is equal to... ...Expression 2.
Numeric Equa tions Contain Onl y Number s Equations Only Numbers Numeric equations contain only numbers and operations. For example, 3 + 2 = 5 and (6 • 4) + 3 = 31 – 4 are both numeric equations. If both sides of the equation do have the same value, then the equation is said to be true, or balanced. Example
1
Prove that (10 • 4) + 4 = (8 • 11) ÷ 2 is a true equation. Solution
Don’t forget: The order of operations is Parentheses, Exponents, Multiplication and Division, then Addition and Subtraction — PEMDAS.
To show that this is a true equation, you need to evaluate both sides. Treat them as two expressions, and evaluate them both according to the order of operations. Fir st simplify the par entheses First parentheses (10 • 4) + 4 = (8 • 11) ÷ 2 Then complete the w or k 40 + 4 = 88 ÷ 2 wor ork 44 = 44 Both sides equal 44, so the equation is true.
The numbers and operations are different on the left-hand and right-hand sides of the equation. But the value of both sides is the same.
24
Section 1.2 — Equations
Guided Practice Prove that the equations in Exercises 1–6 are true by evaluating both sides. 2. 4 • 5 = 60 ÷ 3 1. 4 + 5 = 9 + 0 3. 4 + 2 • 3 = 3 • 2 + 4
4. 6 – 4 ÷ 2 = 12 ÷ 2 – 2
5. 8 + 6 • 3 = 2(10 + 3)
6. 20 ÷ 4 – 5 = 14 ÷ 2 – 7
Varia ble Equa tions Contain Number s and Varia bles ariab Equations Numbers ariab Variable equations contain variables as well as numbers and operations. There may be variables on either or both sides of the equation. For example, 2x + 2 = 6 and 3x = 2y are both variable equations.
Don’t forget: x is a variable — that means that it could stand for any number. But this equation is only true when it stands for 2. So x = 2 is called a solution of the equation.
The same rules that apply to numeric equations also apply to variable equations. The expressions that make up the two sides of the equation still have to be equal in value. The two equations above are both true if x = 2 and y = 3. 2x + 2 = 6 2•2+2=6 6=6
3x = 2y 3•2=2•3 6=6
Writing Equa tions In volv es Writing Expr essions Equations Inv olves Expressions To write an equation you write two expressions that have the same value and join them with an equals sign. One of the expressions will often just be a number. Example
2
Write an equation to describe the sentence “Eight increased by the product of a number, k, and two is equal to twenty-four.” Solution
The phrase “is equal to” represents the equals sign. It also separates the two expressions that make up the two sides of the equation. One expression is “Eight increased by the product of a number, k, and two.” This turns into the expression 8 + 2k. The other expression is just a number, 24. So the sentence “Eight increased by the product of a number, k, and two is equal to twenty-four” turns into the equation 8 + 2k = 24.
Section 1.2 — Equations
25
Guided Practice Write an equation to describe each of the sentences in Exercises 7–11. 7. Five less than a sixth of m is equal to 40. 8. Five more than the product of six and d is equal to ten. 9. Four increased by the product of three and t is equal to 40. 10. Nine less than the product of six and r is equal to 11. 11. Two times y is equal to y divided by four.
To Write an Equa tion, Identify the K ey Inf or ma tion Equation, Ke Infor orma mation When math problems are described using words, you’ll often be given lots of extra information as part of the question. You need to be able to extract the important information and use it to set up an equation — just like you set up an expression. Example Check it out: It doesn’t matter exactly what Sarah is selling here, or what she buys with her profit. The important thing is how the numbers in the situation relate to each other — and that’s what the equation is describing.
3
Sarah has been selling lemonade. The lemonade cost her $9 to make, and she sold each glass for $0.75. She made $20 profit, which she is going to use to buy a necklace. Write an equation to describe this information. Use x to represent the number of glasses she sold. Solution
Sarah made $20 profit. So the price of one glass ($0.75) multiplied by the number of glasses she sold (x), minus the amount the lemonade cost her to make ($9), is equal to 20. So you can write 0.75x – 9 = 20.
Guided Practice Write an equation to describe each of the situations in Exercises 12–15. 12. Javier spent $20 at the gas station. He bought a drink for $3 and spent the rest, $d, on gas. 13. Jane is wrapping a parcel. She needs 15 feet of string to tie it up. A roll of string is p feet long. She uses exactly three rolls. 14. Sam takes 12 sheets of paper to write an essay on. The essay is 2h pages long. He has k spare sheets left to put back. 15. A telephone company charges $0.05 a minute for local calls and $0.10 a minute for long-distance calls. Asuncion makes one local and one long-distance call. Each call is y minutes long. Her calls cost a total of $4.
26
Section 1.2 — Equations
AF or mula is an Equa tion Tha tes a R ule For orm Equation hatt Sta States Rule A formula is a specific type of equation that sets out a rule for you. It explains how some variables are related to each other. For example: Area of a rectangle.
Example
A = l•w
Product of length and width.
4
Write a formula for the perimeter of a square. Solution
Check it out: You can use any letters or symbols you like for the variables. But make sure you use the same ones all through your work, and remember to say what they stand for.
To calculate the perimeter of a square you multiply its side length by 4. Choose variables to use: Let P = perimeter of the square, and let s = side length. So the formula becomes P = 4 • s. The formula shows the relationship between a square’s side length and perimeter. The formula works for any square at all — if you are given the value of one of the variables you can always calculate the other.
Guided Practice Use the formula to calculate the missing values in Exercises 16–18. 16. Rectangle area = length • width length = 4 cm, width = 0.5 cm 17. Speed = distance ÷ time
distance = 8 miles, time = 2 h
18. Length in cm = 2.54 • (length in in.)
length in in. = 10
Independent Practice 1. Which of a) and b) is an expression? Which is an equation? How do you know? a) 2w – 6 = 21 b) 5c + 3 Now try these: Lesson 1.2.2 additional questions — p432
2. Given that Distance = Speed • Time, calculate the distance traveled when a car goes 55 mi/h for 8 hours. Write an equation to describe each of the sentences in Exercises 3–5. 3. Six more than x is equal to four. 4. The product of h and two is equal to 40 decreased by h. 5. Ten increased by the result of dividing five by t is equal to nine. 6. Mike earned $100 working for h hours in a restaurant. He earns $10 an hour, and received $30 in tips. Write an equation using this information.
Round Up The equals sign in an equation is very important — it tells you that both sides of the equation have exactly the same value. In the next Lesson you’ll see how to solve an equation that you’ve written. Section 1.2 — Equations
27
Lesson
1.2.3
Solving One-Ste p Equa tions One-Step Equations
CA Standard California Standards: covered:
Solving an equation containing a variable means finding the value of the variable. It’s all about changing the equation around to get the variable on its own.
Alg ebr a and Functions 4.1 Alge bra Solve Solv e two-step linear equa tions and inequalities in equations one vvaria aria ble o ver the ariab ov ra tional n umber s, interpret number umbers the solution or solutions in the context from which they What it means arose, and verify thefor you: You’ll learn whatofanthe equation reasonableness results. is, and how it’s different from an expression. What it means for you: You’ll learn how to solve an equation to find out the value of an unknown variable.
Do the Same to Both Sides and Equa tions Sta y Tr ue Equations Stay The equals sign in an equation tells you that the two sides of the equation are of exactly equal value. So if you do the same thing to both sides of the equation, like add five or take away three, they will still have the same value as each other. 4+6=9+1 Add 5 to both sides. 4+6+5=9+1+5 Then simplify.
Key words:
15 = 15
• solve • isolate • inverse
All three are balanced equations.
You Can Use This to Find the Value of a Varia ble ariab To get a variable in an equation on its own you need to do the inverse operation to the operation that has already been performed on it.
Don’t forget: This is using the inverse properties of addition and multiplication. a + –a = 0 a•
1 a
=1
You came across these in Lesson 1.1.4.
• If a variable has had a number added to it, subtract the same number from both sides. + Æ – • If a variable has had a number subtracted from it, add the same number to both sides. – Æ + • If a variable has been multiplied by a number, divide both sides by the same number. × Æ ÷ • If a variable has been divided by a number, multiply both sides by the same number. ÷ Æ × For example:
y – 5 = 33 y – 5 + 5 = 33 + 5
y has had 5 subtracted from it, so add 5 to both sides.
y + 0 = 38 y = 38 28
Section 1.2 — Equations
You’ve got the variable alone on one side of the equation, so now you know its value.
Rever se Ad dition b y Subtr acting erse Addition by Subtracting When a variable has had something added to it, you can undo the addition using subtraction. Example
1
Find the value of x when x + 15 = 45. Solution
x + 15 = 45 x + 15 – 15 = 45 – 15 x = 30
Subtr act 15 fr om both sides Subtract from Simplify to find x
Rever se Subtr action b y Ad ding erse Subtraction by Adding When a variable has had something taken away from it, you can undo the subtraction using addition. Example
2
Find the value of k when k – 17 = 10. Solution
k – 17 = 10 k – 17 + 17 = 10 + 17 k = 27
Example Check it out: Here the variable is on the right-hand side of the equals sign. But that doesn’t matter — as long as it’s on its own. You’ve still found its value by isolating it.
Add 17 to both sides Simplify to find k
3
Find the value of g when –10 = g – 9. Solution
–10 = g – 9 –10 + 9 = g – 9 + 9 –1 = g
Add 9 to both sides Simplify to find g
Guided Practice Find the value of the variable in Exercises 1–8. 1. x – 7 = 14 2. 70 = t + 41 3. f + 13 = 9 4. g – 3 = –54 5. y – 14 = 30 6. 22 = 14 + d 7. 4.5 = 9 + v 8. –6 = b – 4
Section 1.2 — Equations
29
Rever se Multiplica tion b y Di viding erse Multiplication by Dividing When a variable in an equation has been multiplied by a number, you can undo the multiplication by dividing both sides of the equation by the same number. y has been multiplied by 2, so start by dividing both sides by 2.
2y = 18 2y ÷ 2 = 18 ÷ 2 Then simplify. y=9
Example
4
Find the value of b when 20b = 100. Solution
20b = 100 20b ÷ 20 = 100 ÷ 20 b =5
Di vide both sides b y 20 Divide by Simplify to find b
Rever se Di vision b y Multipl ying erse Division by Multiplying
Check it out: Fractions represent divisions. So 1 ÷ 2 and
1 2
When a variable in an equation has been divided by a number, you can undo the division by multiplying both sides of the equation by the same number. d 2
mean
exactly the same thing. You can write either.
d 2
= 50
d has been divided by 2, so start by multiplying both sides by 2.
• 2 = 50 • 2
Then
simplify.
d = 100 Example
5
Find the value of t when t ÷ 4 = 6. Solution
t÷4 =6 t÷4•4 =6•4 t = 24
30
Section 1.2 — Equations
Multipl y both sides b y4 Multiply by Simplify to find t
Guided Practice Find the value of the variables in Exercises 9–16. 9. 3k = 18
10. b ÷ 3 = 4
11. h ÷ 5 = –3 13. q ÷ 8 =
1 2
15. d ÷ –2 = –4
12. –9y = 99 14. 10t = –55 16. 240 = 8m
Independent Practice 1. The Sears Tower in Chicago is 1451 feet tall, which is 405 feet taller than the Chrysler Building in New York. Use the equation C + 405 = 1451 to find the height of the Chrysler Building. Find the value of the variable in Exercises 2–7. 3. c + 10 = –27 2. k + 7 = 10 5. 70 = 5 + b 4. s + 4 = –7 7. 32 = 11 + a 6. h + 0 = 14 8. The Holland Tunnel in New York is 342 feet longer than the 8216-foot-long Lincoln Tunnel. Use the equation H – 342 = 8216 to find the length of the Holland Tunnel. Find the value of the variable in Exercises 9–14. 9. x – 7 = 13 10. 41 = m – 35 11. p – 13 = –82 12. t – 27 = 37 13. 100 = g – 18 14. –7 = y – 2 15. Marlon buys a sweater for $28 that has $17 off its usual price in a sale. Write an equation to describe the cost of the sweater in the sale compared with its usual price. Then solve the equation to find the usual price of the sweater. Now try these: Lesson 1.2.3 additional questions — p432
Find the value of the variable in Exercises 16–21. 16. 5c = 80 17. v ÷ 7 = 3 18. 22x = –374 19. h ÷ –2 = 4 20. –3k = –24 21. –27 = f ÷ 3 22. The tallest geyser in Yellowstone Park is the Steamboat Geyser. Reaching a height of 380 feet, it is twice as high as the Old Faithful Geyser. Use the equation 2F = 380 to find the height reached by the Old Faithful Geyser.
Round Up Solving an equation tells you the value of the unknown number — the variable. To solve an equation all you need to do is the reverse of what’s already been done to the variable. That way you can isolate the variable. Just remember that you need to do the same thing to both sides. That’s what keeps the equation balanced. Section 1.2 — Equations
31
Lesson
1.2.4
Solving Two-Ste p Equa tions -Step Equations
California Standards:
When you have an equation with two operations in it, you need to do two inverse operations to isolate the variable. But other than that, the process is just the same as for solving a one-step equation.
Alg ebr a and Functions 4.1 Alge bra Solve two-ste o-step Solv e tw o-ste p linear equa tions and inequalities in equations one vvaria aria ble o ver the ariab ov ra tional n umber s , interpret number umbers the solution or solutions in the context from which they arose, and verify the reasonableness of the results..
Two-Ste p Equa tions Ha ve Two Oper a tions o-Step Equations Hav Opera A two-step equation is one that involves two different operations.
What it means for you: You’ll learn how to solve an equation that involves more than one operation to find out the value of an unknown variable.
Key words: • solve • isolate • inverse
7 • x + 3 = 17 second operation
first operation
You need to perform two inverse operations to isolate the variable. It’s easiest to undo the operations in the opposite order to the way that they were done. It’s like taking off your shoes and socks. You normally put on your socks first and then your shoes. But when you’re removing them you go in the reverse order — you take your shoes off first, and then your socks. Here x is first multiplied by seven, and then the product has three added to it.
Don’t forget: You need to remember to think of PEMDAS or GEMA. That way you’ll know what order the operations have been done in — and what order to go in to reverse them.
So to isolate the variable, first subtract three from both sides... ...and then divide both sides by seven.
Example
7x + 3 = 17 7x + 3 – 3 = 17 – 3 7x = 14 7x ÷ 7 = 14 ÷ 7 x=2
1
Find the value of d when 4d + 6 = 38. Solution
In the equation 4d + 6 = 38 the variable d has first been multiplied by 4, and 6 has then been added to the product. So to isolate the variable you must first subtract 6 from both sides, and then divide both sides by 4. 4d + 6 = 38 4d + 6 – 6 = 38 – 6 4d = 32 4d ÷ 4 = 32 ÷ 4 d= 8
32
Section 1.2 — Equations
Subtr act 6 fr om both sides Subtract from Di vide both sides b y 4 Divide by
Example
2
Find the value of h when 3h – 11 = 25. Solution
In the equation 3h – 11 = 25 the variable h has first been multiplied by 3, and 11 has then been subtracted from the product. So to isolate the variable you must first add 11 to both sides, and then divide them both by 3. 3h – 11 = 25 3h – 11 + 11 = 25 + 11 3h = 36 3h ÷ 3 = 36 ÷ 3 h = 12
Add 11 to both sides Di vide both sides b y3 Divide by
Guided Practice Find the value of the variables in Exercises 1–6. 1. 2x + 8 = 12
2. 10 + 5y = 25
3. 5t – 6 = 34
4. 7f – 19 = 30
5. 60 = 8b + 12
6. 34 = 4p – 10
Follo w the Same Pr ocedur e with All the Oper a tions ollow Procedur ocedure Opera You can use this method for any two-step equation. Just perform the inverse of the two operations in the opposite order to the order in which they were done. Sometimes the order in which the operations are performed is less obvious, and you’ll need to think more carefully about it. Example
3
Find the value of r when r ÷ 4 – 6 = 13. Solution
Check it out: This equation could have been written as (r ÷ 4) – 6 = 13. But division takes priority over subtraction, so the parentheses aren’t needed.
In the equation r ÷ 4 – 6 = 13, the variable r has first been divided by 4, and 6 has then been subtracted from the quotient. So to isolate the variable you must first add 6 to both sides, and then multiply both sides by 4. r ÷ 4 – 6 = 13 Add 6 to both sides r ÷ 4 – 6 + 6 = 13 + 6 r ÷ 4 = 19 Multipl y both sides b y4 r ÷ 4 • 4 = 19 • 4 Multiply by r = 76
Section 1.2 — Equations
33
Example
4
Find the value of v when (v + 2) ÷ 7 = 3. Check it out: The parentheses are needed here because addition doesn’t take priority over division. If the parentheses were left out, this equation would have a different solution — because of the order of operations rules.
Solution
In the equation (v + 2) ÷ 7 = 3, the variable v and 2 have first been added together, and then their sum has been divided by 7. So to isolate the variable you must first multiply both sides by 7, and then subtract 2 from both sides. (v + 2) ÷ 7 = 3 (v + 2) ÷ 7 • 7 = 3 • 7 v + 2 = 21 v + 2 – 2 = 21 – 2 v = 19
Multiply both sides by 7 Subtr act 2 fr om both sides Subtract from
Guided Practice Find the value of the variables in Exercises 7–12. 7. x ÷ 2 + 8 = 9
8. d ÷ 7 + 4 = 6
9. k ÷ 3 – 15 = 30
10. y ÷ 4 – 3 = 12
11. 9 = g ÷ 2 – 6
12. (j + 20) ÷ 5 = 3
Independent Practice Now try these: Lesson 1.2.4 additional questions — p432
In Exercises 1–4, say which order you should undo the operations in. 1. x ÷ 3 + 7 = 20 2. 21x – 12 = 44 3. 11 = x ÷ 10 – 5 4. 14 = 2 • (2 + x) Find the value of the variables in Exercises 5–16. 6. 2r + 11 = –13 5. 4h + 2 = 22 8. 5w – 15 = 10 7. 10b – 5 = 55 10. –10 = 2n – 2 9. 14 = 2 + 3c 12. d ÷ 2 + 9 = –9 11. m ÷ 4 + 6 = 11 14. f ÷ 3 – 17 = –20 13. p ÷ 7 – 4 = 2 16. –20 = q ÷ 2 – 12 15. 10 = 5 + a ÷ 10
Round Up Solving a two-step equation uses the same techniques as solving a one-step equation. The important thing to remember with two-step equations is to do the inverse operations in the reverse of the original order. This same method applies to every equation, no matter how many steps it has. Later in this Section you’ll use this technique to solve real-life problems. 34
Section 1.2 — Equations
Lesson
1.2.5
Mor e Two-Ste p Equa tions More o-Step Equations
California Standards:
When you have a fraction in an equation, you can think of it as being two different operations that have been merged together. That means it can be solved in the same way as any other two-step equation.
Alg ebr a and Functions 4.1 Alge bra Solv e tw o-ste p linear Solve two-ste o-step equa tions and inequalities in equations one vvaria aria ble o ver the ariab ov ra tional n umber s , interpret number umbers the solution or solutions in the context from which they arose, and verify the reasona bleness of the easonab r esults esults..
Fractions Can Be R ewritten as Two Se par a te Ste ps Re Separ para Steps Fractions can be thought of as a combination of multiplication and division. You might see what is essentially the same expression written in several different ways. For example:
What it means for you:
3x 4
3 x 4
You’ll learn how to deal with fractions in equations, and how to check that your answer is right.
(3 • x) ÷ 4 1 • 3x 4
Key words: • fraction • isolate • check
3•
1 •x 4
All five expressions are the same.
Deal with a F tion as Two Ste ps Frraction in an Equa Equation Steps Because a fraction can be split into two steps, an equation with a fraction in it can be solved using the two-step method. Using the example above: 3 x 4
Check it out: Another way to do this is to multiply both sides by the reciprocal of the fraction. Multiplying a fraction by its reciprocal gives a product of 1 — so it “gets rid of” the fraction. To find the reciprocal of a fraction you invert it. So the reciprocal of
2 3
is
3 . 2
2 a=6 3 3 2 3 • a= 6 • 2 3 2 a=9
For more on reciprocals see Lesson 1.1.4.
=6
3x ÷ 4 = 6 3x = 24
First split the expression into two separate operations: here x is first multiplied by 3, and then divided by 4. Then solve as a two-step equation.
x=8 Example
1
Find the value of a when
2 a 3
= 6.
Solution 2 a 3
=6
2a ÷ 3 = 6 2a = 18 a= 9
Split the e xpr ession into tw o oper a tions expr xpression two opera Solv e as a tw o-ste p equa tion Solve two-ste o-step equation
Section 1.2 — Equations
35
Don’t forget: The number on top of a fraction is called the numerator. 2 5
Here is another example — this one has a more complicated numerator. Example
2
Find the value of h when Solution
The number on the bottom of a fraction is called the denominator.
h+2 4
=3
(h + 2) ÷ 4 = 3 h + 2 = 12 h = 10
h+2 4
=3
The whole expression h + 2 is being divided by 4 — the fraction bar “groups” it. Put it in parentheses here to show that this operation originally took priority.
Split the e xpr ession into tw o oper a tions expr xpression two opera Solv e as a tw o-ste p equa tion Solve two-ste o-step equation
Guided Practice Find the value of the variables in Exercises 1–6. 1
1. 2 a = 2 2 v 3
=4
5. 6 =
2 s 5
3.
2.
3 q 4
= 33
4
4. 1 r = –8 6.
2c 3
=6
Chec k Your Ans wer b y Substituting It Bac k In Check Answ by Back When you’ve worked out the value of a variable you can check your answer is right by substituting it into the original equation. Once you’ve substituted the value in, evaluate the equation — if the equation is still true then your calculated value is a correct solution. 3x + 2 = 14 3x + 2 – 2 = 14 – 2 If the equation isn’t true when you’ve substituted in your solution, look back through your work to find the error.
3x ÷ 3 = 12 ÷ 3 x=4 Now substitute the calculated value back into the equation.
3x + 2 = 14, 3(4) + 2 = 14 12 + 2 = 14 As both sides are the same, the value of x is correct.
36
Section 1.2 — Equations
First solve the equation to find the value of x.
3x = 12
Don’t forget:
14 = 14
x=4 Then evaluate the equation using your calculated value.
Example
3
Check that c = 8 is a solution of the equation 10c + 15 = 95. Check it out:
Solution
10c + 15 = 95 10(8) + 15 = 95 80 + 15 = 95 95 = 95
It might seem like needless extra work to check your solution, but it’s always worth it just to make sure you’ve got the right answer.
Substitute 8 into the equa tion equation
The equation is still true, so c = 8 is a solution of the equation 10c + 15 = 95.
Guided Practice Solve the equations below and check your answers are correct. 7. 12m + 8 = 56
8. 22 + 3h = 34
9. 56 = 18 + 19v
10. 16 – 4g = –28
11. 3 – 6x = 9
12. 5y – 12 = 28
Independent Practice Find the value of the variables in Exercises 1–6. 4
3
2. 5 k = 8
1. 4 d = 24 3.
–
2 b 3
5. 22 = n • Now try these: Lesson 1.2.5 additional questions — p433
4. 27 =
= 14 2 5
6.
5t 10
3 w 2
=4
Solve the equations in Exercises 7–10 and check your solution. 8. 3r – 6 = –12 7. 2x + 4 = 16 9. 6 = v ÷ 4 + 2 10. 3 c = 15 4
11. For each of the equations, say whether a) y = 3, or b) y = –3, is a correct solution. Equation 1: 10 – 2y = 16 Equation 2:
–
2y 3
= –2
For each equation in Exercises 12–14, say whether the solution given is a correct one. 12. x ÷ 2 + 4 = 9, x = 10. 13. 3x – 9 = 12, x = 4. 14. 8 = 5x – 7, x = 3.
Round Up You can think of a fraction as a combination of two operations. So a fraction in an equation can be treated as two steps. And don’t forget — when you’ve found a solution, you should always substitute it back into the equation to check that it’s right. Section 1.2 — Equations
37
Lesson
1.2.6
A pplica tions of Equa tions pplications Equations
California Standards:
Equations can be really useful in helping you to understand real-life situations. Writing an equation can help you sort out the information contained in a word problem and turn it into a number problem.
Alg ebr a and Functions 4.1 Alge bra Solv e tw o-ste p linear Solve two-ste o-step equa tions and inequalities in equations one vvaria aria ble o ver the ariab ov ra tional n umber s, inter pr et number umbers interpr pret the solution or solutions in om w the conte xt fr hic h the y context from whic hich they ar ose erify the arose ose,, and vverify reasona bleness of the easonab r esults esults.. Ma thema tical R easoning 2.1 Mathema thematical Reasoning erify the estimation Use estima tion to vverify reasona b leness of easonab calcula ted rresults esults calculated esults..
Equa tions Can Describe R eal-Lif e Situa tions Equations Real-Lif eal-Life Situations An equation can help you to model a real-life situation — to describe it in math terms. For example: • You’ve just had your car repaired. The bill was $280. • You know the parts cost $120. • You know the mechanic charges labor at $40 per hour. • You want to know how long the mechanic worked on your car.
What it means for you: You’ll see how to use equations to help solve real-life math problems, and how to check if your answer is sensible.
Key words: • • • •
model check reasonable sensible
Let h = number of hours worked by mechanic. 40h + 120 = 280 40h = 160 h=4
1. Choose a variable. 2. Write an equation. 3. Solve the equation.
So you know the mechanic must have worked on your car for 4 hours. You can use an equation to help you describe almost any situation that involves numbers and unknown numbers. Example
1
At the school supply store, Mr. Ellis bought a notebook costing $3 and six pens. He spent $15 in total. Find the price of one pen, p. Solution
First write out the information you have: Total spent = $15 Cost of notebook = $3 Cost of six pens = 6p
Check it out: When you’ve solved your equation you’ll need to decide if the solution needs units. In this example you’re figuring out the price of a pen in dollars, so your answer is $2. You’ll see more about how to find the right units for your answer in Lesson 1.2.7.
38
Section 1.2 — Equations
You know that six pens and the notebook cost a total of $15. So you can write an equation with the cost of each of the items bought on one side, and the total spent on the other. 6p + 3 = 15 Now you have a two-step equation. You can find the cost of one pen by solving it. 6p + 3 = 15 6p = 12 p=2
One pen costs $2.
Guided Practice Write an equation to describe each of the situations in Exercises 1–3. Then solve it to find the value of the variable. 1. Emily is seven years older than Ariela. The sum of their ages is 45. How old is Ariela? 2. A sale rack at a store has shirts for $9 each. Raul has $50 and a coupon for $4 off any purchase. How many shirts can he buy? 3. The price for renting bikes is $15 for half a day, then $3 for each additional hour. How many hours longer than half a day can you keep a bike if you have $24?
You Need to Chec k Tha wer is R easona ble Check hatt Your Ans Answ Reasona easonab When you’ve solved an equation that describes a real-life problem, you need to look at your answer carefully and see if it is reasonable. Here are two important things to think about:
1) Does Your Ans wer Mak e Sense? Answ Make You must always check that the answer makes sense in the context of the question. For example:
Don’t forget: You might need to round your answer up or down, depending on the question. In this example, you calculate that you can afford 7.5 bags — but you can only buy whole ba gs bags gs. You can’t afford to buy 8 bags — so it’s sensible to round your answer down to 7. If instead you were working out how many bags you needed for a recipe and your answer came out as 7.5, then you would round it up. You would buy 8 bags, because you want a t least 7.5. There’s a lot more about rounding, and how to round reasonably, in Section 8.3.
An orchard charges $1.10 for a pound of apples. You have $8.25. How many pounds of apples can you buy? • Set up an equation to describe the problem: Number of pounds = 8.25 ÷ 1.10 = 7.5 Æ This is a reasonable answer as the orchard will happily sell you half a pound of apples. But if you change the problem slightly: A store charges $1.10 for a bag of apples. You have $8.25. How many bags of apples can you buy? • Number of bags = 8.25 ÷ 1.10 = 7.5. Æ
This is no longer a reasonable answer — the store wouldn’t sell you half a bag of apples. You could only buy 7 bags.
2) Is Your Ans wer About the Right Siz e? Answ Size? The size of your answer has to make sense in relation to the question that is being asked. For example: Æ If you’re finding the height of a mountain, and your answer is 5 feet, it’s not reasonable. Æ If you’re finding the height of a person, and your answer is 5000 feet, that’s not reasonable either. If the size of your answer doesn’t seem reasonable then it’s really important to go back and check your work to see if you’ve made an error somewhere.
Section 1.2 — Equations
39
Example
2
Kea is going to walk 1.5 miles at a steady speed of 3 miles per hour. She works out how long it will take using the work shown. Is her answer reasonable? Check it out: If Kea walked at 3 miles per hour for 1 hour she’d go 3 miles. So to cover half that distance would take her half the time — 0.5 hours.
Distance = 1. 5 miles Speed = 3 mi/hour Time = 1.5 × 3 = 4.5 hours
Solution
Given that Kea’s walk is only 1.5 miles long and she walks at 3 mi/h, 4.5 hours is not a reasonable answer — it is much too long. (Kea multiplied the distance of the walk by her speed. She should divide the distance by the speed instead: Time = 1.5 ÷ 3 = 0.5 hours.)
Guided Practice 4. Pete is buying trading cards. One card costs 20¢. He says 10 cards will cost $20. Is this a sensible answer? Explain why or why not. 5. Six friends earn $87 washing cars. How much will each one get if they split it evenly? Is your answer reasonable in the context of the question? 6. A yard has a 150-foot perimeter. Fencing is sold in 40-foot rolls. Write an equation to describe the number of rolls, n, you need to buy to fence the yard. Solve the equation. Is your answer reasonable in the context of the question? 5
7. Ana is 6 as tall as T.J., who is 174 cm tall. Write an equation to describe Ana’s height, A. Solve it. Is the size of your answer reasonable?
Independent Practice
Now try these: Lesson 1.2.6 additional questions — p433
Write an equation to describe each situation in Exercises 1–2, and solve the equation to answer the question. 1. Don has spent $474 ordering sticks for his hockey team. A stick costs $50. Shipping costs $24. How many did he buy? 2. Tiana is saving up to buy a fishing rod. The rod costs $99 with tax. She already has $27, and can afford to save another $12 each week. How long will it take her to save enough for the rod? 3. Joy went to the fabric store to buy ribbon. She got f feet, and spent $5. The ribbon cost 80¢ a foot. Write an equation to describe how much she got. Solve it. Is your answer reasonable in the context of the question? 1 4. Mike is asked to multiply 5 by 2 . He says the answer is 10. Is this reasonable in the context of the question? Explain why or why not. 5. Two friends run a dog walking service, each walking the same number of dogs. Write and solve an equation to show how many dogs, d, each friend walks if they walk nine dogs between them. Is your answer reasonable?
Round Up Equations can help you to understand situations. They can also help you to describe a real-life math problem involving an unknown number and come up with a solution. But don’t forget to always think carefully about whether the answer is a reasonable one in relation to the question. 40
Section 1.2 — Equations
Lesson
1.2.7
Under standing Pr ob lems Understanding Prob oblems
California Standards:
Math problems are full of all kinds of details. The challenge is to work out which bits of information you need and which bits you don’t need. To be able to do this you need to understand exactly what the question is asking.
Alg ebr a and Functions 4.1 Alge bra Solv e tw o-ste p linear Solve two-ste o-step equa tions and inequalities in equations one vvaria aria ble o ver the ariab ov ra tional n umber s, inter pr et number umbers interpr pret the solution or solutions in hic h the y the conte xt fr om w hich they context from whic ar ose erify the arose ose,, and vverify reasona bleness of the easonab r esults esults.. Ma thema tical R easoning 1.1 Mathema thematical Reasoning Anal yz e pr ob lems b y Analyz yze prob oblems by identifying rrela ela tionships elationships tionships,, distinguishing rrele ele v ant fr om elev from ir v ant inf or ma tion, irrr ele elev infor orma mation, identifying missing inf or ma tion, sequencing infor orma mation, and prioritizing inf or ma tion, infor orma mation, and obser ving pa tter ns observing patter tterns ns..
What it means for you: You’ll learn how to spot which pieces of information are important in answering a question, and how to check that your answer has the correct units.
You Can e a Pr ob lem with Inf or ma tion Missing Can’’ t Solv Solve Prob oblem Infor orma mation Sometimes a piece of information needed to solve a real-life problem will be missing. You need to be able to read the question through and identify exactly what vital piece of information is missing. Example
1
Brian’s mechanic charged $320 to fix his car. The bill for labor was $157.50. How many hours did the mechanic work on the car? Solution
The question tells you that Brian’s total bill for labor was $157.50. But to use this piece of information to work out how many hours the mechanic worked on the car you would also need to know what the mechanic’s hourly rate was, as hours worked = bill for labor ÷ hourly rate. You can’t solve the problem as the mechanic’s hourly rate is missing.
Guided Practice Key words: • relevant • irrelevant • unit
In Exercises 1–4 say what piece of information is missing that you need to solve the problem. 1. Samantha is 20 inches taller than half Adam’s height. How tall is Samantha? 2. A coffee bar charges $2 for a smoothie. Sol buys a smoothie and a juice. How much is his check? 3. Erin has $36 and is going to save a further $12 a week. How many weeks will it take her to save enough for a camera? 4. A box contains 11 large tins and 17 small tins. A large tin weighs 22 ounces. What is the weight of the box?
Some Inf or ma tion in a Question Ma y Not Be R ele vant Infor orma mation May Rele elev You will often come across real-life problems that contain more information than you need to find a solution. Information that you don’t need to solve a problem is called irrelevant information. You need to be able to sort out the information you do need from the information you don’t. A good example of this is a question where you have to pick out the information that you need from a table. Section 1.2 — Equations
41
Example
2
At the hardware store Aura spent $140 on paint. She bought four cans of blue paint and spent the rest of the money on green paint. Use the table below to calculate how many liters of green paint she bought. Aura only bought blue paint and green paint. Æ So you only need the circled data in these two rows to answer Æ the question.
Color of paint
Volume of can (l)
Price of can ($)
Blue
1
20
Yellow
2
35
Red
1
20
Green
1.5
30
Solution
To answer the question you need the price of a can of blue paint, and the volume and price of a can of green paint. The volume of cans of blue paint is irrelevant, as is the information about red and yellow paint. • First work out how much Aura spent on blue paint. You know that she bought four cans of blue paint that cost $20 each. So she spent $80 on blue paint. That means she spent $140 – $80 = $60 on green paint. • Each can of green paint is $30. So she bought $60 ÷ $30 = 2 cans. • A can of green paint is 1.5 liters. So she bought 1.5 • 2 = 3 liters.
Guided Practice Use the table from Example 2 in Exercises 5–7. 5. Eduardo bought one can of yellow paint and three liters of blue paint. How much did he spend? 6. Lamarr bought 2 cans of green paint and some yellow paint. He spent $165. How many liters of yellow paint did he buy? 7. Amber spent $120. She bought twice as much red paint as blue paint. How many cans of red paint did she buy?
Ans wer s Should Al ways Ha ve the Cor Answ ers Alw Hav Corrrect Units When you work out the answer to a problem, you need to think about the right units to use. If you apply the same operations to the units as you do to the numbers, you’ll find out what units your answer should have. Example Check it out: When you’re writing units, remember that km/hour means the same as (km ÷ hours), and person-days means the same as (persons × days).
42
Section 1.2 — Equations
3
Laura drives her car 150 km in 2 hours. Use the formula speed = distance ÷ time to calculate her average speed. Solution
speed = distance ÷ time speed = 150 ÷ 2 = 75 Now do the same operations to the units of the numbers: km ÷ hours = km/hour. So the average speed of the car is 75 km/hour.
You can do this with any calculation to find the correct units for the answer. Example
4
The power consumption of a computer is 0.5 kilowatts. If the computer is running for 4 hours, how much energy will it use? Use the equation: Power Consumption • Time Used = Energy Used. Solution
First do the numerical calculation. Power Consumption • Time Used = Energy Used 0.5 • 4 = 2 Check it out: A kilowatt-hour is a measure of energy consumption.
Then work out the units. kilowatts • hours = kilowatt-hours The computer will use 2 kilowatt-hours of energy.
Guided Practice Check it out: You can use the / symbol to mean “divided by” when you are writing units.
Say what units the answers will have in Exercises 8–11. 8. 40 miles ÷ 2 hours = 20 ? 9. 5 newtons • 3 meters = 15 ? 10. 6 persons • 4 days = 24 ? 11. $25 ÷ 5 hours = 5 ?
Independent Practice 1. The sale bin at a music store has CDs for $4 each. Eric buys four CDs and some posters, and uses a coupon for $2 off his purchase. He pays $26. How many posters did he buy? Say what information is missing from the question that you would need to solve the problem. 2. Liz meets Ana to go ice-skating at 7 p.m. Admission is $8 and coffee costs $1.50. Liz has $14 and wants to buy some $2 bottles of water for her and Ana to drink afterwards. Calculate how many bottles of water Liz can buy. What information are you given that isn’t relevant ?
Now try these: Lesson 1.2.7 additional questions — p433
3. Sean has $60 to buy books for math club. A book costs $9.95. He orders them on a Monday. Shipping costs $10 an order. How many books could he buy? What information are you given that isn’t relevant? Say what units the answers will have in Exercises 4–7. 4. 4 persons • 4 hours = 16 ? 5. 100 trees ÷ 10 acres = 10 ? 6. 6 meters • 7 meters = 42 ? 7. 21 meters/second ÷ 7 seconds = 3 ?
Round Up When you’re solving a math problem, you need to be able to pick out the important information. Then you can use the relevant bits to write an equation and find the solution. Always remember to check what units your answer needs to be written in too. Section 1.2 — Equations
43
Lesson
Section 1.3
1.3.1
Inequalities
California Standards:
Inequalities are a lot like equations. But where an equation has an equals sign, an inequality has an inequality symbol. It tells you that the two sides may not be equal or are not equal — that’s why it’s different from an equation.
Alg ebr a and Functions 1.4 Alge bra minolo gy alge braic terminolo minolog Use alg ebr aic ter (e .g ., variable, equation, term, (e.g .g., coefficient, inequality inequality, expression, constant)) cor y. corrr ectl ectly Alg ebr a and Functions 1.5 Alge bra ti ve present quantitati tiv Repr esent quantita r ela tionships g y elationships grra phicall phically and interpret the meaning of a specific part of a graph in the situation represented by the graph.
An Inequality Does Not Ha ve to Balance Hav In the last Section you saw that an equation is a balanced math sentence. The expressions on each side of the equals sign are equal in value. An inequality is a math sentence that doesn’t have to be balanced. The expression on one side does not have to have the same value as the expression on the other.
What it means for you: You’ll learn what an inequality is, and how to show one on a number line.
Key words: • • • •
inequality greater than less than equal to
x £ 5, 10 > 3y, 4h ≥ 19, and k < 5 are all inequalities. Inequalities are made up of two expressions that are separated by one of the four inequality symbols: The Inequality Symbols < means “Less than.”
> £ ≥
means “Greater than.” means “Less than or equal to.” means “Greater than or equal to.”
The symbol that you use explains how the two expressions relate to each other. So 2 < 10 means “two is less than ten” and x ≥ 5 means “x is greater than or equal to five.” The smaller end of the symbol always points to the smaller number. So x < 2 and 2 > x are telling you the same thing — that the variable x is a number less than 2.
Guided Practice Fill in the blanks in the statements in Exercises 1–4.
44
Section 1.3 — Inequalities
1. If a < b then b__a.
2. If m ≥ n then n__m.
3. If c > d then d__c.
4. If j £ k then k__ j.
Plot the Solutions to an Inequality on a Number Line An inequality has an infinite number of solutions. When you solve an inequality you are describing a group or set of solutions. For example, for the inequality x < 9, any number that is less than 9 is a solution of the inequality. The solutions are not limited to just whole numbers or positive numbers. Check it out: A number line is a line that represents every number, with the numbers increasing in value along the line from left to right. To give the line a scale, some numbers are shown as labeled points spaced out evenly along the line.
All the possible solutions of an inequality can be shown on a number line. The following number line shows the solution of the inequality c > 1. The set of possible solutions for c is all the numbers greater than 1. The circle shows the point on the number line where the solution to the inequality starts.
-2 -1 0
1
2
The shading stretches away to infinity in the direction of the arrow. This is showing that c can be any number that’s greater than 1.
3
Here the circle is empty (open), because 1 is not included in the solution.
Don’t forget: It’s important to remember that there are an infinite amount of decimal numbers between each labeled point in the solution set. All of these are solutions too.
The number line below shows the solution of the inequality y £ 2. The set of possible solutions for y is 2 and all the numbers less than 2.
The shading stretches away to infinity in the direction of the arrow. This is showing that y can be any number that’s less than or equal to 2.
The circle shows the point on the number line where the solution to the inequality starts.
-2 -1 0
1
2
3
Here the circle is filled -in (closed), because 2 is included in the solution.
To plot an inequality on a number line: 1) Draw a circle on the number line around the point where the set of solutions starts. It should be a closed circle if the point is included in the solution set, but an open circle if it isn’t. Don’t forget: A ray is just a straight line that begins at a point and goes on forever in one direction.
2) Draw a ray along the number line in the direction of the numbers in the solution set. Add an arrowhead at the end to show its direction (and that it goes on forever).
Section 1.3 — Inequalities
45
Example
1
Plot the solution to the inequality y ≥ –1 on a number line. Don’t forget: If the sign is ≥ or £ then you should use a closed circle to show that the start point is included in the solution. If the sign is > or < then you should use an open circle to show that the start point isn’t included in the solution.
Solution
First place a closed circle at –1 on the number line to show that –1 is included in the solution set. -2 -1 0
1
2
3
y is greater than or equal to –1, so add a ray with an arrowhead pointing along the number line to the right of the circle.
-2 -1 0
1
2
3
Guided Practice Plot the inequalities in Exercises 5–8 on the number line. 5. h < 2
6. –1 < a
7. k ≥ 1
8. t £ 0
Independent Practice In Exercises 1–4, give a number that is part of the solution set of the inequality. 1. m < –3 2. k £ 12 3. w ≥ 9 4. h > –9 Now try these: Lesson 1.3.1 additional questions — p434
Match the inequalities 5–8 with the number line plots A–D. 5. x > –4 A B 6. x ≥ 2 0 1 2 3 4 5 6 -2 -1 0 1 2 7. x < 1 C D 8. x ≥ –2 -5 -4 -3 -2 -1 0
1
-5 -4 -3 -2 -1 0
1
9. Explain why the solution of an inequality must be graphed on a number line and not listed. Plot the inequalities in Exercises 10–11 on number lines. 10. –1 £ x 11. x < 0.5
Round Up An inequality is like an unbalanced equation — its two expressions can have different values. You can show all the possible solutions of an inequality by graphing it on a number line. Don’t forget the four symbols — you’ll need them to write and solve inequalities later. 46
Section 1.3 — Inequalities
Lesson
1.3.2
Writing Inequalities
California Standards:
In Lesson 1.2.2 you saw how to write an equation from a word problem. To write an inequality you use exactly the same process — but this time instead of joining the two expressions with an equals sign, you use an inequality symbol.
Alg ebr a and Functions 1.1 Alge bra ariab Use vvaria aria bles and a ppr opria te oper a tions to ppropria opriate opera write an expression, an equation, an inequality inequality, or a system of equations or epr esents a thatt rre presents inequalities tha verbal description (e.g., three less than a number, half as large as area A).
What it means for you: You’ll learn how to turn a word problem into an inequality.
Key words: • • • • •
inequality under over minimum maximum
You Need to Spot Whic h Symbol is Being Described hich To write an inequality you write two expressions that have (or can have) different values and join them with one of the four inequality symbols. You need to be able to recognize phrases that describe the four symbols. > means “greater than” or “more than” or “over.” < means “less than” or “under.” ≥ means “greater than or equal to” or “a minimum of ” or “at least.” £ means “less than or equal to” or “a maximum of ” or “no more than.” Example
1
Write an inequality to describe the sentence, “Four times a number, y, is less than 27.” Solution
The phrase “is less than” is represented by the less than symbol, <. It separates the expressions that make up the two sides of the inequality. One expression is “Four times a number, y.” This turns into the expression 4y. Don’t forget:
The other expression is a number, 27.
The inequality 27 > 4y means exactly the same thing as 4y < 27.
So the sentence, “Four times a number, y, is less than 27,” turns into the inequality 4y < 27. Example
2
Write an inequality to describe the sentence, “A number, h, increased by two is at least 16.” Solution
The phrase “at least” is represented by the greater than or equal to symbol, ≥ . One expression is “A number, h, increased by two.” This turns into the expression h + 2. Don’t forget: The inequality 16 £ h + 2 means exactly the same thing as h + 2 ≥ 16.
The other expression is a number, 16. So the sentence, “A number, h, increased by two is at least 16,” turns into the inequality h + 2 ≥ 16. Section 1.3 — Inequalities
47
Guided Practice Write an inequality to describe each of the sentences in Exercises 1–5. 1. A number, x, increased by five is more than 12. 2. Twice a number, k, is greater than or equal to two. 3. Fifteen decreased by a number, g, is no more than six. 4. A number, p, divided by two, is under four. 5. Negative two is less than the sum of a number, m, and five.
Inequalities ar e Often Used in R eal-Lif e Situa tions are Real-Lif eal-Life Situations You’ll come across lots of inequality phrases in real life. ELEVATOR
“Height” > 4 feet
Maximum capacity 10 people
“Number of people” £ 10
You must be over 4 feet tall to ride
Child tickets for 8 years and under.
“Age” £ 8
In math you might be asked to write an inequality to represent the information given in a word problem. Writing inequalities from word problems is a lot like writing equations from word problems. You need to spot key information and use it to write expressions — but you also need to work out which inequality symbol to use. Example
3
Your local conservation group runs a junior award program. To get a gold award you must complete a minimum of 50 hours’ conservation work. You have already done 17 hours. Write an inequality to represent the additional amount of work you need to do to gain your gold award. Solution
First define a variable: let the additional amount of hours you need to complete = H. The amount of hours you need to complete to get your award is 50. So one expression is just 50. The other expression describes the number of hours you have already done plus the additional number you need to spend — the variable H. So the expression is 17 + H. The phrase “minimum” tells you that the number of hours you complete has to be greater than or equal to 50. So the inequality is 17 + H ≥ 50. 48
Section 1.3 — Inequalities
Guided Practice Write an inequality to describe each of the situations in Exercises 6–9. 6. Lauren is three years younger than her friend Gabriela, who is k years old. Lauren is under 20. Don’t forget: You can write all of these inequalities in the reverse direction just by changing the sign you use. For example: a > b is the same as b < a.
7. Erin has visited b states. Kieran has visited two more states than Erin, and figures out that he has visited at least 28. 8. The number of boys enrolled at a university is half the number of girls, g, who are enrolled. The number of boys enrolled is more than 2000. 9. Pedro has set aside a maximum of $100 in order to buy gifts for his family. He wants to spend the same amount, $x, on each of his 3 family members.
Independent Practice
Now try these: Lesson 1.3.2 additional questions — p434
Write an inequality to describe each of the sentences in Exercises 1–5. 1. A number, m, decreased by seven is less than 16. 2. Nine more than a number, d, is at least 11. 3. The product of ten and a number, j, is a maximum of six. 4. Four is more than a number, y, divided by five. 5. The sum of a number, f, and six is less than negative one. 6. Explain whether the two statements “six more than a number is at least four” and “six more than a number is more than four” mean the same. Write an inequality to describe each of the situations in Exercises 7–9. 7. Alex and Mallory both spent time cleaning the house. Alex spent y minutes cleaning. Mallory spent 15 minutes less than Alex, but over 55 minutes, cleaning. 8. Rebecca has a maximum of $40 to spend on her cat. She buys a collar for $17 and then spends $d on cat food. 9. Alejandra’s collection of baseball cards is twice the size of Jordan’s. Alejandra has collected at least 2000 cards, and Jordan has collected c baseball cards. For each sentence in Exercises 10–12 say which inequality symbol would be used. 10. Maximum weight on this bridge is six tons. 11. The play park is for people under ten years old only. 12. This toy is for children aged three years and up.
Round Up When you turn a word problem into an inequality, the key thing is to figure out which of the four inequality symbols is being described. Then just write out the two expressions and join them with the correct symbol. You’ll see how to solve inequalities like the ones you’ve written in Chapter 4.
Section 1.3 — Inequalities
49
Lesson
1.3.3
Two-Ste p Inequalities o-Step
California Standards:
An inequality with two different operations in it is called a two-step inequality. To write a two-step inequality, follow the same steps that you learned in the last Lesson. The only difference this time will be that one of your expressions could have two operations in it.
Alg ebr a and Functions 1.1 Alge bra ariab Use vvaria aria bles and a ppr opria te oper a tions to ppropria opriate opera write an expression, an equation, an inequality inequality, or a system of equations or epr esents a thatt rre presents inequalities tha verbal description (e.g., three less than a number, half as large as area A).
Two-Ste p Inequalities Ha ve Two Oper a tions o-Step Hav Opera A two-step inequality is one that involves two different operations.
What it means for you:
inequality symbol
You’ll learn how to turn a word problem into a two-step inequality.
4 ÷ x + 9 > 10 first operation
Key words: • • • • •
inequality under over minimum maximum
second operation
It has the same structure as a two-step equation, but with an inequality symbol instead of an equals sign.
Writing a Two-Ste p Inequality o-Step Writing a two-step inequality is like writing a one-step inequality. You still need to write out your two expressions and join them with the correct inequality symbol — but now one of the expressions will contain two operations instead of one. That also means that you need to remember to use PEMDAS — the order of operations. Example
1
Write an inequality to describe the sentence, “Six more than the product of four and a number, h, is under 42.” Solution
The phrase “is under” represents the less than symbol. It also separates the two expressions that make up the two sides of the inequality. Don’t forget: The four inequality symbols: > is greater than. < is less than. ≥ is greater than or equal to. £ is less than or equal to.
50
Section 1.3 — Inequalities
One expression is, “Six more than the product of four and a number, h.” This tells you to multiply four by h and add six to the product. It turns into the expression 4h + 6. The other expression is the number 42. So the sentence, “Six more than the product of four and a number, h, is under 42,” turns into the inequality 4h + 6 < 42.
Guided Practice Write an inequality to describe each of the sentences in Exercises 1–6. 1. Five increased by the product of ten and a number, m, is more than 11. 2. Two plus the result of dividing a number, k, by six is at least five. 3. One subtracted from the product of a number, y, and nine is less than or equal to 33. 4. Ten subtracted from half of a number, t, is under –1. 5. A third of a number number, r, plus nine, is no greater than –4. 6. Double the sum of a number, x, and two is less than 20.
An Inequality Can Describe a Wor d Pr ob lem ord Prob oblem To write a two-step inequality from a word problem, just follow the same rules as for a one-step inequality: • Identify the important information you have been given. • Spot which operation phrases are being used. • Work out what the two expressions are. • Join them using the correct inequality symbol. Example
2
Hector needs to save at least $250 to buy a new bicycle. He already has $80, and receives $10 each week for mowing the neighbor’s lawn. Write an inequality to represent the number of weeks that Hector will need to save for. Solution
First define a variable: let the number of weeks Hector needs to save for = w. Don’t forget: To make sure your expression is right you can check it with a simple number using mental math. For example: after 2 weeks Hector will have $80 plus two times $10. This is $100. Then put w = 2 into your expression. 80 + 10(2) = 100. So your expression is likely to be right.
The minimum amount of money Hector needs to save is $250. So one expression is just 250. The other expression describes the amount of money he will have after w weeks. This will be the $80 he already has plus the number of weeks multiplied by the $10 he earns each week. So the expression is 80 + 10w. The phrase “at least” is telling you that the amount Hector needs to save has to be greater than or equal to 250. So the inequality is 80 + 10w ≥ 250.
Section 1.3 — Inequalities
51
Guided Practice Write an inequality to describe each of the following situations. 7. Mrs. Clark parks by a meter that charges $2 for the first hour and $0.50 for each additional hour parked. She spends no more than $10, and parks for the first hour and h additional hours. 8. Luis collects seashells. He has four boxes, each containing s shells. He gives 40 shells to Jon, and still has more than 200 shells in his collection. 9. Marcia is buying new shirts that cost $15 each for x people in her Little League team. She has a coupon for $12 off her order, and a maximum of $150 to spend. 10. Daniel’s teacher tells him that to be considered low-fat, a meal must contain less than three grams of fat. Daniel prepares a low-fat breakfast of yogurt topped with pumpkin seeds for a school project. A cup of yogurt contains y grams of fat. Daniel uses half a cup, and tops it with pumpkin seeds containing a total of 1 gram of fat.
Independent Practice Write an inequality to describe each of the sentences in Exercises 1–4. 1. Twenty more than twice a number, f, is less than 35. 2. Fifty subtracted from a quarter of a number, n, is at least 77. 3. Eight increased by the product of four and a number, d, is no more than 13. 4. Negative eighteen is more than a number, a, divided by 41, minus six. Now try these: Lesson 1.3.3 additional questions — p434
5. Vanessa has ordered a meal that costs under $12. She is having a baked potato, costing $7, and d salads costing $2 each. Write an inequality to describe this information. 6. Filipa and her four friends are looking for a house to rent for a vacation. The price of an airplane ticket is $200, and they will split the cost of the house rental, $r. Filipa has budgeted $600 for the airplane ticket and house. Write an inequality to determine the maximum rental price that the house can be. 7. Tom and his two friends run a babysitting service that makes $p a month income. Each month they spend $20 to advertise, then split the remaining money evenly. Tom wants to earn at least $80 a month. Write an inequality to describe how much income the service must make each month for this to happen.
Round Up Writing a two-step inequality is just like writing a one-step inequality. You still need to look for the key information in the question, identify the operation phrases, and spot which inequality symbol is needed. But this time one of the expressions might contain two operations. You’ll see how to solve two-step inequalities in Chapter 4. 52
Section 1.3 — Inequalities
Chapter 1 Investigation
Whic h Phone Deal is Best? hich You can write expressions to model real-life situations. In this Investigation, you’ll write expressions to represent different cell phone plans, and by evaluating your expressions you’ll find out which is the best value plan for different users. Two phone companies are offering different family plan deals to attract new customers. Company A
Company B
$30 a month for 500 minutes $0.02 for each additional minute
$10 a month for 500 minutes $0.04 for each additional minute
Part 1: Write expressions for Companies A and B that could be used to represent the price of one month’s phone bill. Which company offers the better plan for a family using 1000 minutes a month? Part 2: How many minutes does a family have to talk so that Company A offers a better deal than Company B? Things to think about: • How can you compare the prices of both companies for different numbers of minutes? • The basic price for Company B’s plan is $20 more than Company A’s plan. Thinking just about cost, why would a family choose Company A’s plan instead of Company B’s? Extensions 1) Write an inequality that could be used to calculate the number of minutes a family could talk with Company A if they want to spend under $35 a month. 2) The Sutro family uses Company A and talks an average of 800 minutes a month. How much will they save over a year by switching to Company B? Open-ended Extensions 1) Is it possible for the price of Company A’s plan to be double the price of Company B’s plan? Assume calls are charged to the nearest minute. Make an organized list or table to compare them. 2) Company C wants to charge a flat per minute fee and have their price lie between Company A’s and Company B’s prices when between 500 and 750 minutes are used. What per minute fees could Company C charge to accomplish this goal?
Round Up A number that can change is called a variable and is represented with a letter. In the cell phone plans, the variable was the number of minutes used. By evaluating expressions with different values for the variable, you can find the prices when different numbers of minutes are used. Cha pter 1 In vestig a tion — Which Phone Deal is Best? Chapter Inv estiga
53
Chapter 2 Rational and Irrational Numbers Section 2.1
Rational Numbers ....................................................... 55
Section 2.2
Absolute Value ............................................................ 65
Section 2.3
Operations on Rational Numbers ................................ 71
Section 2.4
More Operations on Rational Numbers....................... 93
Section 2.5
Exploration — Basic Powers ..................................... 105 Basic Powers ............................................................. 106
Section 2.6
Exploration — The Side of a Square ........................ 119 Irrational Numbers and Square Roots ...................... 120
Chapter Investigation — Designing a Deck ...................................... 130
54
Lesson
Section 2.1
2.1.1
Ra tional Number s Numbers
California Standards:
Pretty much all the numbers you’ve met so far are rational — positive and negative integers and fractions are all rational, as are most decimals. The only decimals that aren’t rational are the ones that go on and on forever, without having a repeating pattern of digits.
Number Sense 1.3 actions to Conv ertt fr fractions Con ver decimals and percents and use these representations in estimations, computations, and applications. Number Sense 1.4 een Difffer erentia entiate between Dif entia te betw ra tional and ir irrra tional n umber s. umbers Number Sense 1.5 w tha ver y rra a tional Know thatt e ev ery Kno number is either a ter mina ting or a rre epea ting termina minating peating decimal and be able to convert terminating decimals into reduced fractions.
All R a tional Number s Can Be Written as F Ra Numbers Frractions You get all sorts of numbers in the rational set, for example 1.05, 0.3333..., 1 , and 6 are all rational. Rational numbers have all got one thing in 2 common — they can each be written as a simple fraction, with an integer on the top and a nonzero integer on the bottom. In formal math: a
A rational number is a number that can be written as b , where both a and b are integers (and b is not equal to 0).
What it means for you:
There are basically three types of rational numbers.
You’ll meet rational numbers and see which kinds of numbers that you’ve already met fall into this category.
All Inte ger s ar eR a tional Integ ers are Ra Any integer can be written as a fraction over 1.
Key words: • • • • • •
The integers are the numbers in the set {..., –4, –3, –2, –1, 0, 1, 2, 3, 4, ...}.
rational number irrational number fraction decimal terminating repeating
So integers are all rational. For example: 7 = Example
7 1
−18 =
–18 1
1
Show that 5 is a rational number. Solution 5
5 can be written as 1 . This fits the above definition, so 5 must be rational.
Don’t forget: A fraction can be thought of 5
as a division. So 1 says the same thing as 5 ÷ 1, and when you divide a number by 1, it doesn’t change.
Check it out: In the next Lesson you’ll see how to turn terminating decimals into fractions.
All Ter mina ting Decimals ar e R a tional ermina minating are Ra Numbers like 1.2, 5.689, –3.72, and –0.69245 are known as terminating decimals — they all have definite ends. All terminating decimals can be converted to fractions of the form
a , b
where a and b are both integers. So all terminating decimals are rational. 6
1
For example, 1.2 is equivalent to 5 , 0.125 is equivalent to 8 , 3
and 0.75 is equivalent to 4 .
Section 2.1 — Rational Numbers
55
All R epea ting Decimals ar eR a tional Re peating are Ra Check it out:
0.09090909... is a repeating decimal. It will go on forever repeating the same digits (09) over and over again.
In Lesson 2.1.3 you’ll see how to turn repeating decimals into fractions.
Repeating decimals can always be converted to the form
a b
where a and
b are both integers — so they are always rational. 0.0909090909... =
1 11
Other examples of repeating decimals are: 0.33333...
⎛ 1 ⎞⎟ ⎜⎜ ⎟ , ⎜⎝ 3 ⎟⎠
⎛ 5 ⎞ and 0.045045045... ⎜⎜⎜ ⎟⎟⎟ . ⎝111⎠
The usual way to show that decimals are repeating is to put a small bar above them. The bar should cover one complete set of the repeated digits, so 0.15 means 0.151515..., but 0.151 means 0.151151151...
Ne ver -Ending epea ting Decimals ar e Ir Nev er-Ending -Ending,, Nonr Nonre peating are Irrra tional Check it out: Irrational numbers are covered in more detail in Section 2.6.
A number that cannot be written in the form integers is an irrational number.
a b
where a and b are both
Irrational numbers are always nonrepeating, nonterminating decimals. p is an irrational number: p = 3.141592653...
goes on forever, never repeats
Guided Practice Show that the numbers in Exercises 1–6 are rational. 1. 2 2. –8 3. 0.5 4. 0.25 5. –0.1 6. 1.5 7. Luis does a complicated calculation and his 10-digit calculator screen shows the result 1.123456789. Can you say whether the answer of Luis’s calculation is rational? 8. Is 2p rational?
Fractions Can Be Con ver ted into Decimals b y Di vision Conv erted by Division
Check it out: b can’t be zero — you can’t divide by zero; it’s undefined.
56
All integers, terminating decimals, and repeating decimals are rational, so they can be written as fractions. The opposite is also true — every fraction can be converted into an integer, a terminating decimal, or a repeating decimal. a b
can be read as the instruction “a divided by b.”
If you divide the integer a by the integer b you’ll end up with an integer, a repeating decimal, or a terminating decimal.
Section 2.1 — Rational Numbers
AR emainder of Zer o Means a Ter mina ting Decimal Remainder Zero ermina minating When you’re dividing the numerator of a fraction by the denominator, you might get to a point where you have no remainder left — that means that it’s a terminating decimal. Example Convert
7 8
2
into a decimal.
Solution
Don’t forget: When you add 0s to the end of a number after a decimal point, as with the 7 in Example 2, the value of the number doesn’t change. So 7.0000 = 7.0 = 7.
Divide 7 by 8. 0.875 8 7.0000 64 60 56 no remainder left, 40 so this is a 40 terminating decimal 00
So,
7 8
as a decimal is 0.875.
Guided Practice Convert the fractions given in Exercises 9–12 into decimals without using a calculator. 9.
3 6
10.
4 5
11.
6 4
12.
5 32
AR epea ted R emainder Means a R epea ting Decimal Re peated Remainder Re peating If you get a remainder during long division that you’ve had before, then you have a repeating decimal. Don’t forget: The quotient is what you get when you divide one number by another.
repeated remainder
0.13 15 2.000 15 50 45 50
The repeating digits are only the ones that you worked out since the last time you saw the same remainder. repeating digit Æ In this example you’ve had a remainder of 0.13 50 before. Since the last time you had this 15 2.000 remainder, you’ve found the digit 3. 15 50 That means 3 is the repeating part of repeated 45 the decimal. remainder 50 So
2 15
= 0.1333... with the 3 repeating forever.
Which you can write as
2 15
= 0.13 .
Section 2.1 — Rational Numbers
57
Example Convert
5 22
3 into a decimal.
Solution
Divide 5 by 22. repeating digits
repeated remainder
0.227 22 5.000 44 60 44 160 154 60
So,
5 22
as a decimal is 0.227 .
Guided Practice Convert the fractions given in Exercises 13–16 into decimals, without using a calculator. 5 27
13.
14.
6 41
15.
15 7
16.
7 12
Independent Practice 1. Read statements a) and b). Only one of them is true. Which one? How do you know? a) All integers are rational numbers. b) All rational numbers are integers. Now try these: Lesson 2.1.1 additional questions — p435
Show that the numbers in Exercises 2–7 are rational. 2. 4 3. 1 4. –2 5. 0.2 6. 1.25 7. – 0. 3 Convert the fractions given in Exercises 8–13 to decimals without using a calculator. Say whether they are terminating or repeating decimals. 8.
1 9
11.
9. 5 11
8 5
12.
15 8
10.
11 16
13.
5 22
Round Up Rational numbers can all be written as fractions, where the top and bottom numbers are integers, and the bottom number isn’t zero. You already know how to write integers as fractions, and you’ll see how to convert terminating decimals and repeating decimals to fractions in the next two Lessons. 58
Section 2.1 — Rational Numbers
Lesson
Con ver ting Ter mina ting Conv erting ermina minating Decimals to F Frractions
2.1.2
California Standards: Number Sense 1.5 Know that every rational number is either a terminating or a repeating decimal and be a ble to con ver mina ting conv ertt ter termina minating decimals into rreduced educed fr actions fractions actions..
This Lesson is a bit like the opposite of the last Lesson — you’ll be taking decimals and finding their equivalent fractions. This is how you can show that they’re definitely rational numbers.
What it means for you:
If you read decimals using the place-value system, then it’s more straightforward to convert them into fractions. For example, a number
You’ll see how to change terminating decimals into fractions that have the same value.
Decimals Can Be Tur ned into F urned Frractions
like 0.15 is said “fifteen-hundredths,” so it turns into the fraction
15 . 100
You need to remember the value of each position after the decimal point: decimal point
Key words:
0.1234
• fraction • decimal • terminating
tenths hundredths
ten-thousandths thousandths
Then when you are reading a decimal number, look at the position of the last digit. For example: 0.1 is one-tenth, which is the fraction one-hundredth, which is the fraction Example
0.01 is
1
Convert 0.27 into a fraction.
Don’t forget: You can ignore any extra 0s at the end of the decimal, because they don’t change its value. For example: 0.270000 = 0.27 =
1 . 100
1 . 10
27 . 100
Solution
0.27 is twenty-seven hundredths, so it is
Example
27 . 100
2
Convert 0.3497 into a fraction. Solution
0.3497 is 3497 ten-thousandths, so it is
3497 . 10, 000
Guided Practice Convert the decimals in Exercises 1–12 into fractions without using a calculator. 1. 0.1
2. 0.23
3. 0.17
4. –0.87
5. 0.7
6. 0.35
7. 0.174
8. –0.364
9. 0.127
10. 0.9827
11. 0.5212
12. –0.4454
Section 2.1 — Rational Numbers
59
Some F Frractions Can Be Made Simpler When you convert decimals to fractions this way, you’ll often get fractions 5 that aren’t in their simplest form. For instance, could be written more Don’t forget: Dividing the top and bottom of a fraction by the same thing doesn’t change the value of the fraction.
1
simply as 2 , and
75 100
10
3
could be written more simply as 4 .
If an answer is a fraction, you should usually give it in its simplest form. This is how to reduce a fraction to its simplest form: 1) Find the biggest number that will divide into both the numerator and the denominator without leaving any remainder. This number is called the greatest common factor, or GCF. 2) Then divide both the numerator and the denominator by the GCF. Example
3
Convert 0.12 into a fraction. Solution
• 0.12 is twelve-hundredths. As a fraction it is
12 . 100
• The factors of 12 are 1, 2, 3, 4, 6, and 12. The biggest of these that also divides into 100 leaving no remainder is 4. So the greatest common factor of 12 and 100 is 4. • Divide both the numerator and denominator by 4. 12 ÷ 4 3 = . 100 ÷ 4 25
So 0.12 as a fraction in its simplest form is
3 . 25
If the greatest common factor is 1 then the fraction is already in its simplest form. Example
4
Convert 0.7 into a fraction. Solution
• 0.7 is seven-tenths so, it is
7 . 10
• The greatest common factor of 7 and 10 is 1, so this fraction is already in its simplest form.
Guided Practice Convert the decimals in Exercises 13–20 into fractions and then simplify them if possible. 13. 0.25
14. 0.65
15. –0.02
16. 0.256
17. 0.0175
18. –0.84
19. 0.267
20. 0.866
21. Priscilla measures a paper clip. She decides that it is six-eighths of an inch long. Otis measures the same paper clip with a different ruler and says it is twelve-sixteenths of an inch long. How can their different answers be explained? 60
Section 2.1 — Rational Numbers
Decimals Gr ea ter than 1 Become Impr oper F Grea eater Improper Frractions
Don’t forget: A proper fraction is a fraction whose numerator is smaller than its denominator. For example:
1 2
and
9 . 10
An improper fraction is a fraction whose numerator is equal to or larger than its denominator. For example:
3 2
and
27 . 4
When you convert a decimal number greater than 1 into a fraction it’s probably easier to change it into a mixed number first. Then you can change the mixed number into an improper fraction. Example
5
Convert 13.7 into a fraction. Solution
• Convert 0.7 first — this becomes
7 . 10 7
Don’t forget:
• Add on the 13. This can be written as 13 10 .
A mixed number is a number made up of an integer and a fraction. For example:
• Now turn 13 10 into an improper fraction.
1 2
2 3
4 5
1 , 2 , and 10 .
A mixed number.
7
13 whole units are equivalent to
13 10 130 i = . 1 10 10
⎛13 10 ⎞⎟ 7 130 7 137 ⎜⎜ • ⎟ + = + = ⎜⎝ 1 10 ⎟⎠ 10 10 10 10
So add
7 10
to this:
A quicker way of doing this is: (13i10) + 7 137 = 10 10
Guided Practice Convert the decimals in Exercises 22–33 into fractions without using a calculator. 22. 4.3
23. –1.03
24. 15.98
25. –1.7
26. 9.7
27. –4.5
28. 12.904
29. –13.142
30. –8.217
31. 0.3627
32. 1.8028
33. 4.1234
Independent Practice Now try these: Lesson 2.1.2 additional questions — p435
Convert the decimals given in Exercises 1–20 to fractions in their simplest form. 1. 0.3
2. 0.2
3. 0.4
4. 0.30
5. 0.26
6. 0.18
7. –0.34
8. –1.34
9. 0.234
10. 2.234
11. 9.140
12. 3.655
13. –0.121
14. –0.655
15. –10.760
16. 5.001
17. 0.2985
18. 2.3222
19. –9.3452
20. –0.2400
Round Up The important thing when converting a decimal to a fraction is to think about the place value of the last digit. Then read the decimal and turn it into a fraction. If the decimal is greater than 1, ignore the whole number until you get the decimal part figured out. Take your time, do each step carefully, and you should be OK. Section 2.1 — Rational Numbers
61
Lesson
2.1.3
Con ver ting R epea ting Conv erting Re peating Decimals to F Frractions
California Standards: You’ve seen how to convert a terminating decimal into a fraction. But repeating decimals are also rational numbers, so they can be represented as fractions too. That’s what this Lesson is all about — taking a repeating decimal and finding a fraction with the same value.
Number Sense 1.5 Kno w tha ver y rra a tional Know thatt e ev ery number is either a ter mina ting or a rre epea ting termina minating peating decimal and be a ble to ab con ver conv ertt terminating decimals into rreduced educed fr actions fractions actions..
Repea ting Decimals Can Be “Subtr acted Away” peating “Subtracted
What it means for you:
Look at the decimal 0.33333..., or 0. 3 .
You’ll see how to change repeating decimals into fractions that have the same value.
Key words: • fraction • decimal • repeating
If you multiply it by 10, you get 3.33333..., or 3. 3 . In both these numbers, the digits after the decimal point are the same. So if you subtract one from the other, the decimal part of the number “disappears.” Example
1
Find 3. 3 − 0. 3 . Solution
Don’t forget: To multiply a decimal by 10, move the decimal point one place to the right. So 0.3333... × 10 = 3.3333...
The digits after the decimal point in both these numbers are the same, since 0. 3 = 0.3333... and 3. 3 = 3.3333... So when you subtract the numbers, the result has no digits after the decimal point. 3.3 3.3333...
−0.3333... 3.0000...
or
−0.3 3.0
So 3.3 − 0.3 = 3 . This idea of getting repeating decimals to “disappear” by subtracting is used when you convert a repeating decimal to a fraction. Example
2
If x = 0. 3 , find: (i) 10x, and (ii) 9x. Use your results to write x as a fraction in its simplest form. Solution
Don’t forget: The greatest common factor of 3 and 9 is 3. So: 3 3÷3 1 = = 9 9÷3 3
62
(i) 10x = 10 × 0. 3 = 3. 3 . (ii) 9x = 10x – x = 3. 3 − 0. 3 = 3 (from Example 1 above). You now know that 9x = 3. So you can divide both sides by 9 to find x as a fraction: 3 1 x = , which can be simplified to x = . 9 3
Section 2.1 — Rational Numbers
Guided Practice In Exercises 1–3, use x = 0.4 . 1. Find 10x. 2. Use your answer to Exercise 1 to find 9x. 3. Write x as a fraction in its simplest form. In Exercises 4–6, use y = 1. 2 . 4. Find 10y. 5. Use your answer to Exercise 4 to find 9y. 6. Write y as a fraction in its simplest form. Convert the numbers in Exercises 7–9 to fractions. 8. 4. 1 9. −2. 5 7. 2. 5
You Ma y Need to Multipl yb y 100 or 1000 or 10,000... May Multiply by If two digits are repeated forever, then multiply by 100 before subtracting. Example
3
Convert 0.23 to a fraction. Solution
Call the number x. There are two repeating digits in x, so you need to multiply by 100 before subtracting.
Don’t forget: This means that
0.23 =
23 99
100x = 23.23 Now subtract: 100x – x = 23.23 – 0.23 = 23. 23 So 99x = 23, which means that x = . 99 If three digits are repeated forever, then multiply by 1000, and so on. Example
Check it out:
4
n
In fact, you multiply by 10 , where n is the number of repeating digits. So where there’s one repeating digit, you multiply by 101 = 10. Where there are two repeating digits, you multiply by 102 = 10 × 10 = 100, and so on.
Convert 1.728 to a fraction in its simplest form. Solution
Call the fraction y. There are three repeating digits in y, so you need to multiply by 1000 before subtracting. 1000y = 1728.728 Now subtract: 1000y – y = 1728.728 – 1.728 = 1727. So 999y = 1727, which means that y =
1727 . 999
Section 2.1 — Rational Numbers
63
Guided Practice For Exercises 10–15, write each repeating decimal as a fraction in its simplest form. 10. 0.09 11. 0.18 12. 0.909 13. 0.123 14. 2.12 15. 0.1234
The Numer a tor and Denomina tor Must Be Inte ger s Numera Denominator Integ ers You won’t always get a whole number as the result of the subtraction. If this happens, you may need to multiply the numerator and denominator of the fraction to make sure they are both integers. Example Check it out: Make the repeating digits line up to make the subtraction easier (and remember that the decimal points have to line up too). So write 10x as 34.33 rather than 34.3 . Then the subtraction becomes: 34.33 −3.43 30.9 Which is an easier subtraction to do than 34.3 − 3.43 .
5
Convert 3.43 to a fraction. Solution
Call the number x. There is one repeating digit in x, so multiply by 10. 10x = 34.33 (using 34.33 rather than 34.3 makes the subtraction easier). Subtract as usual: 10x – x = 34.33 − 3.43 = 30.9 . 30.9 . So 9x = 30.9, which means that x = 9 But the numerator here isn’t an integer, so multiply the numerator and denominator by 10 to get an equivalent fraction of the same value.
x=
30.9×10 309 103 = , or more simply, x = 9×10 90 30
Guided Practice For Exercises 16–18, write each repeating decimal as a fraction in its simplest form. 16. 1.12 17. 2.334 18. 0.54321
Independent Practice Now try these: Lesson 2.1.3 additional questions — p435
Convert the numbers in Exercises 1–9 to fractions. Give your answers in their simplest form. 1. 0. 8 2. 0. 7 3. 1. 1 4. 0.26 5. 4.87 6. 0.246 7. 0.142857 8. 3.142854 9. 10.0 1
Round Up This is a really handy 3-step method — (i) multiply by 10, 100, 1000, or whatever, (ii) subtract the original number, and (iii) divide to form your fraction. 64
Section 2.1 — Rational Numbers
Lesson
Section 2.2
2.2.1
Absolute Value
California Standards:
You can think of the number “–5” as having two parts — a negative sign that tells you it’s less than zero, and “5,” which tells you its size, or how far from zero it is. The absolute value of a number is just its size — it’s not affected by whether it’s greater or less than zero.
Number Sense 2.5 Understand the meaning of the absolute value of a number; interpret the absolute value as the distance of the number from zero on a number line; and determine the absolute value of real numbers.
What it means for you: You’ll learn how to find the absolute value of a number, and use it in calculations.
Key words: • absolute value • distance • opposite
Absolute Value is Distance from Zero
The absolute value of a number is its distance from 0 on the number line. The absolute value of a number is never negative — that’s because the absolute value describes how far the number is from zero on the number line. It doesn’t matter if the number is to the left or to the right of zero — the distance can’t be negative.
Opposites Have the Same Absolute Value Opposites are numbers that are the same distance from 0, but going in opposite directions. Opposites have the same absolute value. –5 and 5 are opposites:
Check it out: Because you want to know the distance from the origin but don’t care about direction, the absolute value of a number is always just the number without its original sign.
Distance of 5
–6 –5 –4 –3 –2 –1 0
Distance of 5
1
2
3
4
5
6
So they each have an absolute value of 5. A set of bars, | |, are used to represent absolute value. So the expression |–10| means “the absolute value of negative ten.” Example
1
What is |3.25|? What is |–3.25|? Solution
3.25 and –3.25 are opposites. They’re the same distance from 0, so they have the same absolute value. Distance of 3.25
Distance of 3.25
Check it out: A number and its opposite (additive inverse) always have the same absolute value. For example: | 2 | = | –2 | = 2
–6 –5 –4 –3 –2 –1 0
1
2
3
4
5
6
So, | 3.25|| = | –3.25|| = 3.25
Section 2.2 — Absolute Value
65
Guided Practice Find the values of the expressions in Exercises 1–8. 1. |12|
2. |–9|
3. |16|
4. |–1|
5. |1.7|
6. |–3.2|
7. |– 2 |
1
8. |0|
In Exercises 9–12, say which is bigger. Check it out:
9. |17| or |16|
10. |–2| or |–5|
Find the value of each absolute value expression, then compare them.
11. |–9| or |8|
12. |–1| or |1|
Absolute Value Equations Often Have Two Solutions Think about the equation |x| = 2. The absolute value of x is 2, so you know that x is 2 units away from 0 on the number line, but you don’t know in which direction. x could be 2, but it could also be –2. You can show the two possibilities like this:
–2 –1
Example
2
Solve |z| = 3. Solution
z can be either 3 or –3. 3 units
–6 –5 –4 –3 –2 –1 0
3 units
1
2
3
4
5
6
Guided Practice Give the solutions to the equations in Exercises 13–16. 13. |a| = 1 14. |r| = 4 15. |q| = 6 16. |g| = 7
Treat Absolute Value Signs Like Parentheses You should treat absolute value bars like parentheses when you’re deciding what order to do the operations in. Work out what’s inside them first, then take the absolute value of that.
66
Section 2.2 — Absolute Value
Example
3
What is the value of |7 – 3| + |4 – 6|? Solution
|7 – 3| + |4 – 6| = |4| + |–2| Simplify whatever is inside the absolute value signs =4+2 Find the absolute values =6 Simplify the expression
Guided Practice Evaluate the expressions in Exercises 17–22. Check it out: The absolute value bars only affect whatever’s inside them. So the value of the whole expression can be negative. For example –| a + 7 | will be negative, unless a = –7.
17. |1 – 3| – |2 + 2|
18. |2 – 7| + |0 – 6|
19. –|5 – 6|
20. |–8| × |2 – 3|
21. 2 × |4 – 6|
22. |7 – 2| ÷ |1 – 6|
Independent Practice Evaluate the expressions in Exercises 1–4? 1. |–45|
2. |6|
5 6
3. |–0.6|
4. | |
5. Let x and y be two integers. The absolute value of y is larger than the absolute value of x. Which of the two integers is further from 0? Show the solutions of the equations in Exercises 6–13 on number lines. 6. | u | = 3
7. | d | = 9
8. | x | = 5
9. | w | = 15
10. | y | = 4
11. | v | = 1
12. | k | = 16
13. | z | = 7
In Exercises 14–19, say which is bigger. 14. |–6| or |–1| 15. |3| or |–5|
16. |2 – 2| or |5 – 8|
17. |6 – 8| or |2 – 1|
19. |11 + 1| or |–2 – 8|
18. |3 – 2| or |–5|
Evaluate the expressions in Exercises 20–25. Now try these:
20. |3 – 5| + |2 – 5|
21. |0 + 5| + |0 – 5|
22. |5 – 10| – |0 – 2|
Lesson 2.2.1 additional questions — p436
23. |–1| × |3 – 3|
24. 8 × |1 – 4|
25. |2 – 8| ÷ |4 – 1|
26. What is the sum of two different numbers that have the same absolute value? Explain your answer. 27. Is it always true that | y | < 2y when y is an integer?
Round Up The absolute value of a number is its distance from zero on a number line. Absolute values are always positive. So if a number has a negative sign, get rid of it; if it doesn’t, then leave it alone. If you see absolute value bars in an expression, work out what’s between them first — just like parentheses. Section 2.2 — Absolute Value
67
Lesson
2.2.2
Using Absolute Value
California Standards:
You use absolute value a lot in real life — often without even thinking about it. For example, if the temperature falls from 3 °C to –3 °C you might use it to find the overall change. This Lesson looks at some of the ways that absolute value can apply to everyday situations.
Number Sense 2.5 Understand the meaning of the absolute value of a number; interpret the absolute value as the distance of the number from zero on a number line; and determine the absolute value of real numbers. Algebra and Functions 1.1 Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represents a verbal description (e.g., three less than a number, half as large as area A).
What it means for you: You’ll use absolute value to find the difference between two numbers and see how absolute values apply to real-life situations.
Key words: • absolute value • comparison • difference
Absolute Values Help Find Distances Between Numbers To find the distance between two numbers on the number line you could count the number of spaces between them.
2
6
Using subtraction is a quicker way — just subtract the lesser number from the greater. 8 – 6 = 2, so 8 and 6 are 2 units apart.
7
8
If you did the subtraction the other way around you’d get a negative number — and distances can’t be negative. But if you use absolute value bars you can do the subtraction in either order and you’ll always get a positive value for the distance. | 6 – 8|| = | –2|| = 2 and ||8 – 6|| = 2
For any numbers a and b: The distance between a and b on the number line is |a – b|. This is particularly useful when you’re finding the distance between a positive and negative number. Example
1
What is the distance between –3 and 5? Solution
The distance between –3 and 5 is | –3 – 5 | = | –8 | = 8. Check it out: If |a – b| is small then a and b are close to each other. If |a – b| is large then a and b are far away from each other. For example, –1 and –3 are closer together than 6 and –6, so you’d expect |(–1) – (–3) | to be smaller than |6 – (–6)|.
Distance of 8
–6 –5 –4 –3 –2 –1 0
1
2
3
4
5
6
Guided Practice Find the distance between the numbers given in each of Exercises 1–8. 1. 1, 5 2. –3, –8 3. 6, –9 4. –1, 10 5. 3, –5 6. 5, –1 7. –1.2, 2.3 8. –0.3, 2.7 9. At 1 p.m., Amanda was 8 miles east of her home. She then traveled in a straight line west until she was 6 miles west of her home. How many miles did she travel?
68
Section 2.2 — Absolute Value
Absolute Values are Used to Compare Things You can use absolute values to compare numbers when it doesn’t matter which side of a fixed point they are. Example
2
Find how far point A is above point B.
Point A = 50 m (above sea level)
Sea level = 0 m Point B = –35 m (below sea level)
Solution
It doesn’t matter that B is below sea level and A is above. It’s the distance between them that’s important. You can find this by working out | 50 m – (–35 m)|| = | 85 m|| = 85 m. You’d get the same answer if you did the subtraction the other way around: | –35 m – 50 m|| = | –85 m|| = 85 m.
Guided Practice 10. A miner digs the shaft shown on the right. What distance was he from the top of the crane when he finished digging?
20 ft.
11. The top of Mount Whitney is 14,505 ft above sea level. The bottom of Death Valley is 282 ft below sea level. How much higher is the top of Mount Whitney than the bottom of Death Valley?
0 ft. –10 ft. –20 ft. –30 ft. –40 ft. –50 ft.
Absolute Values Can Describe Limits Another use of absolute values is to describe the acceptable limits of a measurement. It might not be important whether something is above or below a set value, but how far above or below it is. Example
Check it out: If your body temperature goes too far from the normal value in either direction it can be dangerous, so this is an important use of absolute value.
3
The average temperature of the human body is 98.6 °F, but in a healthy person it can be up to 1.4 °F higher or lower. The difference between a person’s temperature, x, and the average healthy temperature can be found using the expression | 98.6 – x |. Aaron is feeling unwell so measures his temperature. It is 100.2 °F. Is Aaron’s temperature within the healthy range? Solution
The difference between Aaron’s temperature and the average healthy temperature is | 98.6 – 100.2 | = | –1.6 | = 1.6 °F. Aaron’s temperature is outside of the normal healthy range.
Section 2.2 — Absolute Value
69
Example Check it out: Lots of manufacturers have tolerance limits for measurements. A tolerance limit is how different in size something is allowed to be from the size it should be. For example, size 6 knitting needles should have a diameter of 4 mm. Only very small differences from this size, such as 0.004 mm, will be allowed.
4
A factory manufactures wheels that it advertises as no more than 1 inch away from 30 inches in diameter. They use the expression |30 – d| to test whether wheels are within the advertised size (where d is the diameter). Apply the expression to wheels of diameters 31, 29, and 35 inches, and say whether they meet the advertised standard. Solution
• Wheel of diameter 31 inches: | 30 – 31|| = ||–1|| = 1 inch. This wheel is within the standard. • Wheel of diameter 29 inches: | 30 – 29|| = ||1|| = 1 inch. This wheel is within the standard. • Wheel of diameter 35 inches: | 30 – 35|| = ||–5|| = 5 inches. This wheel is not within the standard.
Guided Practice 12. The height of a cupboard door should be no more than 0.05 cm away from 40 cm. The expression |40 – h| is used to check whether a door of height h cm fits the size requirement. If a door measures 40.049 cm, is it within the correct range? 13. Ms. Valesquez’s car needs a tire pressure, p, of 30 psi. It should be within 0.5 psi of the recommended value. She uses the expression |30 – p| to test whether the pressure is acceptable. Is a pressure of 29.4 psi acceptable?
Independent Practice Find the distance between the numbers given in each of Exercises 1–4. 1. 9, –9 2. 3, –4 3. 5, 6 4. –32, –52 5. The table below shows the temperature at different times of day. How much did the temperature change by between 7 a.m. and 8 a.m.? Check it out: It might help you to draw diagrams to go with Exercises 5 and 6.
Now try these: Lesson 2.2.2 additional questions — p436
Time
7 a.m.
8 a.m.
Temperature
–5 °C
1 °C
6. A person stands on a pier fishing. The top of their rod is 20 feet above sea level. The line goes vertically down and hooks a fish 13 feet below sea level. How long is the line? 7. Priscilla tries to keep the balance of her checking account, b, always less than $50 away from $200. She uses the expression |200 – b| to check that it is within these limits. Is a balance of $242.50 acceptable?
Round Up Absolute values are used to find distances between numbers. They’re also useful when measurements are only allowed to be a certain distance away from a set value. In these situations, it doesn’t matter if the numbers are above or below the set point — it’s how far away they are that’s important. 70
Section 2.2 — Absolute Value
Lesson
2.3.1
Section 2.3
Ad ding and Subtr acting Adding Subtracting Inte ger s and Decimals Integ ers
California Standards: Number Sense 1.2 Add, subtract, Ad d, subtr act, multiply, and umber s number umbers divide ra tional n inte ger s , fractions, and integ ers (inte ter mina ting decimals termina minating decimals) and take positive rational numbers to whole-number powers.
What it means for you: You’ll practice adding and subtracting integers on a number line, and then you’ll go on to working with decimals.
You’ve had plenty of practice adding and subtracting integers in earlier grades — and you’ve probably done some adding and subtracting of decimals too. The same rules apply to decimals — you just have to be careful where you put the decimal point.
Ad d Inte ger s Using a Number Line Add Integ ers A number line is a really good way to show integers in order. Adding and subtracting just involves counting left or right on the number line. Example
1
Calculate 10 + 2 using a number line. Key words:
Solution
Find the first number on the number line.
• integer • decimal
8
Don’t forget: Integers are all the numbers that don’t involve a decimal or a fraction. They can be positive or negative.
9 10 11 12 13 14
Count the number of positions given by the second number. Go right if it’s a positive number.
8
9 10 11 12 13 14
Then just read off the number you’ve ended up at. So 10 + 2 = 12 If you’re adding a negative number, you count to the left. Example
2
Calculate 2 + (–3) using a number line. Solution
So 2 + (–3) = –1 –4 –3 –2 –1 0
1
2
3
Subtractions can be turned into additions — then you can use the methods above for subtractions. Subtracting a positive number is the same as adding a negative one. For example, 5 – 6 = 5 + (–6) And subtracting a negative number is the same as adding a positive one. For example, 5 – (–6) = 5 + 6 Example
3
Calculate: (i) 2 – 3
(ii) 10 – (–2)
Solution
(i) 2 – 3 is the same as 2 + (–3). So 2 – 3 = –1 (using Example 2). (ii) 10 – (–2) equals 10 + 2. So 10 – (–2) = 12 (using Example 1). Section 2.3 — Operations on Rational Numbers
71
Guided Practice Don’t forget: a–b a – (–b)
a + (–b) a+b
Use a number line to do the calculations in Exercises 1–8. 1. 10 + 3 2. 6 + 1 3. –3 + 5 4. –9 – 7 5. 7 + (–6) 6. 19 – (–5) 7. 3 + (–2) 8. –1 – (–9)
Decimal Ad dition Needs a Decimal Number Line Addition Addition with decimals isn’t any tougher than with integers — you’ve just got to remember to draw a number line that includes decimals. Example
4
Calculate 0.9 – 0.3 using a number line. Solution
0.9 – 0.3 is a subtraction. But because subtracting a positive number is the same as adding a negative one, it can be written as: 0.9 – 0.3 = 0.9 + (–0.3) You’re dealing with decimals, so you need a number line that shows decimal values. Find 0.9, and count 0.3 to the left because you’re adding a negative number. So 0.9 – 0.3 = 0.6
0.5 0.6 0.7 0.8 0.9 1.0
Guided Practice Use a number line to do the calculations in Exercises 9–17. 9. 0.6 + 0.4 10. 0.1 + 0.8 11. 1.3 + 0.7 12. 2.3 + (–0.4) 13. 3.1 – 0.7 14. –0.7 – (–0.4) 15. –0.9 – 0.3 16. –1.2 – (–0.5) 17. 0.3 – 0.5
You Don ways Ha ve to Use a Number Line Don’’ t Al Alw Hav It’s hard to add big numbers on a number line, so you need to be able to add and subtract without a number line. Example
5
Calculate 432 + 34 without using a number line. Solution
You can break the calculation down into hundreds, tens, and ones.
432 + 34
Four hundreds Three Two tens ones
You need to add the ones of each number together, then the tens, and so on. Write the numbers on top of each other with the ones lined up. 432 Work out the sum of the ones column first, +34 then the tens, then the hundreds.
466
So 432 + 34 = 466 72
Section 2.3 — Operations on Rational Numbers
Three Four tens ones
432 +34
The last example was easier than some calculations because all the column sums came to less than 10. If they go over 10 then you have to carry the extra numbers to the next column. Example
6
Calculate 567 + 125. Solution
First you need to write the sum out vertically with 567 the ones, the tens, and the hundreds lined up. +125
567 +125 1 2
10 is the same as 1 ten — so carry a 1 to the tens column
The ones calculation is 7 + 5 = 12. 12 is the same as saying “one ten and two ones” — so write 2 under the ones, and carry the 10 to the tens column.
When you add up the next column, remember to add the 1 that you carried. So 567 + 125 = 692
567 +125 1 692
So the tens column is now 6+2+1=9
You can use a similar method for decimals, but there are a few things to remember. The digits after a decimal point show parts of a whole — so 24.56 means “2 tens, 4 ones, 5 tenths, and 6 hundredths.” Don’t forget: “Hundreds,” “tens,” “ones,” “tenths,” “hundredths,” and so on are called place values.
two tens
four units
six hundredths
24.560 five tenths
zero thousandths
You can also add extra zeros onto the end of a decimal and it doesn’t change the value.
You should always only add digits with the same place values. So when you’re adding decimals, line the values up by the decimal point. Example
7
Calculate 13.93 + 5.2. Solution
Write the sum out vertically with the decimal points lined up.
Make sure the decimal points are lined up
13.93 + 5.20
Add zeros to get the same number of decimal places
Then work out the sum of each column in turn, 13.93 starting with the right-hand side. Don’t forget + 5.20 Carry numbers to 1 the next column 19.13 the decimal point in the answer. on the left just like before. So 13.93 + 5.2 = 19.13
Guided Practice Calculate the following sums without using a number line. 18. 210 + 643 19. 613 + 117 20. 1264 + 527 21. 33.7 + 12.4 22. 14.8 + 16.2 23. 55.82 + 34.81 24. 75.1 + 14.31 25. 62.4 + 31.99 26. 2.29 + 9.92 Section 2.3 — Operations on Rational Numbers
73
Some Subtr actions In volv e “Bor Subtractions Inv olve “Borrr o wing” Doing column subtractions seems tough if the top number in a column is smaller than the number underneath — but there’s a handy method. Example
8
Calculate 30 – 18. Solution
30 If you write this in columns, the upper number in –18 the ones column is smaller than the number below. You can break down 30 into different parts:
30 = 20 + 10 Don’t forget: If you “borrow” a 10 when you’re subtracting, remember that the number you borrowed from will now be 1 ten lower.
You could say this as “2 tens plus 1 ten” or you could say “2 tens plus 10 ones.”
All you’ve done is separate one of the tens from the number. You can show this in column notation: 2 10 “Borrow” 10 from the So 30 – 18 = 12 Example
30 –1 8 12
tens column, so now the column subtractions are 10 – 8 and 2 – 1.
9
Calculate 65.37 – 31.5. Solution
65.37 Add a zero to make them the same number of decimal places: – 31.50
If the top number in a column is smaller than the bottom one, use the same method of “borrowing” from the next column.
4 13
65.37 – 31.50 33.87
You need to borrow from the ones column. A one is ten tenths — so the 3 in the tenths column becomes 13.
So 65.37 – 31.5 = 33.87
Guided Practice Calculate the following sums without using a number line. 27. 50 – 26 28. 62 – 18 29. 76 – 49 30. 941 – 46 31. 62.4 – 31.2 32. 68.83 – 11.3 33. 29.42 – 13.18 34. 46.11 – 21.95 35. 42.38 – 36.45
Independent Practice Now try these: Lesson 2.3.1 additional questions — p436
Draw number lines to answer Exercises 1–4. 1. 7 + 9 2. 7 – (–9) 3. 9 + 7
4. –9 – 7
Calculate the following without using a number line. 5. 0.99 + 0.45 6. 1.86 + 3.33 7. 15.64 + 3.67 8. 45.64 + 13.88 9. 164.31 + 251.3 10. 32.12 – 12.1 11. 112.13 – 38.7 12. 19.4 – 5.37 13. 64.11 – 44.7
Round Up Adding and subtracting integers and decimals — it sounds like a lot to learn, but you use the same methods over and over. You’ll need to add and subtract decimals when you’re doing real-life math too. 74
Section 2.3 — Operations on Rational Numbers
Lesson
2.3.2
Multipl ying and Di viding Multiplying Dividing Inte ger s Integ ers
California Standards: Number Sense 1.2 y, and Add, subtract, multipl ultiply di vide rra a tional n umber s divide number umbers inte ger s , fractions, and (inte integ ers terminating decimals) and take positive rational numbers to whole-number powers.
What it means for you: You’ll practice multiplying and dividing integers on a number line, and then using other methods.
Multiplying and dividing are important skills, both in math and real life. In this Lesson you’ll see how multiplication and division can be modeled.
Pictur e Multiplica tion and Di vision on a Number Line Picture Multiplication Division A number line is just a way of showing the order of numbers — so you can use it for any kind of calculation, including multiplication and division. Multiplying by positive integers looks like a set of “hops” away from zero. Example
1
Calculate using the number line: Key words:
Solution –2
3 × (–2) –2
(i) 2 × 4, (ii) 3 × (–2) 2×4
–2
4
4
• integer
–6 –5 –4 –3 –2 –1
0 1
2
3 4
5 6 7
8 9
(i) Multiplying 4 by 2 means you need to move 2 groups of 4 away from zero. So 2 × 4 = 8 (ii) Multiplying –2 by 3 means you need to move 3 groups of –2 from zero. So 3 × (–2) = –6 Dividing by a positive integer looks like you’re breaking a number down into equally sized parts. Example
2
Calculate using the number line: Solution –2
–6 ÷ 3 –2
(i) 8 ÷ 2, (ii) –6 ÷ 3 8÷2
–2
–6 –5 –4 –3 –2 –1
4
0 1
2
4
3 4
5 6 7
8 9
(i) Dividing 8 by 2 means finding 8 on the number line, and then splitting the distance between 8 and 0 into 2 equal parts. So 8 ÷ 2 = 4
Don’t forget: You learned about the rules for multiplying by positive and negative numbers in grade 6.
Don’t forget: The “sign” of a number just means whether it is positive or negative.
(ii) Dividing –6 by 3 means you find –6 on the number line, and then split the distance between –6 and 0 into 3 equal parts. So –6 ÷ 3 = –2 Sometimes it’s hard to tell whether your answer will be positive or negative. The table below shows what the sign of the answer will be: positive ×/÷ positive = positive positive ×/÷ negative = negative negative ×/÷ positive = negative negative ×/÷ negative = positive
So for example, 8 ÷ (–2) = –4 while –2 × (–4) = 8
Section 2.3 — Operations on Rational Numbers
75
Guided Practice Use a number line to work out the problems given in Exercises 1–12. 1. 4 × 3 2. 2 × (–5) 3. –4 × 6 4. –4 × (–3) 5. –6 × 4 6. –3 × (–3) 7. 20 ÷ 5 8. 18 ÷ 6 9. 22 ÷ 2 10. –15 ÷ 3 11. 16 ÷ (–4) 12. –14 ÷ (–7)
You Must Kno w Ho w to Multipl y Without a Calcula tor Know How Multiply Calculator You can only really use a number line for simple multiplications. Here are two good ways of multiplying any two numbers together: Example
3
Calculate 12 × 14. Solution
Picture each number as the length of a side of a rectangle — but break each number down into Total length = 14 tens and ones. Then you can do the 10 4 multiplication “part by part.” Total The total area of the rectangle is 10 40 100 12 × 14, and you can see this equals length = 12 100 + 40 + 20 + 8 = 168 20 8 2
4 × 10
4×2
Another method is to write the numbers on top of each other.
14 × 12
First write the numbers as a vertical calculation.
28 + 140
Multiply the top number by the ones digit of the bottom number. 2 × 14 = 28 Then multiply the top number by the tens of the bottom number: 10 × 14 = 140
168
Then add them together, just like in the “rectangle” method. This is known as long multiplication. Notice how the above two methods are very similar. The first line of work in the long multiplication is the same as the area of the bottom part of the rectangle, and the second line of work in the long multiplication is the same as the area of the top part of the rectangle. In both methods you then add these together to get the overall result.
Guided Practice Use the methods in Example 3 for Exercises 13–20. 13. 12 × 13 14. 22 × 16 15. 32 × 18 16. –14 × 37 17. –46 × 42 18. 25 × 58 19. 52 × 67 20. 85 × 95 76
Section 2.3 — Operations on Rational Numbers
Wor k Thr ough Long Di vision fr om Left to Right ork hrough Division from Long division is a good way of writing down and solving tricky division problems. It involves dividing big numbers bit by bit, by breaking them into collections of smaller numbers. Check it out: Long division means breaking a division down into small problems like figuring out that 9 × 5 goes into 46 (with 1 left over). But, in Example 4, remember that the 4 is in the hundreds column and the 6 is in the tens column, so you’re really working out that 90 × 5 goes into 460 (with 10 left over). Long division is just a way of breaking up complicated division problems.
Example
4
Calculate 467 ÷ 5.
9 4 5 6 7 You need to divide the whole of – 45 467 by 5, but you can work through it bit by bit. 17
Solution
The number you’re dividing by goes on the left. Work from the left to find numbers that divide by 5.
93 5 467 – 450 17 Keep going until you’ve divided –15 the whole number. 2
The answer to this division is 93 with remainder 2. The standard way of writing this is 467 = (93 × 5) + 2.
5 doesn't divide into 4, but it divides into 46 (this is actually 460). The maximum number of times 5 divides into 46 is 9, so put 9 in the tens column. You now know that the answer is in the 90s. 9 × 5 = 45, so put 45 underneath, lined up with 46. Subtract 45 from 46, and bring the 7 down to find the amount left. There's 17 remaining. Now divide 17 by 5. 5 goes into 17 a maximum of 3 times. Write this in the units column.
3 × 5 = 15, so put 15 underneath, lining up the units. There's just 2 left. 5 won't go into 2, so it's a remainder.
9 3 R2 5 467
Guided Practice Write out and solve these calculations using long division. 21. 72 ÷ 6 22. 104 ÷ 4 23. 105 ÷ 7 24. 274 ÷ 13 25. 1955 ÷ 8 26. 5366 ÷ 13
Independent Practice Check it out: A quotient is what you get when you divide one number by another. A product is what you get when you multiply numbers.
Now try these: Lesson 2.3.2 additional questions — p437
Show the multiplications in Exercises 1–3 on a number line. 1. 3 × 1 2. 2 × 5 3. 4 × –2 Use the area method to find the products in Exercises 4–6. 4. 13 × 15 5. 12 × 65 6. 33 × 56 Find the products in Exercises 7–12. 7. 11 × 18 8. 13 × 22
9. 25 × 21
10. –33 × 12
12. –23 × –51
11. 16 × –15
Find the quotients in Exercises 13–18. 13. 710 ÷ 5 14. 138 ÷ 6
15. 1248 ÷ –8
16. 190 ÷ 3
18. 172 ÷ 5
17. 274 ÷ 4
Round Up Multiplying and dividing large numbers without a calculator seems like quite a task. But if you break the numbers up into ones, tens, and hundreds, then they’re much simpler to handle. Section 2.3 — Operations on Rational Numbers
77
Lesson
2.3.3
Multipl ying F Multiplying Frr actions
California Standards:
You’ve multiplied fractions in other grades — but it’s still not an easy topic. In this Lesson you’ll get plenty more practice at it.
Number Sense 1.2 y , and Add, subtract, m ultipl ultiply umber s divide ra tional n number umbers actions (integers, fr fractions actions, and terminating decimals) and take positive rational numbers to whole-number powers.
What it means for you:
Multiplying a fraction by another fraction means working out parts of a part. For example,
×
2 3
means “one-fifth of two-thirds.”
You can show this graphically — it’s called an area model. You need to start by drawing a rectangle. 1 • Shade in
1 5
5
of the rectangle in one direction:
Key words: • area model • mixed numbers
1 5
{
You’ll practice multiplying fractions, and you’ll extend this to multiplying fractions by integers and mixed numbers.
Ar ea Models Sho w F tion Area Show Frraction Multiplica Multiplication
• Then shade in
2 3
of it in the other direction,
2 3
using a different color: 1 5
The part showing
×
2 3
{
is the part that represents one-fifth of two-thirds
— this is the part that’s shaded in both colors. There are 2 squares shaded out of a total of 15, so Example Calculate
3 4
1 5
×
2 3
=
2 15
.
1 ×
1 3
using the area model method.
Solution
You need to work out three-quarters of one-third — so shade in the rectangle in one direction, and
1 3
in the other direction:
The answer here is correct, although it could be simplified. See the next page (or Section 2.1) for more information.
{
Check it out:
3 4
1 3
There are 3 out of 12 squares shaded in both colors, so
3 4
×
1 3
=
3 12
{
.
Guided Practice Calculate these fraction multiplications by drawing area models: 1.
78
1 3
×
2 5
Section 2.3 — Operations on Rational Numbers
2.
3 4
×
1 5
3.
3 4
×
1 6
3 4
of
You Can Multipl y F awing Dia g rams Multiply Frractions Without Dr Dra Diag Don’t forget:
5
denominator
You’ve already seen the area model for
1 5
×
2 3
1 5
{
The top number in a fraction is called the numerator, and the bottom number is called the denominator. 2 numerator
When you draw an area model, the total number of squares is always the same as the product of the denominators of the fractions you’re multiplying. :
2 3
Multiply the denominators: • The total number of squares is 5 × 3 = 15.
{
Also, the number of squares shaded in both colors is always the same as the product of the numerators. Multiply the numerators: • The total number of squares shaded in both colors is 1 × 2 = 2. That means you can work the product out without drawing the area model. Example Calculate
3 4
2 ×
1 3
without drawing a diagram.
Solution
Multiply the numerators: 3 × 1 = 3 Multiply the denominators: 4 × 3 = 12 Now write this as a fraction: 3
12
So
Don’t forget:
3 4
×
1 3
=
3 . 12
The numerator and denominator can both be divided by 3...
That means you could write
A fraction is in its simplest form if 1 is the only number that divides exactly into both the numerator and denominator.
denominator
The solution to Example 2 could be simplified a bit more. Simplifying just means writing the solution using smaller numbers, but so that the fraction still means the same thing.
You can simplify fractions using the greatest common factor (GCF) — there’s a lot more about this in Section 2.1.
Don’t forget:
numerator
3 4
÷3
3 12 ×
÷3 1 3
=
1 4
1
...so 4 represents the same amount, but simplified.
1 . 4
Guided Practice Calculate these fraction multiplications without drawing area models. Simplify your answer where possible. 4.
5 6
×
1 2
5.
2 3
×
1 3
6.
1 3
7.
3 8
×
1 4
8.
5 7
×
3 5
9.
11 12
× ×
6 7 6 7
Section 2.3 — Operations on Rational Numbers
79
Fir st Con ver ed Number s to F First Conv ertt Whole or Mix Mixed Numbers Frractions Don’t forget: A mixed number contains an integer and a fraction. 1
For example, 3 2 is a mixed number.
To multiply fractions by mixed numbers, you can just write out the mixed numbers as a single fraction and carry on multiplying as normal. The same is true if you need to multiply a fraction by an integer — you can write the integer as a fraction and use the multiplication method from before. Example
Don’t forget:
3
The whole number 3 is
Solution
6 2 equivalent to 3 × = 2 . 2
(i) Convert 3
2 2
1 2
Calculate: (i) 3 1 2
×
1 4
,
4 5
(ii) 1 2
×8
(3× 2) +1
7
= 2 2 Then just multiply out the fractions as normal:
is the same as one whole.
3
1 2
×
1 4
=
7 2
to a fraction: 3 =
×
1 4
=
7 8
(ii) The integer 8 can be written as
8 1 4 5
. 4 5
8 1
So you can multiply as normal: ×8 = × =
32 5
Guided Practice Calculate the following, and simplify your solutions where possible. 1 3
×
13. 3 ×
2 7
10. 1
Don’t forget: If you’re multiplying negative numbers, remember the rules on p75 for working out the sign of the answer.
Don’t forget: You should usually write answers as mixed numbers instead of improper fractions — this keeps the numbers simpler.
Now try these: Lesson 2.3.3 additional questions — p437
1 5
11.
1 3
14. 1
×2 1 4
×
1 3 1 5
2 3 5 15. 1 7
12. 1
×
2 3
×1
1 2
Independent Practice Find the product and simplify each calculation in Exercises 1–3. 1.
2 3
×
7 10
2. –
4 9
×
3 15
3. –
5 12
×2
1 3
4. A positive whole number is multiplied by a positive fraction smaller than one. Explain how the size of the product compares to the original whole number. 5. A rectangular patio measures 8 14 feet wide and 12 12 feet long. What is the area of the patio? 6. A recipe for 12 muffins calls for 3 14 cups of flour. How many cups of flour are needed to make 42 muffins?
Round Up Multiplying fractions is OK because you don’t need to put each fraction over the same denominator. If you need to multiply by integers or mixed numbers, just turn them into fractions too. 80
Section 2.3 — Operations on Rational Numbers
Lesson
2.3.4
California Standard:
Di viding F Dividing Frractions Dividing fractions is hard to grasp — but once you learn a couple of useful techniques it’ll all seem a lot easier. The most important thing to learn is how to use reciprocals.
Number Sense 1.2 Add, subtract, multiply, and di vide rra a tional n umber s divide number umbers actions (integers, fr fractions actions, and terminating decimals) and take positive rational numbers to whole-number powers.
You Can Sho w F vision on a Number Line Show Frraction Di Division
What it means for you:
out how many halves are in three.” You can show it on a number line.
You’ll learn about the reciprocal of a fraction and how to use it when dividing by fractions.
A problem like 3 ÷
1 2
1 2
1 2
0
1 2
1 2
1 2
1
can be hard to imagine. In words, it means “work
1 2
2
You can see that So 3 ÷
3
1 2
1 2
fits into 3 six times.
= 6.
Key words:
You can use a number line to show that the number of halves in any number will always be double that number.
• reciprocal • divisor
For example, there are 8 halves in 4, so 4 ÷
1 2
= 4 × 2 = 8.
Di viding is the Same as Multipl ying b y the R ecipr ocal Dividing Multiplying by Recipr eciprocal Don’t forget: The reciprocal (or multiplicative inverse) of a number is what you have to multiply the number by to get 1. There’s more about reciprocals in Lesson 1.1.4.
Multiplication and division are really closely linked. In fact, you can write any division problem as a multiplication problem using a reciprocal. Dividing by a number is the same as multiplying by its reciprocal. The product of a number and its reciprocal is 1. For example, the reciprocal of 2 is by
1 2
1 . 2
This means that dividing something
is the same as multiplying by 2.
Example
1 1 4
Calculate 3 ÷ . Solution
The reciprocal of
1 4
is 4.
So you can rewrite this as a multiplication: 3 ÷ 1 4
So 3 ÷
1 4
= 3 × 4 = 12.
= 12.
Guided Practice Don’t forget: Also, since the reciprocal of 1 2
is 2, dividing something by
2 is the same as multiplying by
Calculate these by converting each division problem into a multiplication problem. Give your solutions in their simplest form.
1 . 2
1. 3 ÷
1 3
2. 8 ÷
4. 3 ÷
3 5
5.
There’s more about
this on the next page.
5 6
1 4
÷5
3. 8 ÷ 6.
11 12
2 3
÷ 11
Section 2.3 — Operations on Rational Numbers
81
Solv e F ecipr ocals Too Solve Frraction ÷ F Frraction Using R Recipr eciprocals You can turn any division into a multiplication using the reciprocal of the divisor (the thing you’re dividing by). It makes dividing fractions by fractions much easier than it seems at first. Example Don’t forget: 1 9 and are reciprocals. 9 1 They multiply to give 1. 1 9 1× 9 9 × = = =1 9 1 9 ×1 9
Don’t forget: To multiply two fractions, you first multiply the numerators to get the numerator of your answer. You then multiply the two denominators, and this becomes the denominator of your answer. See the previous Lesson for more information.
Calculate
2 3
2 1 9
÷ .
Solution
Dividing by a number is the same as multiplying by its reciprocal. The reciprocal of
1 9
9 1
Fraction:
Reciprocal:
1 9
9 1
is 9, or . 2 3
9 1
So you need to work out × . This is
2 9 2× 9 18 × = = =6 3 1 3×1 3 2
1
To convince yourself that ÷ really does equal 6, remember that in 3 9 words, the problem means, “how many ninths are in two-thirds?” Look at the square on the right. Two-thirds of it has been colored in. The square has then been divided into ninths — there are six ninths in the colored two-thirds. In other words,
2 3
÷
1 9
1 9 1 9 1 9
= 6.
1 9 1 9 1 9
1 9 1 9 1 9
Guided Practice Find the reciprocal of the following fractions: 7.
1 5
8.
2 3
9.
5 7
Calculate Exercises 10–15 by converting each division problem into a multiplication problem. Give your solutions in their simplest form.
82
10.
1 3
÷
1 5
11.
2 5
÷
1 6
12.
13.
1 2
÷
3 8
14.
1 3
÷
9 1
15.
Section 2.3 — Operations on Rational Numbers
1 5
6 7
÷ ÷
3 4 1 4
Con ver ed Number s into F Conv ertt Mix Mixed Numbers Frractions Fir st First
If you have a division problem involving mixed numbers, you need to write them out as fractions before you start to divide.
Example
3 6 7
1 6
Calculate −4 ÷ 2 . Solution
Don’t forget: 7 1 can be written as . 7
So 4 is equal to 4 ×
7 . 7
First convert the mixed numbers to fractions. (4×7) + 6 (2×6) +1 13 34 6 1 =− = −4 = − and 2 = 7 6 7 7 6 6 Write the division as a multiplication using the reciprocal of the divisor. The divisor is the number that you’re dividing by. 34 13 34 6 ÷ =− × 7 6 7 13 Do the multiplication in the normal way. −4
−
6 1 ÷2 7 6
=−
34 6 34 ×6 204 22 × =− =− , or −2 91 7 13 7×13 91
Guided Practice Calculate the answers to the following divisions. Write each solution as a mixed number or an integer. 16. 4 ÷ 2
1 2
1 3
17. 5 ÷
4 9
1 2
18. −9 ÷−3
1 8
Independent Practice
Now try these: Lesson 2.3.4 additional questions — p437
Calculate the following. 3 1. ÷ 5 2. 4 1 1 4. ÷ 5. 2 3 1 1 7. 2 3 ÷ 8. 2
2 ÷7 3 2 3 ÷ 5 4 5 4÷ 6
9 ÷2 13 7 2 6. ÷ 9 18
3.
9. 8
9 1 ÷7 10 6
10. A one-pound bag of sugar is equal to 2 25 cups. A recipe for cornbread requires 23 of a cup of sugar. How many batches of cornbread can be made with the bag of sugar? 11. The area of a rectangular floor is 62 43 square feet. The length of the room is 10 14 feet. What is the width of the room?
Round Up The most important thing to remember here is that you can convert any fraction division into a multiplication using the reciprocal. Remember that little fact, and you’ll make your life a lot easier. Section 2.3 — Operations on Rational Numbers
83
Lesson
2.3.5
Common Denomina tor s Denominator tors
California Standards:
Adding and subtracting fractions isn’t always straightforward. Before you can add or subtract fractions, they need to be over a common denominator, and one way to find a common denominator is to find the least common multiple (LCM) of the denominators.
Number Sense 1.1 Read, write, and compare rational numbers in scientific notation (positive and negative powers of 10), compar e rra a tional n umber s compare number umbers in g ener al. gener eneral. Number Sense 1.2 Add, subtract, Ad d, subtr act, multiply, and umber s divide ra tional n number umbers actions fractions actions, and (integers, fr terminating decimals) and take positive rational numbers to whole-number powers. Number Sense 2.2 Ad d and Add by using common
subtr act fr actions subtract fractions factoring to find denomina tor s. denominator tors
Prime F actoriza tion — the Fir st Ste p to Finding an L CM Factoriza actorization First Step LCM Finding the prime factorization of a number involves writing it as a product of prime factors (prime numbers multiplied together). Example
1
Write 36 as a product of prime factors. Solution
One way to do this is to look for factors of 36, then look for factors of those factors, and then look for factors of those factors, and so on. You can arrange your factors in a “tree.” Break down 36 into two factors and write them underneath. Then look for factors of your factors.
What it means for you:
Key words: • • • •
prime factorization least common multiple common denominator
It doesn’t matter which factors you start with...
36
You’ll learn the first step of adding and subtracting fractions — putting fractions over a common denominator. You’ll also see how this allows you to compare fractions to find which is greater.
4
9
2 2 3 3
36 2
...the numbers at the ends of the branches will always be the same.
18 2
9 3 3
Each branch ends in a prime number — these can’t be broken down further. So the prime factorization of 36 is 36 = 2 × 2 × 3 × 3 Example
2
Write the following as products of prime factors: a) 35, b) 37 Solution
Don’t forget: A number that has no factors except itself and 1 is called a prime number.
a)
35 5
7
b) 37 Don’t forget:
5 and 7 are both prime, so the tree stops here. The prime factorization of 35 is 35 = 5 × 7 37 is prime, so the tree has no branches. 37 doesn’t factor.
Factors of a number are any whole numbers that divide exactly into it.
Guided Practice Write the following as products of prime factors. 1. 15 2. 30 3. 17 4. 16 5. 24 6. 50 84
Section 2.3 — Operations on Rational Numbers
Don’t forget:
The L CM is the Least Common Multiple LCM
Multiples are what you get in multiplication tables. For example, the multiples of 3 are 3, 6, 9, 12, 15, 18... and so on.
You can use prime factorizations to find the least common multiple of two numbers, which is the smallest number that both will divide into evenly. Example
3
Find the least common multiple of 8 and 14. Solution
There are two ways of doing this. 1. You can start listing all the multiples of each number, and the first one that they have in common is the least common multiple: • Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72... • Multiples of 14: 14, 28, 42, 56... So the least common multiple is 56. This method is fine, but it can get a bit tedious when the LCM is high. 2. You can first write each number as the product of its prime factors. 8 = 2 × 2 × 2 and 14 = 2 × 7. 8
Check it out: The least common multiple is useful when finding a common denominator for two fractions.
2
2
2
Then make a table showing these factorizations. 14 2 7 Wherever possible, put each factor of the second number in a column with the same factor. Start a new column if the number’s not already there — 8 2 2 2 so in this example, start a new column for the 7. 2 7 14 Now make a new row — the LCM row. LCM 2 2 2 7 For each box in the LCM row, write down the number in the boxes above. Finally, multiply together all the numbers in the LCM row to find the least common multiple. So the least common multiple of 8 and 14 is 2 × 2 × 2 × 7 = 56. Example
4
Find the least common multiple of 10 and 15. Solution
• Prime factorizations: 10 = 2 × 5 and 15 = 3 × 5. • Make a table: • Multiply the numbers in the LCM row. So the least common multiple of 10 and 15 is 2 × 5 × 3 = 30.
10
2
5
15 LCM 2
5 5
3 3
Guided Practice Find the least common multiple of each pair of numbers in Exercises 7–15. 7. 6 and 15 8. 10 and 12 9. 8 and 9 10. 4 and 10 11. 8 and 16 12. 20 and 30 13. 13 and 7 14. 9 and 81 15. 42 and 30 Section 2.3 — Operations on Rational Numbers
85
Compar e F tor Compare Frractions Using a Common Denomina Denominator Check it out: You don’t have to use the least common multiple here — any common multiple would be fine. For example, you could use 8 × 14 = 112. However, using the least common multiple will mean all the numbers will be smaller, which can be very useful.
Don’t forget: When you multiply both the numerator and denominator of a fraction by the same number, you don’t change its value. This is because you are really just multiplying the fraction by 1, since
7 7
=
4 4
= 1.
Least common multiples can be used as common denominators. Example
5
By using a common denominator, find which is greater,
5 9 or . 8 14
Solution
It’s not easy to say which is the larger fraction when they have different denominators. You need to find fractions equivalent to each of these that have a common denominator. You can use any common multiple of 8 and 14 as the common denominator. Here, we’ll use the least common multiple (which is 56 — see Example 3). You need to decide how many 8s and how many 14s make 56 — and then multiply the top and bottom of each fraction by the right number: 9 9× 4 36 5 5× 7 35 = = = = 14 14 × 4 56 8 8× 7 56 36 35 9 5 is greater than . You can now see that is greater than , since 56 56 14 8
Guided Practice In Exercises 16–21, put each pair of fractions over a common denominator to find which is the greater in each pair. 16.
1 2 and 4 10
17.
1 2 and 3 7
18.
3 2 and 10 7
19.
4 5 and 9 11
20.
11 12 and 15 17
21.
6 2 and 9 3
Independent Practice Write the numbers in Exercises 1–6 as products of prime factors. 1. 21 2. 100 3. 32 4. 50 5. 49 6. 132
Now try these:
Find the least common multiple of each pair in Exercises 7–12. 7. 6 and 8 8. 10 and 25 9. 48 and 21 10. 32 and 50 11. 100 and 49 12. 49 and 132
Lesson 2.3.5 additional questions — p438
Find the greater fraction in each pair in Exercises 13–15. 13.
10 51 and 21 100
14.
50 13 and 132 49
16. Order these fractions from least to greatest:
15.
33 32 and 50 49
2 14 13 3 16 , , , , 3 17 16 4 21
Round Up Finding common denominators is something you should get real comfortable with doing, because you need to do it a lot in math — in the next few Lessons, for example. 86
Section 2.3 — Operations on Rational Numbers
Lesson
Ad ding and Subtr acting Adding Subtracting Fr actions
2.3.6
California Standards: Number Sense 1.2 Add, subtract act, multiply, and Ad d, subtr act umber s divide ra tional n number umbers actions (integers, fr fractions actions, and terminating decimals) and take positive rational numbers to whole-number powers. Number Sense 2.2 Ad d and subtr act fr actions Add subtract fractions by using factoring to find common denomina tor s. denominator tors
Adding and subtracting fractions can be quick, or it can be quite a long process — it all depends on whether the fractions already have a common denominator, or whether you have to find it first.
You Can Ad d F tor Add Frractions with a Common Denomina Denominator If fractions have a common denominator (their denominators are the same), adding them is fairly straightforward. To find the numerator of the sum, you add the numerators of the individual fractions. The denominator stays the same. Example
What it means for you: You’ll see how to add and subtract fractions.
Find
1
2 3 + . 7 7
Solution
Key words: • denominator • numerator • common denominator
These two fractions have a common denominator, 7. So 7 will also be the denominator of the sum. The numerator of the sum will be 2 + 3 = 5. So
Don’t forget: The numerator is the top line of a fraction. The denominator is the bottom line of a fraction.
2 3 2+3 5 + = = . 7 7 7 7
You subtract fractions with a common denominator in exactly the same way. Example
Don’t forget: You can think of adding fractions as adding parts of whole shapes.
2 7 2 7
+
3 7 3 7
=
7 2 − . 9 9
Solution
The denominator of the result will be 9 (the fractions’ common denominator). The numerator of the result will be 7 – 2 = 5.
==
+ +
Find
2
5 7 5 7
So
7 2 7−2 5 − = = . 9 9 9 9
Guided Practice Find the sums and differences in Exercises 1–8. 1.
1 2 + 5 5
2.
3 4 + 11 11
3.
4 1 − 5 5
4.
10 2 − 21 21
5.
7 4 + 15 15
6.
23 19 + 50 50
7.
9 7 − 25 25
8.
5 1 + 17 17
Section 2.3 — Operations on Rational Numbers
87
You Ma y Need to Find a Common Denomina tor Fir st May Denominator First Fractions with unlike denominators cannot be directly added or subtracted. You must first find equivalent fractions with a common denominator. Example Find Don’t forget: A fraction whose numerator and denominator are the same is equal to 1 — so multiplying another fraction 6
9
by, say, 6 or 9 doesn’t change its value. This is why you can multiply the numerator and denominator by the same amount, and leave the value of the fraction unchanged.
Don’t forget: To find an LCM, you can use a table like this one. (See Lesson 2.3.5 for more information.)
9 6 2 LCM 2
3
3
3 3
3
Check it out: Again, any common multiple of 15 and 20 can be used as a common denominator — including their product 15 × 20 = 300. But this will mean the numbers in the calculation will be bigger than necessary. For example: 11 113 3 22022045 45 − −= = − − 15 1520 20300300300300 175175 = = 300300 7 7 = = 12 12
Check it out: Even if you use the LCM as your common denominator, you may still be able to simplify the answer at the end.
88
3
7 5 + . 9 6
Solution
The denominators are different here. This means you need to find two fractions equivalent to them, but with a common denominator. The common denominator can be any common multiple of 9 and 6. • You could use 9 × 6 = 54 as your common denominator. Then the equivalent fractions will be: 7 6 7×6 42 × = = 9 6 9×6 54
5 9 5× 9 45 × = = 6 9 6 × 9 54 42 45 87 + = Now you can add these fractions: , which you can 54 54 54 29 simplify to by dividing the numerator and denominator by 3. 18 • Or you could find the LCM (least common multiple) using prime factorizations. Since 9 = 32 and 6 = 2 × 3, the LCM is 2 × 3 × 3 = 18.
This time, the equivalent fractions are:
5 3 5×3 15 7 2 7× 2 14 = . × = = and × = 6 3 6 ×3 18 9 2 9× 2 18 14 15 29 + = . Now you can add these to get 18 18 18 You can use the LCM or any other common multiple as your common denominator — you’ll end up with the same answer. But using the LCM means that the numbers in your fractions are smaller and easier to use. Example
4
By putting both fractions over a common denominator, find
11 3 − . 15 20
Solution
Use a table to find the LCM of 15 and 20 — this is is 5 × 3 × 2 × 2 = 60. Find equivalent fractions with denominator 60: 11 11× 4 44 3 3×3 9 = = = = and 15 15× 4 60 20 20 ×3 60 So rewriting the subtraction gives: This can be simplified to
Section 2.3 — Operations on Rational Numbers
15
5
3
20 5 LCM 5
3
11 3 44 9 35 − = − = 15 20 60 60 60
35 7 = . 60 12
2 2
2 2
Guided Practice Calculate the answers in Exercises 9–11. 14 2 7 2 − + 10. 9. 15 5 10 3
2 4 11. − + 3 7
Be Extr a Car eful if Ther e ar e Ne ga ti ve Signs Extra Careful here are Neg tiv As always in math, if there are negative numbers around, you have to be extra careful. Don’t forget: Remember... subtracting a negative number is exactly the same as adding a positive number.
Don’t forget: You could also swap the terms around to form a subtraction: −3 2 2 3 + = − 5 7 7 5
Don’t forget: −11 11 11 , − , and are all 35 35 −35 equal.
Example
5
3 ⎛ −2 ⎞ Find − − ⎜⎜⎜ ⎟⎟⎟ . 5 ⎝ 7 ⎠ Solution
This looks tricky because of all the negative signs. So take things slowly and carefully. −3 2 + — it means exactly the same. This sum can be rewritten as 5 7 The LCM of 5 and 7 is 5 × 7 = 35. So put both these fractions over a common denominator of 35. −3 −3× 7 −21 2 2×5 10 = = = = 5 5× 7 35 7 7×5 35 Now you can add the two fractions in the same way as before. −3 ⎛⎜ −2 ⎞⎟ −21 10 −21 + 10 −11 11 − ⎜ ⎟⎟ = + = = or − 5 ⎜⎝ 7 ⎠ 35 35 35 35 35
Guided Practice Calculate the answers in Exercises 12–14. Simplify your answers.
2 ⎛ 3 ⎞⎟ 12. − − ⎜⎜⎜− ⎟⎟ 3 ⎝ 8⎠
13.
7 ⎛⎜ 4 ⎞⎟ − ⎜− ⎟ 6 ⎜⎝ 9 ⎟⎠
14. –
25 ⎛⎜ 7 ⎞⎟ – ⎜– ⎟ 48 ⎜⎝ 16 ⎟⎠
Independent Practice Now try these: Lesson 2.3.6 additional questions — p438
Calculate the following. Give all your answers in their simplest form. 1.
2 8 + 7 7
2.
8 2 − 5 5
3.
7 2 − 9 9
5.
19 7 + 30 20
6.
5 5 + 7 8
7. −
3 5 − 16 17
4.
7 3 − 16 8
8.
19 ⎛⎜ 20 ⎞⎟ + ⎜− ⎟ 20 ⎜⎝ 21 ⎟⎠
Round Up Keep practicing this until it becomes routine... (i) find a common denominator, (ii) put both fractions over this denominator, (iii) do the addition or subtraction, (iv) simplify your answer if possible. You’ll see this again next Lesson. Section 2.3 — Operations on Rational Numbers
89
Lesson
Ad ding and Subtr acting Adding Subtracting Mix ed Number s Mixed Numbers
2.3.7
California Standards: Number Sense 1.2 Add, subtract act, multiply, and Ad d, subtr act umber s divide ra tional n number umbers actions (integers, fr fractions actions, and terminating decimals) and take positive rational numbers to whole-number powers. Number Sense 2.2 Ad d and subtr act fr actions Add subtract fractions by using factoring to find common denomina tor s. denominator tors
What it means for you: You’ll learn to add and
This Lesson covers nothing new — it just combines things that you’ve learned before — converting mixed numbers to fractions, and adding and subtracting fractions. So if they seemed tricky the first time around, this is a good chance to get a bit more practice.
Ad ding Mix ed Number s — Con ver st Adding Mixed Numbers Conv ertt to F Frractions Fir First 1
7
Mixed numbers are things like 4 and − 6 , with both an integer and a 2 8 fraction part. It’s easiest to add the numbers together if you convert them to fractions first. Once they’re written as improper fractions you can add them by finding the least common multiple as before.
1
subtract numbers like 2 2 and 3
5 . 16
Example 1 2
1 2 3
Find 1 + 2 .
Key words:
Solution
• mixed number • common denominator
Convert both the numbers to fractions. (2×3) + 2 8 (1× 2) +1 3 2 1 = 2 = = 1 = 3 2 3 3 2 2 Put both fractions over a common denominator of 2 × 3 = 6. 3 3×3 9 = = 2 2×3 6
Don’t forget: You can convert 2
2 3
to a
fraction by remembering that: (i)
2 means (2 × 1) + 3 . 3 can be written as 3 .
2 2 3
(ii) 1
⎛ 3⎞ 2 2 So 2 = ⎜⎜2 × ⎟⎟⎟ + ⎜⎝ 3 3⎠ 3 2× 3 2 = + 3 3 (2× 3) + 2 = 3
8 8× 2 16 = = 3 3× 2 6
9 16 25 Do the addition: 1 1 + 2 2 = + = 2 3 6 6 6
Example 1 2
2 7 8
Find 4 − 6 . Solution
Convert both the numbers to fractions. (4× 2) +1 9 (6×8) + 7 55 1 7 = = 4 = 6 = 2 8 2 8 2 8 Put both fractions over a common denominator of 8. 9 9× 4 36 55 , and is already over a denominator of 8. = = 2 2× 4 8 8 36 55 36 − 55 19 Do the subtraction: 4 1 − 6 7 = − = =− 2 8 8 8 8 8
90
Section 2.3 — Operations on Rational Numbers
Guided Practice For Exercises 1–6, give your answers as fractions. 1. 3 + 2
3 4
3 4
2. 8 + 1
3 4
3 5
5. 1
4. 2 − 4
1 2
2 3
3. 6
1 5 −2 7 8
2 1 − 3 2 4 5
6. 4 −1
1 6
Tak e Extr a Car e With Ne ga ti ve Signs ake Extra Care Neg tiv If there are a lot of negative signs, you should take extra care. Example 2 3
3
( 67 ) .
Find −3 − −2 Solution
Convert the mixed numbers to fractions. ⎛(3×3) + 2 ⎞⎟ ⎛(2× 7) + 6 ⎞⎟ 11 20 2 6 ⎟⎟ = − ⎟⎟ = − −3 = −⎜⎜⎜ and − 2 = −⎜⎜⎜ 3 7 ⎟ ⎟ ⎜⎝ 3 7 3 7 ⎠ ⎝⎜ ⎠ Now find a common denominator — you can use 3 × 7 = 21. 11 11× 7 77 20 20 ×3 60 − =− =− and − = − =− 3 3× 7 21 7 7×3 21 Don’t forget: Subtracting a negative number is the same as adding a positive number.
So the calculation becomes 77 ⎛ 60 ⎞ 77 60 −77 + 60 17 − − ⎜⎜− ⎟⎟⎟ = − + = =− 21 ⎜⎝ 21⎠ 21 21 21 21 Also take extra care if the calculation has more than two terms — you have to make sure that all the fractions have the same denominator. Example Find
4
5 3 2 − +2 . 3 3 5
Solution
Don’t forget: Always do additions and subtractions from left to right.
Convert the mixed number to a fraction.
2 3
2 =
(2×3) + 2 3
=
8 3
Now find a common denominator — you can use 5 × 3 = 15. 5 5×5 25 3 3×3 9 8 8×5 40 = = = = = = 3 3×5 15 5 5×3 15 3 3×5 15 So the calculation becomes
25 9 40 25 − 9 + 40 56 − + = = 15 15 15 15 15
Section 2.3 — Operations on Rational Numbers
91
Guided Practice Do each of the calculations in Exercises 7–9. Give your answers as fractions. 7. −4
5 2 −2 6 3
8. −5
(
5 11 − −4 6 12
)
1 5 1 9. 2 2 − + 1 12 6
Simplify Your Ans wer s if P ossib le Answ ers Possib ossible It’s usually a good idea to simplify your answers if possible. Example 2 3
5 5 6
Find 1 + 2 . Solution
Convert the mixed numbers to fractions. (1×3) + 2 5 (2×6) + 5 17 2 5 = and 2 = = 1 = 3 6 3 6 3 6 You can use 6 as a common denominator. Find a fraction equivalent to Don’t forget: To simplify a fraction you divide the numerator and denominator by the same number (a common factor).
2 3
5 6
Do the addition: 1 + 2 =
5 3
with a denominator of 6:
5 5× 2 10 = = . 3 3× 2 6
10 17 10 + 17 27 + = = 6 6 6 6
Simplify this to get your final answer:
27 9 Di vide the n umer a tor Divide numer umera = tor b y 3 by denominator 6 2 and denomina
Guided Practice For Exercises 10–12, give your answers in their simplest form. 3 8
10. 2 + 3
5
1 8
1
11. 3 16 −1 16
12. 1
3 5 +2 8 10
Independent Practice Calculate the following. Give all your answers in their simplest form. Now try these: Lesson 2.3.7 additional questions — p438
1. 2
2 1 +3 3 3
5. 5
7 3 −12 8 16
2. 3
1 1 −1 4 2
6. 8
3. 2 1 7 −1 4 11
2 4 −1 7 5
4. 2 7. 2
3 2 3 +3 −5 4 3 8
Round Up You can hopefully see how you can use the same old routine over and over... Convert mixed numbers to fractions, find a common denominator, do the calculation. 92
Section 2.3 — Operations on Rational Numbers
3 4 −9 5 15
Lesson
2.4.1
Section 2.4
Fur ther Oper a tions with Further Opera Fr actions
California Standards: Number Sense 1.2 ultipl y, and Add, subtract, multipl ultiply Ad d, subtr act, m di vide rra a tional n umber s divide number umbers (inte ger s, fr actions (integ ers fractions actions, and terminating decimals) and take positive rational numbers to whole-number powers. Number Sense 2.2 Ad d and subtr act fr actions Add subtract fractions by using factoring to find common denomina tor s. denominator tors
The problems start to get more and more complicated now. There are all kinds of calculations in this Lesson. But you just have to remember what you’ve learned before, and use it carefully.
Or der of Oper a tions: Multiplica tion Bef or e Ad dition Order Opera Multiplication Befor ore Addition Math problems with fractions can involve combinations of operations — for example, you might need to do a multiplication and addition. Example
What it means for you: You’ll see how doing complex calculations involving fractions uses exactly the same ideas you’ve seen earlier in this Section.
Key words: • order of operations • common denominator
Don’t forget: LCM means least common multiple — see Lesson 2.3.5.
Don’t forget: You can use PEMDAS to help you remember the correct order of operations. Parentheses, Exponents, Multiplication and Division, Addition, and Subtraction.
Calculate
1
2 5 7 + × . 5 2 3
Solution
Remember — if there are no parentheses, you do multiplication before addition. So this calculation becomes 5× 7 2 35 2 + = + 2×3 5 6 5 The LCM of 6 and 5 is 6 × 5 = 30. Use this as the common denominator. 35 35×5 175 2 2×6 12 = = and = = 6 6 ×5 30 5 5×6 30 Find the result. 175 12 175 + 12 187 + = = 30 30 30 30 Example Find
2
30 2 3 ÷ −1 . 4 45 3
Solution
Don’t forget: Dividing by a fraction is the same as multiplying by its reciprocal.
Don’t forget: A raised dot . means multiplication. 4 5
So 2 ⋅ 2 = 2
4 ×2 5
Do the division first.
30 2 30 3 30 ×3 90 ÷ = × = = =1 45 3 45 2 45× 2 90
Then the subtraction.
4 7 4−7 3 3 1 −1 = − = =− 4 4 4 4 4
Guided Practice Calculate the following. Simplify your answers where possible. 1.
1 3 5 + × 2 2 6
3
2. 4 4 ÷ 2 −
1 8
7 8
3. 1 ⋅ 2 + 2
4 ⋅2 5
Section 2.4 — More Operations on Rational Numbers
93
Remember PEMD AS with R eall y Comple x Expr essions PEMDAS Reall eally Complex Expressions With calculations that look complicated, take things slowly. Example
(
3 2
1
)
1
Calculate 4 3 + 2 × 6 . Solution
First convert the mixed number to a fraction so that you can do the calculation more easily. 2 3
4 =
(4×3) + 2 3
=
14 3
Then calculate the sum inside the parentheses using 6 as the common denominator. ⎛14 1 ⎞⎟ 1 ⎛ 28 3 ⎞⎟ 1 ⎜⎜ + ⎟ × = ⎜⎜ + ⎟ × ⎜⎝ 3 2 ⎟⎠ 6 ⎜⎝ 6 6 ⎟⎠ 6 ⎛ 31⎞ 1 = ⎜⎜ ⎟⎟⎟ × ⎜⎝ 6 ⎠ 6 Finally, do the multiplication. 31 1 31 × = 6 6 36
Example
4
⎛ 3 1⎞ 1 Calculate ⎜⎜⎜3 4 + ⎟⎟⎟ − 3 ⋅ ⎝ 4⎠ 4 Solution
Don’t forget: It’s a good idea to simplify fractions whenever you can — don’t wait until you’ve got your final answer. This way, you’re making the numbers smaller and easier to use in the rest of the calculation.
94
(3× 4) + 3
15 4 4 1 15 1 16 3 Evaluate the expression in the parentheses. 3 + = + = = 4 4 4 4 4 4 1 Do the multiplication. 3 ⋅ = 3 4 4 3 4
First turn the mixed number into a fraction. 3 =
⎛ 3 1⎞ 1 3 Now you have: ⎜⎜3 + ⎟⎟ – 3 ⋅ = 4 – ⎜⎝ 4 4 ⎟⎠ 4 4 13 16 3 = – = Finally, do the subtraction. 4 4 4
Section 2.4 — More Operations on Rational Numbers
=
Example
5
1 1 Calculate 21 × 1 + 4 3 3
(
)
3
Solution
Don’t forget: Do divisions and multiplications from left to right.
Don’t forget: 1 3
3 =
(3 × 3) + 1 3
=
10 3
Parentheses first: To add the numbers, put them over a common denominator of 3. 3 (4 ×3) + 1 3 13 16 1 = + = 1+ 4 = + 3 3 3 3 3 3 10 1 The division is next, but you’ll need to convert 3 to first. 3 3 1 2 = 1 ÷ 10 = 1 × 3 = 1×3 = 3 1 2 3 2 10 2×10 20 3 3
You can now do the multiplication. 1 2 × 1 + 4 1 = 3 ×16 = 3×16 = 48 1 3 20 3 20 ×3 60 3
(
)
3
The final step is to simplify the answer:
48 4 = 60 5
Guided Practice Calculate the expressions in Exercises 4–5. Give your answers in their simplest form.
4.
1 2 5 1 7 8 ⋅ − ⋅ + ⋅ 6 7 7 2 12 7
5 +8 5. 1 1 4 2 3
1 8
Independent Practice Calculate the following. Give all your answers in their simplest form. Now try these: Lesson 2.4.1 additional questions — p439
1.
3 1 5 ⋅ + 4 6 12
2.
3.
5 9 3 5 5 1 ⋅ + ⋅ − ⋅ 12 2 4 3 8 3
4.
8 3 1 27 − ⋅ + 15 5 3 30
3 2 + 5 15 + 2 2 − 1 3 2 2 3
Round Up Some of the expressions in this Lesson were really complicated. When you have a tricky expression to evaluate, you just need to take your time, remember PEMDAS, and work through it very carefully. Section 2.4 — More Operations on Rational Numbers
95
Lesson
Multipl ying and Di viding Multiplying Dividing Decimals
2.4.2
California Standards: Number Sense 1.2 y, and Add, subtract, multipl ultiply di vide rra a tional n umber s divide number umbers (integers, fractions, and ter mina ting decimals termina minating decimals) and take positive rational numbers to whole-number powers.
What it means for you: You’ll practice multiplying and dividing decimals.
The trickiest thing about multiplying and dividing decimals is working out where the decimal point should go. Once you understand the method, it’s a lot more straightforward though.
Modeling Decimal Multiplica tion Multiplication You can use an area model to show what happens when you multiply two decimal numbers. Example
Key words:
1
Calculate 0.2 × 0.6.
• integer • decimal
{
0.2
Solution
0.2 is two-tenths — that’s how much of the square on the right is shaded. 0.6 is six-tenths — this is shaded in the other direction on the diagram below.
Check it out:
were multiplying
1 5
3
and 5 . These fractions are equivalent to the decimals 0.2 and 0.6.
The part that’s been shaded twice is 0.2 × 0.6. The square’s now divided into 100, so each small square is 1 ÷ 100 = 0.01.
0.2
{
This is really similar to the fraction multiplication area model in Lesson 2.3.3. You would have shaded exactly the same areas if you
There are 12 squares that have been shaded twice and each represents 0.01. 0.6
So 0.2 × 0.6 = 0.12
{
Check it out: In Example 1, multiplying two decimals each with 1 decimal place gives you a solution with 2 decimal places. In fact, you always count the decimal places in both factors to find out how many decimal places the solution has.
Guided Practice Use an area model to show the following multiplications. 1. 0.4 × 0.3
96
2. 0.8 × 0.2
Section 2.4 — More Operations on Rational Numbers
3. 0.1 × 0.9
Multiplying Decimals Multiplying decimals is just like multiplying integers — only you have to make sure you put the decimal point in the correct place. You can rewrite a decimal multiplication as a fraction calculation. Example
2
Calculate: 2 × 1.6 Solution
16 10
2 × 1.6 = 2 ×
Write the decimal as a fraction...
=
2 ×16 10
=
32 = 32 ÷ 10 = 3.2 10
...which can be rewritten like this.
Now divide by the 10 to get the decimal answer.
This works if both numbers are decimals too: Example
3
Calculate: 2.1 × 0.04 Solution
2.1 × 0.04 = Check it out: Another way of multiplying decimals is to just ignore the decimal point, do an integer multiplication, then figure out where the decimal point goes. For example: 2.1 × 0.04 Ignore the decimal points and do 21 × 4 = 84. The factors had a total of 3 decimal places: 2.1 (1 decimal place) × 0.04 (2 decimal places)
21 4 × 10 100
=
21× 4 10 ×100
=
84 = 84 ÷ 1000 = 0.084 1000
84
0 084
Multiply them.
Now divide by the 1000 to get the decimal answer.
Al ways Chec k the P osition of Your Decimal P oint Alw Check Position Point You can check you have the correct number of decimal places in the product by making sure it’s the same as the total number of decimal places in the factors. For example,
So you know the product must have 3 decimal places, so move the decimal point 3 places to the left:
Write both decimals as fractions.
2.1 × 3.67 = 7.707 1 digit + 2 digits = 3 digits
Watch out for calculations that give decimals with zeros at the end — you have to count the final zeros. For example, multiplying 1.5 by 1.2 gives 1.80. You must count up the decimal places before rewriting this as 1.8.
Guided Practice Don’t forget:
Write out and solve these calculations.
Multiplying a positive by a negative gives a negative — see page 75 for more.
4. 1.2 × (–1.1)
5. 4.5 × 5.9
6. 1.6 × (–8.2)
7. 6.31 × 6.4
8. (–2.77) × (–7.3)
9. 9.1 × 2.44
Section 2.4 — More Operations on Rational Numbers
97
Di viding Decimals — Tak e Car e with the Decimal P oint Dividing ake Care Point You can think about decimal division in a similar way. Example
4
Calculate: 46.5 ÷ 0.05 Solution
465 5 ÷ 10 100
Write the decimals as fr actions fractions
=
465 100 × 10 5
To di vide b y a fr action, m ultipl y b y its divide by fraction, multipl ultiply by r ecipr ocal eciprocal
=
4650 = 4650 ÷ 5 = 930 5
46.5 ÷ 0.05 =
Don’t forget: You can do integer division using long division. Check back to Lesson 2.3.2.
Another way is to ignore the decimal points and just use integer division. Then you have to find the correct position for the decimal point: • Count the number of digits after the decimal point in the first number and move the decimal point this many places to the left. Check it out: You could also do the division 46.5 ÷ 0.05 in Example 4 by doing integer division, then working out the decimal places: 465 ÷ 5 = 93 93. There’s 1 decimal place in 46.5, and 2 decimal places in 0.05. So move 1 decimal places to the left, and 2 to the right right. So the decimal point moves one place right overall: 46.5 ÷ 0.05 = 930
Check it out: 0.5 is 10 times smaller than 5. This means that the result of dividing something by 0.5 will be 10 times bigger than the result of dividing the same thing by 5. To make a result 10 times bigger, you move the decimal point one place to the right.
Now try these: Lesson 2.4.2 additional questions — p439
• Count the number of digits after the decimal point in the second number and move the decimal point this many places to the right. Example
5
If 5 ÷ 25 = 0.2, calculate 0.005 ÷ 2.5. Solution
0.005 is the first number — it has 3 decimal places. 2.5 is the second number — it has 1 decimal place. So you have to move 3 places to the left, and then 1 to the right.
0.0 0 0.2
0.0.0 0 2
0.005 ÷ 2.5 = 0.002
Guided Practice Solve these divisions. 10. 273 ÷ 1.3 11. 195.2 ÷ 8 13. 1.56 ÷ 3 14. 1.44 ÷ 3 16. 3.84 ÷ 1.2 17. 4.64 ÷ 1.6
12. 53.56 ÷ 1.3 15. 6.55 ÷ 5 18. 33.58 ÷ 2.3
Independent Practice In Exercises 1–6, find the product of each pair of numbers. 1. 1.03 × 0.5 2. 4.781 × –6.0 3. –3.11 × (–9.14) 4. 7.2 × 0.6 5. 1.04 × 4 6. 10.5 × 0.07 Find the quotient in Exercises 7–9. 7. –7.25 ÷ 0.25 8. –16.8 ÷ (–0.04)
9. 12.48 ÷ –1.2
Round Up Wow — that was a lot of information about multiplying and dividing. But don’t panic — when you’re multiplying and dividing with decimals, you just need to take it slow, working bit by bit. 98
Section 2.4 — More Operations on Rational Numbers
Lesson
2.4.3
Oper a tions with Opera Fractions and Decimals
California Standards: Number Sense 1.2 Ad d, subtr act, multipl y, and Add, subtract, ultiply umber s di vide ra tional n number umbers divide ( integers, fr actions fractions actions,, and ter mina ting decimals termina minating decimals) and take positive rational numbers to whole-number powers.
What it means for you: You’ll practice adding, subtracting, multiplying, and dividing fractions and decimals together.
Key words: • • • •
fraction decimal multiply divide
You’ve probably now seen nearly all the ideas, techniques, and tricks you’ll ever need for almost any kind of number problem. Now you’re going to put them all into action...
Calcula tions Can In volv e Both F Calculations Inv olve Frractions and Decimals To do a calculation involving a fraction and a decimal, you either have to make them both fractions or both decimals. 3 For example, if you needed to find 0.3 × , you could... 5 ⎛3⎞ 3 3 9 • Convert 0.3 to a fraction ⎜⎜⎜ ⎟⎟⎟ and find × = . ⎝10 ⎠ 10 5 50 3 • Convert to a decimal (0.6) and find 0.3 × 0.6 = 0.18. 5 Both methods are correct, and they both give the same answer (you can check by calculating 9 ÷ 50, which gives 0.18). Example
1
4 by: 5 a) converting 0.25 to a fraction, 4 b) converting to a decimal. 5
Calculate 0.25 ×
Solution
a) 0.25 is b)
25 1 = . So 0.25× 4 = 1 × 4 = 4 = 1 100 4 5 4 5 20 5
4 4 is 4 ÷ 5 = 0.8. So 0.25 × = 0.25 × 0.8 = 0.2 5 5
The best idea is to choose the method that makes the calculation easiest. Example 2 1 Calculate − 0.5 . 3 Don’t forget: 0.3333... can also be written as 0.3 . (But that doesn’t make it any easier to subtract 0.5 from.)
Solution
If you convert
1 3
to a decimal, you get 0.333333333..., which is difficult
to subtract 0.5 from. So it’s best to use fractions here. So
1 1 1 2 3 1 − 0.5 = − = − = − 3 3 2 6 6 6
Section 2.4 — More Operations on Rational Numbers
99
Guided Practice For Exercises 1–6, find the answers to the calculations by: (i) converting all the numbers to fractions, (ii) converting all numbers to decimals. Check that both your answers for each exercise are equivalent. 1 1 1 1. + 0.2 2. − 0.2 3. + 0.4 2 4 4 1 1 1 4. × 0.3 5. ×1.2 6. × 2.5 10 3 6 For Exercises 7–9, use the easiest method to do each calculation. ⎛1 ⎞ 1 3 1 7. + 8. 0.15 ÷ 9. 0.3 − ⎜⎜⎜ × 4.5⎟⎟⎟ ⎝3 ⎠ 7 98 2
You Can Think of F tion Another Way Frraction Multiplica Multiplication There’s another useful way to think about multiplying by fractions. 3 5
Check it out:
When you multiply by the fraction , it is the same as either:
Multiplying by
(i) multiplying by 3 and then dividing by 5, or (ii) dividing by 5 and then multiplying by 3.
1 is the same 5 as dividing by 5.
Example
3
2 Calculate 6.93× . 3 Solution
This would be difficult to do using decimals (6.93 × 0.6666...). ⎛ 693 ⎞⎟ ⎟. And 6.93 doesn’t convert to an “easy” fraction ⎜⎜⎜ ⎝100 ⎟⎠ 2 But multiplying by 3 means dividing by 3 and then multiplying by 2. This is much quicker. Dividing by 3 gives: Then multiplying by 2 gives: 2 So, 6.93× = 4.62 3 Example Calculate Check it out: You can do the multiplication and division in either order — so choose whichever will be easier.
6.93 ÷ 3 = 2.31 2.31 × 2 = 4.62
4
2 ×1.5 . 7
Solution
You’re multiplying 1.5 by
2 , 7
so multiply it by 2, then divide by 7.
So 1.5 × 2 = 3, and then 3 ÷ 7 =
100
Section 2.4 — More Operations on Rational Numbers
3 3 2 . So, ×1.5 = 7 7 7
Example Calculate
5
1 (6.35 − 3.02) . 3
Solution
As always, work out the parentheses first. 6.35 – 3.02 = 3.33.
Don’t forget: 1 is the same 3 as dividing by 3. Multiplying by
Then you can divide by 3. 3.33 ÷ 3 = 1.11
Guided Practice Calculate the value of the expressions in Exercises 10–14. 10.
3 × 0.24 4
11.
7 ×1.5 15
⎛3 1⎞ 13. 3.5×⎜⎜ − ⎟⎟⎟ ⎜⎝ 5 5 ⎠
12. 0.9× 14.
8 9
5 ×(8.03 + 0.97) 9
Independent Practice Calculate the following. 1.
1 × 0.8 2
4. 1.25 + Don’t forget: Dividing by
27 13
is the same as 13
multiplying by its reciprocal, 27 .
7. 2.5×
2.
7 4
5. 0.8 −
2 5
8. 1.3×
10. (4.57 + 3.53)×
1 9
Now try these: Lesson 2.4.3 additional questions — p439
1 ×1.2 3
12. (2.89 −18.89) ÷
3.
1 5
1 + 0.25 2
6. −1.3 −
10 13
9. 2.7 ÷
27 13
11. (−5.36 −1.64)×
8 5
3 10
2 7
(5.67 +12.53) 13.
2 3
Round Up The theme that’s been running through the last few Lessons is this — although questions might look complicated, you just need to take things real slow, and use all those things you learned earlier in the Section. Don’t try to hurry — that makes you more likely to make a mistake. Section 2.4 — More Operations on Rational Numbers
101
Lesson
2.4.4
Pr ob lems In volving Prob oblems Inv Fractions and Decimals
California Standards: Number Sense 1.2 Ad d, subtr act, m ultipl y, and Add, subtract, multipl ultiply di vide rra a tional n umber s divide number umbers actions (inte ger s, fr fractions actions, and (integ ers terminating decimals) and take positive rational numbers to whole-number powers. Ma thema tical R easoning 2.2 Mathema thematical Reasoning A ppl y str a te gies and rresults esults pply stra tegies from simpler problems to mor e comple x pr ob lems more complex prob oblems lems..
What it means for you: You’ll use ideas from previous Lessons in this Section to solve some real-life problems.
This is the last Lesson in this Section. And again, it’s all about using the skills you’ve already learned. But this time before you can do the math, you have to write the problem down in math language using a description of a real-life situation.
Fir st Write R eal-Lif e Pr ob lems as Ma th First Real-Lif eal-Life Prob oblems Math Example
1 1
The length of a rectangular room is 54 feet, while its width 3 is 33.3 feet. What is the area of the room? Solution
You find the area of a rectangle by multiplying its length by its width. So the area of this room is given by 54
Key words: • word problem • units
Very often, writing your answer in a sensible way just means adding units. You should also always check that it’s realistic — see Chapter 1 for more information.
× 33.3.
Now you need to work through all the steps from the previous Lessons. 163 1 54 ×33.3 = ×33.3 Con ver ed n umber to a fr action Conv ertt mix mixed number fraction 3 3 = (33.3 ÷ 3)×163 Rewrite as “÷ then ×” = 11.1×163
Check it out:
1 3
Do the di vision division
You could use a calculator here, but you don’t really need to. To do this multiplication without a calculator, you can rewrite it using the ideas in Section 2.3.2. 11.1 × 163 = (10 × 163) + (1 × 163) + (0.1 × 163) = 1630 + 163 + 16.3 = 1809.3 With real-life problems you must always think about what your answer means, and then write it in a sensible way. Here, you need to add units. So the area of the room is 1809.3 square feet.
Guided Practice 1. A rectangular dance floor is 28.6 feet wide and 15
1 2
feet long.
What is the area of the dance floor? 2. A rectangular playing field is 16
2 3
What is the area of the playing field? 102
Section 2.4 — More Operations on Rational Numbers
yards wide and 30.9 yards long.
Dr awing a Dia g ram Can Mak e Things Clear er Dra Diag Make Clearer Sometimes it’s not doing the math that’s the hardest thing in a problem. It’s working out what math to do in the first place. Example
2
Some public sewer lines are being installed along 8
1 4
miles of road.
The supervisor says they will be able to complete 0.75 of a mile a day. How long will the project take? 0.75 miles
Solution
Don’t forget: Your diagram doesn’t have to be accurate — just good enough to get an idea of what you need to do.
If you can’t see how to answer a question, draw a picture.
1 8 miles 4
Sewer
Road 1 4
You need to find out how many times 0.75 goes into 8 . In math language, this is a division — so you need to solve 8 1 ÷ 0.75 . 4
Now that the problem is written in math language, you can use the techniques from previous Lessons. 1 4
3 4 33 3 = ÷ 4 4 33 4 = × 4 3 33× 4 = 4 ×3 33 = = 11 3 1 4
8 ÷ 0.75 = 8 ÷
Check it out: You could work out the 132
fraction here as 12 , and then simplify it. But here, the factor of 4 in the numerator and the factor of 4 in the denominator have been canceled straightaway.
33 × 4 4×3
Con ver action Conv ertt the decimal to a fr fraction Con ver ed n umber to a fr action Conv ertt the mix mixed number fraction Rewrite the di vision as a m ultiplica tion division multiplica ultiplication Multipl y the fr actions Multiply fractions Simplify
Don’t forget: about units... The project will take 11 days.
Guided Practice 3. A piece of rope 17.25 feet long must be divided up into smaller 3 lengths of 4 of a foot long. How many lengths can be made from this piece of rope? 4. A birthday cake needs 2 1 cups of raisins. However, because a big 2 party is planned, several cakes are made by increasing the amount of all the ingredients used. In fact, 8 3 cups of raisins are used. 4
How many cakes is this enough raisins for? Section 2.4 — More Operations on Rational Numbers
103
Example
3
Aisha worked 40.5 hours in one particular week. Three-fifths of these hours she was in meetings, while the rest of the time was spent traveling. How many hours did Aisha spend traveling during the week? Solution
40.5 hours
Again, a diagram might help. You need to work out how many hours the red part of the bar represents.
3 5
= meetings
3
Check it out: Here, you could do the math as fractions... 2 81 2 40.5 × = × 5 2 5 81 = 5 = 16
1
If the blue part of the bar is 5 of the total hours, then the red part must 3 2 represent: 1− = 5 5 2 So you need to work out 5 of 40.5 hours 2 — this can be written as a multiplication: 40.5× 5 Now you can do the math. 2 Rewrite the fr action m ultiplica tion as a fraction multiplica ultiplication 40.5× = (40.5 ÷ 5)× 2 di vision, f ollo w ed b y a m ultiplica tion follo ollow by multiplica ultiplication division, 5 = 8.1× 2 Do the di vision division ultiplica tion Do the m multiplica ultiplication = 16.2 Now say what your answer means. Aisha spent 16.2 hours traveling during the week.
5
...or decimals... 40.5 × 0.4 = 16.2 You get the same answer whatever method you use.
Guided Practice 5. A small town has a sports field with a total area of 1533.75 square yards. One third of this area is used only by parents with small children, while the rest can be used by anyone. How many square yards are for anyone’s use?
Independent Practice 1. The floor area of a room is 99 Now try these: Lesson 2.4.4 additional questions — p440
The length of the room is
1 10 4
15 16
square feet.
feet. What is the width of the room?
1
2. Approximately of the students in a high school take an advanced 5 placement class. Of those students it is thought 50% will receive a scholarship for college. What fraction of the students in the high school are expected to receive a scholarship?
Round Up That’s the end of a Section containing a lot of little bits of information. An important thing to remember from it all is that you must work through complex problems slowly and carefully. 104
Section 2.4 — More Operations on Rational Numbers
Section 2.5 introduction — an exploration into:
Basic P ower s Po ers You can find the powers of numbers by repeatedly folding a piece of paper. Each extra fold you make produces more rectangles. You’ll see how the numbers of rectangles produced are powers. 2 rectangles
Take a sheet of paper and fold it into two equal sections. When you open it up, you’ll see there are two rectangles.
fold
fold
4 rectangles open out
fold again
open out
fold
If you fold the paper into two again, then fold it once more, you’ll have four rectangles when you open it up.
folds
The number of rectangles produced by each fold represents the powers of 2. 20 = 1
21 = 2
22 = 4
Exercises 1. Continue to fold the paper into two equal sections each time. Write the total number of rectangles produced after the given number of folds. a. 3 b. 4 c. 5 2. Write your answers to 1a, b, and c as powers of 2. Now take a sheet of paper and make two folds, so that it forms three equal sections. 3 rectangles fold into three sections
open out folds
Repeat this by folding the paper into three equal sections each time.
fold into three sections
9 rectangles
open out folds
Exercises
3. Fold a piece of paper repeatedly into three, in the way described above. A long piece of paper makes this part easier. What is the total number of rectangles produced from the: a. second set of folds? b. third set of folds? c. fourth set of folds? 4. Write your answers to 3a, b, and c as powers of 3. 5. With a new piece of paper, experiment with this process to make 25 rectangles. 6. Is it possible to make 10 rectangles of equal size by using the process above?
Round Up When you repeatedly fold a piece of paper into two, you repeatedly multiply the number of rectangles by two. And when you repeatedly fold a piece of paper into three, you repeatedly multiply the number of rectangles by three. This gives you the powers of 2 and the powers of 3 — because powers are produced by repeated multiplication. Section 2.5 Explor a tion — Basic Powers 105 Explora
Lesson
Section 2.5
2.5.1
Power s of Inte ger s ers Integ ers
California Standards:
A power is just the product that you get when you repeatedly multiply a number by itself, like 2 • 2, or 3 • 3 • 3. Repeated multiplication expressions can be very long. So there’s a special system you can use for writing out powers in a shorter way — and that’s what this Lesson is about.
Number Sense 1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and tak e positi ve rra a tional take positiv number s to w hole-n umber umbers whole-n hole-number po wer s. pow ers Alg ebr a and Functions 2.1 Alge bra Inter pr et positi ve w holeInterpr pret positiv wholenumber po wer s as rre epea ted pow ers peated multiplica tion and negative ultiplication whole-number powers as repeated division or multiplication by the multiplicative inverse. Simplify and evaluate expressions that include exponents.
A P ower is a R epea ted Multiplica tion Po Re peated Multiplication A power is a product that results from repeatedly multiplying a number by itself. For example: 2•2 = 4, or “two to the second power.” 2•2•2 = 8, or “two to the third power.” 2 • 2 • 2 • 2 = 16, or “two to the fourth power.” So 4, 8, and 16 are all powers of 2.
What it means for you: You’ll learn how to write repeated multiplications in a shorter form.
Key words: • • • •
power base exponent factor
You Can Write a P ower as a Base and an Exponent Po If every time you used a repeated multiplication you wrote it out in full, it would make your work very complicated. So there’s a shorter way to write them. For example:
2 • 2 • 2 • 2 = 24
Check it out: Raising something to the second power is called squaring it. Raising something to the third power is called cubing it. There’s more on this in Lesson 2.5.3.
onent — it tells you This is the exp exponent w many times the base number is a how ho fac factt or in the multiplication expression.
er 2 is the b ase — it’s the numb number that’ s bbeing eing multiplied that’s
For example, the expression 10 • 10 can be written in this form — the base is 10 and since 10 occurs twice, the exponent is 2.
Don’t forget:
exponent
A factor is one of the terms in a multiplication expression.
Check it out: You can describe a number as being “raised to” or “taken to” a power. So 24 could be read as, “two to the fourth power,” or “two raised to the fourth power,” or “two taken to the fourth power.”
106
base
10
2
You can rewrite any repeated multiplication in this form. So any number, x, to the nth power can be written as:
Section 2.5 — Basic Powers
base
x
n
exponent
Example
1
Write the expression 3 • 3 • 3 • 3 • 3 in base and exponent form. Solution
The number that is being multiplied is 3. So the base is 3. 3 occurs as a factor five times in the multiplication expression. So the exponent is 5. So 3 • 3 • 3 • 3 • 3 = 35. If a number has an exponent of 1 then it occurs only once in the expanded multiplication expression. So any number to the power 1 is just the number itself. For example: 51 = 5, 1371 = 137, x1 = x.
Guided Practice Check it out: When you write a negative number raised to a power, you need to put parentheses around the number. For example, (–2)4 tells you to raise negative two to the fourth power — it has a value of 16.
Write each of the expressions in Exercises 1–8 as a power in base and exponent form. 1. 8 • 8
2. 2 • 2 • 2
3. 7 • 7 • 7 • 7 • 7
4. 5
5. 9 • 9 • 9 • 9
6. 4 • 4 • 4 • 4 • 4 • 4 • 4
7. –5 • –5
8. –8 • –8 • –8 • –8 • –8
Ev alua te a P ower b y Doing the Multiplica tion Evalua aluate Po by Multiplication Evaluating a power means working out its value. Just write it out in its expanded form — then treat it as any other multiplication calculation. Example
2
Evaluate 54. Solution
54 means “four copies of the number five multiplied together.” 54 = 5 • 5 • 5 • 5 54 = 625.
Section 2.5 — Basic Powers
107
Example Check it out: A negative number raised to an odd power always gives a negative answer. For example: (–2)3 = –2 • –2 • –2 = –8 A negative number raised to an even power always gives a positive answer. For example: (–2)4 = –2 • –2 • –2 • –2 = (–2 • –2) • (–2 • –2) = 4 • 4 = 16
3
Evaluate (–2)2. Solution
(–2)2 means “two copies of the number negative two multiplied together.” (–2)2 = –2 • –2 (–2)2 = 4.
Guided Practice Evaluate the exponential expressions in Exercises 9–16. 9. 102
10. 53
11. 71
12. 36
13. 471
14. (–15)1
15. (–3)2
16. (–4)3
Independent Practice Write each of the expressions in Exercises 1–6 in base and exponent form. 2. 9 • 9 1. 4 • 4 • 4 4. 5 • 5 • 5 • 5 • 5 • 5 3. 8 6. –3 • –3 • –3 • –3 5. –4 • –4 • –4 7. Kiera and 11 of her friends are handing out fliers for a school fund-raiser. Each person hands out fliers to 12 people. How many people receive a flier?
Now try these: Lesson 2.5.1 additional questions — p440
Evaluate the exponential expressions in Exercises 8–13. 8. 152 9. 43 1 10. 8 11. 18 12. (–5)3 13. (–5)4 14. A single yeast cell is placed on a nutrient medium. This cell will divide into two cells after one hour. These two cells will then divide to form four cells after another hour. The process continues indefinitely. a) How many yeast cells will be present after 1 hour, 2 hours, and 6 hours? b) Write exponential expressions with two as the base to describe the number of yeast cells that will be present after 1 hour, 2 hours, and 6 hours. c) How many hours will it take for the yeast population to reach 256?
Round Up If you need to use a repeated multiplication, it’s useful to have a shorter way of writing it. That’s why bases and exponents come in really handy when you’re writing out powers of numbers. You’ll see lots of powers used in expressions, equations, and formulas. For example, the formula for the area of a circle is pr2 where r is the radius. So it’s important you know what they mean. 108
Section 2.5 — Basic Powers
Lesson
2.5.2
Power s of R ational Number s ers Ra Numbers
California Standards:
In just the same way that you can raise whole numbers to powers, you can also raise fractions and decimals to powers.
Number Sense 1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and tak e positi ve rra a tional take positiv number s to w hole-n umber umbers whole-n hole-number po wer s. pow ers Alg ebr a and Functions 2.1 Alge bra pr et positi ve w holeInterpr pret positiv wholeInter number po wer s as rre epea ted pow ers peated multiplica tion and negative ultiplication whole-number powers as repeated division or multiplication by the multiplicative inverse. Simplify and evaluate expressions that include exponents.
You Can R aise a F ower Raise Frraction to a P Po A fraction raised to a power means exactly the same as a whole number raised to a power — repeated multiplication. But now the complete fraction is the base. exponent base
This expression means
2 3
•
2 . 3
When you raise a fraction to a power, you are raising the numerator and the denominator separately to the same power. For example:
What it means for you: You’ll learn how to take fractions and decimals to powers.
Key words: • • • • •
power exponent base decimal fraction
2 2 3
2 ⎛ 2 ⎞⎟ ⎜⎜ ⎟ = 2 2 ⎜⎝ 3 ⎟⎠ 3 2
This makes evaluating the fraction easier. You can evaluate the numerator and the denominator separately. Example
1
⎛1⎞ Evaluate ⎜⎜⎜ ⎟⎟⎟ . ⎝4⎠ Solution 3
13 = 1 • 1 • 1 = 1
3 ⎛ 1 ⎞⎟ ⎜⎜ ⎟ = 1 ⎜⎝ 4 ⎟⎠ 43 3
Raise both the numerator and the denominator to the third power.
43 = 4 • 4 • 4 = 64
13 1 = 3 4 64
Example Check it out: If you are raising a negative fraction to a power, just keep the minus sign with the numerator all the way through your work. For example: ⎛ 3 ⎞⎟ (–3)4 ⎜⎜ – ⎟ = ⎜⎝ 4 ⎟⎠ 44 4
2 1
⎛ 2⎞ Evaluate ⎜⎜⎜ – ⎟⎟⎟ ⎝ 5⎠ Solution
2
2 ⎛ 2 ⎞⎟ ⎜⎜ – ⎟ = (–2) ⎜⎝ 5 ⎟⎠ 52 2
(–2)2 = –2 • –2 = 4 Raise both the numerator and the denominator to the second power. 52 = 5 • 5 = 25
(−2)2 4 = 2 5 25 Section 2.5 — Basic Powers
109
Guided Practice Evaluate the exponential expressions in Exercises 1–6. 1.
⎛ 1 ⎞⎟ ⎜⎜ ⎟ ⎜⎝ 2 ⎟⎠
2.
⎛ 1 ⎞⎟ ⎜⎜ ⎟ ⎜⎝ 2 ⎟⎠
3.
⎛ 5 ⎞⎟ ⎜⎜ ⎟ ⎜⎝ 3 ⎟⎠
4.
⎛ 9 ⎞⎟ ⎜⎜ ⎟ ⎜⎝ 7 ⎟⎠
4
2
3
1
⎛
⎛
3⎞
2
3⎞
3
6. ⎜⎜⎜⎝ – 10 ⎟⎟⎟⎠
5. ⎜⎜⎜⎝ – 10 ⎟⎟⎟⎠
You Can R aise a Decimal to a P ower Raise Po A decimal raised to a power means exactly the same as a whole number raised to a power — it’s a repeated multiplication. The decimal is the base.
exponent base
0.24
2
This expression is the same as saying 0.24 • 0.24. When you evaluate a decimal raised to a power, you multiply the decimal by itself the specified number of times. The tricky thing when you’re multiplying decimals is to get the decimal point in the right place — you saw how to do this in Section 2.4.
Example
3
Evaluate (0.3)3. Solution
The multiplication you are doing here is (0.3)3 = 0.3 • 0.3 • 0.3. 3 3 3 × × 10 10 10
write the decimals as fr actions fractions
=
3× 3× 3 10 ×10 ×10
m ultipl y the fr actions ultiply fractions
=
27 = 27 ÷ 1000 = 0.027 1000
0.3 × 0.3 × 0.3 = Check it out: To check the number of decimal places your answer should have, just count the number of decimal places in all of the factors and add them together. For example, in the expression 0.3 • 0.3 • 0.3, you have three factors, and each contains one decimal place. So your answer should contain three decimal places, which 0.027 does.
110
Section 2.5 — Basic Powers
So, (0.3)3 = 0.027
Example Check it out: When you put the decimal point back into your answer, put in 0s to fill up any places between the decimal point and the numerical part of the answer. For example, here the numerical part of your answer is 529, but you need four decimal places in your answer. So use a 0 before the numerical part — so the answer is 0.0529.
4
Evaluate (0.23)2. Solution
0.23 × 0.23 =
=
23 23 × 100 100
write the decimals as fr actions fractions
529 = 529 ÷ 10,000 = 0.0529 multiply 10, 000
Guided Practice Evaluate the exponential expressions in Exercises 7–12. 7. (0.5)2 8. (0.2)3 9. (0.78)1 10. (0.12)2 11. (0.15)3 12. (0.08)2
Independent Practice Write each of the expressions in Exercises 1–4 in base and exponent form. 1. 3.
1 2 1 7
• •
1 2 1 7
• •
1 2 1 7
2. 0.25 • 0.25 •
1 7
4.
–
1 4
•
–
1 4
•
–
1 4
Evaluate the exponential expressions in Exercises 5–10. 5. ⎜⎜⎜⎝ 9 ⎟⎟⎟⎠
⎛1⎞
6. ⎜⎜⎜⎝ 4 ⎟⎟⎟⎠
⎛ 2⎞
8. ⎜⎜⎜⎝ 8 ⎟⎟⎟⎠
⎛1⎞
10. ⎜⎜⎜⎝ – 5 ⎟⎟⎟⎠
2
3
7. ⎜⎜⎜⎝ 3 ⎟⎟⎟⎠
5
9. ⎜⎜⎜⎝ 5 ⎟⎟⎟⎠ Now try these: Lesson 2.5.2 additional questions — p440
⎛1⎞
3
⎛9⎞
1
⎛ 2⎞
3
11. Mark is feeding chickens. He divides 135 g of corn into thirds. Each portion is then divided into thirds again to give small portions. What fraction of the original amount is in each small portion? How much does each small portion weigh? Evaluate the exponential expressions in Exercises 12–17. 12. (0.4)2 13. (0.1)4 14. (0.21)2 15. (0.97)1 2 16. (0.02) 17. (0.25)3
Round Up To raise a fraction to a power, you raise the numerator and the denominator separately to the same power. To raise a decimal to a power, you use the decimal as the base and raise it to a power as you would a whole number — just by multiplying it by itself the correct number of times. Section 2.5 — Basic Powers
111
Lesson
2.5.3
Uses of P ower s Po ers
California Standards:
You’ll come across powers a lot both in math and real-life situations. That’s because you use them to work out areas and volumes. They’re also handy when you need to write out a very big number — you can use powers to write these numbers in a shorter form.
Number Sense 1.1 e write,, and compar compare Read, write ra tional n umber s in number umbers scientific nota tion (positi ve notation (positiv wer s of 10), pow ers and negative po compare rational numbers in general. Number Sense 1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and tak e positi ve rra a tional take positiv number s to w hole-n umber umbers whole-n hole-number po wer s. pow ers
What it means for you:
Exponents ar e Used in Some F or mulas are For orm Exponents are used in the formulas for the areas of squares and circles. In this Lesson you’ll see how exponents are used in finding the area of a square. In the next Chapter you’ll use a formula to find the area of a circle. The formula for the area of a square is Area = s • s = s2, where s represents the side length of the square.
You’ll see how you can use exponents to work out areas of squares and volumes of cubes, and learn about a shorter way to write very large numbers.
4 3 2 1
2
1
Key words: • squared • cubed • scientific notation
12 = 1
4
3
22 = 4
32 = 9
42 = 16
When you find the area of a square, the side length is used as a factor twice in the multiplication. So raising a number to the second power is called squaring it.
Example
1
Find the area of the square shown below.
Check it out: The areas of all shapes, not just squares, are measured in square units. These could be cm2, m2, inches2, feet2, or even miles2. You’ll learn more about this in Chapter 3.
112
Section 2.5 — Basic Powers
1 cm
Solution
Each small square is 1 cm wide. So the side length of the whole square is 2 cm. The area of the whole square is 2 cm • 2 cm = 4 cm2. You can see that this is true, because it is made up of four smaller 1 cm2 squares.
Example
2
A square has a side length of 11 inches. Find its area. Don’t forget:
Solution
When you’re working out the units that go with a calculation, use the unit analysis method you saw in Chapter 1 (p42). Just apply the same operations to the units as you did to the numbers.
Area = (side length)2 Area = 112 = 11 • 11 = 121 Units: inches • inches = in2 Area = 121 in2
Guided Practice Find the areas of the squares in Exercises 1–4. 3 miles
1.
2. Square of side length 6 feet.
3. Square of side length 3.5 m.
4.
5 mm
Exponents ar e Used tto o Find Volume s of Some Solid s are olumes Solids Exponents are also used in formulas to work out volumes of solids, like cubes, spheres, and prisms. The formula for the volume of a cube is Volume = s • s • s = s3, where s represents the side length of the cube. 2 1
2
1 2
1
13 = 1
23 = 8
When you find the volume of a cube, the side length is used as a factor three times in the multiplication. So raising a number to the third power is called cubing it. Example
3
Check it out: The volumes of all solids, not just cubes, are measured in cubed units. For example, the units could be cm3, m3, inches3, feet3, or even miles3. You’ll learn more about this in Chapter 7.
A cube has a side length of 5 cm. Find its volume. Solution
Volume = (side length)3 Volume = 5 • 5 • 5 = 53 = 125 Units: cm • cm • cm = cm3 Volume = 125 cm3. Section 2.5 — Basic Powers
113
Guided Practice Find the volumes of the cubes in Exercises 5–8. 4m
5.
6. Cube of side length 5 feet.
4m 4m
7. Cube of side length 7.5 mm.
8.
1 in.
Use Scientific Nota tion to Write Big Number s Notation Numbers Sometimes in math and science you’ll need to deal with numbers that are very big, like 570,000,000. To avoid having to write numbers like this out in full every time, you can rewrite them as a product of two factors. For instance: 570,000,000 = 5.7 × 100,000,000 The second factor is a power of ten. You can write it in base and exponent form. 5.7 × 100,000,000 = 5.7 × 108. So 5.7 × 108 is 570,000,000 written in scientific notation.
Check it out: To work out what power of ten the second factor is, just count the zeros in it. For example, 10 is 101, 1000 is 103, and 10,000,000 is 107.
Scientific Notation To write a number in scientific notation turn it into two factors: Æ the first factor must be a number that’s at least one but less than ten. Æ the second factor must be a power of 10 written in exponent form.
Example
4
Write the number 128,000,000,000 in scientific notation. Solution
128,000,000,000 = 1.28 × 100,000,000,000 = 1.28 × 1011
114
Section 2.5 — Basic Powers
Split the number into a decimal between 1 and 10 and a power of ten.
Write the number as a product of the two factors.
Check it out: If the number you were putting into scientific notation was 51,473,582, then you would probably round it before putting it into scientific notation. You’ll see more about how to round numbers in Chapter 8.
Example
5
The number 5.1× 107 is written in scientific notation. Write it out in full. Solution
5.1 × 107 = 5.1 × 10,000,000 = 51,000,000
Write out the power of ten as a factor in full. Multiply the two together: move the decimal point as many places to the right as there are zeros in the power of ten.
Guided Practice Write the numbers in Exercises 9–12 in scientific notation. 10. 32,800 9. 6,700,000 12. 1,040,000,000 11. –270,000 Write out the numbers in Exercises 13–16 in full. 13. 3.1 × 103 14. 8.14 × 106 7 15. –5.05 × 10 16. 9.091 × 109
Independent Practice Find the areas of the squares in Exercises 1–4. 1. Square of side length 2 cm. 2. Square of side length 8 yd. 3. Square of side length 13 m. 4. Square of side length 5.5 ft. 5. Maria is painting a wall that is 8 feet high and 8 feet wide. She has to apply two coats of paint. Each paint can will cover 32 feet2. How many cans of paint does she need? Find the volumes of the cubes in Exercises 6–9. 7. Cube of side length 6 yd. 6. Cube of side length 3 ft. 9. Cube of side length 1.5 in. 8. Cube of side length 9 cm. 10. Tyreese is tidying up his little sister’s toys. Her building blocks are small cubes, each with a side length of 3 cm. They completely fill a storage box that is a cube with a side length of 15 cm. How many blocks does Tyreese’s sister have? Now try these: Lesson 2.5.3 additional questions — p441
Write the numbers in Exercises 11–14 in scientific notation. 11. 21,000 12. –51,900,000 13. 42,820,000 14. 31,420,000,000,000 Write out the numbers in Exercises 15–18 in full. 15. 8.4 × 105 16. 2.05 × 108 17. –9.1 × 104 18. 3.0146 × 1010 19. In 2006 the population of the USA was approximately 299,000,000. Of those 152,000,000 were female. How many were male? Write your answer in scientific notation.
Round Up When you’re finding the area of a square or the volume of a cube, your calculation will always involve powers. That’s because the formulas for both the area of a square and the volume of a cube involve repeated multiplication of the side length. Powers also come in useful for writing very large numbers in a shorter form — that’s what scientific notation is for. Section 2.5 — Basic Powers
115
Lesson
2.5.4
Mor e on the Or der More Order of Oper a tions Opera
California Standards: Number Sense 1.2 ultipl y, and Add, subtract, multipl ultiply Ad d, subtr act, m di vide rra a tional n umber s divide number umbers actions (inte ger s, fr fractions actions,, and (integ ers ter mina ting decimals) and termina minating tak e positi ve rra a tional take positiv number s to w hole-n umber umbers whole-n hole-number s. po wer ers pow
What it means for you: You’ll learn how to use the PEMDAS rules with expressions that have decimals, fractions, and exponents.
In Chapter One you saw how the order of operations rules help you to figure out which operation you need to do first in a calculation. This Lesson will review what the order is, and give you practice at applying it to expressions with exponents in them.
PEMD AS Tells You Wha der to F ollo w PEMDAS hatt Or Order Follo ollow When you come across an expression that contains multiple operations, the PEMDAS rule will help you to work out which one to do first. For example: Parentheses
Key words: • PEMDAS • operations • exponent
Exponents
= 4 + 6 • (6)2 – 10 ÷ 2
Multiplication and Division
= 4 + 6 • 36 – 10 ÷ 2
Addition and Subtraction
= 4 + 216 – 5
Example Check it out:
4 + 6 • (2 + 4)2 – 10 ÷ 2
= 215
Multiplic ation and division Multiplication have equal priorit priorityy in PEMDAS. o right You work them out from lef leftt tto right.
A ddition and subtr ac tion subtrac action have equal priorit priorityy too — work o right them out from lef leftt tto right.
1
Evaluate the expression 52 – 16 ÷ 23 • (3 + 2).
If you see parentheses with an exponent, the exponent applies to the whole expression inside them. So (1 + 2)2 is (1 + 2) • (1 + 2). Everything in the parentheses is the repeated factor in the multiplication. So to follow the PEMDAS rules you need to simplify the contents of the parentheses first, and then apply the exponent to the result. So (1 + 2)2 = 32
Solution
52 – 16 ÷ 23 • (3 + 2) = 52 – 16 ÷ 23 • 5
Do the ad dition in the par entheses addition parentheses
= 25 – 16 ÷ 8 • 5
Then e valua te the tw oe xponents ev aluate two exponents
= 25 – 2 • 5 = 25 – 10
Ne xt it’ sm ultiplica tion and di vision — Next it’s multiplica ultiplication division do the di vision fir st, as it comes fir st, division first, first, then do the m ultiplica tion multiplica ultiplication
= 15
Finall y do the subtr action Finally subtraction
Guided Practice Evaluate the expressions in Exercises 1–6.
116
Section 2.5 — Basic Powers
1. 6 – 10 • 32
2. (5 – 3)3 + 43 ÷ 8
3. 24 + (3 • 2 – 10)2
4. 5 + 64 ÷ (6 – 2)1
5. (36 ÷ 12 – 24)2
6. (10 • 2 – 5)2 – (4 ÷ 2)3 • 3
Check it out: • When you multiply a negative number by a negative number, the result is positive. • When you multiply a positive number and a negative number, the result is negative. • So if you raise a negative number to an even power, the result will be positive. • But if you raise a negative number to an odd power, the result will be negative. • For example: (–2)3 = –2 • –2 • –2 = 4 • –2 = –8 (–2)4 = –2 • –2 • –2 • –2 = 4 • 4 = 16
Check it out: If you have parentheses inside parentheses, for example, (3 + (4 + 2)), you should start with the innermost parentheses and work outward.
Tak e Car e with Expr essions Tha ve Ne ga ti ve Signs ake Care Expressions hatt Ha Hav Neg tiv When an expression contains a combination of negative numbers and exponents, you need to think carefully about what it means. For example:
–(22) = –(2 • 2) = –4 (–2)2 = –2 • –2 = 4 Example
2
Evaluate the expression (–(32) • 5) + (–3)2. Solution
(–(32) • 5) + (–3)2 = (–9 • 5) + (–3)2 = –45 + (–3)2 = –45 + 9 = –36
Ev alua te the e xponent in the inner par entheses Evalua aluate exponent parentheses Do the m ultiplica tion in the par entheses multiplica ultiplication parentheses Ev alua te the e xponent Evalua aluate exponent Finall y do the ad dition Finally addition
Guided Practice Evaluate the expressions in Exercises 7–12. 8. (–4)2 7. –(42) 9. –(22) • 5 + 1
10. (–4)2 ÷ 2 – 4
11. 10 + (2 • –(52)) + (–7)2
12. 12 + (–(22) + (–2)2) ÷ 2
The Or der Applies to Decimals and F Order Frractions Too When you’re working out a problem involving decimals or fractions you follow the same order of operations. Example
3
⎛1⎞ 1 Evaluate the expression ⎜⎜⎜ ⎟⎟⎟ + • (10 – 7)2. ⎝ 2⎠ 16 Solution 4 ⎛ 1 ⎞⎟ ⎜⎜ ⎟ + 1 • (10 – 7)2 ⎜⎝ 2 ⎟⎠ 16 4
⎛1⎞ 1 action in the par entheses = ⎜⎜⎜ ⎟⎟⎟ + • 32 Do the subtr subtraction parentheses 4
⎝ 2⎠
16
1 1 + •9 16 16 1 9 = + 16 16
=
=
10 = 5 16 8
Then e valua te the tw oe xponents ev aluate two exponents Perf or m the m ultiplica tion erfor orm multiplica ultiplication
Finall y do the ad dition and simplify Finally addition
Section 2.5 — Basic Powers
117
Example
4
Evaluate the expression 0.25 + 7.75 ÷ 3.1 – (0.3)4. Solution
0.25 + 7.75 ÷ 3.1 – (0.3)4 = 0.25 + 7.75 ÷ 3.1 – 0.0081 = 0.25 + 2.5 – 0.0081 = 2.75 – 0.0081 = 2.7419
Ev alua te the e xponent Evalua aluate exponent Then perf or m vision perfor orm the di division Do the ad dition fir st, as it comes fir st addition first, first Finall y do the subtr action Finally subtraction
Guided Practice Evaluate the expressions in Exercises 13–20. ⎛1
3⎞
3
13. ⎜⎜⎜⎝ 2 + 2 ⎟⎟⎟⎠ –
Don’t forget: If your calculation involves a mixture of fractions and decimals, convert everything to either fractions or decimals first. For a reminder of how to do this, see Section 2.1.
14.
15. 0.1 + (0.25)2 – 0.2 ÷ 2 ⎛3
⎛2 1⎞
1⎞
2
⎛ 1 ⎞⎟ ⎜⎜ ⎟ ⎜⎝ 4 ⎟⎠
2
1 8
•3+4÷
1 2
16. (0.72 + 0.08) ÷ 16 + (0.4)2
3
17. ⎜⎜⎜⎝ 4 ÷ 2 ⎟⎟⎟⎠ + ⎜⎜⎜⎝ 3 • 2 ⎟⎟⎟⎠
18. 0.5 • (1 + 0.25)2 + 1.2 ⎛ 1 ⎞⎟ ⎜⎜ ⎟ ⎜⎝ 5 ⎟⎠
2
⎛ 1 ⎞⎟ ⎜⎜ ⎟ ⎜⎝ 2 ⎟⎠
2
19. 2 •
–2
+ (5 ÷ 10)2 • 4
20. (5 • 0.1 + 0.2) •
Independent Practice Evaluate the expressions in Exercises 1–6. 1.
12 + 23 5
2. (42 – 23) ÷ 22 + 81
3. (10 + 24 • 3) + (52 – 15)2 5. (–6)3 • 3 – 122
4. –33 • 22 + 9 6. (43 – 34)2 ÷ (17)2
7. In the expression (x – y2 • z)6, x, y, and z stand for whole numbers. If you evaluate it, will the expression have a positive or a negative value? (The expression is not equal to zero.) Explain your answer. Evaluate the expressions in Exercises 8–13. ⎛1⎞
2
2 27 2 ⎛ 1 ⎞⎟ ⎜ ⎟ + ⎜⎜⎝ 6 ⎟⎠
8. ⎜⎜⎜⎝ 3⎟⎟⎟⎠ + 2 • 10.
⎛ 2 4 ⎞⎟ ⎜⎜ ÷ ⎟ ⎜⎝ 3 5 ⎟⎠
12.
⎛ ⎞ ⎜⎜0.5 i 6 ⎟⎟ ⎜⎝ 8 ⎟⎠
2
Now try these: Lesson 2.5.4 additional questions — p441
2
9. (0.5)2 + 0.8 ÷ (0.1)3 •4
⎛ 1 ⎞⎟ ⎜⎜ ⎟ ⎜⎝ 4 ⎟⎠
3
–
11. (0.5 + 1.8)2 • 1.5 + 0.065 ⎛ 1 ⎞⎟ ⎜⎜ ⎟ ⎜⎝ 2 ⎟⎠
3
13. (0.2 • 4 – 0.3)2 +
•2
5
14. Lakesha is making bread. She has 4 lb of flour, which she splits into two equal piles. Then she splits each of these in half again. She adds three of the resulting piles to her mixture. How much flour has she added to her mixture? Give your answer as a fraction.
Round Up When you have an expression containing exponents, you must follow the order of operations to evaluate it. You use the same order with expressions that contain fractions and decimals too. 118
Section 2.5 — Basic Powers
Section 2.6 introduction — an exploration into:
The Side of a Squar e Square A perfect square has sides whose lengths are whole numbers. You’ll be given square tiles and be asked to construct larger squares with particular areas — you’ll be able to produce some of the larger squares, but not others. The lengths of the sides of the squares are the square roots of the areas. You’ll see that the areas of some squares have whole number square roots, but others don’t. Each small square has an area of 1 square unit. Example Make a square with an area of 4 square units. Then write down the square root of 4. 2
Solution
With 4 square tiles:
2
This a perfect square — it’s got an area of 4 square units and sides of 2 units. So, 2 is the square root of 4.
Exercises 1. Use the tiles to make squares with the given areas. When you have made a square, write the lengths of the sides. a. 9 square units b. 16 square units c. 25 square units
d. 36 square units
2. What are the square root of the following? Use your answers to Exercise 1 to help you. a. 9 b. 16 c. 25 d. 36 You can use the tiles to estimate the square root of a number that is not perfect square. Example Use tiles to estimate the square root of 8. Solution
This is the closest shape you can make to a square using 8 tiles — It’s bigger than a 2 by 2 square, but smaller than a 3 by 3 square. As it’s closer to a 3 by 3 square, you can estimate that the square root of 8 is about 3.
Looks more like this one.
Exercises 3. Construct a figure that is as close to a square as possible. Use this to estimate the square roots of these numbers. a. 5
b. 14
c. 22
Round Up Some numbers are perfect squares — like 4, 9, 16, 25. These numbers have square roots that are whole numbers. If you make a square with a perfect square area, its sides will be whole numbers. Section 2.6 Explor a tion — The Side of a Square 119 Explora
Lesson
2.6.1
California Standards: Number Sense 2.4 ver se rrela ela tionship inv erse elationship Use the in betw een rraising aising to a po wer between pow and e xtr acting the rroot oot of a extr xtracting perf ect squar e inte ger er; for perfect square integ an integer that is not square, determine without a calculator the two integers between which its square root lies and explain why.
What it means for you: You’ll see what a square root is and how to find the square root of a square number.
Section 2.6
Perf ect Squar es and erfect Squares Their R oots Roots If you multiply the side length of a square by itself you get the area of the square. You can do the opposite too — find the side length of the square from the area. That’s called finding the square root.
The Squar e of an Inte ger is a P erf ect Squar e Number Square Integ Perf erfect Square Raising a number to the power two is called squaring it. That’s because you find the area of a square by multiplying the side length by itself. So, the area of a square = s • s = s2, where s is the side length. 2 3
2 22 = 4
Key words: • • • •
perfect square square root positive root negative root
4
3 4 32 = 9
2
4 = 16
All the numbers in red are the squares of the numbers in blue. The square of an integer is called a perfect square. Perfect squares are always integers too.
Example
1•1=1 2•2=4 4 • 4 = 16
These numbers ar aree perfect squares
3.5 • 3.5 = 12.25 5.1 • 5.1 = 26.01
en’t perfect squares These numbers ar aren’t
1
Is the number 81 a perfect square? Solution
9 • 9 = 81 As 9 is an integer, 81 is a perfect square.
Guided Practice Give the square of each of the numbers in Exercises 1–6.
120
1. 4
2. 7
3. 12
4. 1
5. –2
6. –12
Section 2.6 — Irrational Numbers and Square Roots
The Opposite of Squaring iis s Finding the Squar eR oot Square Root You might know the area of a square and want to know the side length. You find the side length of a square by finding the square root.
Area =4
Area = 16
2 2=4 Side length is 2 2 is the square root of 4
4 4 = 16 Side length is 4 4 is the square root of 16
So, to find the square root of a square number, you have to find the number that multiplied by itself gives the square number. Check it out: Your calculator should have a square root button on it that looks like this: ÷ . To find the square root of 2 on your calculator press:
÷
2
=.
On some types of calculators you have to press: 2 ÷
Check it out: You can’t find the square root of a negative number. To square a number you have to multiply it by itself. • If you multiply a positive number by a positive number the result is always positive. • If you multiply a negative number by a negative number the result is also positive.
For example: 5 • 5 = 25. 25 is a square number. 5 is a square root of 25. The symbol
is used to show a square root.
So you can say that
25 = 5.
Unless the number you’re finding the square root of is a perfect square, the square root will be a decimal — and may well be irrational. (There’s more on this in Lesson 2.6.2.)
All P ositi ve Number s Ha ve Two Squar eR oots Positi ositiv Numbers Hav Square Roots Every positive number has one positive, and one negative, square root. This is because 4 • 4 = 16 and –4 • –4 = 16 — so the square root of 16 could be either 4 or –4. A square root symbol,
, by itself means just “the positive square root.”
16 = 4 — this is the positive square root of 16. – 16 = –4 — this is the negative square root of 16.
So if you multiply any number by itself the answer is always positive.
Guided Practice Evaluate the square roots in Exercises 7–14.
Check it out: These numbers are all perfect square numbers. So their square roots will all be whole numbers.
7. 36
8. – 64
9. 100
10. – 144
11. 121
12. – 169
13. 1
14. 400
Section 2.6 — Irrational Numbers and Square Roots
121
Writing Squar eR oots with F Square Roots Frractional Exponents In Section 2.4 you saw that an exponent means a repeated multiplication. When you square a number it is repeated as a factor two times in the multiplication expression. So you can write x • x = x2.
Check it out: If you multiply a square root by itself you get back to the original number.
x
•
x
= x.
1 2
So,
1 2
1 2
9 = 9 and 16 = 16 .
Guided Practice
– Don’t forget: 1 2
Taking the square root of a number is the reverse process to squaring it. Because it undoes squaring we can also write x as x .
Evaluate the expressions in Exercises 15–18. 1 2
–100 means –(100 ). It can’t be (–100) because that would be the square root of a negative number. 1 2
1
1
15. 4 2
16. – 25 2
17. 49
1 2
18. –100
1 2
Independent Practice Give the square of each of the numbers in Exercises 1–6. 1. 6 2. 11 3. 16 4. –10 5. –13 6. –15 7. Marissa is making patterns with small square mosaic tiles. She has 50 tiles. Can she arrange them to make one larger square, using all the tiles? Explain your answer.
Now try these: Lesson 2.6.1 additional questions — p441
8. This year’s senior class will have 225 students graduating. The faculty wants the chairs to be arranged in the form of a square. How many chairs should be put in each row? Evaluate the square roots in Exercises 9–16. 10. – 25 9. 25 11. 64 13. 9
12. – 49 14. 289
1
1
15. 3612
16. –49 2
17. Give the square roots of 16. 18. Give the square roots of 81. 19. A square deck has an area of 81 feet2. Paul is planning to enlarge the deck by increasing the length of each side by 2 feet. How much will the area of the deck increase by?
Round Up If you multiply any integer by itself you will get a perfect square number. The factor that you multiply by itself to get a square number is called its square root. Every square number has one positive and one negative square root. And don't forget — negative numbers don’t have square roots. 122
Section 2.6 — Irrational Numbers and Square Roots
Lesson
2.6.2
Ir s Irrrational Number Numbers
California Standards:
If you find the square root of 2 on your calculator, you get a number that fills the display, and none of the digits repeat. In this Lesson you’ll learn what makes numbers like that special.
Number Sense 1.4 een Difffer erentia entiate between Dif entia te betw ra tional and ir irrra tional n umber s. umbers
What it means for you: You’ll learn what irrational numbers are, and how they’re different from rational numbers.
Ra tional Number s Can Be Written as F Numbers Frractions a
In Section 2.1 you saw that any number that can be written in the form b where a and b are both integers, and b isn’t 0, is called a rational number. For example: can be written as 2 2 1 3.7
Key words: • • • • •
rational irrational integer terminating decimal repeating decimal
0.81
can be written as can be written as
You can write all of these as fractions as described above, so they are all rational numbers.
37 10 9 11
All fractions, integers, terminating decimals, and repeating decimals are rational numbers. You can add square roots of perfect squares to that list too, because they are always integers.
Guided Practice Prove that the numbers in Exercises 1–4 are rational by writing each one as a fraction in its simplest form. 1. 6 2. 0.8 3. 0.3
Check it out: You could never write an irrational number out in full because it would go on forever.
4. 16
Ir s Can Irrra tional Number Numbers Can’’ t Be Written as F Frractions • Any number that can’t be written as a ratio of two integers is called an irrational number. • Irrational numbers are nonterminating, nonrepeating decimals. 0.123456789101112131415161718192021...
Check it out: In Chapter 3, you’ll use p to calculate the circumference of a circle. You can never calculate the circumference exactly, because p goes on forever.
5.12112111211112111112111111211111112...
Neither of these decimals terminate or have repeating patterns of digits. They’re both irrational numbers.
The most famous irrational number is p — you can’t write p as a fraction. p starts 3.1415926535897932384626433832795... The value of p has been calculated to over a million decimal places so far — it never ends and never repeats.
Section 2.6 — Irrational Numbers and Square Roots
123
The square roots of perfect squares are integers — so they’re rational. The square root of any integer other than a perfect square is irrational. 1 = 1, 2 , 3 , 4 = 2, 5 , 6 , 7 , 8 , 9 = 3, 10 ... These numbers are rational
These numbers are irrational
Guided Practice Classify the numbers in Exercises 5–10 as rational or irrational. 5.
7 9
6. p
7. 5
8. 100
9. 1.2543
10. 14
Check it out: The “repeat period” of any decimal is the length of the bit of it that repeats. So the repeat period of 0.3 is one digit — the digit 3, the repeat period of 2.945 is three digits — the group of digits 945, and the repeat period of 5.27316 is two digits — the pair of digits, 16.
Check it out: You know that 1 ÷ 7 must be a rational number because it can be expressed as
1 . 7
Some Decimals Ha ve a Long R epea eriod Hav Re peatt P Period Sometimes it might not be obvious straightaway whether a number is rational or irrational. Some decimals that have a large repeat period may look as if they are irrational but are actually rational. For example, when you divide 1 by 7 on your calculator you get a decimal number. From the number on your calculator display, you can’t tell if that decimal ever ends or repeats. 1 ÷ 7 = 0.1428571... To show that the decimal does in fact repeat, work out 1 ÷ 7 using long division:
0.14285714... 7 1.00000000... 7 30 28 20 14 60 56 40 35 now the same cycle 50 begins again — this is a 49 repeating decimal 10 07 30 28
The repeat period starts here. So you can write this decimal as 0.142857
You can see that the same long division cycle begins again. This means the decimal does repeat.
124
Section 2.6 — Irrational Numbers and Square Roots
It’s really hard to prove a number’s irrational — because it could just be a decimal with a really really long repeat pattern. The only way you can know that a number is irrational is if it has a pattern that you know will never repeat — there’s an endless set of irrational numbers like this to choose from: Æ They might have a pattern of digits that you could generate with a formula. 0.1248163264128256512... This pattern of digits represents the powers of 2: 20, 21, 22, 23, 24, etc. It goes on forever without repeating.
Æ They could have any pattern of digits where each time the pattern is repeated the number of copies of each digit increases. 0.12112211122211112222... In this pattern the numbers 1 and 2 alternate, but each time they are repeated the number of copies of the digits increases by 1.
Guided Practice Check it out: A bar above some digits in a decimal number shows that these digits are repeated over and over again.
Classify the numbers in Exercises 11–15 as rational or irrational. 11. 0.369121518... where the pattern of digits are the multiples of 3 12. 3.122468
13. 0.2647931834
14. 0.14114411144414
15. 0.123112233111222...
Independent Practice
Now try these: Lesson 2.6.2 additional questions — p442
Prove that the numbers in Exercises 1–4 are rational by expressing them as fractions in their simplest form. 1. 14 2. 121 3. 2.6 4. 1.6 5. Read statements a) and b). Only one of them is true. Which one? How do you know? a) All fractions can be written as decimals. b) All decimals can be written as fractions. Classify the numbers in Exercises 6–13 as rational or irrational. 14 6. 10 7. 5 8. 0.497623 10. 9.129587253648
9. 3p
12. 225
13. 22.343344333444...
11. 27
14. Write any four irrational numbers between zero and five.
Round Up Irrational numbers can’t be written as fractions where the numerators and denominators are both integers. Irrational numbers are always nonterminating and nonrepeating decimals. Section 2.6 — Irrational Numbers and Square Roots
125
Lesson
2.6.3
Estima ting Ir oots Estimating Irrra tional R Roots
California Standard:
You saw in the last Lesson that all square roots of integers that aren’t perfect squares are irrational numbers. That means that you could never write their exact decimal values, because the numbers would go on forever. But you can use an approximate value instead.
Number Sense 2.4 Use the inverse relationship between raising to a power and extracting the root of a perfect square integer; for an inte ger tha e, integ thatt is not squar square deter mine without a determine calcula tor the tw o inte ger s calculator two integ ers een w hic h its squar e betw whic hich square between root lies and e xplain w hy . explain wh
Squar eR oots of Nonperf ect Squar es ar e Ir Square Roots Nonperfect Squares are Irrra tional Perfect square numbers have square roots that are integers.
Ma thema tical R easoning 2.7 Mathema thematical Reasoning Indicate the relative advantages of exact and approximate solutions to ve ans wer s giv answ ers problems and gi to a specified de g ree of deg accur ac y. accurac acy
2 The area of this square is 4 units. So its side length must be 4 units = 2 units — which is rational.
2 2
2 =4
What it means for you:
Numbers that are not perfect squares still have square roots.
You’ll learn how to find the approximate square root of any number without using a calculator.
Square roots of integers that are not perfect squares are always irrational numbers. ÷5
Key words:
The area of this square is 5 units. So its side length must be 5 units — which is irrational.
÷5
• irrational • perfect square • square root
2
(÷5) = 5
Guided Practice Say whether each number in Exercises 1–6 is rational or irrational. 1.
9
2.
2
3. 12
4. 16
5. 169
6. 140
Don’t forget: Rounding a number makes it shorter and easier to work with. How much you round a number depends on how accurate you need to be. For example, you might round 99.26 up to 100, down to 99, or up to 99.3 — depending on how accurate you want to be. There are rules for rounding — they’re explained in Chapter 8.
126
You Can A ppr oxima te Ir e R oots ppro ximate Irrra tional Squar Square Roots If you are asked to give a decimal value for 2 , or for any other irrational number, you would have to give an approximation. You could never give an exact answer because the exact answer goes on forever.
Section 2.6 — Irrational Numbers and Square Roots
Example
1
Find the approximate value of
2 using a calculator.
Solution
You should have a button on your calculator that has the square root symbol on it. It will look like this: ÷ . Check it out: Some calculators need you to press
2
÷
To find the square root of 2, press the square root button, then the number 2, and then the equals button, like this: ÷ 2 = . You will get an answer on the screen that looks something like this: 1.414213562
Even though the answer on your screen stops, it’s not the exact answer. It’s just an approximation based on how many digits can fit on the screen. Check it out: ª ” means “is The symbol “ª approximately equal to.” You can use it in situations where you can’t write the exact value of a number because it’s irrational.
Check it out: When you round a decimal answer you will usually be told how many decimal places to round it to. That’s just how many digits there are after the decimal point. If you’re rounding a number to six decimal places, you have to look at the digit in the seventh decimal place. If it’s 5 or more, then round the sixth digit up. If it’s less than 5, then round the sixth digit down. For example, 6 22... 28 = 5.2915026 seventh digit The seventh digit is more than 5, so the sixth digit is rounded up to 3. So,
So you should write your answer like this: Or like this: 2 ª 1.41 Then you’ve shown that you know it’s an approximate answer.
Guided Practice Use your calculator to approximate the square roots in Exercises 7–12. Give the values to six decimal places. 7. 5
8.
9. 10
10.
11.
12. 160
47
6 29
Estima te Squar eR oots Using a Number Line Estimate Square Roots Estimating the square root of a number without using a calculator involves working out which two perfect square numbers it lies between. For example, the square roots of all the numbers between 4 and 9 lie between 2 and 3 on the number line.
4
5
6
7 8 9
x 2 3
28 = 5.291503 to 6 d.p.
There’s more on how and when to round numbers in Section 8.3.
State the accuracy you’ve given the approximate answer to.
2 = 1.414213562 (to 9 decimal places)
9 10 11 12 13 14 15 16
x 7
x x
3
4 13
The square roots of all the numbers between 9 and 16 lie between 3 and 4 on the number line.
Section 2.6 — Irrational Numbers and Square Roots
127
There are two steps to finding an approximation of the square root of a number. For example: find the two numbers that
7 lies between on the number line.
1) First find the two perfect square numbers that 7 lies between on the number line. 7 lies between 4 and 9
4
5
6
8
7
perfect square number
9 10
perfect square number
2) Find the square roots of these two perfect square numbers. The square root of 7 must be between these two square roots.
÷9 ÷16 ÷25
÷1 ÷4 4 = 2,
9 = 3.
1
2
÷7
3
4
5
So 7 must lie between 2 and 3. Example
2
Find the two numbers that 14 lies between on the number line. Solution
First find the two perfect squares that 14 lies between on the number line. 14 lies between 9 and 16. 9 = 3 and 16 = 4. So 14 lies between 3 and 4 on the number line.
Example
3
Find the numbers that 18 lies between on the number line. Solution
First find the two perfect squares that 18 lies between on the number line. 18 lies between 16 and 25. 16 = 4 and
25 = 5.
So 18 lies between 4 and 5 on the number line. 128
Section 2.6 — Irrational Numbers and Square Roots
Guided Practice In Exercises 13–20 find the whole numbers that the root lies between. 13.
5
14. 15
15.
24
16.
46
17.
68
18.
98
19. 125
20. 150
Independent Practice Use your calculator to approximate the square roots in Exercises 1–4 to four decimal places. 1. 17
2.
3.
4. 155
73
28
In Exercises 5–10 say which two perfect square numbers the number lies between. 5. 3 6. 29 7. 50 8. 95 9. 125 10. 200 In Exercises 11–18 find the whole numbers that the root lies between.
Now try these: Lesson 2.6.3 additional questions — p442
11.
3
12. 13
13.
22
14.
33
15.
58
16.
93
18.
216
17. 160
19. A square has an area of 85 inches2. What whole-inch measurements does the side length lie between? 20. If Don’t forget: A cube has 6 faces that are all identical squares.
a ª 2.4 then which two perfect squares does a lie between?
21. Latoya has a new office. It is a square room, with a floor area of 230 feet2. She wants to fit a 15 ft desk area along one wall — will this fit along one of the sides? Explain your reasoning. 22. A math class is shown a cube made of card. Pupils are told that the total surface area of the cube is 90 cm2. They are asked to guess the length of each side of the cube in centimeters. Peter guesses 10 centimeters, and John guesses 4 centimeters. Whose guess is the closer?
Round Up You can never write out the exact value of the square root of a nonperfect square number — but you can use an approximation. To figure out which two integers a number’s square root lies between, it’s just a case of knowing which two perfect squares the number lies between, and finding their square roots. Section 2.6 — Irrational Numbers and Square Roots
129
Chapter 2 Investigation
Designing a Deck You have to add, subtract, multiply, and divide numbers to solve lots of real-life problems. Being able to use powers and find square roots can sometimes come in useful too. The Dedona family has a deck on the back of their house that they want to make bigger. The current deck is a rectangle with a length of 12 feet and width of 8 feet. They want the new deck to be 225 square feet in area. 12 feet
8 feet
Deck
3 feet
Railing
House
Part 1: What would the dimensions of the enlarged deck be if it were in the shape of a square? Part 2: Make four different designs for additions to the deck that satisfy the Dedonas’ area requirement. Things to think about: • The new deck must contain the original, rectangular deck. However, the new, enlarged deck doesn’t have to be rectangular itself. Extensions The cost of railing is $18.95 for a 3-foot section. Railing is only sold in 3-foot sections. 1) What design for the new deck provides the required area with the least amount of railing? 2) How many sections of railing do they need to buy for the new deck? What is the total cost for the railing? Open-ended Extensions 1) Propose a design that would satisfy the Dedonas’ area requirements and cost the most money for railings. The narrowest any section of the deck can ever be is 1 foot. How would your solution change if the existing deck was torn down? How would it change if the 3-foot railing sections could not be cut? 2) Would a square design be the least expensive if there was no railing against the house?
Round Up When you’re solving real-life problems, you often have to combine all the operations. You use multiplication to find areas and addition to find the total length around the edge. And if you have a square with a set area, you can find its side length by finding the square root of the area. pter 2 In vestig a tion — Designing a Deck Chapter Inv estiga 130 Cha
Chapter 3 Two-Dimensional Figures Section 3.1
Exploration — Area and Perimeter Patterns ............. 132 Perimeter, Circumference, and Area ......................... 133
Section 3.2
Exploration — Coordinate 4-in-a-row ........................ 149 The Coordinate Plane ............................................... 150
Section 3.3
Exploration — Measuring Right Triangles ................. 158 The Pythagorean Theorem ....................................... 159
Section 3.4
Exploration — Transforming Shapes ........................ 174 Comparing Figures .................................................... 175
Section 3.5
Constructions ............................................................ 196
Section 3.6
Conjectures and Generalizations .............................. 206
Chapter Investigation — Designing a House .................................... 213
131
Section 3.1 introduction — an exploration into:
Ar ea and P erimeter P a tter ns Area Perimeter Pa tterns You can draw shapes which have the same area, but different perimeters. In this Exploration, you’ll look at how to maximize the perimeter for a given area. You’ll also look at shapes that have the same perimeters, but different areas. You can find the area of a shape by counting up the number of unit squares. The perimeter is calculated by finding the sum of all the side lengths. Example 6
Find the area and the perimeter of this shape.
2
2
Solution
Area = 12 square units
Perimeter = 6 + 2 + 6 + 2 = 16 units
6
When given a set area, you can draw shapes with different perimeters — like these: These shapes both have an area of 10 square units.
This one has a perimeter of 22 units...
...but this one has a perimeter of 14 units.
And when given a set perimeter, you can draw shapes with different areas — like these: These shapes both have a perimeter of 12 units. This one has an area of 5 square units...
...but this one has an area of 8 square units.
Exercises 1. Find the area and perimeter of each shape. G A
B
C
D
E
F
I H 2. Look at the areas and perimeters of the following sets of shapes. What do you notice about them? Which type of shape maximizes the perimeter in each set? A and B C and D E and F G, H and I 3. Draw three different rectangles with areas of 12 square units that all have different perimeters. What are the dimensions of the rectangle that has the largest perimeter?
Use only whole-number dimensions for Exercises 3–6.
4. Draw the rectangle with an area of 20 square units that has the largest perimeter possible. 5. Draw two rectangles that have different areas but both have perimeters of 14 units. 6. Draw a rectangle that has a perimeter of 20 units and has the largest area possible.
Round Up A rectangle with a big difference between its length and width measurement will have a large perimeter for its area. It works the other way for maximizing the area of a rectangle with a fixed perimeter — the closer the shape is to a square, the bigger the area will be. a tion — Area and Perimeter Patterns Explora 132 Section 3.1 Explor
Lesson
Section 3.1
3.1.1
Polygons and Perimeter
California Standards:
You’re probably pretty familiar with a lot of shapes — this Lesson gives you a chance to brush up on their names, and shows you how you can use formulas to find the distance around the outside of some shapes.
Measurement and Geometry 2.1 Use formulas routinely for finding the perimeter and area of basic twodimensional figures, and the surface area and volume of basic three-dimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders.
Polygons Have Straight Sides Polygons are flat shapes. They’re made from straight line segments that never cross. The line segments are joined end to end.
This is a polygon
This isn’t a polygon (the lines cross)
What it means for you: You’ll learn about the names of different shapes and use formulas for finding the perimeters of some shapes.
Key words: • • • • • • •
polygon perimeter regular polygon irregular polygon quadrilateral parallelogram trapezoid
This isn’t a polygon (it’s not made from straight lines)
The name of a polygon depends on how many sides it has. 3 sides
Triangle
7 sides
Heptagon
4 s i de s
Quadrilateral
8 sides
Octagon
5 sides
Pentagon
9 sides
Nonagon
6 sides
Hexagon
10 sides
Decagon
Example
1
Identify each of the following shapes.
2.
1.
Solution
1. This shape has 6 sides, so it’s a hexagon. 2. This shape has 7 sides, so it’s a heptagon.
A Quadrilateral is a Polygon with Four Sides Don’t forget: Dashes are used to show that certain lengths are equal. You might see single and double dashes. Sides with double dashes are the same length as each other, but not the same as those with single dashes.
A quadrilateral is any shape that has four sides. You need to be able to name a few of them. Rectangle A rectangle has opposite sides of equal length. It also has four right angles.
Parallelogram Its opposite sides are parallel.
Rhombus This is a parallelogram with all sides of the same length.
Square This is a special kind of rectangle. It has four sides all of equal length.
Trapezoid This has only one pair of parallel sides.
Section 3.1 — Perimeter, Circumference, and Area
133
Guided Practice Identify each of the following polygons: 1.
2.
3.
Regular Polygons Have Equal Sides and Angles Regular polygons have equal angles, and sides of equal length. Irregular polygons don’t have all sides and angles equal.
Check it out: A rhombus has all sides of equal length, but the angles aren’t all the same. So a rhombus is always irregular. A square, however, has all sides and all angles equal, so a square is always regular.
All sides and angles are the same.
rregular egular
irr egular irregular
Neither sides nor angles are all the same.
All sides and angles are the same. The sides are the same lengths but the angles aren’t the same.
rregular egular Example
irr egular irregular
2
Decide whether this polygon is regular or irregular. Solution
The shape has all angles equal. But the lengths of the sides are not the same, so it is an irregular polygon.
Guided Practice
Check it out: Polygons occur a lot in everyday life — for example:
Decide whether each of the following shapes is a regular polygon, an irregular polygon, or not a polygon at all.
The Pentagon
4. 2 in.
2 in.
2 in.
2 in.
5.
6.
octagon
7. triangle
3 ft
4.2 ft 6.4 ft
134
8.
4 ft
Section 3.1 — Perimeter, Circumference, and Area
Perimeter is the Distance Around a Polygon The perimeter is the distance around the edge of a shape.
Don’t forget: The “d” in the formula for the perimeter of a parallelogram stands for the length of the diagonal. Don’t get this confused with the vertical height, which you’ll use when you work out the area in the next Lesson.
You can find the perimeter by adding up the lengths of the sides of a polygon, but some polygons have a formula you can use to find the perimeter more quickly.
P = 2(l + w)
P = 2(b + d)
P = 4s
w
d 5
s
l
b Example
Square:
Rectangle:
Parallelogram:
s
3
Find the perimeter of a rectangle of length 54 cm and width 26 cm. Solution
Don’t forget: Perimeter is a distance and needs a unit. Check what unit the question contains and make sure your answer has the correct unit.
Draw a diagram, and use the formula P = 2(l + w).
26 cm
Substitute the values for l and w, and evaluate.
54 cm
P = 2 × (54 cm + 26 cm) = 2 × 80 cm = 160 cm.
Guided Practice 9. Find the length of the diagonal of a parallelogram that has a base of 4.6 in. and a perimeter of 14 in. 10. Find the perimeter of a square of side 8.3 m.
Independent Practice Find the perimeter of the figures in Exercises 1–4. 1. 3.2 m 2. 6.1 cm 6.1 cm 1.1 m Now try these: Lesson 3.1.1 additional questions — p443
3.
5 ft
5 ft
4.
2 in. 2.1 in.
5 ft 1.8 in.
5 ft 5ft
3 in.
5. Brandy wants to know how many pieces of wood she needs to mark out the boundary of her new house. How many pieces of wood will she need if each piece of wood is 50 in. long and her boundary is a square of side 650 in.?
Round Up There are a few formulas here that make it much quicker to do perimeter calculations. If you can’t apply one of the formulas, remember that you can always just add up the lengths of the sides. Section 3.1 — Perimeter, Circumference, and Area
135
Lesson
3.1.2
Areas of Polygons
California Standards:
Area is the amount of space inside a shape. Like for perimeter, there are formulas for working out the areas of some polygons. You’ll practice using some of them in this Lesson.
Measurement and Geometry 2.1 Use formulas routinely for finding the perimeter and area of basic twodimensional figures, and the surface area and volume of basic three-dimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders.
What it means for you: You’ll use formulas to find the areas of regular shapes.
Area is the Amount of Space Inside a Shape Area is the amount of surface covered by a shape. Parallelograms, rectangles, and squares all have useful formulas for finding their areas. Triangles and other shapes can be a little more difficult, but there are formulas for those too — which we’ll come to next. Rectangle:
Square:
Parallelogram:
A = lw
A = s2
A = bh
w Key words: • • • • • •
area triangle parallelogram trapezoid formula substitution
h
s
l Example
s b
1
Use a formula to evaluate the area of this shape.
7 in.
Solution
Check it out: The area of a parallelogram is exactly the same as the area of a rectangle of the same base length and vertical height.
Use the formula for the area of a rectangle. Substitute in the values given in the question to evaluate the area. A = lw = 7 in. × 2 in. = 14 in2 You can also rearrange the formulas to find a missing length: Example
Check it out: Remember — the height you use to calculate the area of a parallelogram is the vertical height and not the length of the side.
Don’t forget: Use dimensional analysis to make sure your answer has the correct units. You’re multiplying a length by a length — for example, meters × meters. So areas should always be a square unit — such as square meters (m2).
136
2 in.
2
Find the height of a parallelogram of area 42 cm2 and base length 7 cm. Solution
Rearrange the formula for the area of a parallelogram, and substitute. A = bh A b = h=h b b 42 h= = 6 cm 7
h 7 cm
Guided Practice 1. Find the area of a square of side 2.4 m. 2. Find the length of a rectangle if it has area 30 in2, and width 5 in.
Section 3.1 — Perimeter, Circumference, and Area
The Area of a Triangle is Half that of a Parallelogram The area of a triangle is half the area of a parallelogram that has the same base length and vertical height. Don’t forget: The height you use for working out the area of a triangle is the vertical height and not the length of a side. This vertical height is often called the altitude.
height (h)
height (h) base (b)
Area of triangle = = =
Don’t forget: Multiplying by
1 is the same 2
as dividing by 2.
Example
1 2
base (b) • area of parallelogram
1 (base 2
× height)
1
A = 2 bh
1 bh 2
3
Find the base length of the triangle opposite if it has an area of 20 in2 and a height of 8 in.
8 in.
Solution
Rearrange the formula for the area to give an expression b in. for the base length of the triangle. 1 A = bh 2 2 A = bh Multiply both sides by 2 2A 2A h = b = b . So b = Divide both sides by the height (h) h h h Now substitute in the values and evaluate to give the base length. b = (2 × 20) ÷ 8 = 5 in.
Guided Practice 3. Find the area of a triangle of base length 3 ft and height 4.5 ft. 4. Find the base length of a triangle with height 50 m and area 400 m2.
Break a Trapezoid into Parts to Find its Area Check it out: Splitting the trapezoid into two triangles isn’t the only way — it’s just the easiest. You will get the same answer if you split it into two triangles and a rectangle, for example.
The most straightforward way to find the area of a trapezoid is to split it up into two triangles. You then have to work out the area of both triangles and add them together to find the total area. base of triangle 2 (b2)
2
height (h)
1 base of triangle 1 (b1)
Notice that both triangles have the same height but different bases. Section 3.1 — Perimeter, Circumference, and Area
137
So, the area of the trapezoid is the sum of the areas of each triangle.
Check it out: Taking out the common factor uses the distributive property that you learned about in Chapter 1 (see p8).
Area of trapezoid = area of Triangle 1 + area of Triangle 2 b2 1 1 Area of trapezoid = b1h + b2h 2 2 1 h Take out the common factor of h to give: Area of trapezoid =
1 h(b1 2
2
b1
+ b2)
A = 12 h(b1 + b2 ) Example
4 12 ft
Find the area of the trapezoid shown.
8 ft
Solution
Area of trapezoid =
1 h(b1 2
+ b2)
30 ft
Substitute in the values given in the question and evaluate. Area of trapezoid = Don’t forget: It’s always best to draw a diagram before attempting to solve an area question. It’s easy to make mistakes otherwise.
1 × 2
8 ft × (12 ft + 30 ft) =
1 × 2
8 ft × 42 ft = 168 ft2.
Guided Practice Find the areas of the trapezoids in Exercises 5–8, using the formula. 5.
6.
5 in.
7.
20 cm
3 in.
11 cm
10 in.
4 cm
8.
1.1 m
105 ft 80 ft
0.7 m
245 ft
1.5 m
Independent Practice Find the area of each of the shapes in Exercises 1–6. 2. 1. 3. 1.2 ft
2.3 in.
1m 2 in. 1m
1.2 ft
Now try these: Lesson 3.1.2 additional questions — p443
11 cm 4.5 ft
4.
2.5 in.
5.
6. 12 cm
7 in.
3.1 ft
20 cm
7. Miguel wants to know the area of his flower bed, shown opposite. Find the area using the correct 3.1 m formula. 2.4 m
Round Up Later you’ll use these formulas to find the areas of irregular shapes. Make sure you practice all this stuff so that you’re on track for the next few Lessons. 138
Section 3.1 — Perimeter, Circumference, and Area
Lesson
3.1.3
Circles
California Standards:
You’ve already met the special irrational number p or “pi”. Now you’re going to use it to find the circumference and area of circles.
The distance of any point on a circle from the center is called the radius. The distance from one side of the circle to the other, through the center point, is called the diameter. Notice the diameter is always twice the radius.
diameter = 2 • radius Example
Key words: • • • • •
pi (p) radius diameter irrational number circumference
Check it out: Circles are not polygons — they don’t have any straight sides. A circle is formed from the set of all points that are an equal distance from a given center point.
d = 2r
eter
What it means for you: You’ll find the circumference and area of circles using formulas.
Circles Have a Radius and a Diameter
diam
Measurement and Geometry 2.1 Use formulas routinely for finding the perimeter and area of basic twodimensional figures, and the surface area and volume of basic three-dimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders.
radius
1
If a circle has a diameter of 4 in., what is its radius? Solution
4 in
Use the formula: d = 2r. d 2 Substitute d from question: r = 4 ÷ 2 = 2 in.
Rearrange to give r in terms of d, so r =
r
Guided Practice 1. If a circle has a radius of 2 in., what is its diameter? 2. A circle has a diameter of 12 m. What is its radius?
Circumference is the Perimeter of a Circle The circumference is the distance around the edge of a circle. This is similar to the perimeter of a polygon. There’s a formula to find the circumference.
Check it out: There’s a special p button on your calculator that will allow you to do very precise calculations. Otherwise, use the approximate values of 3.14 or
22 7
in your
r
Circumference = p • diameter d
C
C = pd Because the diameter = 2 × radius, Circumference = 2 × p × radius
pr C = 2p
calculations.
Section 3.1 — Perimeter, Circumference, and Area
139
Example Check it out: p is the ratio between the circumference and the diameter of a circle. For any size of circle: circumference ÷ diameter = p.
2
Find the circumference of the circle below. Use the approximation, p ª 3.14. 12 cm
Solution
Use the formula that has diameter in it. Substitute in the values and evaluate with your calculator. C = pd ª 3.14 × 12 cm = 37.68 cm ª 37.7 cm.
Guided Practice Check it out: There are two formulas for the circumference. If you’re given the radius in the question, use C = 2pr; if you’re given the diameter, use C = pd.
Find the circumference of the circles in Exercises 3–6. 10 cm 5. 3. 4. 6. 2m 5 ft
13 in.
7. Find the radius of a circle that has a circumference of 56 ft. 8. Find the diameter of a circle that has a circumference of 7 m.
The Area of a Circle Involves p Too The area of a circle is the amount of surface it covers.
Check it out:
The area of a circle is related to p — just like the circumference. There’s a formula for it:
p is the ratio between the area and the square of the radius. For any size of circle: area ÷ (radius)2 = p
A
r
Area = p • (radius)2 A = p r2
Example
3
Find the area of the circle opposite, using p ª 3.14. Solution
Use the formula: A = p r2 Substitute in the values and evaluate the area
A ª 3.14 × (12 ft)2 = 3.14 × 144 ft2 = 452.16 ft2 ª 452 ft2 140
Section 3.1 — Perimeter, Circumference, and Area
12 ft
If you know the area of a circle you can calculate its radius: Example
4
The area of a circle is 200 cm2. What is the radius of this circle? Use p ª 3.14. Solution
Don’t forget: r2 = r × r, not 2r.
The question gives the area, and you need to find the radius. This means rearranging the formula for the area of a circle to get r by itself. A = pr2 A = r2 p A r= p 200 r= ª 63.7 ª 8 cm p
Divide both sides by p Take the square root of both sides Substitute in the values and find the radius
Guided Practice Don’t forget: You don’t use the diameter in area problems, so it’s useful to make sure you find the radius before attempting any other calculations. Remember, the radius is half the diameter.
9. Find the area of a circle that has a diameter of 12 in. 10. Find the area of a circle that has a radius of 5 m. 11. If a circle has an area of 45 in2, what is its radius?
Independent Practice In Exercises 1–3, find the area of the circles shown. 2.
1. r=8m
3. d = 2 in.
C = 45 cm
4. Find the circumference and area of a circle with a diameter of 6 m. 20 ft Now try these: Lesson 3.1.3 additional questions — p443
5. Lakesha has measured the diameter of her new whirlpool bath as 20 ft. Find its surface area. 6. Find the circumference of the base of a glass with a 1.5 inch radius. 7. Find the area of the base of the glass in Exercise 6. 8. A circle has an area of 36 cm2. Find its radius and circumference.
Round Up This Lesson is all about circles, and how to find their circumferences and areas. There are a few formulas that you need to master — make sure you practice rearranging them. Section 3.1 — Perimeter, Circumference, and Area
141
Lesson
3.1.4
Areas of Complex Shapes
California Standards:
You’ve practiced finding the areas of regular shapes. Now you’re going to use what you’ve learned to find areas of more complex shapes.
Measurement and Geometry 2.2 Estimate and compute the area of more complex or irregular two- and threedimensional figures by breaking the figures down into more basic geometric objects.
What it means for you: You’ll use the area formulas for regular shapes to find the areas of more complex shapes.
Key words:
Complex Shapes Can Be Broken into Parts There are no easy formulas for finding the areas of complex shapes. However, complex shapes are often made up from simpler shapes that you know how to find the area of. To find the area of a complex shape you: 1) Break it up into shapes that you know how to find the area of. 2) Find the area of each part separately. 3) Add the areas of each part together to get the total area.
Shapes can often be broken up in different ways. Whichever way you choose, you’ll get the same total area.
• complex shape • addition • subtraction
Example
1 5 cm
Find the area of this shape.
Don’t forget: Areas should always have a squared unit. Check what it should be and make sure your answer includes it.
2 cm Solution
Split the shape into a rectangle and a triangle.
Check it out:
B
You get the same answer however you split the shape.
3.2 cm
A
You could also have split the shape into a rectangle and a trapezoid.
A
1.8 cm
B Area A is a rectangle. Area A = bh = 5 cm × 2 cm = 10 cm2. Area B is a triangle. 1
Area B = 2 bh =
1 2
× 2 cm × 1.8 cm = 1.8 cm2.
Total area = area A + area B = 10 cm2 + 1.8 cm2 = 11.8 cm2
142
Section 3.1 — Perimeter, Circumference, and Area
4 cm
Guided Practice 1. Find the area of the complex shape below. 50 ft Don’t forget:
12 ft 18 ft
It’s easy to count some pieces of the shape twice. Always draw the shape split into its parts first so you know exactly what you’re dealing with.
12 ft 30 ft
18 ft
12 ft 32 ft
You Can Find Areas by Subtraction Too So far we’ve looked at complex shapes where you add together the areas of the different parts. For some shapes, it’s easiest to find the area of a larger shape and subtract the area of a smaller shape.
Check it out: Most problems can be solved by either addition or subtraction of areas. Use whichever one looks simpler.
Example
2
Find the shaded area of this shape.
2 cm B 10 cm 5 cm
Solution
A
First calculate the area of rectangle A, 20 cm then subtract the area of rectangle B. Area A = lw = 20 × 10 = 200 cm2 Area B = lw = 5 × 2 = 10 cm2 Total area = area A – area B = 200 cm2 – 10 cm2 = 190 cm2 Since there are many stages to these questions, always explain what you’re doing and set your work out clearly. Example
3
Find the shaded area of this shape. 23 ft
Solution
First calculate the area of triangle A, then subtract the area of triangle B. Area A =
A
12 ft B
1 1 bh = × 54 ft × 23 ft = 621 ft2 2 2
7 ft
54 ft
1 1 bh = × 12 ft × 7 ft = 42 ft2 2 2 Total area = area A – area B = 621 ft2 – 42 ft2 = 579 ft2
Area B =
Section 3.1 — Perimeter, Circumference, and Area
143
Guided Practice Use subtraction to find the areas of the shapes in Exercises 2–4. 5 ft 2. 3. 4. 100 in. 4 ft 0.7 m 5 ft 3m 30 in. 20 in. 1m 30 in.
7 ft
3m
Independent Practice Use either addition or subtraction to find the areas of the following shapes. 12 mm 2. 1. 2 cm 11 mm 7 mm 6 mm 3 cm 5 cm
8 cm
100 ft
4.
3. 20 in. 40 in.
50 ft
125 ft
9 in.
23 ft
25 ft
130 ft
5. Damion needs his window frame replacing. If the outside edge of the frame is a rectangle measuring 3 ft × 5 ft and the pane of glass inside is a rectangle measuring 2.6 ft by 4.5 ft, what is the total area of the frame that Damion needs? 6. Aisha has a decking area in her backyard. Find its area, if the deck is made from six isosceles triangles of base 4 m and height 5 m. Now try these: Lesson 3.1.4 additional questions — p444
7. Find the area of the metal bracket opposite.
4 in. 2.6 in. 3.5 in.
5 in.
Round Up You can find the areas of complex shapes by splitting them up into shapes you know formulas for — squares, rectangles, triangles, trapezoids, parallelograms... Take care to include every piece though. 144
Section 3.1 — Perimeter, Circumference, and Area
Lesson
3.1.5
More Complex Shapes
California Standards:
In the last Lesson you found the areas of complex shapes by breaking them down into rectangles and triangles. Complex shapes can sometimes be broken down into other shapes — such as parts of circles or trapezoids. That’s what you’ll practice in this Lesson. You’ll also look at finding the perimeters of complex shapes.
Measurement and Geometry 2.2 Estimate and compute the area of more complex or irregular two- and threedimensional figures by breaking the figures down into more basic geometric objects.
What it means for you: You’ll find the areas of complex shapes by using the area formulas for simple shapes, including those for circles and trapezoids, and you’ll also find perimeters of complex shapes.
Complex Shapes Can Contain Circles Some complex shapes involve circles, or fractions of circles. To calculate the area, you first have to decide what fraction of the full circle is in the shape — for example, a half or a quarter. Once you know what fraction of the circle you want, find the area of the whole circle, and then multiply that area by the fraction of the circle in the shape. For example, a semicircle has half the area of a full circle. Example
Key words: • sector • semicircle • circle • complex shapes • trapezoid
1
Find the area of the complex shape opposite. Solution
6 in.
Split the shape into a semicircle and a rectangle. 5 in. 5 in.
The semicircle has half the area of a full circle, and has a diameter of 5 in. This means its radius is 2.5 in.
Don’t forget: If you’re given the diameter of a circle in an area question, you need to halve it to find the radius before using the area formula.
Area of full circle = pr2 = p × (2.5 in)2 = p × 6.25 = 19.6 in2 Area of semicircle = 0.5 × area of full circle Example 1 = 0.5 × 19.6 in2 = 9.8 in2 Area of rectangle = lw = 5 in. × 6 in. = 30 in2 Total area = area of semicircle + area of rectangle = 9.8 in2 + 30 in2 Total area = 39.8 in2
Guided Practice Don’t forget:
1. Find the area of the complex shape opposite.
1 in.
A little square in a corner means that the angle is a right angle.
5 in.
Section 3.1 — Perimeter, Circumference, and Area
145
Look Out for Trapezoids and Parallelograms Too Some complex shapes need to be broken into more parts than others. Often it’s not obvious what the best way to break them up is. If you only look for triangles and rectangles you could miss the easiest way to solve the problem — look out for trapezoids and parallelograms too. Example Don’t forget: If you see a problem that looks too tricky, don’t panic! Break it down into small manageable chunks, instead of attempting the whole thing right away.
2
Find the area of the complex shape below by breaking it down into trapezoids. 4 ft 2.5 ft
2.5 ft
A B
0.3 ft
C
3.5 ft D
2.5 ft
1 ft 0.7 ft 0.8 ft
Solution
The equation for the area of a trapezoid is A =
1 h(b1 2
+ b2 ).
Area of trapezoid A =
1 2
× 0.3 ft × (2.5 ft + 4 ft) = 0.975 ft2
Area of trapezoid B =
1 2
× 1 ft × (4 ft + 2.5 ft) = 3.25 ft2
Area of trapezoid C =
1 2
× 0.7 ft × (2.5 ft + 3.5 ft) = 2.1 ft2
Area of trapezoid D =
1 2
× 0.8 ft × (3.5 ft + 2.5 ft) = 2.4 ft2
Total area = 0.975 ft2 + 3.25 ft2 + 2.1 ft2 + 2.4 ft2 = 8.725 ft2
Guided Practice 2. Find the area of the complex shape below. 4 cm 15 cm 10 cm 7 cm 3 cm
146
Section 3.1 — Perimeter, Circumference, and Area
17 cm
Finding the Perimeter of Complex Shapes The perimeter is the distance around the edge of a shape. To find the perimeter of a complex shape, you need to add the lengths of each side. It’s likely that you won’t be given the lengths of all the sides, so you may need to find some lengths yourself — for example, the circumference of a semicircle, which is half the circumference of a full circle. Example
3
Find the perimeter of the complex shape opposite. Don’t forget: Remember — don’t include the lengths of sides that don’t form the outline of the final shape. For example, the fourth side of the rectangle isn’t included here.
Solution
6 in
The circumference of a semicircle is half the circumference of a full circle of the same radius. Circumference of semicircle =
1 2
5 in • circumference of full circle
= 12 pd =
1 2
× p × 5 in. = 7.85 in.
Perimeter of 3 sides of rectangle = 6 in. + 5 in. + 6 in. = 17 in. Total perimeter = circumference of semicircle + perimeter of rectangle. Total perimeter = 7.85 in. + 17 in. = 24.85 in.
Don’t forget: Perimeter has units of length (for example, feet). Area has units of length squared (for example, ft2).
Guided Practice 3. Find the perimeter of the shape below. 3 in. 1 in. 4.5 in.
2 in. 1 in.
5 in. 4. Davina has made a flower shape out of some wood by cutting out a regular pentagon and sticking a semicircle to each side of the pentagon, as shown. She wants to make a border for her shape by sticking some ribbon all around the edge. Find the length of ribbon that Davina will need.
10 cm
Section 3.1 — Perimeter, Circumference, and Area
147
Independent Practice 1. Kia’s swimming pool is rectangular in shape with a circular wading pool at one 10 m corner, as shown. Find the total surface area of Kia’s pool and the distance around the edge.
20 m
15 m
2. Find the area of the face of the castle below (don’t include the windows). Assume that all the windows are the same size and that all the turrets are the same size, and evenly spaced. 100 ft 70 ft
50 ft 50 ft
220 ft 50 ft
Check it out: If there is a rectangle overlapping a circle, always try to find the area of the rectangle, then the area of the rest of the circle. You’ll get in a mess if you try to solve it the other way around!
440 ft 3. Find the area of the button shown below if each hole has a diameter of 0.1 in. and the button has a diameter of 1.2 in.
4. T.J. has five friends coming to his 13th birthday party. He bakes a cake that is 12 inches in diameter. At the party, T.J. and his friends divide the cake equally between them, into identically shaped slices, as shown. Find the perimeter and area of the base of each slice of cake. 5. Find the perimeter and area of the shape below. Now try these: Lesson 3.1.5 additional questions — p444
3 cm 5 cm 2 cm 2.3 cm
Round Up Now you know everything you need to know about finding the area and perimeter of complex shapes. The first step is to look at the shape and decide on the easiest way to break it up into simple shapes. 148
Section 3.1 — Perimeter, Circumference, and Area
Section 3.2 introduction — an exploration into:
Coor dina te 4-in-a-r ow Coordina dinate 4-in-a-ro Ordered pairs are used to represent points on the coordinate plane. The goal of this game is to get as many points in a line as possible — the lines can be vertical, horizontal, or diagonal. You score for each row of four or more points that you make — the scoring system is below. Scoring System 4-in-a-row 1 point 5-in-a-row 2 points 6-in-a-row 3 points (or more) Example You need part of a coordinate plane, like shown. Players take turns calling out coordinates.
y
10 9 8
For example: Player 1 ( Player 2 ( Player 1 ( Player 2 ( Player 1 (
7 6
): (4, 2) — right 4, up 2. ): (9, 3) — right 9, up 3. ): (5, 3) — right 5, up 3. ): (6, 4) — right 6, up 4. ): (3, 1) — right 3, up 1.
5 4 3 2
Now, Player 1 needs to get (2, 0) to get 4-in-a-row and score 1 point. But it’s Player 2’s turn next, and if they choose (2, 0), they’ll block the point.
1 0 0
1
2
3
4
5
6
7
8
9
10
x y 10 9
After a while, your coordinate plane will look a bit like this:
8 7
Player 1 ( ) has 4 points in total so far and Player 2 ( ) has 3 points so far. So Player 1 is winning at the moment.
6
3 points
2 points
5
1 point
4
If a player calls out a point that’s already taken, or plots a different one to that which they called, they lose their turn.
3 2 1 0 0
1
2
3
4
5
6
7
8
9
1 point
10
x
Exercises 1. Play the game with another person. The person whose birthday is next goes first. 2. Play the game again. This time, players are allowed to pick two points in each turn.
Round Up You can pinpoint a certain place on the coordinate plane using a pair of coordinates, and by plotting several points you can form a straight line. Section 3.2 Explor a tion — Coordinate 4-in-a-row 149 Explora
Lesson
Section 3.2
3.2.1
Plotting Points
California Standards:
When you draw a graph you draw it on a coordinate plane. This is a flat grid that has a horizontal axis and a vertical axis. You can describe where any point on the plane is using a pair of numbers called coordinates.
Measurement and Geometry 3.2 Understand and use coordinate graphs to plot simple figures, determine lengths and areas related to them, and determine their image under translations and reflections.
What it means for you:
You Plot Coordinates on a Coordinate Plane The coordinate plane is a two-dimensional (flat) area where points and lines can be graphed. y-axis y
4
You’ll see how to use a grid system to plot numbered points.
3 2 1
x-axis origin
–4 –3 –2 –1 0 –1
1
2
3
4 x
–2 –3 –4
Key words: • • • •
coordinate x-axis y-axis quadrant
The plane is formed by the intersection of a vertical number line, or y-axis, and a horizontal number line, or x-axis. They cross where they are both equal to 0 — a point called the origin.
Coordinates Describe Points on the Plane The x and y coordinates of a point describe where on the plane it lies. The coordinates are written as (x, y). Coordinates are sometimes called ordered pairs. This means that the order of the numbers matters — (1, 2) is different from (2, 1).
y-coordinate
x-coordinate
Check it out:
(2, –3) When you plot points on the coordinate plane you plot them in relation to the origin, which has coordinates of (0, 0). • The x-coordinate tells you how many spaces along the x-axis to go. Negative values mean you go left. Positive values mean you go right. • The y-coordinate tells you how many spaces up or down the y-axis to go. Positive values mean you go up. Negative values mean you go down.
y
4 3 2 1 –4 –3 –2 –1
So a point with the coordinates (2, –3) will be two units to the right of the origin, and three units below.
150
Section 3.2 — The Coordinate Plane
1
2
3
4 x
0 –1 –2 –3 –4
(2, –3)
Example
1
Plot the point with the coordinates (3, 4). Solution
Step 1: start at the origin, (0, 0). y
Check it out: A positive x-coordinate tells you to move right along the x-axis, while a negative x-coordinate tells you to move left along the x-axis. A positive y-coordinate tells you to move up the y-axis, while a negative y-coordinate tells you to move down the y-axis.
4
Step 2: move right along the x-axis 3 units.
3 2 1 –4 –3 –2 –1
1
2
3
4 x
0 –1
Step 3: now move straight up 4 units and plot the point.
–2
When you are reading the coordinates of a point on a graph you can use the same idea. Example
2
What are the coordinates of point A?
y
4 3 2
A
1 –4 –3 –2 –1
1
2
3
4 x
0 –1 –2
Solution
Start at (0, 0). To get to point A on the graph you need to move 2 units to the left. So the x-value of your coordinate is –2. Then you need to go 2 units straight up. So the y-value is 2. The coordinates of the point A are (–2, 2).
Guided Practice Plot and label each of the coordinate pairs in Exercises 1–6 on a coordinate plane. 1. (1, 4)
2. (2, –3)
3. (–1, –2)
4. (–4, 2)
5. (0, 3)
6. (–4, 0)
Section 3.2 — The Coordinate Plane
151
The Coordinate Plane is Divided into Four Quadrants The x-axis and y-axis divide the coordinate plane into four sections. Each of these sections is called a quadrant. The quadrants are represented by Roman numerals, and are labeled counterclockwise. y
4
II (–, +)
I
3 2
(+, +)
1 –4 –3 –2 –1
1
2
III (–, –)
4
IV
–2 –3 –4
3
x
0 –1
(+, –)
The signs of the x and y values are different in each quadrant. For instance, in quadrant I both the x and y values are positive. But in quadrant II the x value is negative and the y value is positive. You can tell which quadrant a point will fall in by looking at the signs of the x and y coordinates.
Example
3
Which quadrant is the point (1, –4) in? Solution
Check it out: A point that is on either the x-axis or the y-axis is not in any of the quadrants.
The x-value is 1. This is positive, so the point must be in quadrant I or IV. The y-value is –4. This is negative, so the point must be in quadrant IV. The point (1, –4) is in quadrant IV.
Example
4
Which quadrant is the point (–3, –6) in? Solution
Both coordinates are negative, so the point (–3, –6) is in quadrant III.
152
Section 3.2 — The Coordinate Plane
Guided Practice In Exercises 7–14 say which quadrant the point lies in. 7. (1, 3)
8. (–2, –4)
9. (7, –2)
10. (–3, 6)
11. (–2, –2)
12. (2, 2)
13. (–1.5, 2.5)
14. (1, –1)
Independent Practice In Exercises 1–6 say which quadrant the point is in. 1. (1, 1) 2. (–1, –1) 3. (–1, 2) 4. (–2, 1) 5. (3, –2) 6. (–61, 141) 7. Do the coordinate pairs (–3, 4) and (4, –3) correspond to the same point on the plane? Plot each of the points in Exercises 8–13 on a coordinate plane. 9. (3, 2) 8. (0, 0) 11. (–3, 4) 10. (–2, –3) 13. (0, 4) 12. (1, –2) 14. Sophie and Jorge are playing a game. Sophie marks out a coordinate plane on the beach, and buries some objects at different points. Jorge has to use the map below to find the objects. y
2
Now try these: Lesson 3.2.1 additional questions — p445
1 –3 –2 –1
1 0
2
3
x
–1 –2 –3 –4
a) What are the coordinates of the keys? b) What are the coordinates of the baseball cap? c) What are the coordinates of the apple? d) How many units are there between the keys and the flashlight? e) How many units are there between the baseball mitt and the keys?
Round Up Coordinates allow you to describe where points are plotted — they’re written as pairs of numbers, such as (1, –5). The first number tells you the horizontal or x-coordinate. The second tells you the vertical or y-coordinate. Plotting points on coordinate planes is a big part of drawing graphs, and will be used a lot in the rest of this Chapter and in the next Chapter. Section 3.2 — The Coordinate Plane
153
Lesson Lesson
3.2.2 3.2.2
Drawing Shapes on the Coordinate Plane
California Standards: Measurement and Geometry 3.2 Understand and use coordinate graphs to plot simple figures, determine lengths and areas related to them, and determine their image under translations and reflections.
What it means for you: You’ll see how to draw shapes on a grid by plotting points and joining them.
Key words: • coordinate • area • perimeter
In the last Lesson you saw how to plot points on the coordinate plane. If you plot several points and then join them up, you get a shape.
You Can Make Shapes by Joining Points You can draw a shape on the coordinate plane by plotting points and joining them together. The coordinates are the corners of the shape. Example
1
Plot the shape ABCD on the coordinate plane, where A is (3, 2), B is (3, –3), C is (–2, –3), and D is (–2, 2). Name the shape you have drawn. y (3, 2) D (–2, 2) Solution Step 1: Plot and label the points A, B, C, and D.
Check it out: You have to join the points in order. So pentagon DEFGH must have D joined to E, E joined to F, F joined to G, etc... If you joined the points in a different order you’d get a completely different shape.
A
2 1
–3 –2 –1
1
2
3
0
x
–1
Step 2: Join the points in order. So A joins to B, B joins to C, C joins to D, and D joins to A.
(–2, –3)
C
–2
(3, –3)
–3 –4
B
The shape has four sides of equal length and four right angles — so it’s a square.
Check it out: You can tell something about the shape you’re drawing just from the number of coordinates you’re given. If you are only given three points to plot, the shape must be a triangle. If you are given four points, then it’s probably a quadrilateral (though it could be a triangle — if three of the points are in a line).
154
Guided Practice Plot the shapes in Exercises 1–4 on the coordinate plane and name the shapes. 1. JKL
J(3, 1) K(2, 3) L(1, 1)
2. EFG
E(1, 2) F(2, –3) G(–3, –2)
3. RSTU
R(–2, 1) S(2, 1) T(3, –1) U(–3, –1)
4. PQRS
P(–1, 3) Q(3, 3) R(1, 0) S(–3, 0)
Section 3.2 — The Coordinate Plane
Use the Shape’s Properties to Find Missing Points Sometimes you might be given a shape to graph with the coordinates of a corner missing. You can use the properties of the shape to work out the missing pair of coordinates. Example
2
VWXY is a square on the coordinate plane, where V is (2, 2), W is (2, –1), and X is (–1, –1). What are the coordinates of point Y? Solution
First plot points V, W, and X and join them in order. y
2
V
1 –2 –1
1
2x
0 –1
X
W
You know that VWXY is a square. So it must have four equal-length sides that meet at right angles. The lines VW and WX are both 3 units long. So point Y must be 3 units left of V, and 3 units above X. Add it to the graph, and form the square. y
Y
2
V
1 –2 –1
1
2x
0
X
–1
W
Now read the coordinates of Y from the graph: point Y is at (–1, 2).
Don’t forget: An isosceles triangle is one that has at least two sides the same length and at least two angles of equal measure.
Guided Practice In Exercises 5–8 find the missing point. 5. Square KLMN
K(–2, 2) L(2, 2) M(2, –2) N(?, ?)
6. Rectangle CDEF
C(1, 3) D(1, –1) E(–2, –1) F(?, ?)
7. Parallelogram ABCD
A(–3, –2) B(1, –2) C(3, 1) D (?, ?)
8. Isosceles triangle RST
R(1, –2) S(0, 1) T(?, ?)
Section 3.2 — The Coordinate Plane
155
Find Lengths Using Absolute Value Once you’ve plotted a shape on the coordinate plane you can find out its area or perimeter using the formulas that you saw in Section 3.1. Don’t forget: If you need a reminder, then all the perimeter formulas that you might need are in Lesson 3.1.1. All the area formulas that you might need are in Lesson 3.1.2.
But first you’ll need to find some lengths on the coordinate plane — such as the side lengths of the shapes. You could do this by counting squares on the diagram. Another way of doing this is to use x- and y-coordinate values. This is shown in the example below. Example
3
Plot the rectangle WXYZ on the coordinate plane, where W is (–1, 1), X is (4, 1), Y is (4, –2), and Z is (–1, –2). What are the perimeter and area of WXYZ? Solution
First plot WXYZ on the coordinate plane.
y
(–1, 1) W
1
2
3
–1
Z
–2
Y (4, –2)
The sides WX and YZ give the length. They’re the same, so find either. To find the length (l) using side WX, subtract the x-coordinate of W from the x-coordinate of X: l = 4 − (−1) = 5. Sides WZ and XY give the width. They’re the same, so find either. To find the width (w) using side XY, subtract the y-coordinate of Y from the y-coordinate of X: w = 1 – (–2) = 3. Now just plug the length and width values into the formulas for perimeter and area. Perimeter of WXYZ = 2(l + w) = 2(5 + 3) = 16 units Area of WXYZ = l • w = 5 • 3 = 15 units2
156
4 x
0
(–1, –2)
The bars around the calculation here show that the absolute value is being calculated. This means it doesn’t matter which coordinate you subtract from which — you’ll always get the correct, positive answer.
X (4, 1)
1
–4 –3 –2 –1
To find the perimeter and area, you need to know the width and the length.
Don’t forget:
2
Section 3.2 — The Coordinate Plane
Guided Practice 9. What is the perimeter of square ABCD, where A is (1, 1), B is (3, 1), C is (3, 3), and D is (1, 3)? 10. What are the perimeter and area of rectangle EFGH, where E is (–2, 1), F is (3, 1), G is (3, –2), and H is (–2, –2)? 11. What is the area of triangle JKL, when J is (–1, –3), K is (3, –3), and L is (1, 0)?
Independent Practice Plot and name the shapes in Exercises 1–3 on the coordinate plane. 1. ABC A(2, 3) B(3, –3) C(–2, –1) 2. TUVW T(4, –1) U(0, –1) V(0, 2) W(4, 2) 3. EFGH
E(–2, –2) F(–1, 0) G(1, 0) H(2, –2)
4. Anthony is marking out a pond in his yard. It is going to be perfectly square. He is marking it out on a grid system, and has put the first three marker stakes in at (–1, –3), (–1, 1), and (3, 1). At what coordinates should he put in the last stake? In Exercises 5–7, find the missing pair of coordinates. 5. Square CDEF 6. Rectangle TUVW 7. Parallelogram KLMN
C(1, 2) D(4, 2) E(4, –1) F(?, ?) T(–3, 3) U(–2, 3) V(–2, –2) W(?, ?) K(1, 0) L(–2, 0) M(–1, 2) N(?, ?)
8. What is the perimeter of rectangle BCDE, where B is (–2, 4), C is (3, 4), D is (3, 2), and E is (–2, 2)? 9. What is the area of triangle JKL, where J is (1, 2), K is (4, –1), and L is (7, –1)? Now try these: Lesson 3.2.2 additional questions — p445
10. What are the area and perimeter of rectangle PQRS, where P is (0, 0), Q is (3, 0), R is (3, –2), and S is (0, –2)? 11. A school has decided to set aside an area of their playing field as a nature reserve. A plan is made using a grid with 10-feet units. The coordinates of the corners of the area set aside are (0, 0), (4, 0), (2, –2), and (–2, –2). What area will the nature reserve cover?
Round Up Drawing shapes on the coordinate plane just means plotting their corners from coordinates and joining them together. You can even use the known properties of some shapes to figure out the coordinates of any missing corners. Once you’ve got the shapes plotted, you can use the standard formulas to work out their perimeters and areas. Section 3.2 — The Coordinate Plane
157
Section 3.3 introduction — an exploration into:
Measuring Right Triang les riangles
There’s a special relationship between the leg-lengths and the hypotenuse-length in a right triangle. The purpose of the Exploration is to discover this relationship. The hypotenuse of a right triangle is the side directly across from the right-angle. The other sides are called legs. Some right triangles are shown below with their hypotenuses labeled. hypotenuse hypotenuse
hypotenuse
Example On grid paper, draw a right triangle. Measure the length of each leg and the length of the hypotenuse. Solution
You can draw any triangle, as long as it has a right-angle. Legs = 2 cm and 3 cm Hypotenuse = 3.6 cm
3 cm
3.6 cm
hypotenuse
2 cm
Exercises 1. Draw 5 right triangles on grid paper. Label them A-E. Then label the hypotenuse on each. 2. For each right triangle, measure the length of each leg and the length of the hypotenuse. Measure in centimeters and record your measurements in a copy of this table.
Tr i an g l e
L eg 1 (c m )
L eg 2 (c m )
Hy p o t e n u s e ( c m )
Ex am p l e
2
3
3.6
A B C D E
3. Explain how the length of a right triangle’s hypotenuse compares to the lengths of its legs. 4. Explain how the sum of the legs of each right triangle compares to the hypotenuse length. 5. Add three new columns to your table, like this: Tr i an g l e Ex am p l e
L eg 1 (c m ) L eg 2 (c m ) Hy p o t en u s e (c m ) L eg 1 s q u ar ed 2
3
3.6
4
L eg 2 s q u ar ed Hy p o t en u s e s q u ar ed 9
13
Complete these columns, and then compare the squared side lengths for each triangle. What patterns do you notice?
Round Up You should now have discovered how the leg-lengths of a right triangle are related to the hypotenuselength. This is known as the Pythagorean Theorem — you’ll be using it in this Section. a tion — Measuring Right Triangles Explora 158 Section 3.3 Explor
Lesson
Section 3.3
3.3.1
The Pytha gor ean Theor em Pythag orean heorem
California Standards:
You will have come across right triangles before — they’re just triangles that have one corner that’s a 90° angle. Well, there’s a special formula that links the side lengths of a right triangle — it comes from the Pythagorean theorem.
Measur ement and Measurement Geometr y 3.3 Geometry Kno w and under stand the Know understand Pytha gor ean theor em and Pythag orean theorem its converse and use it to find the length of the missing side of a right triangle and the lengths of other line segments and, in some situa tions y situations tions,, empiricall empirically verify the Pytha gor ean Pythag orean theor em b y dir ect theorem by direct measur ement. measurement.
What it means for you: You’ll learn about an equation that you can use to find a missing side length of a right triangle.
The Pytha gor ean Theor em is About Right Triang les Pythag orean heorem riangles A right triangle is any triangle that has a 90° angle (or right angle) as one of it corners. You need to know the names of the parts of a right triangle: o ttenuse enuse is the The hyp hypo longest side of the triangle. It’s the side directly opposite the right angle.
c b 90º
a Right angle
Key words: • • • • •
Pythagorean theorem right triangle hypotenuse legs right angle
Check it out: A right angle is an angle of exactly 90°. In diagrams, a right angle is shown as a small square in the corner like this:
The other two sides of the triangle are called the legs legs.
In diagrams of right triangles, the hypotenuse is usually labeled as c, and the two legs as a and b. It doesn’t matter which leg you label a, and which you label b.
Guided Practice Complete the missing labels on the diagram.
1.
90º
Any other angle is shown as a piece of a circle like this:
3. 2.
In Exercises 4–7 say which side of the right triangle is the hypotenuse. 5.
4. I
I
II
II III
III
6.
7.
I
II
I
II III
III
Section 3.3 — The Pythagorean Theorem
159
The Theor em Links Side Lengths of Right Triang les heorem riangles Pythagoras was a Greek mathematician who lived around 500 B.C. A famous theorem about right triangles is named after him. It’s called the Pythagorean theorem:
For any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two legs. This all sounds very complicated, but it’s not so bad once you know what it actually means. Look again at the right triangle. Now add three squares whose side lengths are the same as the side lengths of the triangle:
2
Area = c
2
c
b
Area = b
a Area = a
2
What the Pythagorean theorem is saying is that the area of the red square is the same as the area of the blue square plus the area of the green square.
c2
=
a2
+
b2
So this is what the Pythagorean theorem looks like written algebraically:
For any right triangle: c2 = a2 + b2 It means that if you know the lengths of two sides of a right triangle, you can always find the length of the other side using the equation.
160
Section 3.3 — The Pythagorean Theorem
You Can Chec k the Theor em Using a Right Triang le Check heorem riangle Don’t forget: It’s really important to remember that the y Pythagorean theorem onl only works on right triangles. It won’t work on any other type.
You can check for yourself that the theorem works by measuring the side lengths of right triangles, and putting the values into the equation. Example
1
Use the right triangle below to verify the Pythagorean theorem.
5 units 4 units 3 units Solution
a = 3 units
b = 4 units
c = 5 units
c 2 = a2 + b2 52 = 32 + 42 25 = 9 + 16 25 = 25
Guided Practice Use the right triangles in Exercises 8–11 to verify the Pythagorean theorem. 8.
9. 15 units
13 units 12 units 12 units 5 units
9 units 10.
11. 4.1 units
4 units
0.7 units
2.5 units 2.4 units
0.9 units
Section 3.3 — The Pythagorean Theorem
161
Independent Practice In Exercises 1–3 say whether the triangle is a right triangle or not.
1.
2.
3.
In Exercises 4–6 say which side of the right triangle is the hypotenuse. Don’t forget: The hypotenuse is always the longest side of a right triangle.
II 4.
5.
III
I
III
6.
I
III
I II
II
Use the triangles in Exercises 7–10 to verify the Pythagorean theorem. 7.
8. 10 units
17 units
8 units
15 units
6 units 8 units
9.
10. 1.25 units
1 unit
2 units
0.75 units
1.6 units
1.2 units
Now try these: Lesson 3.3.1 additional questions — p445
11. Victor used the triangle shown on the right to try to verify the Pythagorean theorem. Explain why his work is wrong. Victor’s work: 92 = 122 + 152 81 = 144 + 225 81 = 369
15 units
12 units
9 units
Round Up The Pythagorean theorem describes the relationship between the lengths of the hypotenuse and the legs of a right triangle. It means that when you know the lengths of two of the sides of a right triangle, you can always find the length of the third side. You’ll get a lot of practice at using it in the next few Lessons. 162
Section 3.3 — The Pythagorean Theorem
Lesson
3.3.2
California Standards: Measur ement and Measurement Geometr y 3.2 Geometry Under stand and use Understand coor dina te g coordina dinate grra phs to plot mine simple figures, deter determine lengths and areas r ela ted to elated them them, and determine their image under translations and reflections. Measur ement and Measurement Geometr y 3.3 Geometry Kno w and under stand the Know understand Pytha gor ean theor em and Pythag orean theorem its converse and use it to find the length of the missing side of a right triangle and the lengths of other line se gments and, in segments some situations, empirically verify the Pythagorean theorem by direct measurement.
What it means for you:
Using the Pytha gor ean Pythag orean T heor em heorem In the last Lesson, you met the Pythagorean theorem and saw how it linked the lengths of the sides of a right triangle. In this Lesson, you’ll practice using the theorem to work out missing side lengths in right triangles.
Use the Pytha gor ean Theor em to Find the Hypoten use Pythag orean heorem Hypotenuse If you know the lengths of the two legs of a right triangle you can use them to find the length of the hypotenuse. The theorem says that c2 = a2 + b2, where c is the length of the hypotenuse, and a and b are the lengths of the two legs. So if you know the lengths of the legs you can put them into the equation, and solve it to find the length of the hypotenuse. Example
1
Use the Pythagorean theorem to find the length of the hypotenuse of the right triangle shown below.
You’ll see how to use the Pythagorean theorem to find missing side lengths of right triangles.
c cm
6 cm
Key words: • • • •
Pythagorean theorem hypotenuse legs square root
Don’t forget: Finding the square root of a number is the opposite of squaring it. The symbol: ÷ represents the positive square root of the number inside it. For more about square roots, see Section 2.6.
8 cm
Solution
c 2 = a 2 + b2
Fir st write out the equa tion First equation
c2 = 62 + 82
Substitute in the side lengths tha ou kno w thatt yyou know
c2 = 36 + 64 c2 = 100
Simplify the e xpr ession xpression expr
c = 100
Tak e the squar e rroot oot of both sides ake square
c = 10 cm
A lot of the time your solution won’t be a whole number. That’s because the last step of the work is taking a square root, which often leaves a decimal or an irrational number answer.
Section 3.3 — The Pythagorean Theorem
163
Example
2
Use the Pythagorean theorem to find the length of the hypotenuse of the right triangle shown. Check it out: Before you start, it’s very important to work out which side of the triangle is the longest side — the hypotenuse. Otherwise you’ll get an incorrect answer.
cm
1m
Solution
c2 c
= a2 + b2
2
2
=1 +1
c2
=1+1
c2
=2
c=
2
Fir st write out the equa tion First equation
1m
Substitute in the side lengths tha ou kno w thatt yyou know
Simplify the e xpr ession expr xpression
2m
Cancel out the squaring b y taking the squar e rroot oot by square
If you do this calculation on a calculator, you’ll see that approximately equal to 1.4 m.
2 m is
The Pythagorean theorem is also useful for finding lengths on graphs that aren’t horizontal or vertical. Example
3
y
Find the length of the line segment KL.
L
4 3
Solution
2 1
Draw a horizontal and vertical line on the plane to make a right triangle — then use the method above. c 2 = a2 + b2 Fir st write out the equa tion First equation c2
= 32 + 2 2
c2
= 9 + 4 = 13 Simplify the eexpr xpr ession xpression
0
3 units K 2 units 1
2
3
Substitute in the side lengths tha ou kno w thatt yyou know
c = 13 units ª 3.6 units Cancel out the squaring by taking the squar e rroot oot square
Guided Practice Use the Pythagorean theorem to find the length of the hypotenuse in Exercises 1–3. 1. 2. 3. 15 units
8 units
c ft
c mm
3.6 mm
12 ft 1.5 mm
c units 5 ft y
4. Use the Pythagorean theorem to find the length of the line segment XY.
2 Y 1 –2 –1
1
X
164
Section 3.3 — The Pythagorean Theorem
2 x
0 –1
4
x
You Can Use the Theor em to Find a Le g Length heorem Leg If you know the length of the hypotenuse and one of the legs, you can use the theorem to find the length of the “missing” leg. You just need to rearrange the formula: Don’t forget: To keep an equation balanced, you have to do the same thing to both sides. Here b2 is subtracted from each side. For a reminder on equations see Section 1.2.
a2 + b2 = c2 a2 = c2 – b2
Subtract b2 from both sides to get the a2 term by itself.
Remember that it doesn’t matter which of the legs you call a and which you call b. But the hypotenuse is always c. Now you can substitute in values to find the missing leg length as you did with the hypotenuse. Example
4
Find the missing leg length in this right triangle.
÷58 cm
3 cm a
Solution
c2 = a2 + b2
Fir st write out the equa tion First equation
a2 = c2 – b2
Rear e it earrrang ange
a2 = ( 58 )2 – 32
w Substitute in the side lengths tha thatt yyou know ou kno
a2 = 58 – 9 a2 = 49 a =
Simplify the e xpr ession expr xpression
49
Tak e the squar e rroot oot of both sides ake square
a = 7 cm
Guided Practice Use the Pythagorean theorem to calculate the missing leg lengths in Exercises 5–8.
6.
5. 20 cm
a ft
1.6 ft
16 cm
3.4 ft a cm
7.
a units
136 units
8. 10 units
89 units
a units
5 units
S
