HAESE
&
HARRIS PUBLICATIONS
Core Skills Mathematics
6 Helen Hall Mandy Spiers Stan Pulgies
CORE SKILLS MATHEMATICS 6 Helen Hall Mandy Spiers Stan Pulgies
B.Ed., Dip.T. Dip.T. M.Ed., B.Ed., Grad.Dip.T.
Haese & Harris Publications 3 Frank Collopy Court, Adelaide Airport SA 5950 Telephone: (08) 8355 9444, Fax: (08) 8355 9471 email:
[email protected] web: www.haeseandharris.com.au National Library of Australia Card Number & ISBN 1 876543 74 4 © Haese & Harris Publications 2004 Published by Raksar Nominees Pty Ltd, 3 Frank Collopy Court, Adelaide Airport SA 5950 First Edition
2004
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FOREWORD
We have written this book to provide a sound course in mathematics that Year 6 students will find easy to read and understand. Our particular aim was to cover the core skills in a clear and readable way, so that every Year 6 student can be given a sound foundation in mathematics that will stand them in good stead as they near the end of their primary-level education. Units are presented in easy-to-follow, double-page spreads. Attention has been paid to sentence length and page layout to ensure the book is easy to read. The content and order of the thirteen chapters parallels the content and order of the thirteen chapters in Mathematics for Year 6 (second edition) also published by Haese & Harris Publications and that book could be used by teachers seeking extension work for students at this level. Throughout this book, as appropriate, the main idea and an example are presented at the top of the left hand page; graded exercises and activities follow, and more challenging questions appear towards the foot of the right-hand page. With the support of the interactive Student CD, there is plenty of explanation, revision and practice. We hope that this book will help to give students a sound foundation in mathematics, but we also caution that no single book should be the sole resource for any classroom teacher. We welcome your feedback. Email:
[email protected] Web: www.haeseandharris.com.au HH MS SP
Active icons – for use with interactive student CD By clicking on the CD-link icon you can access a range of interactive features, including: ! spreadsheets ! video clips ! graphing and geometry software ! computer demonstrations and simulations.
CD LINK
5
TABLE OF CONTENTS
TABLE OF CONTENTS
Chapter 1 WHOLE NUMBERS AND MONEY
Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6
Different number systems Our number system Place value Rounding numbers Number lines and number puzzles Review of chapter 1
8 10 12 14 16 18
Chapter 2 POINTS, LINES, ANGLES AND POLYGONS
Unit 7 Unit 8 Unit 9 Unit 10 Unit 11 Unit 12
Points, lines and polygons Angles Triangles and quadrilaterals Angles in a triangle Angles in a quadrilateral Review of chapter 2
20 22 24 26 28 30
Chapter 3 NUMBER FACTS
Unit 13 Unit 14 Unit 15 Unit 16 Unit 17 Unit 18 Unit 19 Unit 20
Addition and subtraction Multiplication and division by powers of 10 Multiplication Division and problem solving Estimation and approximation Factors, divisibility rules and zero Multiples, LCM and operations Review of chapter 3
32 34 36 38 40 42 44 46
Review of chapters 1, 2 and 3
48
TEST YOURSELF Chapter 4 FRACTIONS
Unit 21 Unit 22 Unit 23 Unit 24 Unit 25 Unit 26 Unit 27 Unit 28 Unit 29
Fractions Fractions of quantities Finding the whole and ordering fractions Equivalent fractions and lowest terms Mixed numbers and improper fractions Addition of fractions Subtracting fractions and problem solving Ratio Review of chapter 4
50 52 54 56 58 60 62 64 66
Chapter 5 DECIMALS
Unit 30 Unit 31 Unit 32 Unit 33 Unit 34 Unit 35 Unit 36 Unit 37 Unit 38
Representing decimals Using a number line and the value of money Place value Adding and subtracting decimal numbers Multiplying and dividing by 10, 100, 1000 Conversions and percentage Multiplication and division Money and rounding decimals Review of chapter 5
68 70 72 74 76 78 80 82 84
6
TABLE OF CONTENTS
Chapter 6 MEASUREMENT
Unit 39 Unit 40 Unit 41 Unit 42 Unit 43 Unit 44 Unit 45 Unit 46 Unit 47
TEST YOURSELF
Length Converting length units and perimeter Perimeter continued Area Area of rectangles Composite areas and problem solving Volume Capacity Review of chapter 6
86 88 90 92 94 96 98 100 102
Review of chapters 4, 5 and 6
104
Chapter 7 LOCATION AND POSITION
Unit 48 Unit 49 Unit 50 Unit 51 Unit 52 Unit 53 Unit 54
Introduction to scales and grids Grids Maps Direction Plans Coordinates Review of chapter 7
106 108 110 112 114 116 118
Chapter 8 SOLIDS AND MASS
Unit 55 Unit 56 Unit 57 Unit 58 Unit 59 Unit 60 Unit 61
Solids Polyhedra Drawing solids Making solids from nets Different views of objects Mass Review of chapter 8
120 122 124 126 128 130 132
Chapter 9 DATA COLLECTION AND REPRESENTATION
Unit 62 Unit 63 Unit 64 Unit 65 Unit 66 Unit 67 Unit 68 Unit 69
Organising data Pictographs and strip graphs Column and bar graphs Line graphs Pie charts and media graphs Interpreting data Measuring the middle of a data set Review of chapter 9
134 136 138 140 142 144 146 148
Chapter 10 TIME AND TEMPERATURE
Unit 70 Unit 71 Unit 72 Unit 73 Unit 74 Unit 75 Unit 76 Unit 77
Time lines Units of time A date with a calender Reading clocks and watches Clockwise direction and using a stopwatch Timetables Speed and temperature Review of chapter 10
150 152 154 156 158 160 162 164
Review of chapters 7, 8, 9 and 10
166
TEST YOURSELF
TABLE OF CONTENTS
Chapter 11 PATTERNS AND ALGEBRA
Unit 78 Unit 79 Unit 80 Unit 81 Unit 82 Unit 83 Unit 84 Unit 85 Unit 86
Number patterns (sequences) Dot and matchstick patterns Rules and problem solving Graphing patterns and tables Using word formulae Converting words to symbols Algebraic expressions and equations Graphing from a rule Review of chapter 11
168 170 172 174 176 178 180 182 184
Chapter 12 TRANSFORMATIONS
Unit 87 Unit 88 Unit 89 Unit 90 Unit 91 Unit 92
The language of transformations Tessellations Line symmetry Rotations and rotational symmetry Enlargements and reductions Review of chapter 12
186 188 190 192 194 196
Chapter 13 CHANCE AND PROBABILITY
Unit 93 Unit 94 Unit 95 Unit 96
Describing chance All possible results Probability Review of chapter 13
198 200 202 204
TEST YOURSELF
Review of chapters 11, 12 and 13
206
ANSWERS
208
Correlation Chart: R-7 SACSA Mathematics Teaching Resource
236
INDEX
240
7
CHAPTER 1
8
WHOLE NUMBERS AND MONEY
Unit 1
Different number systems
Roman numerals 1 I
2 II
3 III
4 IV
5 V
6 VI
7 VII
8 VIII
9 IX
10 X
20 XX
30 XXX
40 XL
50 L
60 LX
70 LXX
80 LXXX
90 XC
100 C
500 D
Look for Roman numerals on clocks and watches, at the end of movies when the credits are being shown, on plaques and on the top of buildings, and as chapter numbers in novels.
1000 M
Roman numerals must be written in order. Notice that whereas
IV VI
stands for 1 before 5, stands for 1 after 5,
There are also larger numerals, though we rarely use these today.
i.e., 4 i.e., 6
5000 10 000 50 000 100 000 500 000 1 000 000 V X L C D M
These examples show how Roman numerals are written: 49 is |{z} XL |{z} IX 40 + 9
540 is |{z} D XL |{z} 500 + 40
1999 is
M |{z}
So 49 is XLIX
So 540 is DXL
So 1999 is MCMXCIX
CM |{z}
XC |{z}
IX |{z}
1000 + 900 + 90 + 9
Exercise 1 1 What numbers are represented by the following symbols? a XII b XX c LI e XXXI f LXXXIII g CXXIV 2 Write these numbers in Roman numerals: a 18 b 34 c 279 d 3
902
e
1046
d h
f
XVI MCCLVII
2551
a Which Roman numeral less than one hundred has the greatest number of symbols? b What is XI + VIII? c Write the year 2004 using Roman symbols.
4 Use Roman numerals to answer these questions. a Each week Octavius sharpened CCCLIV swords for the General. i How many swords is this? ii How many would he need to sharpen if the General doubled his order? Write your answer in Roman numerals. b What would it cost Claudius to finish his courtyard if he needed to pay for CL pavers at VIII denarii each and labour costs of XCIV denarii? c
Julius and his road building crew were expected to build MDC metres of road before the end of spring. If they completed CCCLX metres in summer, CDLXXX in autumn, and CCCXV in winter, how much more did they need to complete in the last season?
Denarii was the unit of currency used by the Romans.
9
WHOLE NUMBERS AND MONEY (CHAPTER 1)
Ancient Greek or Attic System Numbers 1
2
3
20
4
5
30
700
1000
50
6
7
60
8
100
9
10
400
Can you see what the smaller D, H, and X symbols do to the symbol when they are joined?
500
5000
²
² is our number 1324.
is our number 6781.
1000
6000
300
700
20
80
4 1324
1 6781
5 Change these Ancient Greek numerals into numbers in our system: a
b
c
d
e
f
6 Write these Hindu-Arabic numbers as ancient Greek numerals: a 14 b 31 c 99 d 555
4082
e
Research
5601
f
Other ways of counting Find out: 1 how the Ancient Egyptians and Mayans represented numbers larger than 1000 2 whether the Egyptians used a symbol for zero
3 how to write the symbols 1 to 10 using Chinese or Japanese characters 4 what larger Braille numbers feel like 5 how deaf people ‘sign’ numbers 6 what the Roman numerals were for the two Australian Olympiads.
Numbers in Braille 1
2
3
4
5
6
7
8
9
0
Investigation
Birth dates in Roman
What to do:
Use a calendar to help you with this investigation. 1 In Roman numerals write: a your date of birth, for example, XXI-XI-MCMXLVI b what the date will be when you are i XV ii L iii XXI iv C
VI XIII XX XXVII
2 Ignoring any leap year what will be the date: a XIV days after your XVI birthday b LX days after your XIX birthday?
I VII XIV XXI
XXVII
VIII XV XXII
I XX IX
II IX XVI XXIII
III X XVII XXIV
IV XI XVIII XXV
V XII XIX
XXVI
10
WHOLE NUMBERS AND MONEY (CHAPTER 1)
Unit 2
Our number system
We use the Hindu-Arabic number system.
Most of the time we are not worried about the difference between ‘numeral’ and ‘number’. We usually use the word number.
We can use the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 to write any number. We use the digit 0 or zero to show an empty space.
5
6
7
9
this 6 represents 600 ¡000
its un
For example, in 567 942:
hu thondre us d ten ands th o us an th o ds us an ds hu nd red s ten s
Each digit in a number has a place value.
2
4
this 4 represents 40
Exercise 2 1 What number is represented by the digit 7? a 27 b 74 e 1971 f 7635 i 80 007 j 72 024
c g k
567 3751 87 894
d h l
758 27 906 478 864
2 What is the place value of the digit 5? a 385 b 4548
c
32 756
d
577 908
3 Write down the place value of the 2, the 4 and the 8: a 1824 b 32 804 c
80 402
d
248 935
4
a Use the digits 9, 5 and 7 once only to make the smallest number you can. b Write the largest number you can, using the digits 3, 2, 0, 9 and 8 once only. c What is the largest 6 digit number you can write using each of the digits 1, 4 and 7 twice? d What are the different numbers you can write using the digits 6, 7 and 8 once only?
5 Place these numbers in order, beginning with the smallest (ascending order): a 62, 26, 20, 16, 60 b 67, 18, 85, 26, 64, 29 c 770, 70, 700, 7, 707 d 2808, two thousand and eight, 2080, two thousand eight hundred 6 Place these numbers in order, beginning with the largest (descending order): a 17, 21, 20, 16, 32 b 77, 28, 95, 36, 64, 49 c 880, 800, 80, 808, 8 d 2606, two thousand and six, 2060, two thousand six hundred
In Mathematics, we often use symbols instead of words. For example: = means is equal to 6 = means is not equal to > means is greater than < means is less than 7 Replace the ¤ with =, > or < to make a correct statement: a 7¤9 b 9¤7
c
2+2 ¤ 4
d
3¡1 ¤ 9¡7
e
6+1 ¤ 5
f
7¡3 ¤ 9¡5
g
16 ¤ 5
h
5 ¤ 16 ¡ 9
i
12 ¤ 24 ¥ 2
j
11 £ 2 ¤ 44 ¥ 2
k
15 ¡ 9 ¤ 2 £ 3
l
7 + 13 ¤ 5 £ 4
m
118 ¡ 17 ¤ 98 + 3
n
7900 ¤ 9700
o
7900 ¤ 7090
p
25 £ 4 ¤ 99
q
99 ¤ 25 £ 4
r
345 678 ¤ 345 687
WHOLE NUMBERS AND MONEY (CHAPTER 1)
11
We can write numbers in several ways. Example:
3908
² ² ²
in simplest form or numeral form is 3908 in expanded form is 3 £ 1000 + 9 £ 100 + 8 in words is three thousand nine hundred and eight.
8 Express these in simplest form: a 4 £ 10 + 9 b 7 £ 100 + 4 c 3 £ 100 + 8 £ 10 + 6 d 2 £ 1000 + 6 £ 100 + 3 £ 10 + 4 e 6 £ 10 000 + 5 £ 100 + 8 £ 10 + 3 f 9 £ 10 000 + 3 £ 1000 + 8 g 3 £ 100 + 4 £ 10 000 + 7 £ 10 + 6 £ 1000 + 5 h 2 £ 10 + 9 £ 100 000 + 8 £ 1000 + 3 i 2 £ 100 000 + 3 £ 100 + 7 £ 10 000 + 8 9 Write in expanded form: a 486 e 24 569
b f
340 39 804
10 Write in numeral form: a thirty six c thirty e nine hundred g five hundred and twenty i six thousand and fourteen k fourteen thousand and four m fifteen thousand eight hundred and sixty nine o seven hundred and eight thousand one hundred 11 Write these numbers a 66 e 4389 i 15 040 m 50 500
in words: b 660 f 6010 j 44 444 n 505 000
c g k o
DEMO
c g
2438 400 308
b d f h j l n and
seventy eighteen nine thousand five hundred and two six thousand four hundred and forty forty thousand and forty ninety five thousand three hundred and eleven ninety eight.
715 90 000 408 804 500 500
d h l p
d h
888 38 700 246 357 50 050
4083 254 372
The number 102¡675 is said 102 thousand 675.
12 Write these operations and their answers in numerical form: a four more than forty b six greater than eleven c three less than two hundred d eight fewer than eighty e eighteen fewer than six thousand f three thousand reduced by two hundred g an additional fifty to eleven thousand h 38 more than five hundred and nine thousand 13 With the number 543: a i place a zero between the 4 and 3 ii write the new number in words
In 5801, the 0 is a place value holder for tens. 5801 is not the same as 581.
b
i place two zeros between the 5 and 4 ii write the new number in words
c
i place a zero between the 5 and 4 and two zeros between the 4 and 3 ii write the new number in words
d
i place two zeros after the 3 ii write the new number in words
12
WHOLE NUMBERS AND MONEY (CHAPTER 1)
Unit 3
Place value
This diagram of MA blocks represents the number three thousand five hundred and forty nine (3549):
Exercise 3 1 What number is represented by these diagrams? a
2 Draw representations of:
a
1024
b
b 4186
3 Write as a whole number: a 60 000 + 700 + 90 + 8 c 200 + 6000 + 80 + 30 000 e 80 + 2 + 70 000 + 600 + 200 000
b d f
300 000 + 7000 + 200 + 3 7000 + 5 + 90 000 + 70 3 000 000 + 2000 + 90 000
4 Write numbers which are: a one less than 600 000 c one more than 3 thousand e one hundred less than five thousand
b d f
50 more than 20 000 6000 more than 32 000 ten less than twenty eight thousand
5 Write in numerals and words the a largest b smallest six digit number you can make with the digits 0 to 9 (not repeated). You should not use 0 to start your numbers. 6 In numerals and words, what is the difference between the largest and smallest number in 5? 7 What is the sum of the largest and smallest numbers in 5? 8 Start with the smallest number that can be made with all the digits 0 to 3 using them once only. List in ascending order all the numbers that can be formed. Do not include numbers starting with 0. 9 Using the abacus Each column of this abacus represents a place value. Using whole numbers, the unit is the far right column. The beads in the example represent 251 463 which we say as 251 thousand 463. What numbers do the beads in a and b represent? Example:
a
b
Ascending means going up.
WHOLE NUMBERS AND MONEY (CHAPTER 1)
In this chart, one cent represents the unit, ten cents represents the tens, a dollar represents the hundreds, ten dollars represents the thousands and one hundred dollars represents the ten thousands place. Ten thousands
Thousands
Hundreds
Tens
ten thousand cents
one thousand cents
one hundred cents
Units not in use today
ten cents
one cent
One dollar is 100 cents, so $111 is 11¡100 cents. 10 Write the place values for these amounts: a
b
11 What would be the monetary values of a 8765 cents b 24 075 cents d fifteen thousand four hundred and forty cents e ninety eight thousand three hundred and seven cents f the sum of the money in 10 a and b?
56 908 cents
c
12 In the place-value card game for the highest number, which hand of each pair of hands “wins”? Note that an ace represents 1. 9ª ª 2 5 9§ § 8 8 5 2 a A 4 B 4 § § § ªª
7
A A
B
§ § § § § § §
7
§ § § § §§§ § §
7
ªª ªªª ªª
A
7
2
9
§ § §
§
9 9
5 9
7
9
8 7
4
7
§
7
9
4
A
§§§ § § § §
8
A
A
ª ªª
5
A
ª
2
b
ª
ªª ªªª ªª
A
In words, write the i sum of ii difference between each pair of hands. Find the sum of each column (A and B). 13 Using a calculator, key in the numbers shown. a Now subtract 5678. Say the number and write down the digit which appears in the thousands place. Repeat this subtraction process 5 more times, that is, say the number and write the digit.
123609
b What number is left after you have subtracted 6 times? c What is the highest number you can make with the digits you have written? d What is the lowest number you can make with the digits you have written? e In the highest number, what digit appears in the thousands place? 14
a
b d
Key in the number 23, then multiply it by 3 and write down your answer. Multiply your new answer by 3, say the number then write it down. Keep on repeating this pattern until you have a 4 digit in the hundred thousands place. How many times did you multiply by 3? c What is the number? In your answers, how many times did the 7 digit appear in the hundreds place?
13
14
WHOLE NUMBERS AND MONEY (CHAPTER 1)
Unit 4
Rounding numbers
Often we are not really interested in the exact value of a number. We just want a reasonable estimate of it. For example: if there may be 38 948 spectators at a football match, a good approximation of the number would be 40 000 spectators. Rules for rounding off are: ²
If the digit after the one being rounded off is less than 5 (i.e., 0, 1, 2, 3 or 4), we round down.
²
If the digit after the one being rounded off is 5 or more (i.e., 5, 6, 7, 8, 9), we round up.
Examples: ² Round 38 to the nearest 10. 38 is approximately 40. 8 is greater than 5, so round up.
The symbol t or + means ‘is approximately equal to’.
² Round 650 to the nearest 100. 650 + 700. We round up for the digit 5. ² Round 141 678 to the nearest 10 000. 141 568 + 140 000. 1 is less than 5, so round down. DEMO
DEMO
Exercise 4 1 Round off to the nearest 10: a 23 b e 347 f i 3015 j m 2895 n
65 561 2856 9995
c g k o
68 409 3094 30 905
d h l p
97 598 8885 49 895
2 Round off to the nearest 100: a 81 b e 349 f i 999 j
671 982 13 484
c g k
617 2111 99 199
d h l
850 3949 10 074
3 Round off to the nearest 1000: a 834 b 695 e 7800 f 6500 i 13 095 j 7543
c g k
1089 9990 246 088
d h l
5485 9399 499 359
DEMO
4 Round off to the nearest 10 000: a 18 124 b 47 600 e 89 888 f 52 749
c g
54 500 90 555
d h
75 850 99 776
DEMO
5 Round off to the nearest 100 000: a 181 000 b 342 000 e 139 888 f 450 749
c g
654 000 290 555
d h
709 850 89 512
6 Round off to the accuracy given: a $187:45 (to nearest $10) b $18 745 (to nearest $1000) c 375 km (to nearest 10 km) d $785 (to nearest $100) e 995 cm (to nearest metre) f 8945 litres (to nearest kilolitre) g the population of a town is 29 295 (to nearest one thousand) h the cost of a house was $274 950 (to nearest $10 000) i the number of sheep on a farm is 491 560 (nearest 100 000)
One kilolitre is one thousand litres.
WHOLE NUMBERS AND MONEY (CHAPTER 1)
15
Rounding money Rounding to the nearest 5 cents
Because we no longer use 1 cent and 2 cent coins, amounts of money to be paid in cash must be rounded to the nearest 5 cents. For example, a supermarket bill and the bill for fuel at a service station must be rounded to the nearest 5 cents. ² ² ² ² ²
If the number of cents ends in: 0 or 5, the amount remains unchanged. 1 or 2, the amount is rounded down to 0. 3 or 4, the amount is rounded up to 5. 6 or 7, the amount is rounded down to 5. 8 or 9, the amount is rounded up to 10.
Examples: $5:95 remains unchanged. $1:42 would be rounded down to $1:40 $12:63 would be rounded up to $12:65 $3:16 would be rounded down to $3:15 $24:99 would be rounded up to $25:00
Rounding to the nearest whole dollar
For the purposes of estimation, money is rounded to the nearest whole dollar. If the number of cents in an amount is from 0 to 49, the cents are left off and the number of dollars is unchanged. If the number of cents is 50 or more, the amount is rounded to the next whole dollar. So $4:37 is rounded down to $4:00 and $16:85 is rounded up to $17:00. 7 Round these amounts to the nearest 5 cents: a 49 cents b $2:74 e $34:00 f $25:05 i $13:01 j $102:23 8
c g k
$1:87 $16:77 $430:84
d h l
$1:84 $4:98 $93:92
a Rachel paid cash for her supermarket bill of $84:72. How much did she pay? b Jason filled his car with petrol and the amount shown at the petrol pump was $31:66. How much did he pay in cash? c Nicolas used the special dry-cleaning offer of ‘3 items for $9:99’. How much money did he pay?
9 For the purpose of estimation, round these amounts to the nearest whole dollar: a $3:87 b $9:28 c $4:39 d $11:05 f $19:45 g $19:55 h $39:45 i $39:50 10 Estimate the total cost (by rounding the prices to the nearest dollar) of a one icecream, a packet of crisps, a health bar and a drink b 5 licorice ropes, 4 icecreams, 2 honeycomb bars and 4 drinks
e j
$7:55 $61:19
To estimate the cost of 28 health bars at $1.95 each, we use 28¡´¡$1.95¡+¡28¡´¡$2¡+¡$56.
c 3 ice blocks, 2 packets pineapple lumps, 4 chocolate bars and 3 cheese snacks d 10 health bars, 4 icecreams, 6 jubes and 3 licorice ropes e 19 ice blocks, 11 drinks, 12 packets cheese snacks and 9 packets pineapple lumps f 21 packets crisps, 18 chocolate bars, 28 health bars and 45 drinks g 4 dozen drinks, half a dozen packets of pineapple lumps and a dozen health bars h 192 honeycomb bars, 115 icecreams, 189 packets crisps and 237 drinks i 225 licorice ropes, 269 drinks, 324 honeycomb bars and 209 ice blocks.
Ice blocks $0.85 Licorice rope $0.75 Chocolate bar $1.30 Jubes $1.20 300mL drink $1.15
Cheese snacks $1.30 Ice cream $2.10 Crisps $1.05 Health bar $1.95 Pineapple lumps $1.80 Honeycomb bar $0.95
16
WHOLE NUMBERS AND MONEY (CHAPTER 1)
Unit 5
Number lines and number puzzles
The number line The number line shows the order and position of numbers. Points representing whole numbers are equally spaced along the line. We can show the numbers 9, 15, 3 and 6 with dots on a number line like this: 3
6
0
9
5
15 10
15
20
Number lines can also be used to show the four basic operations of adding, subtracting, multiplying and dividing. For example:
3
3+8¡6=5
²
0
5
10
4 lots of 3
4 £ 3 + 2 = 14
²
Remember the multiples of 4 are 1¡´¡4, 2¡´¡4, 3¡´¡4, 4¡´¡4, ... etc.
+8 -6
0
5
DEMO
15
+2 10
15
Exercise 5 1 Use a c e
dots to show these numbers on a number line: 9, 4, 8, 2, 7 70, 30, 60, 90, 40 250, 75, 200, 25, 125
b d f
14, 19, 16, 18, 13 multiples of 4 below 40 4000, 3000, 500, 2500, 1500
2 What operations do these number lines show? Give a final answer. a b 0
5
10
15
20
c
0
5
10
15
20
d 0
10
20
30
40
50
e
0
100
200
300
400
500
600
700
f 0
10
20
30
40
50
60
70
0
5
10
3 Draw a number line and show these operations. Give a final answer. a 9+8¡6 b 2+4+8¡2 d 55 + 60 + 75 ¡ 40 e 3£9¡8 4 Use a number line to solve this question:
Eric is in a building. He is 26 m above the ground. He climbs up 4 flights of stairs, each of which is 3 m high. How high is Eric now above the ground?
26 m
c f
15
20
40 + 70 + 90 ¡ 50 4£6+5
WHOLE NUMBERS AND MONEY (CHAPTER 1)
17
Number puzzles 5 In the eleven squares write all the numbers from 1 to 11 so that every set of three numbers in a straight line adds up to 18.
Draw three triangles like the one shown. Using each number once only, place the numbers 2 to 7 in the squares so that each side of the triangle adds up to a 12 b 13 c 14
6
7 Draw three triangles like the one shown. Using each number once only, place the numbers 11 to 19 in the squares so that each side of the triangle adds up to a 57 b 59 c 63 8
Draw three shapes as shown. Using each number once only, place the numbers 1 to 10 in the circles so that each line leading to the centre adds up to a 19 b 21 c 25 PRINTABLE TEMPLATE
9 Copy the grid below and then complete these cross numbers. There is only one way in which all the numbers will fit.
2 digits eighty six, ninety, twenty five, seventy eight, forty five, forty one, seventy five, forty two, forty three, seventy two, eighty five 3 digits 739, 246, 208, 267, 846, 540
6
4 digits 9306, 9346, 4098, 8914, 2672, 1984, 2635, 8961
5 digits fifty six thousand three hundred and eighty four, 53 804, forty four thousand nine hundred and sixty seven, 36 495.
Activity
The language of number Mathematics is more than just working with numbers. It also has symbols and language. Numbers have a language of their own to tell us “how many”.
For example:
two apples
a couple of people
a duet
twins
a pair of socks
Three people playing musical instruments or singing together are a trio. Some other words meaning three include: hat-trick, thrice, triad, triple, trilogy and trifecta. In groups, brainstorm a list of as many words as you can which can mean four or five. Check your list for the correct spelling and meanings using a dictionary or thesaurus.
18
WHOLE NUMBERS AND MONEY (CHAPTER 1)
Unit 6
Review of chapter 1
Review set 1A 1 Give the numbers represented by the Roman symbols:
a VIII
b LIV
2 Write these numbers in Roman symbols:
a
23
b 110
3 Give the number represented by the digit 2 in
a
253
b 12 467
4 Place these numbers in order beginning with the smallest: 201, 121, 102, 211, 21, 112 5
a Express 3 £ 10 000 + 4 £ 100 + 5 £ 10 + 9 in simplest form. b Write 9407 in expanded form.
6 Write the smallest whole number you can make with the digits 6, 3, 1, 1, 2. 7 Write fifty three thousand and seventy two in numerical form. 8 Replace 2 with =, > or < to make these statements correct: a 7790 2 7709 b 30 £ 5 2 200 ¡ 50 9 Write the operation and answer in numerical form: eleven.
c
63 2 6 £ 12
three hundred and six more than four thousand and
10 Write numbers which are: a
50 more than 7 thousand
b
1 less than 20¡000
11 Write the place values for the sum of these amounts: a
b
12 Write the value in dollars and cents of: a 175 cents b
2545 cents
c
650¡000 cents
13 Round the following: a c
64 762 to the nearest 10 000 $1:98 to the nearest 5 cents
b
1976 grams to the nearest kilogram
14 Write the numbers represented by the following: a
b
15 Write $1620 as cents. 16 Show the first six even numbers as dots on a number line. 17 What operations does the number line show? Give a final answer.
0
5
10
15
20
WHOLE NUMBERS AND MONEY (CHAPTER 1)
Review set 1B 1 Give the numbers represented by the Roman symbols:
a XIX
b
2 Write these numbers using Roman symbols:
a 11
b 43
3 Give the number represented by the digit 9 in
a
59 632
XXXV
b 691 025
4 Place these numbers in order beginning with the largest: 4258, 582, 8254, 558, 8425, 4825 5 Express 9 £ 10 000 + 5 £ 1000 + 4 £ 100 + 6 in simplest form. 6
a Write the largest whole number possible with the digits 0, 2, 3, 7, 9. b Write 37 029 in expanded form.
7
a Write the number 51 602 in words. b Write fifty thousand six hundred and ten in numerical form. c Write the operation and answer in numerical form: increase 863 by 794.
8 Replace 2 with =, > or < to make these statements correct: a 8000 2 80¡000 b 0¥9 2 9£0 9 Round these: a 265 to the nearest 10 c 375 cm to the nearest metre 10 Estimate the cost of
a
c
b 52 794 to the nearest 1000 d $4:92 to the nearest 5 cents
23 iceblocks at $1.05 each
120 pies at $1.85 each
b
11 Write the numbers represented by the following: a
12
1194 2 1419
b
a
Dave paid cash for his supermarket bill of $23.16. How much did he pay in cash?
b
The petrol for Kirsty’s car cost $45.93. How much did she pay if she paid in cash?
13 Write 6005 cents as dollars. 14 Show the odd numbers between 4 and 14 as dots on a number line. 15 What operations do these number lines show? Give a final answer. a
b 0
5
10
15
0
16 Show the operations 4 £ 3 ¡ 8 on a number line. Give a final answer.
50
100
150
19
20
POINTS, LINES, ANGLES AND POLYGONS
Unit 7
Points, lines and polygons
CHAPTER 2
Points and lines The arrows show that the lines are parallel.
A small dot is used to mark the position of a point.
a point
B
A line extends forever in each direction.
B
A
D
A C
A line segment is a part of a straight line.
B
A
C
DEMO
A ray extends forever in one direction.
B
A
B X
A
D
Parallel lines are always the same distance apart. AB k CD means that line AB is parallel to line CD.
Lines that are not parallel will intersect. X is the point of intersection of lines AB and CD. DEMO
Exercise 7 1 Draw freehand diagrams to represent: a a triangle b c line segment RS d X
2
For a b c d
B
A
D C Y
DEMO
a straight line through P and Q line segments MN and PQ meeting at Y
the given diagram, name: the lines which are parallel the intersection of AB with BC the line segment connecting the parallel lines the line segments which are parallel.
3 For the given diagram answer True or False to the N following: P a Lines LM and NO are parallel. W b Line segments WX and YZ are parallel. c Lines PQ and RS are parallel. d Line segments WZ and XY are parallel. R e Z is the point of intersection of lines RS and LM. X f Lines YR and PO intersect at X. O A
4 E
B
B
Q Z
S Y M
Name the figure according to its number of sides.
b
Name the line segments that make up the sides of the figure.
c
If we draw the line segment AC, we have drawn one of the diagonals of the figure. List all the diagonals of this figure.
5 Give all possible ways of naming the following lines: a b A
L
a
D
C
Look around the classroom. Can you see any parallel lines?
L K
c X
Y
Z
POINTS, LINES, ANGLES AND POLYGONS (CHAPTER 2)
Naming polygons
In geometry a plane is a flat surface.
Polygons are closed figures with straight sides that do not cross.
triangle (3 angles)
quadrilateral (4 sides)
pentagon (5 sides)
hexagon (6 sides)
A heptagon has 7 sides, an octagon has 8 sides, a nonagon has 9 sides and a decagon has 10 sides.
Regular polygons A regular polygon is a polygon with all sides the same length and all angles the same size.
Equal sides are shown with small markings across them and equal angles are shown with the same markings.
The polygons below are all marked to show that they are regular.
equilateral triangle
square
regular pentagon
This figure is not a regular polygon. It has equal sides but its angles are not all equal. 6 Give examples in the classroom of objects whose boundaries are polygons. 7 Draw sketches of: a a quadrilateral d a regular pentagon
b e
a regular quadrilateral a hexagon
c f
a pentagon a regular octagon
8 Name several objects in the classroom which have planes (flat surfaces). 9 True or false? a a square is a regular quadrilateral c a pentagon has six sides
b d
a rectangle is a regular quadrilateral a nonagon has nine sides
10 Give a reason why these are not polygons: a b
c
11 Give a reason why these are not regular polygons: a b
c
12 Name the polygon which has a 5 sides d 8 sides
c f
b e
6 sides 9 sides
7 sides 10 sides
21
22
POINTS, LINES, ANGLES AND POLYGONS (CHAPTER 2)
Unit 8
Angles
Angles are formed whenever two lines meet. DEMO
Measuring angles
A
The size of an angle is a measure of the amount of turn needed to move one of the lines making up the angle onto the other one. One revolution measures 360o .
angle
The diagram shows the angle between AB and BC. B
C
We can classify angles according to the amount of turn or their degree measure: Revolution
Straight angle
Right angle
360o One complete turn
180o 1 2 turn
90o 1 4 turn
Acute angle
Obtuse angle
Reflex angle
betwen 0o and 90o less than a 14 turn
between 90o and 180o between a 14 turn and a 12 turn
between 180o and 360o between a 12 turn and a full turn
P O vertex
Q
We use 3-point notation to name angles.
indicates 90o
This is ]POQ or ]QOP.
D
C
The vertex is the middle letter.
Perpendicular lines meet at right angles (90o ).
A
B
AB ? CD means line AB is perpendicular to line CD.
Exercise 8 1 Draw a diagram which illustrates a an acute angle d an obtuse angle
b e
a straight angle a right angle
c f
a full turn a reflex angle
2 Name the following angles as acute, right angle, obtuse, straight angle or reflex: a b c d
e
f
3 Draw an angle which could be named a ]QRS b ]XYZ
g
c
h
reflex angle KLM
POINTS, LINES, ANGLES AND POLYGONS (CHAPTER 2)
Measuring angles
42°
In the diagram PQ is lined up along the base of the protractor and the line QR passes through 42o .
70
80 90 100 110
120
40
50
60
13 0 0 14
30
170 180 160 150
]PQR measures 42o .
DEMO
10 2 0
)
R
0
P
Q
centre
base line of protractor
4 For the diagrams below: i name the angle in terms of the three points ii find the measure of the angle using a protractor iii classify the angle according to its degree measure (for example, obtuse). a b L PRINTABLE D
K
TEMPLATE
F
E
M
c
DEMO
d U
R S
V
T W
5
a Without using a protractor, draw an angle which you estimate to be i 55o ii 120o iii 240o b Measure the angle you have drawn to see how accurate your estimate is.
6 List in 3 point notation all angles in the following figures: a b R
I
M
T
S
7 Use your protractor or set square to sketch: a line PQ is perpendicular to line LM 8
True or false? a AC ? BC b CB ? AB c BC ? BA
J
K
b
BC ? RS
A
B
9 State which lines are perpendicular in the given figure.
C A
B rectangle
D
C
23
24
POINTS, LINES, ANGLES AND POLYGONS (CHAPTER 2)
Unit 9
Triangles and quadrilaterals
Triangles A triangle is a three-sided polygon. Triangles can be classified according to the lengths of their sides and the sizes of their angles as follows:
scalene all three sides are different in length all three angles are different in size
isosceles two sides are equal in length two angles are equal in size
equilateral all three sides are equal in length all three angles are equal in size
Quadrilaterals A quadrilateral is a four-sided polygon.
Quadrilaterals can be classified according to the lengths of their sides and the number of sides which are parallel. ² A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel and equal in length.
² A rectangle is a parallelogram with all angles right angles.
² A rhombus is a quadrilateral with all sides equal in length.
² A square is a rhombus with all angles right angles.
² A kite is a quadrilateral which has two pairs of adjacent sides equal in length.
Exercise 9 1 Use a ruler to measure the lengths of the sides of these triangles. Classify them as scalene, isosceles or equilateral: a b c
d
2 Draw an example of a: a rectangle
d
3 Name these quadrilaterals: a
b
rhombus b
c
square c
parallelogram
POINTS, LINES, ANGLES AND POLYGONS (CHAPTER 2)
d
e
f
4 Draw a a square b parallelogram. In each of these figures join one vertex to the vertex opposite. (This line is called a diagonal.) Name the triangles formed as scalene, isosceles or equilateral. 5 True or false? Give a reason for your answer. a A square is a special rhombus. c A square is a parallelogram.
A square is a special rectangle. A rectangle is a parallelogram.
b d
6 Name the triangles as scalene, isosceles or equilateral: a b
c
70°
87°
35°
40° 53°
d
70°
40°
115°
e
45°
30°
f 60° 35°
60°
45°
110° 35°
60°
Drawing polygons with ruler and protractor To accurately draw triangle ABC with AB 5 cm long, angle CAB measuring 55o and angle CBA measuring 45o , follow these steps. C
At end A use your protractor to draw an angle of 55o .
Step 3:
At end B use your protractor to draw an angle of 45o .
Step 4:
Where the two lines meet mark C.
50
60
70
80 90 100 1
40
Step 2:
30
With the ruler draw AB exactly 5 cm long.
10 2 0
Step 1:
B
0
1 Using a ruler and protractor draw accurately:
A
5 cm
a triangle ABC with AB 6 cm long, angle CAB measuring 65o and angle CBA measuring 40o b triangle PQR with PQ 4 cm long, angle PQR measuring 80o and QR 5 cm long c the quadrilateral which is roughly drawn
C
B 4 cm A
d
110°
5 cm 40°
110°
5 cm
5 cm
70° 6 cm
80°
the kite which is roughly drawn.
D
DEMO
25
26
POINTS, LINES, ANGLES AND POLYGONS (CHAPTER 2)
Unit 10
Angles in a triangle
Investigation
Angles in a triangle
What to do:
PRINTABLE TEMPLATE
1 Using a protractor, accurately measure the sizes of ]ABC, ]BCA and ]CAB. 2 Copy the table below. Use the measurements you obtained in 1 to complete a. Use measurements from 3 other triangles of your choice and complete b, c and d. Make sure at least one of these triangles has an obtuse angle:
]ABC
]BCA
]CAB
B
sum of 3 angles
a b c d
A C
3 Based on your results in the table, what do you suspect about the sum of the angles in any triangle?
From the investigation you may have discovered that: The sum of the angles of any triangle is 180o . In this triangle:
110°
the two marked angles add to 150o. f40o + 110o = 150o g So the unmarked angle is 180o ¡ 150o = 30o .
40°
Exercise 10 1 Find the third angle of these triangles without using a protractor: a b
c
86°
50°
d
45°
70°
e
50°
35°
f
85°
28° 62°
50°
g
h
16°
130°
j
i
19°
18°
k 20°
20°
143°
l 61°
61° 78°
29°
2 The angles of an equilateral triangle are equal in size. What is the measure of each angle?
POINTS, LINES, ANGLES AND POLYGONS (CHAPTER 2)
27
In this triangle: the two equal angles add to 180o ¡ 46o = 134o .
46°
So each angle is 134o ¥ 2 = 67o .
3 In these triangles, equal angles are marked. Find their measure. a b
c
150°
28°
d
e
f 60°
78° A
4
5 cm 60°
60°
C
a b c
Find the size of ]BAC. What can be said about triangle ABC? Using b, find the lengths of AB and BC.
B 5 Use your ruler to draw AB exactly 3 cm long. Use your protractor to draw an angle of 50o at A. Draw AC 3 cm long. Join BC.
C 3 cm
a What type of triangle is ¢ABC? b Using your protractor, find the size of ]ACB and ]ABC. c Check that the sum of the angles of the triangles is 180o . 6 Use your ruler to draw AB exactly 4 cm long. Use your protractor to draw an angle of 90o at A and an angle of 45o at B. Complete the triangle at point C.
A
50°
B
3 cm
C
a What is the name of this type of triangle? b Calculate the size of ]ACB. Check your answer by measuring. c Measure the length of BC to the nearest millimetre.
A
Investigation
45°
4 cm
Facts about figures
What to do: 1 Click on the icon and print off the triangle and quadrilateral pages. 2 Measure all sides and all angles of each figure and mark these on the figure as you find them. 3
a For an equilateral triangle, what can be said about its angles? b For an isosceles triangle, what can be said about its angles? c For a parallelogram, what can be said about its opposite sides and opposite angles? d For a rectangle, what can be said about its opposite sides and opposite angles? e For a rhombus, what can be said about its opposite angles? f For a kite, what can be said about its opposite angles?
B
PRINTABLE TEMPLATE
28
POINTS, LINES, ANGLES AND POLYGONS (CHAPTER 2)
Unit 11
Angles in a quadrilateral
Activity
Angles in a quadrilateral What to do: 1 Click on the icon to print a page of different quadrilaterals.
WORKSHEET
2 Copy and complete the table below from accurate angle measurements.
]PQR
]QRS
]RSP
]SPQ
sum of angles
a b c d 3 Based on your results in the table what do you suspect about the sum of the angles in any quadrilateral?
From the investigation you may have discovered that: The sum of the angles of any quadrilateral is 360o . In this quadrilateral:
81°
the three marked angles add to
90o
+ 81o
+ 107o
=
278o
360o ¡ 278o = 82o :
So the unmarked angle is
107°
Exercise 11 1 Find the measure of the missing angle without using a protractor. The figures have not been drawn accurately. a b c 109° 78° 102° 84°
82°
111°
78°
d
e
f 110°
105°
79°
110°
2 Extension: In these quadrilaterals, equal angles are marked with a dot. Without using a protractor, find the size of the equal angles. The figures have not been drawn accurately. a b c
80°
70°
60°
POINTS, LINES, ANGLES AND POLYGONS (CHAPTER 2)
Activity
29
Tangrams What to do: 1 On a piece of card mark out a 20 cm by 20 cm square. Then copy these lines onto it and cut along each line. You should have seven different pieces.
PRINTABLE TEMPLATE
2 Each of the following shapes can be made up using all seven pieces of your tangram each time. See how many you are able to complete: a
bridge
b
puppy
c
person running
Game
d
cat
Guessing angles A game for two people to draw two angles, guess their size, and then measure their size to check the guess.
What to do: 1 2 3 4
One of the players draws an angle. Each player then guesses the size of the angle. Players measure the actual size of the angle. Repeat steps 1 to 3 ten times taking it in turns to draw an angle. After each turn both players record the results in a table like this:
Guess .. . .. .
Actual .. . .. .
Difference .. . .. .
5 When 10 angles have been drawn and measured, add the numbers in the difference column. The winner is the person who has the smaller total. DRAWING USING To improve your estimation of angle skills, click on the icon TECHNOLOGY ANGLES and follow the instructions.
Click on the icon for an activity on Drawing using Technology.
30
POINTS, LINES, ANGLES AND POLYGONS (CHAPTER 2)
Unit 12
Review of chapter 2
Review set 2A P
A
1 Using the diagram given: a Find a pair of parallel lines.
² Q C
b Find the intersection of line AB with line segment QR. d What type of angle (according to size) is the angle marked ²? 2 Measure the given angles: a
B
R
c Name the angle marked ² using three point notation.
D
S
b
3 Name these triangles as isosceles, scalene or equilateral: a b c
4 Sketch: a a line AB parallel to line PQ
b
d
a rhombus
c
a regular quadrilateral
5 Find the unknown angle(s) in the triangles given. They have not been drawn accurately. a b 102° 44° 46°
6 Sketch: a an equilateral triangle c a rhombus 7
P
S
Q
b d
a hexagon line AB perpendicular to line BC
a b c
What special figure is PQRS? Name two lines that are parallel. Name two lines that are equal in length.
R
8 Find the unknown angle of the quadrilateral. It has not been drawn accurately.
75°
85°
POINTS, LINES, ANGLES AND POLYGONS (CHAPTER 2)
Review set 2B 1 Use the diagram given to answer these questions:
X
B
A
a What type of polygon is ABCD? b Name parallel line segments. c Name two equal line segments. d Name the intersection of line segment AB with the line XY.
²
D
e Name the angle marked ² using 3 point notation.
Y
2 Measure the given angle.
X
3
a What type of polygon is WXYZ? b What type of triangle is XYZ? c Find the size of ]XWZ.
60° W
Y
d What can you say about triangle WXZ?
Z
4 Find the unknown angle(s) in the triangles given. They have not been drawn accurately. a b
47°
5
42°
64°
a Write in symbols: Line AB is perpendicular to line XY. b Sketch an acute angle AOB. c Sketch an octagon. d Sketch a parallelogram. Mark any equal sides on your diagram.
6
a What special figure is ABCD?
A
B
D
C
b Name two lines that are equal in length. c Name two lines that are parallel.
7 Find the unknown angles of the quadrilateral. It has not been drawn accurately.
8 Name the quadrilaterals which have: a all angles equal to 90o b two pairs of opposite sides that are equal in length c all sides equal in length.
120°
C
110°
31
32
NUMBER FACTS
CHAPTER 3
Unit 13
Addition and subtraction
²
To find the sum of two or more numbers, we add them. For example, the sum of 6 and 18 is 6 + 18 = 24.
²
To find the difference between two numbers, we subtract the smaller from the larger. For example, the difference between 6 and 18 is 18 ¡ 6 = 12.
Examples:
subtraction
addition 375 1964 +43469
1 2 10 0
Change 2 tens 0 units to 1 ten 10 units.
2 3 9 011 1
Change 3 hundreds 0 tens 1 unit to 2 hundreds, 9 tens, and 11 units.
¡ 1
1 2 1
45808 DEMO
DEMO
6 4
¡ 1 4 3 1 5 8
Check your answers by addition: 158 + 143 = 301 X
Exercise 13 1 Calculate a 36 + 45 d
413 274 + 62
b
217 + 38
c
934 + 628
e
2134 916 + 8406
f
508 1989 + 9009
2 Calculate a 423 + 96 d 9234 + 47 261 + 823 + 924
b e
9124 + 376 + 82 82 + 3 + 9274 + 837
Arrange the same place values underneath each other .
c f
412 + 3964 + 82 62 + 6320 + 41 + 92 314
3 Simplify, by changing the order to make it easy where possible: a 4 + 38 + 16 b 3 + 25 + 47 c 11 + 79 + 9 d 6 + 46 + 94 e 80 + 136 + 20 f 75 + 155 + 25 g 48 + 73 + 52 h 145 + 95 + 55 i 163 + 164 + 136 4 Calculate a 63 ¡ 21
b
52 ¡ 19
c
40 ¡ 23
Adding numbers can be done in any order.
d
623 ¡ 125
e
240 ¡ 138
f
200 ¡ 76
g
201 ¡ 127
h
3000 ¡ 147
i
9000 ¡ 2346
4372 and 29 586
c
7342 and 4453
what these number sentences are equal to: 1+2+3+4+5+6+7+8+9 123 + 456 + 789 12 + 3 ¡ 4 + 5 + 67 ¡ 8 + 9
b d f
12 + 34 + 56 + 789 1234 + 56¡789 123 ¡ 45 + 67 ¡ 89
5 Find the difference between a 19 and 32 b 6 Find a c e
To find the difference, subtract the smaller number from the larger number.
NUMBER FACTS (CHAPTER 3)
7 Calculate a 364 + 17 + 845 d 39 740 + 30 974 + 39 704 g 3749 + 5293 ¡ 5864 j 96 843 ¡ 38 472 ¡ 21 528
b e h k
4828 ¡ 3618 13 870 ¡ 9648 6000 ¡ 2987 621 536 + 278 365
8 Solve: a Find the sum of 3, 4 and 8. b What is the difference between 17 and 55? c To three hundred and fifty six, add five hundred and four. d Find the sum of the first 6 even numbers. e How much greater than 426 is 751? f What is the sum of the first 4 multiples of 7 greater than 30? g How much more needs to be added to 349 to reach 638?
c f i
33
189 + 41 748 + 997 50 824 + 36 + 4908 823 + 348 ¡ 986
Challenge: Which five of the six numbers should be added to give the total 3171? 632 719 627 883 448 + 745 3171
9 Solve: a How many animals altogether: three cats, four dogs and eight elephants? b If Jane has 17 sweets and Bronwyn has 55, how many more sweets than Jane does Bronwyn have? c Rachel has three hundred and fifty six dollars in her bank. If she sells her CD collection and gets five hundred and four dollars, what will her balance be? d Jessica and Michael were playing a game of cards. Jessica scored 426 and Michael scored 751. How much more than Jessica did Michael score? e What is the sum of the first four multiples of 8 that are greater than 20? f The Lister family is travelling in a campervan. They will cover 638 km while they are away from home. If they have travelled 349 km so far, how many more kilometres do they still have to travel? g By how much does the total of 617 and 386 exceed 946? h By how many kilometres is Australia’s 3750 kilometre Murray and Darling River system shorter than Africa’s 6695 kilometre Nile River? i A couple has $3200. How much more do they need to buy a $959 fridge, a $1349 washing machine and a $1769 TV? j By how much is the sum of 855 and 555 greater than the difference between 855 and 555? 10 Place addition signs amongst the numbers 1 2 3 4 5 6 7 8 9 so that the sum equals a e
117 666
b f
225 855
c g
8028 963
d h
19 134 1080
11 Place addition and/or subtraction signs amongst the numbers 1 2 3 4 5 6 7 8 9 so the number sentence gives these results: a 1712 b 122 667 c 379 d 756
123+456+7+89 =675
34
NUMBER FACTS (CHAPTER 3)
Unit 14
Multiplication and division by powers of 10
Examples: Multiplication fone zero in 10 ) add one zero to the first numberg
²
58 £ 10 = 580
²
58 £ 100 = 5800
²
58 £ 1000 = 58 000
fthree zeros in 1000 ) add three zeros to the first numberg
²
58 £ 10 000 = 580 000
ffour zeros in 10 000 ) add four zeros to the first numberg
ftwo zeros in 100 ) add two zeros to the first numberg
Investigation
Multiplying by 10, 100, 1000 and 10000
What to do: 1 Use a calculator to work out answers to: a
i 5 £ 10 = ii iv 96 £ 10 = v vii 342 £ 10 = viii Can you see a quicker way of multiplying
b
i iv vii What
29 £ 10 = 103 £ 10 = 1687 £ 10 = by 10?
6 £ 100 = ii 9 £ 100 = 35 £ 100 = v 97 £ 100 = 804 £ 100 = viii 5962 £ 100 = happens to a number when you multiply it by 100?
iii vi ix
58 £ 10 = 217 £ 10 = 2748 £ 10 =
iii vi ix
18 £ 100 = 342 £ 100 = 3049 £ 100 =
2 Work these out without using a calculator. a
i
b c
i i
23 £ 10 = 12 £ 100 = 7 £ 1000 =
ii ii ii
46 £ 10 = 24 £ 100 = 37 £ 1000 =
iii iii iii
91 £ 10 = 57 £ 100 = 48 £ 1000 =
iv iv iv
308 £ 10 = 218 £ 100 = 304 £ 1000 =
3 Predict what the product will be when you multiply these by 10 000: a
i
9
6
ii
iii
12
iv
24
v
50
b Use your calculator to check your predictions.
Exercise 14 1 Without using a calculator, find a e i m
63 £ 10 238 £ 10 37 £ 100 504 £ 1000
b f j n
63 £ 100 238 £ 100 37 £ 10 000 504 £ 10
c g k o
63 £ 1000 238 £ 1000 37 £ 10 504 £ 10 000
d h l p
63 £ 10 000 238 £ 10 000 37 £ 1000 504 £ 100
2 Without using a calculator, find a d g j m p s
96 £ 100 10 000 £ 84 5972 £ 100 83 £ 20 £ 5 10 £ $78 10 000 £ $17 1000 £ 842 litres
b e h k n q t
304 £ 1000 1000 £ 80 10 £ 3090 243 £ 50 £ 2 $597 £ 100 48 £ 10 cents 438 £ 100 cm
3 Check your answers to question 2 using your calculator.
c f i l o r u
10 £ 632 34 £ 10 000 26 £ 5 £ 2 368 £ 20 £ 50 864 £ $1000 352 £ 100 km 58 £ 1000 grams
NUMBER FACTS (CHAPTER 3)
35
Examples: Division ²
19 000 ¥ 10 = 1900
² ²
19 000 ¥ 100 = 190
fone zero in 10
) remove one zero from the first numberg
ftwo zeros in 100
) remove two zeros from the first numberg
fthree zeros in 1000 ) remove three zeros from the first numberg
19 000 ¥ 1000 = 19
Investigation
Dividing by 10, 100 and 1000
What to do: Use a calculator to solve these: a
i iv vii x
60 ¥ 10 940 ¥ 10 89 480 ¥ 10 342 800 ¥ 10
90 ¥ 10 2150 ¥ 10 94 370 ¥ 10
ii v viii
iii vi ix
760 ¥ 10 3470 ¥ 10 138 270 ¥ 10
iii vi ix
2400 ¥ 100 54 200 ¥ 100 301 900 ¥ 100
What rule did you use?
Can you see a quicker way of dividing by 10? b
i iv vii x
300 ¥ 100 5700 ¥ 100 36 100 ¥ 100 87 000 ¥ 100
800 ¥ 100 90 600 ¥ 100 42 700 ¥ 100
ii v viii
What happens to the number when it is divided by 100?
4 Without a calculator, find a e i m
150 ¥ 10 1100 ¥ 100 56 000 ¥ 1000 6800 ¥ 10
b f j n
850 ¥ 10 50 000 ¥ 1000 5600 ¥ 100 2200 ¥ 100
c g k o
29 000 ¥ 1000 5000 ¥ 100 56 000 ¥ 10 8400 ¥ 10
38 000 ¥ 100 34 000 ¥ 1000 56 000 ¥ 100 10 000 ¥ 1000
d h l p
5 Without a calculator, find a d g j m p
9000 ¥ 100 6000 ¥ 1000 307 000 ¥ 1000 $280 ¥ 10 940 kilometres ¥ 10 240 000 litres ¥ 1000
b e h k n q
79 000 ¥ 1000 5070 ¥ 10 91 000 ¥ 100 $92 000 ¥ 1000 68 000 grams ¥ 100 $2705 ¥ 100
c f i l o r
450 ¥ 10 65 000 ¥ 10 $6400 ¥ 100 $10 000 ¥ 100 690 cents ¥ 10 $38 470 ¥ 1000
6 To change the first (Left) calculator display to the second (Right) what operation would you need to perform? a
640
d
3600
g
518
64
b
83
8300
36 000
e
90 900
518 000
h
570 000
909 570
c
203
2030
f
7400
74
i
804 000
804
7 Change each dollar amount to cents. ($1 = 100 cents) a
$3
b
$30
c
$300
d
$3000
e
$492
d
200 km
e
7500 km
8 Change each distance to metres. (1 km = 1000 metres) a
2 km
b
5 km
c
50 km
36
NUMBER FACTS (CHAPTER 3)
Unit 15
Multiplication The word product is used to represent the result of multiplication. For example, the product of 3 and 4 is 3 £ 4 = 12.
Example: To find 387 £ 4 Set the numbers out with units underneath each other: 387 £ 4
7 £ 4 = 28, so put 8 in the units place and carry the 2 to the tens place: £
387 2 4 8
8 £ 4 = 32, plus the carried 2 makes 34. Put 4 in the tens place and carry the 3 to the hundreds place: £
3 £ 4 = 12, plus the carried 3 makes 15. Put 5 in the hundreds place and carry the 1 to the thousands place:
387 3 2 4 48
£
If we had 387 £ 40, the answer would be 10 times larger than the answer to 387 £ 4. So, we add a 0 to the end. That is, 1548 £ 10 = 15¡480.
387 1 3 2 4 1548 DEMO
Exercise 15 1 Find the product: a
24 £7
b
32 £5
c
624 £8
d
921 £6
e
523 £ 4
f
2461 £ 7
2 Find the product: a 56 £ 30 e 193 £ 80
b f
231 £ 70 308 £ 50
c g
93 £ 60 290 £ 90
d h
462 £ 40 4908 £ 30
3 Multiply these by 30: a 20 e 60
b f
200 600
c g
22 66
d h
34 27
b f j
8 £ 50 30 £ 7 60 £ 11
c g k
8 £ 500 300 £ 7 60 £ 110
d h l
80 £ 500 300 £ 700 600 £ 11
4 Find the products: a e i
8£5 3£7 6 £ 11
5 Simplify, taking short cuts where possible: a d g j 6
a b c d
2 £ 18 £ 5 4 £ 30 £ 25 50 £ 17 £ 2 20 £ 64 £ 50
b e h k
5 £ 35 £ 2 25 £ 19 £ 4 40 £ 23 £ 25 8 £ 8 £ 125
What is the product of thirteen and ten? Multiply sixty three by nine. What is the product of 38 and 40? How many hours in a week?
c f i l
16 £ 2 £ 10 60 £ 2 £ 50 100 £ 48 £ 10 50 £ 54 £ 200
When multiplying, sometimes it makes it easier to change the order, for example,
2 £ 38 £ 50 = 2 £ 50 £ 38 = 100 £ 38 = 3800
NUMBER FACTS (CHAPTER 3)
37
Example: ² £ 1 17 18 7 Find: a e i m q u
5 3 1 4 5
8 2 6 0 6
8 £ 19 19 £ 39 63 £ 21 26 £ 14 83 £ 29 79 £ 36
this is 58 £ 2 this is 58 £ 30 adding
17 £ 32 21 £ 58 43 £ 43 82 £ 41 46 £ 53 29 £ 26
b f j n r v
8 Calculate: a 42 £ 28 e 19 £ 81
b f
58 £ 12 116
86 £ 49 87 £ 36
So 58 £ 30 = 1740.
95 £ 48 28 £ 31 60 £ 40 92 £ 11 84 £ 84 81 £ 17
c g k o s w
84 £ 67 42 £ 73
c g
58 £ 23 174
d h l p t x
EXTRA PRACTICE
65 £ 54 37 £ 33
d h
18 £ 51 99 £ 42 18 £ 45 49 £ 66 76 £ 14 88 £ 22
9 Do these multiplications, then use a calculator to check your answer: a e
800 £ 79 237 £ 62
b f
407 £ 90 690 £ 42
c g
508 £ 71 369 £ 58
d h
Activity
436 £ 88 555 £ 39
Calculator use Over the next few years you will use your calculator to help you save time with calculations.
² You need to understand what to do.
Note:
² You need to key in correct digits and operations. ² Not all calculators work the same way. You will need to check how your calculator makes each type of operation. ² Making an estimation helps you check how reasonable your calculations were. You will learn about estimation soon. CLEAR KEYS (for most calculators) ² C clears the last entry
²
AC (or CA ) clears everything
What to do:
1 Press the keys in this order and check that you get the correct answer. a
9
+
7
=
14
b
-8 =
c
4
×
9
=
d
15
÷
3
=
2 Press the keys in this order and check that you get the correct answer. 24
×
9
×
4
÷
18
×
235
+
23 486
3 If you have made a “keying” error, such as AC (or CA ) or C ?
×
[Answer is 34 766.]
=
6 instead of
×
8, which key should you press,
4 Using a calculator, find: a c
1£2£3£4£5£6£7£8£9 1 £ 2 £ 3 £ 45 £ 67 £ 89
b d
1 £ 2 £ 3 £ 4 £ 5 £ 6 £ 789 12 £ 3 £ 45 £ 67 £ 8 £ 9
38
NUMBER FACTS (CHAPTER 3)
Unit 16
Division and problem solving
Division The word quotient is used to represent the result of division.
divisor
For example, the quotient of 12 ¥ 3 is 4. Examples: 6
39 ¥ 6 = 6 with 3 over 34 ¥ 6 = 5 with 4 over 48 ¥ 6 = 8
6 5 8 3 9 34 4 8
4 1 2
3
quotient dividend
Multiplication and division are easier if you know your times tables.
So 3948 ¥ 6 = 658 Remainders: Sometimes the division is not exact and a remainder is left. For example,
17 ¥ 3 = 5 with remainder 2 or 5 r 2.
Division by multiples of 10 60 ¡
5 90 5 738 5 400 338 ¡300 38
90 £ 60 = 5400 5 £ 60 = 300
So, 5738 ¥ 60 = 95 r 38
Exercise 16 1 Find a e i m
15 ¥ 5 48 ¥ 6 378 ¥ 3 2247 ¥ 7
b f j n
150 ¥ 5 480 ¥ 6 236 ¥ 4 3975 ¥ 5
c g k o
63 ¥ 2 523 ¥ 3 1549 ¥ 5
b f
79 ¥ 7 927 ¥ 7 1763 ¥ 6
c g
240 ¥ 60 1360 ¥ 40 1635 ¥ 60
b
630 ¥ 70 1180 ¥ 20 3975 ¥ 50
c
1500 ¥ 5 4800 ¥ 6 760 ¥ 5 9423 ¥ 9
d h l p
15 000 ¥ 5 480 000 ¥ 6 552 ¥ 8 48 628 ¥ 2
2 Find a e i
j
k
43 ¥ 9 825 ¥ 4 3005 ¥ 7
d h
450 ¥ 50 5820 ¥ 60 4231 ¥ 30
d
l
65 ¥ 8 823 ¥ 9 2647 ¥ 8
3 Find a e i 4
a b c d e f
f j
g k
What is the quotient of ninety six and eight? Share $660 equally among 5 people. How many dollars in 43 500 cents? How many weeks in 343 days? How many cartons containing 1 dozen eggs can be filled from 4380 eggs? How many 50 cent pieces are needed to make up 4550 cents?
h l
960 ¥ 40 6570 ¥ 90 8521 ¥ 70
NUMBER FACTS (CHAPTER 3)
39
Problem solving 5 Calculate: a c e g i k m o q s
the difference between 10 and 2 10 less 2 10 more than 2 multiply 10 by 2 the total of 10 and 2 the product of 10 and 2 share 10 between 2 10 lots of 2 2 less than 10 exceed 2 by 10
b d f h j l n p r t
the sum of 10 and 2 the quotient of 10 and 2 add 10 and 2 increase 10 by 2 divide 10 by 2 decrease 10 by 2 10 minus 2 10 take away 2 subtract 2 from 10 reduce 10 by 2
6 Solve these problems. a The 32 staff of a school each collect $575 from a lottery syndicate. How much did the syndicate win altogether? b A theatre sold 895 $7 children’s tickets and 256 $11 adults’ tickets. i How many tickets were sold altogether? ii How much money was paid for all the tickets? c Sophia earned $37 948 in her first year in a new job. In her second year she earned $4376 more than in the first year. i How much did she earn in her second year? ii How much did she earn in her first two years? 7 A shearer shears an average of 28 sheep per hour. If he works an eight hour day, how many sheep does he shear? 8 A stamp collector has 45 pages in her album. If each page can hold 36 stamps, how many stamps can she keep in the album? 9 A farmer has 860 sheep to transport. If the trucking company sends 4 trucks to do the job and each truck carries the same number of sheep, how many sheep will be on each truck? 10 Alyssa has $1040 in her bank. She withdraws $320 to pay for a DVD player. What is the balance of her account now? 11 A family pays $45 per month for their television. What is the yearly cost? 12 A movie theatre holds 490 people. How many rows of seats are there if each row has 35 seats? 13 If team A scores 254 and 312 in their two innings and team B scores 176 and 381, who wins and by how many runs? 14 George delivers 46 papers each day from Monday to Saturday. How many papers does he deliver in a week? If he is paid 10 cents for every delivery, what does he earn in a week? 15 The gates of a football ground have the following number of spectators pass through them: 1742, 1029, 2056, 381, 1973, 2624. What was the size of the crowd?
40
NUMBER FACTS (CHAPTER 3)
Unit 17
Estimation and approximation
Estimating is a way of checking our answers. An estimate is not a guess. It is a quick and easy approximation to the correct answer. When estimating we round each number to one digit and put zeros in the other places. This is rounding to one figure. For example, 594 + 600 and 83 + 80 .
Exercise 17 1 Estimate these: a 78 + 42 d 83 + 61 + 59 g 3189 + 4901 j 89 139 ¡ 31 988
b e h k
478 + 242 834 + 615 + 592 6497 ¡ 2981 59 104 + 20 949
196 + 324 815 ¡ 392 34 614 ¡ 19 047 1489 + 2347 + 6618
c f i l
Go back over these questions and compare your estimates with the exact answers.
89 £ 4 + 90 £ 4 + 360
2 For 1 a, c, e, g, i and k, find the difference between the estimate and the exact answer. 3 Estimate these products: a 19 £ 8 b e 87 £ 5 f i 54 £ 7 j
31 £ 7 92 £ 3 36 £ 9
c g k
28 £ 4 39 £ 9 94 £ 5
52 £ 6 88 £ 8 67 £ 3
d h l
4 Multiply and then use estimation to check that your answers are reasonable: a
41 £ 9
b
78 £ 7
c
53 £ 4
d
92 £ 9
e
27 £ 6
f
69 £ 8
g
82 £ 5
h
38 £ 3
b f
197 £ 9 381 £ 4
c g
521 £ 6 2158 £ 7
d h
238 £ 8 3948 £ 5
5 Estimate the products: a 484 £ 3 e 729 £ 8
6 Multiply and then use estimation to check that your answers are reasonable: a
214 £ 9
b
694 £ 3
c
808 £ 7
d
376 £ 8
e
497 £ 6
f
941 £ 4
g
522 £ 5
h
658 £ 7
7 Estimate these products a 49 £ 32 e 519 £ 38 i 58 975 £ 8
using 1 figure approximations: b 83 £ 57 c 58 £ 43 f 88 £ 307 g 728 £ 65 j 31 942 £ 6 k 6412 £ 37
d h l
389 £ 21 921 £ 78 29 £ 7142
8 Round the given data to one figure to find the approximate value asked for: a A large supermarket has 12 rows of cars in its carpark. If each row has approximately 50 cars, estimate the total number of cars in the park. b A school canteen has 11 shelves in its fridge. Estimate the number of drinks in the fridge if there are approximately 21 drinks on each shelf.
427 £ 89 + 400 £ 90 + 36¡000
NUMBER FACTS (CHAPTER 3)
41
If both numbers end in 5, 50 or 500 etc, we round the smaller number up and the larger number down. This gives a closer approximation than rounding both numbers up. For example:
650 £ 25 is approximately 600 £ 30 which is approximately 18 000
9 Estimate the products a 45 £ 15 e 95 £ 95
b f
65 £ 25 750 £ 15
Both numbers end in 5, 50, .... , so round 650 down to 600 and round 25 up to 30.
c g
75 £ 85 950 £ 45
d h
550 £ 35 9500 £ 45
10 Use estimation to find which of these calculator answers is reasonable: a
126 £ 9
1134
14310
11 344
b
93 £ 28
2804
26 404
2604
+ 6000 ¥ 50
c
685 £ 72
49 320
43 920
4392
+ 120
d
897 ¥ 3
2999
209
299
e
79 £ 196
16 684
15 484
160 484
f
3945 £ 32
120 400
12 040
126 240
g
8151 ¥ 19
3209
429
329
5968 ¥ 51
11 Round the given data to one figure to find the approximate value asked for: a Scott reads 19 pages in one hour. At this rate, estimate how long it will take him to read a 413 page novel. b Each student is expected to raise approximately $28 in a school’s spellathon. If 397 students take part, estimate the amount the school could expect to raise. c A school trip needs one adult helper for every 5 students. Approximately how many adults are needed if 95 students are going on the trip? d Estimate the number of students in a school if there are 21 classes with approximately 28 students in each class.
To estimate the number of paper clips on the sheet of paper: 1 2 3
Divide the paper into equal parts as shown. Count the number of paper clips in one part. Multiply the paper clips in one part by the total number of parts. Number of paper clips in 1 part £ number of parts = 7 £ 8 = 56 paper clips Estimate: 56 paper clips are lying on the sheet of paper.
Click on the icon for a worksheet for you to practise estimation.
PRINTABLE WORKSHEET
DEMO
42
NUMBER FACTS (CHAPTER 3)
Unit 18
Factors, divisibility rules and zero
Factors are the numbers which divide exactly into another number. For example, 4 is a factor of 20 because it divides exactly into 20 (5 times), i.e., 4 £ 5 = 20.
DEMO
The factors of 6 are 1, 2, 3 and 6 since 6 can be written as 1 £ 6 or 2 £ 3: We say 6 has 2 pairs of factors, 1 £ 6 and 2 £ 3. To find the factors of a number we divide it by each whole number in order, starting with 1. Remember, factors are whole numbers. For example, the factor pairs of 20 are 1 £ 20, 2 £ 10, 4 £ 5, so the factors of 20 are 1, 2, 4, 5, 10 and 20. When we write the number as a product of factors we say it has been factorised. A whole number is even if it has 2 as a factor (if it is divisble by 2.) ² 2, 4, 6, 8, 10, 36, 98, 242, 3574 and 48 940 are examples of even numbers. A whole number is odd if it does not have 2 as a factor (it is not divisible by 2.) ² 1, 3, 5, 7, 9, 21, 93, 485, 24 689 are examples of odd numbers. We write the factors in order from smallest to largest.
Exercise 18 1 List a e i
the factor pairs for 12 b 15 22 f 24 9 j 25
c g k
18 28 36
d h l
20 30 48
2 List a e i
the factors of 3 9 18
c g k
5 13 63
d h l
8 16 72
3 Odd a e i
or even? 14 108 1001
b f j
4 12 49 b f j
17 123 8793
c g k
43 244 13 454
d h l
64 388 203 673
c g k
16 = 4 £ :::: 56 = 8 £ :::: 121 = 11 £ ::::
d h l
12 = 2 £ :::: 48 = 4 £ :::: 48 = 3 £ ::::
4 Copy and complete these factorisations: a e i 5
a b c d
22 = 2 £ :::: 28 = 14 £ :::: 45 = 9 £ ::::
b f j
32 = 8 £ :::: 50 = 10 £ :::: 45 = 15 £ ::::
List all the factors of 12. List all the factors of 18. List the numbers that are factors of both 12 and 18. These are called common factors. What number is the highest common factor?
6 Find the common factors of these numbers. a e
4 and 24 27 and 45
b f
9 and 36 18 and 27
c g
21 and 49 8, 16 and 32
d h
12 and 16 12, 30 and 54
c g
18 and 42 14, 56 and 21
d h
15 and 20 16, 48 and 72
7 Find the highest common factor of these numbers. a e
10 and 25 24 and 28
b f
8 and 24 45 and 81
NUMBER FACTS (CHAPTER 3)
Divisibility rules ² ² ²
A number is divisible by 2 if the last digit is even or zero. A number is divisible by 3 if the sum of its digits is divisible by 3. A number is divisible by 5 if the last digit is 0 or 5.
We can use divisibility rules to determine whether numbers can be divided by any of the numbers 2, 3, or 5. Examples: ²
230 has last digit 0, so 230 is even and is divisible by 2 and 5.
²
43 is not even, does not end in 5 or 0, and the sum of its digits (4 + 3) = 7 is not divisible by 3, so 43 is not divisible by any of 2, 3 or 5.
²
75 has last digit 5, so 75 is divisible by 5. The sum of its digits (7 + 5) = 12 is divisible by 3, so 75 is also divisible by 3.
²
243 is not even, and does not end in 0 or 5. The sum of its digits (2 + 4 + 3) = 9 is divisible by 3. So, 243 is divisible by 3.
8 Which of the following are divisible by 5? a 35 b 80 c f 389 g 555 h 9 Use a f k
77 1606
d i
123 5200
e j
divisibility rules to find whether these numbers are divisible by any of 2, 3 or 5. 48 b 95 c 29 d 210 e 17 g 123 h 332 i 333 j 34 l 97 m 171 n 31 o
240 12 345 555 51 651
10 Challenge:
I am a 2-digit number. The product of my digits is 18. I am divisible by 3 but not by 2. What number am I?
Operations with zero When adding or subtracting zero the number remains unchanged.
²
For example, 27 + 0 = 27, 27 ¡ 0 = 27. Multiplying by zero produces zero.
²
For example, 14 £ 0 = 0. Division by zero has no meaning; we say it is undefined.
²
For example, 0 ¥ 5 = 0 but 5 ¥ 0 is undefined. 11 Simplify a d g
18 + 0 72 + 0 + 28 35 ¡ 0 ¡ 35
b e h
0 + 18 123 ¡ 24 + 0 56 ¡ 0 + 27
c f i
14 ¡ 0 123 + 0 ¡ 24 403 + 0 ¡ 304
b e h
3£3£0 52 £ 0 0¥1
c f i
9¥0 0 ¥ 52 1¥0
12 Simplify, if possible: a d g
9£0 0£3£6 0£0
43
44
NUMBER FACTS (CHAPTER 3)
Unit 19
Multiples, LCM and operations
Multiples The multiples of 4 are 4, 8, 12, 16, 20, .... These multiples are found by multiplying each of the whole numbers by 4, i.e.,
1£4=4 2£4=8 3 £ 4 = 12 4 £ 4 = 16 5 £ 4 = 20
Some other multiples of 4 are:
48 60 800 1660 20 000
A multiple of a whole number is found when you multiply it by another whole number .
(12 £ 4) (15 £ 4) (200 £ 4) (415 £ 4) (5000 £ 4)
Lowest common multiple The The The The
multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30. multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45, 50. numbers 15 and 30 are multiples of both 3 and 5. We say they are common multiples of 3 and 5. number 15 is the lowest common multiple of 3 and 5.
Exercise 19 1 Complete these to find the multiples: a
8 £ 5 = :::::,
80 £ 5 = :::::,
800 £ 5 = :::::,
880 £ 5 = :::::
b
3 £ 6 = :::::,
30 £ 6 = :::::,
300 £ 6 = :::::,
330 £ 6 = :::::
c
9 £ 7 = :::::,
9 £ 70 = :::::,
9 £ 700 = :::::,
9 £ 770 = :::::
d 3 £ 20 = :::::,
4 £ 20 = :::::,
6 £ 20 = :::::,
30 £ 20 = :::::
2 Complete the patterns of multiples: a 3, 6, 9, ...., 15, 18, ...., ...., ...., 30 c e
b
...., ...., 21, 28, ...., ...., ...., 56, 63, .... 160, 168, 176, ...., 192, 200, ...., ...., 224
d f
6, 12, ...., 24, ...., 36, ...., 48, 54, .... 35, ...., ...., 50, 55, ...., ...., 70, 75, .... 244, 248, 252, ...., ...., 264, ...., 272, ....
3 Continue the number patterns by writing the next 6 multiples. a 20, 30, 40, .... b 27, 36, 45, .... d 700, 800, 900, .... e 120, 150, 180, .... 4 List the first 6 multiples of a 2 b 10
30, 33, 36, .... 15, 30, 45, ....
c f
c
20
d
8
e
11
c
9
d
12
e
50
5 List the first 4 multiples greater than 40 of: a
3
b
8
6 Find the lowest common multiples of a e
3 and 4 3 and 8
b f
6 and 10 4 and 10
c g
4 and 7 10 and 15
d h
6 and 9 8 and 12
c g
8 and 12 6 and 9
d h
5 and 7 10 and 20
c
6 and 10
d
3 and 4
7 Find the common multiples between 30 and 50 of a e
3 and 4 8 and 6
b f
5 and 15 2 and 3
8 Find the lowest common multiple greater than 60 for a
4 and 9
b
5 and 20
45
NUMBER FACTS (CHAPTER 3)
9 True or false? a For any two numbers, the lowest common multiple is less than the larger number? b The LCM of 4 and 20 is 40.
Use your knowledge of lowest common multiples to help you solve these problems. 10 Mark helps his family by walking the dog every second night and cooking dinner every third night. On Monday he walks the dog and cooks dinner. How long will it be before he walks the dog and cooks dinner on the same night again? 11 Anna runs a 400 m lap in 80 seconds. Jessica takes 90 seconds to complete a 400 m lap. If they start running together, after how many seconds will they complete a lap together?
One operation after another
DEMO
Extension in
A number is fed into the number crunching machine. It is multiplied by 3 and then, in the next part, 4 is subtracted.
So, for 7 going in we have: £3
7 times 3
subtract 4 out
21
¡4
in
out
12 Multiply each number by 6 and then subtract 8: a
4
40
b
c
7
d
12
e
100
25
d
200
e
0
d
20
e
50
d
700
e
147
d
90
e
333
13 Multiply each number by four then add six: a
10
15
b
c
14 Divide each number by 2 and then multiply by 6: a
10
2
b
c
200
15 Divide each number by 7 and then multiply by 2: a
21
70
b
c
91
16 Divide each number by 3, then multiply by 4 and then add 9: a
9
27
b
2
£3
6
+4
10
c
4
£3
12
36 +4
16
6
£3
18
+4
22
7
£3
?
+4
?
The operations between the numbers in these boxes are £3 then +4. So the missing numbers in the 4th box are 7 £ 3 = 21 and 21 + 4 = 25. 17 Find out what operations are used in the first 3 boxes then find the missing numbers in the last box. a
3 9 18
10 16 25
17 23 32
23
b
8 5 35
4 1 7
9 6 42
10
c
3 12 13
6 24 25
9 36 37
11
d
7 42 39
8 48 45
12 72 69
20
e
30 6 13
40 8 15
15 3 10
60
f
18 6 60
15 5 50
36 12 120
33
17
46
NUMBER FACTS (CHAPTER 3)
Unit 20
Review of chapter 3
Review set 3A 1 Find the value of a 45 + 219 + 55
b
1475 ¡ 389
2 Find without a calculator: a 34 £ 10
b
9300 ¥ 100
c
$790 £ 100
3 Find these products: a 67 £ 84
b
733 £ 30
c
25 £ 38 £ 4
4 Find the quotients: a 7350 ¥ 7
b
21 375 ¥ 9
c
1400 ¥ 40
5
a b
Find the difference between 804 and 408. Estimate the product of 78 and 32.
6 Without doing division, explain why 679 450 is divisible by 5. 7 List the factors of 24. 8 List the multiples of 3 between 25 and 40. 9 Find the lowest common multiple of a 3 and 9 b
8 and 12
10 On the first day of a test match, 35 432 people attended the cricket. It rained on the second day and only 18 093 people attended. a What was the total attendance for the two days? b By how much did the attendance on the first day exceed the second day’s attendance? c If tickets cost $22 per person per day, estimate how much it would cost for a bus load of 37 people to attend for one day. 11 7 people share first prize in Lotto. If the first prize is $2 346 400, how much did each person receive? 12 Multiply these numbers by 6 and then subtract 12 a 2 b 10
c
11
c
the nearest 10
13 Round 63 728 to a
the nearest 10 000
b
the nearest 1000
14 Estimate: a the difference between 817 and 296 b the product of 41 and 307 c the cost of 42 bunches of flowers at $9:50 each d the cost of 31 calculators at $38:85 each.
d
30
NUMBER FACTS (CHAPTER 3)
Review set 3B 1 Find the value of a 63 + 249 ¡ 33
b
5823 + 986
2 Find the a sum of 18, 4063 and 82 c result when 763 is divided by 7
b d
difference between 177 and 1469 product of 530 and 11
3 Find the value of a 71 £ 1000
c
$2300 ¥ 100
b
4 Work out these products: a 4 £ 363 £ 25 b 5
8559 ¥ 9 49 £ 17
a How much greater than 559 is 693? b Find the sum of the first four multiples of 5. c Find the result of dividing seventy seven by eleven.
6 Multiply these numbers by 3 and then add 2: a 4 b 8
c
12
d
20
7 Estimate the product of 698 and 52. 8 Without performing division, explain how you would know that 65 731 is not divisible by 2. 9 List the factors of 16. 10 Find the lowest common multiple of a 11 and 33 b 20 and 15 11 Fabio gives $13 248 to each of his 8 grandchildren. How much did he give away altogether? 12 If 100 bottles of shampoo weigh 25 000 grams, find the weight of one bottle. 13 Subtract 5 from these numbers and then multiply by 3: a 7 b 27 14 Emily is very fit. She runs every day, swims every second day and goes to the gym every third day. If she did all 3 activities today, how many days will it be before she does them altogether again?
15 Which 5 of these numbers should be added to give the total 2383?
+
579 404 318 267 837 296 2383
47
48
REVIEW OF CHAPTER 1, 2 AND 3
TEST YOURSELF:
Review of chapters 1, 2 and 3
a What do these Roman symbols represent:
1
b Write these as Roman symbols:
17
i
2 What number does the digit 3 represent in
i
XVII
ii
ii
39
iii 148
a 1307
CLIX ?
b 230 000?
3 Place these numbers in order from smallest to largest: 457, 754, 574, 475, 47, 745, 547, 74 a Write 5 £ 100 000 + 6 £ 1000 + 4 £ 100 + 9 £ 10 + 1
4
in simplest form.
b Write 690 307 in expanded form. 5 True or false? a
5769 > 5679
b
25 £ 7 > 200 ¡ 32
c
57 < 7 £ 8
6 Round a c
5765 to the nearest 1000 5765 to the nearest 10
b d
5765 to the nearest 100 $23:87 to the nearest 5 cents
7 What number is represented by a
b
8 Write as cents:
$63
a
b $63:65
9 Write a number which is a
35 more than two thousand
b
63 less than three thousand
10 1765 cents is written as $17:65. Write the following in this form: a
325 cents
b
1480 cents
c
3 698 425 cents
11 Estimate the cost of 29 drinks at $1:95 each. 12 Jack’s supermarket bill came to $38:18. What did he actually pay for the goods? A
13
Find, using a protractor a the measure of i ]ABC ii ]BCA iii ]CAB b the sum of the angle sizes of the triangle.
B
C
14 Name these triangles: a
b
c 3 cm
2.4 cm 2 cm
14 mm 3 cm
3 cm
REVIEW OF CHAPTER 1, 2, AND 3
49
15 Find x in these inaccurately drawn triangles: a
b
c
88°
x° 42° x°
35°
x°
16 Sketch:
a rectangle
a
b
a square
c
a rhombus
d
a pentagon
c
37 000 ¥ 100
17 True or false? a a regular pentagon only needs to have sides equal in length P
b
Q
this angle is ] QRP
R
18 Find the size of the unknown angle of
130° 100°
19 Find the value of:
a
607 + 89
b
2003 ¡ 894
20 Find the value of:
a
67 £ 10
b
$276 £ 100
21 Find the product:
a
27 £ 13
b
86 £ 39
22 Find:
a
369 ¥ 3
b
1248 ¥ 6
c
2576 ¥ 7
c
900 £ 70
d
900 £ 700
23 Find the difference between 379 and 793. 24 If 9 £ 7 = 63, find the value of a
90 £ 7
b
90 £ 70
25 List the factors of 36. 26 List the multiples of 4 between 10 and 30. 27 Find the lowest common multiple of 9 and 12. 28 At a b c
the tennis, the first two days’ attendances were Day 1: 17 642, Day 2: 18 936. What was the total attendance for the two days? What was the difference in attendance for the two days? If tickets were $30 each, how much money was paid for tickets for these two days?
50
FRACTIONS
Fractions
Unit 21 A fraction represents a part of a quantity.
Qe_
The container is two thirds full. Two thirds is written as
CHAPTER 4
Notice that
1 3
2 3
Qe_
We_
indicates the number of filled parts. indicates the number of equal parts in a whole.
Qe_ DEMO
of the container is empty.
Ways of representing fractions: In symbols
3 10
numerator bar denominator
In words
By diagram
three tenths
On a number line
0
1
qD_p_
or
Exercise 21 1 Write the number of shaded parts as a fraction of the total: a b
c
d
e
DEMO
2 Estimate the fraction shaded: a
d
b
c
e
f
3 Copy the following diagrams twice. Use dotted lines to show how the figures could be divided into i 6 equal parts. ii 3 equal parts. a b PRINTABLE TEMPLATE
51
FRACTIONS (CHAPTER 4)
4 Copy and complete the following table:
a
b
Num. Den.
is short for Numerator (top) is short for Denominator (bottom)
Symbol
Words
Num.
Den.
Meaning
1 10
one tenth
1
10
One whole divided into 10 equal parts and one is being considered.
0
One whole divided into 5 equal parts and two are being considered.
0
2 5
two fifths
three eighths
Number line 1 one tenth
1 two fifths
8
c
One whole divided into 6 equal parts and five are being considered. PRINTABLE TEMPLATE
d
0
5 Which of the following shaded shapes shows 34 ? a
b
c
d
e
f
6 Make three copies of each of the following diagrams: a b
c
Use dotted lines to show how the figures could be folded to give i 2 ii 4 iii 8 equal parts. [There may be more than one answer.] 7 Make three copies of each of the following diagrams: a b
i iii
Shade one copy to show 34 . Shade the third copy to show
12 16 .
c
ii
Shade the second copy to show 58 .
iv
Comment on your answers for i and iii.
1
52
FRACTIONS (CHAPTER 4)
Unit 22
Fractions of quantities
Examples: ² There are 27 students in a Year 6 class.
² There are 7 circles. 4 are black. So
4 7
of the circles are black
and
3 7
of the circles are white.
4 7
3 7
=
+
7 7
1 3
of them are more than 10 years old.
1 3
of 27 students = 27 ¥ 3 = 9 students
There are 9 students aged more than 10 years.
=1
Exercise 22 1
a What fraction of this flock of sheep are white?
b What fraction of these letters have stamps? 45 c
45
45 c
45 c
45 c
c
c What fraction of the group i
are wearing caps or hats
ii
have hoops?
2 James scored 23 correct answers in his test of 36 questions. What fraction of his answers were incorrect? 3 Aliki lost 2 pens and broke 3 others. If she had 9 pens to start with, what fraction of her pens remain? 4 Yuka was travelling a journey of 250 km. Her car broke down after 200 km. What fraction of her journey did she still have to travel? 5 David started his homework at 8:05 pm and completed it at 8:53 pm. If he had allowed 60 minutes to do his homework, what fraction of that time did he use? To find Qe_ of 12 we need to divide 12 into 3 equal parts.
6 Rachel spent $1:65 on a drink and $2:70 on chocolates. What fraction of $10 did she spend? 7 Find a d g
1 2 of 10 1 3 of 12 1 10 of 650
b e
grams
h
1 6 1 4 1 2
of 24
c
of 20
f
of $1:20
i
1 4 1 5 1 4
of 16 of 35 of 1 hour (in minutes)
FRACTIONS (CHAPTER 4)
Example:
To find We_ of 84 we need to divide 84 into 3 equal parts and use 2 of them.
There were 84 children at a country school. Only two thirds of them could swim. 1 3
53
of 84 children = 84 ¥ 3 = 28 children 2 3
)
of 84 = 2 £ 28 = 56 children
) 56 children could swim. 8 Richard only won one quarter of the games of chess that he played for his school team. If he played 16 games, how many did he win? 9 One eighth of the apples in a bag were bad. If there were 128 apples in the bag, how many were bad? 10 One sixth of the cars from an assembly line were painted red. If 78 cars came from the assembly line, how many were painted red? 11 Jessica had $120. She spent one third of her money on a new dress. How much did the dress cost? 12 While Bill was on holidays one fifth of the plants in his shadehouse died. If he had 25 plants alive when he went away, how many were still alive when he came home? 13 There are 360o in 1 revolution (one full turn). a Find the number of degrees in i one quarter turn ii iii three quarters of a turn.
a half turn
b What fraction of a revolution is i 30o ii 60o iii 240o ?
14 Find a i c
i
e
5 8
1 5 1 10
of 55 of 610 g
of 640 m
ii
3 5
ii
3 10
of 55 of 610 g
360°
b
i
1 4
of 76
ii
3 4
of 76
d
i
1 8
of $48
ii
3 8
of $48
f
2 3
of 33 days
15 One morning three fifths of the passengers on my bus were school children. If there were 45 passengers, how many were school children? 16 Daniel spent three quarters of his working day installing computers, and the remainder of the time travelling between jobs. If his working day was 10 hours, how much time did he spend travelling? 17 When Emily played netball, she scored a goal with five sixths of her shots for goal. If she shot for goal 18 times in a match, how many goals did she score?
54
FRACTIONS (CHAPTER 4)
Unit 23
Finding the whole and ordering fractions
Finding the whole 1 3
means that we have divided the whole into 3 equal parts and we are considering one part. 1 3
Three thirds make one whole.
1 3
1 3
Examples: ² If 14 of my money is $7, you can find how much I have in this way:
² Two thirds of my barley crop was 176 bags. How many bags was the whole crop? 2 thirds was 176 bags. So, 1 third was 88 bags.
One quarter of my money is $7. Four quarters is the whole of my money. So, I have 4 £ $7 = $28.
The whole crop was 3 thirds = 3 £ 88 bags = 264 bags.
Exercise 23 1 Find the whole amount if a
1 5
is $9
b
1 7
is 110 g
c
1 10
d
1 6
is 6 m
e
1 4
is 3 weeks
f
1 2
is 30 bags is 2 hours 5 minutes
2 Helena spent half her holidays in America. She was in the USA for 17 days. How long were her holidays? 3 Gavin said that one fifth of his supermarket bill was the cost of a leg of lamb at $10:20. What was his supermarket bill? 4 Katy put 16 jars of jam in her pantry. These were 14 of the jars of jam that she had made. How many jars of jam had she made? 5 The deposit on a refrigerator was one eighth of its value. If the deposit was $125, how much did the refrigerator cost?
6 Find the whole amount if a
2 3
is 8 apples
b
3 10
is $171
c
5 8
d
2 5
is 160 g
e
3 4
is 1200 m
f
9 10
7 Three quarters of my dairy herd are grazing in my largest paddock. If this is 90 cows, how many are in my herd? 8 If 25 of a bag of flour weighs 800 g, how much does a bag of flour weigh? 9 Phillip won 23 of the tennis matches that he played this summer. If he won 64 matches, how many matches did he play? 1 10 Briony saved 10 of the money that she earned working at a restaurant. If she spent $927, how much did she earn?
is 10 days is 81 marks
We_ means that we have divided the whole into 3 equal parts and we are considering two of them.
FRACTIONS (CHAPTER 4)
55
Ordering of fractions These diagrams have been shaded to represent the fractions written below: 1 Er_ means 1+ Er_ 2 4
1 4
=
1 2
3 4
4 4
5 4
=1
6 4
= 1 14
7 4
= 1 12
8 4
= 1 34
When two quarters ( 24 ) are shaded, the result is the same as shading the square.
=2
1 2
When four quarters ( 44 ) are shaded, the result is the same as shading the whole square. Counting in quarters we have or it could be written as 11
a Count in halves to four,
i.e.,
b Count in thirds to three. 12
1 2,
1 4
2 4
3 4
4 4
5 4
6 4
7 4
8 4
1 4
1 2
3 4
1
1 14
1 12
1 34
2:
1, 1 12 , .... up to 4.
There are 5 fifths in a whole and 10 fifths in 2 wholes.
c Count in fifths to two.
a Use the circle diagrams given to count in sixths to 2. etc.
b Can you find another way of writing some of these fractions? 13
a Draw circle diagrams to help you count in eighths to 2. b Can you find another way of writing some of these fractions?
14 On a sheet of paper, draw 12 bars, 12 cm long, like the one shown. 1cm
a Count in twelfths up to 1, i.e.,
1 2 3 12 , 12 , 12 ,
....., 1.
b Shade one bar to represent each fraction. Try to find another way to write each fraction.
For example,
2 12
= 16 :
Activity
Shape fitting What to do: 1 Work out how many times each of the shapes i
ii
a
iii
b
will fit into the following.
c
2 Write i
4
’s ii
3
’s iii
4
’s
as a fraction of each of the shapes a, b and c in question 1 above.
56
FRACTIONS (CHAPTER 4)
Unit 24
Equivalent fractions and lowest terms
Equivalent fractions 4 identical squares have been drawn. Half of each square has been shaded.
2 identical parts 1 is shaded
4 identical parts 2 are shaded
8 identical parts 4 are shaded 1 2
The same part of each square is shaded, so, 1 2,
2 4,
4 8,
8 16
2 4
=
4 8
=
16 identical parts 8 are shaded
8 16 .
=
DEMO
are equivalent (equal) fractions.
Multiplying both the numerator and the denominator by the same number produces an equal (equivalent) fraction.
1£2 2£2
For example:
Dividing both the numerator and the denominator by the same number produces an equal (equivalent) fraction.
12 16
For example:
=
2 4
=
12¥4 16¥4
3 4
=
Exercise 24 1 The rectangle has been divided into 16 equal parts. Make three copies of this rectangle, and by shading the fractions given, copy and complete: a
1 2
=
1 4
b
16
=
c
16
4
12 16 .
=
The circle has been divided into 6 equal parts. Make two copies of this diagram.
2
2 6
a
Shade two equal parts. Copy and complete:
b
Shade half the circle. Copy and complete:
c
Copy and complete:
b
3 8
=
3£5 8£2
=
15 40
c
3 7
=
3£2 2£12
=
36 24
f
6=
i
3 15
=
3¥2 15¥3
=
1 5
l
81 9
=
81¥2 9¥9
=
2 1
4
=
3
=
1 2
=
3£2 7£4
=
12 28
6 1
6£3 1£2
3 6
.
.
.
3 Copy and complete a
2 5
d
7 10
=
7£7 10£2
=
49 70
e
3 2
g
22 80
=
22¥2 80¥2
=
11 40
h
27 48
=
27¥3 48¥2
j
18 12
=
18¥6 12¥2
=
3 2
k
77 11
=
77¥2 11¥11
=
2£2 5£8
=
16 40
=
9 16
=
7 1
=7
=
=
=
18 3
=2
4 Express with denominator 12 a
2 3
b
3 4
c
3
d
5 6
14 24
e
5 The equilateral triangle has been divided into four equal parts. a Copy the diagram and add 3 vertical lines to divide it into 8 equal parts. b Use 3 copies of this new diagram to show that i
1 2
=
2 4
=
4 8
ii
1 4
=
2 8
iii
3 4
= 68 .
f
21 36
FRACTIONS (CHAPTER 4)
Fractions to lowest terms 8 12
Can you see that the same shaded part is So,
8 12
=
4 6
=
2 3
4 6
or
or 23 ?
where the last fraction is in lowest terms.
A fraction is in lowest terms if its denominator (bottom number) cannot be made any smaller and remain a whole number. For example:
To reduce a fraction to lowest terms divide the top and bottom number by common factors until the bottom number can be no smaller.
15 20
²
16 18
²
36 60
²
=
15¥5 20¥5
=
16¥2 18¥2
=
36¥12 60¥12
=
3 4
=
8 9
=
3 5
6 Write in lowest terms a
5 10
b
3 9
c
2 8
d
2 6
e
5 20
f
6 15
g
9 30
h
45 100
i
22 55
j
25 60
7 Write the fraction shaded in lowest terms: a b
d
c
e
f
8 Write the fraction shaded in lowest terms: a b
Remember that 1 revolution is 360°.
c 120°
30°
d
e
f 220°
300°
40°
9 Write in lowest terms a
12 20
b
6 24
c
24 36
d
12 18
e
36 72
57
58
FRACTIONS (CHAPTER 4)
Unit 25
Mixed numbers and improper fractions
If Paul ate 23 of a pie and Leslie also ate of a pie, together they ate 43 pies.
2 3
and
But we also see that the amount eaten is 1 whole pie plus 13 of another pie. We write
1 3
1+
as 1 13
make
Paul
and 1 13 =
mixed number
Leslie
4 3
improper fraction
A mixed number has a whole number part and a fraction part. An improper fraction has numerator (top) greater than denominator (bottom).
Converting to improper fractions Examples:
1 23
²
2 15
²
=1+
2 3
=
3 3
2 3
=
5 3
+
=2+
+
=
10 5
=
11 5
1 5
+
+
+
1 5
Exercise 25 1 Copy and complete a
1=
¤ 4
b
1=
¤ 3
c
1=
¤ 10
d
2=
¤ 3
e
2=
¤ 6
f
3=
¤ 4
2 Copy and complete
(or click on the icon for a worksheet)
Diagram
Mixed number
Improper fraction
a b PRINTABLE TEMPLATE
2 23
c d
11 5
e 3 Convert to improper fractions a
1 12
b
1 13
c
1 23
d
1 34
e
2 14
f
2 13
g
2 25
h
2 45
i
3 12
j
3 14
k
3 23
l
3 35
m
4 12
n
4 23
o
5 13
FRACTIONS (CHAPTER 4)
59
Converting to mixed numbers Pizza portions Pizzas were each cut into eight equal pieces for a party. After the party there were 17 pieces of pizza left. How many pizzas were left? Each 8 pieces make up a whole pizza, so how many 8s are in 17? 17 ¥ 8 = 2 + 1 remainder. There are 2 whole pizzas and 1 piece ( 18 of a pizza) remaining. So,
17 8
= 2 18 .
17 8
= 2 18
means 17 ¥ 8 = 2 wholes Examples:
²
7 5
5 5
+
2 5
=1+
2 5
=
= 1 25
or
7 5
1 8
remaining
=7¥5 = 1 with remainder 2 = 1 25
27 4
² =
24 4
+
=6+
3 4
3 4
= 6 34 4 Convert to a mixed number a
3 2
b
5 2
c
4 3
d
8 3
e
17 3
f
5 4
g
13 4
h
21 4
i
8 5
j
11 5
k
32 5
l
7 6
m
23 6
n
19 8
o
31 10
5 Pizzas were each cut into 12 pieces for a party. After the party there were 17 pieces left. How many pizzas were left? 6 Apples were cut into quarters for children to eat at recess time. At the end of recess 15 quarters remained. How many apples was that?
Activity
Shading fractions What to do: 1 Fraction patterns: a What fraction of the largest square on the left is the shaded pattern?
PRINTABLE GRID
b
Make 3 identical copies double the size of the grid on the right.
c
In the first copy, shade the squares to make symmetrical patterns so that 12 of the largest square is shaded. In the second shade symmetrical so that 14 of the largest square is shaded. Shade symmetrical patterns in the third grid.
d
2 This set of diagrams shows just three different ways of shading 1 3 of a nine square grid. a Investigate how many other different ways it is possible to shade 13 of this grid.
b Investigate how many different ways it is possible to shade
2 9
of this grid.
60
FRACTIONS (CHAPTER 4)
Addition of fractions
Unit 26 Addition using diagrams
Fractions can only be added when they have the same denominator (bottom number). Examples:
=
+
= = =
+
=
= =
First write these fractions with the same denominator .
5 7 8 + 8 12 8 1 48 1 12 1 1 2 + 5 5 2 10 + 10 7 10
Exercise 26 1 Use diagrams if necessary to find a
1 5
+
2 5
b
1 8
+
3 8
c
1 4
+
3 4
d
3 10
+
1 10
e
2 7
+
4 7
f
3 5
+
2 5
g
4 9
+
1 9
h
5 12
+
7 12
i
1 14 +
j
2 3
+
2 3
k
3 4
+
3 4
l
3 10
+
9 10
m
5 8
n
2 13 +
o
5 12
p
1 12 + 1 12
1 4
7 8
+
2 3
+
11 12
1 2
+
3 8
2 5
+
1 2
1 4
+
1 3
2 Copy and shade the diagrams. Then complete the addition.
So,
a
b
+
=
+
=
= = So, = = So,
c
= =
=
+
PRINTABLE TEMPLATE
3 Draw and shade to show the following, then complete the addition. 1 4
=
1 8 1 8 + 8
=
8
=
1 10 1 10
a
e
= =
+
10
+
2 5
+
10
2 3
=
1 6 1 6 + 6
=
6
=
3 14 3 14
b
f
+
=
14
=
7
+
1 2
+
14
1 2
=
5 12 5 12 + 12
=
12
c
+
3 4
=
3 8 3 8 + 8
=
8
d
+
=
Print the worksheet by clicking on the icon. This worksheet has additions of fractions from circle diagrams.
PRINTABLE WORKSHEET
FRACTIONS (CHAPTER 4)
Fractions can only be added if they have the same denominator. Always write answers in lowest terms. Write¡ Ry_ as¡ We_\.
Examples: 1 2
²
4 Find a
+
1 6
+ 2 23
=
11 12
+
8 3
=
11 12
+
8£4 3£4
+
32 12
=
1£3 2£3
+
=
3 6
1 6
=
4 6
=
11 12
=
4¥2 6¥2
=
43 12
=
2 3
7 = 3 12
+
1 6
1 6
+
1 2
b
1 4
+
1 2
c
1 10
e
2 3
+
1 6
f
1 4
+
5 8
g
1 6
i
2 5
+
3 10
j
3 14
k
5 12
1 2
+
3 4
b
2 3
+
1 9
c
3 4
e
2 3
+
8 21
f
7 8
+
1 4
g
i
7 10
2 5
j
9 16
+
7 8
k
5 Find a
6 Find a
+
2 7
+
1 5
d
3 10
7 12
h
2 9
+
1 18
2 3
l
5 9
+
5 27
+
1 20
d
1 4
+
7 24
4 5
+
7 15
h
3 4
+
5 8
5 6
+
11 24
l
9 10
+
+
+
+
+
3 5
7 20
1 4
+
1 8
b
1 3
+
1 6
c
1 2
+
3 8
d
1 5
+
3 10
e
1 3
+
2 9
f
3 4
+
1 8
g
1 2
+
7 8
h
3 4
+
5 8
i
5 6
+
2 3
j
3 10
4 5
k
8 15
3 5
l
3 5
+
7 10
c
1 35 +
1 10
d
1 57 +
+
7 Find a 1 25 +
3 10
b
2 38 +
7 24
e
3 25 +
17 25
f
2 45 +
3 20
a
Add 1 23 and 29 .
8
11 12
²
b
+
Find the sum of 2 12 and 34 .
9 Wai-king, William and Ron went hiking. a If William carried the tent 14 of the way and Ron carried it 58 of the way, what fraction of the trip did they carry the tent between them? b Wai-king carried the tent the rest of the way. What fraction of the trip did Wai-king carry the tent? 10 Kelly, Michael and Kim dug a trench. a If Michael dug 13 of the trench and Kelly dug 16 , what fraction of the trench did they dig between them? b What fraction of the trench did Kim dig?
5 28
61
62
FRACTIONS (CHAPTER 4)
Unit 27
Subtracting fractions and problem solving
Subtracting fractions is done in the same way we use to add them, except that instead of adding the numerators, we subtract them. So, we ² ²
make the denominators the same then subtract the new numerators.
Examples:
2 3
²
¡
1 6
=
2£2 3£2
¡
=
4 6
1 6
=
3 ¡ 1 13
² =
3£3 1£3
¡
=
9 3
4 3
3 6
=
5 3
=
3¥3 6¥3
= 1 23
=
1 2
¡
Remember to convert a mixed number to an improper fraction before subtracting.
1 6
¡
4 3
Exercise 27 1 Find a
7 8
b
2 3
¡
1 12
c
2 3
¡
5 9
d
3 4
¡
3 8
e
7 10
¡
3 5
f
5 9
¡
1 3
g
5 6
¡
2 3
h
3 5
¡
1 10
i
11 14
¡
2 7
j
5 6
¡
1 2
k
2 3
¡
5 12
l
3 4
¡
3 16
m
5 6
¡
5 18
n
11 21
o
16 21
2 7
p
7 8
¡
3 4
q
7 8
¡
3 16
r
5 9
s
31 100
t
37 100
u
59 100
v
7 25
w
7 12
x
11 24
¡
1 2
¡
7 20
1 3
¡
¡
1 3
¡
3 50
¡
¡
¡
1 5
1 4
¡
3 10
1 6
¡
2 Find
3
5 8
b
1 14 ¡
1 2
c
2 18 ¡
3 4
d
3 ¡ 1 18
f
2 15 ¡
7 10
g
3 12 ¡
5 8
h
1 59 ¡
Subtract 3 12 from 5.
b
Find 3 15 minus
How much larger than 2 is 4 14 ?
d
Find the difference between 2 14 and
a
1¡
e
1 34 ¡
a c
1 8
7 10 .
4 Find the difference in weight between two parcels weighing 2 14 kg and 3 kg. 5 Ella tipped 3 buckets of water into an empty barrel, then Taylor took 1 58 buckets of water out of the barrel. How many buckets of water remained in the barrel? 6 Carly bought 1 12 kg of potatoes. She cooked dinner. What weight of potatoes was left? 7 Simon’s trailer was
11 12
3 4
of a kg for
full of manure when he left the farm.
When he arrived home the trailer was only 13 full. What fraction of a trailer load did Simon lose on the way home?
2 3
11 12 .
FRACTIONS (CHAPTER 4)
63
Problem solving 8 Two thirds of the offices in a property had been rented. If there were 60 offices in the property, how many were vacant? 9 Marek had to travel 156 km in his car. He stopped to buy fuel when he had travelled How far did he still have to travel?
3 4
of the journey.
10 Brett and Emily each carried 5 litre buckets with water in 3 them. Brett’s bucket was 35 full and Emily’s bucket was 10 full. Would the water all fit in one bucket? Show your working. 11 Sarah spent 25 of her pay cheque on rent. If her rent was $80, how much was she paid? 12 During the football season, Alex scored 14 of the goals and Con scored 18 of the goals for their team. What fraction of the goals did they score between them? 13 Karen was paid a bonus equal to one tenth of her salary. If her salary was $18 600 per year, a how much was the bonus b how much money did she receive that year?
A bonus is extra pay.
14 The minute hand of a clock moves through 5 minutes. a What fraction of an hour has it turned through? b What fraction of a revolution has it turned through? c How many degrees has it turned through? 15 Cindy picked 6 kg of apples. She used 3 12 kg for cooking
and
3 4
kg for eating. The remainder were fed to her horse.
a What weight of apples did Cindy use? b What weight of apples did the horse eat? 16
a Uncle Tom gave his 5 nephews $60 to share equally between them. i What fraction of the money did each receive? ii How much did each receive? b If Uncle Tom had given his 5 nephews $60 with the instruction that the eldest nephew was to receive twice as much as each of the others, i what fraction would the eldest receive ii what fraction would each of the others receive iii how much would the eldest receive?
[Hint: How many equal parts are you dividing $60 into now?] 17 Suzi and Dien each had identical chocolate bars. If Suzi ate 23 of hers and Dien ate of one chocolate bar remained if they put the uneaten portions together?
4 9
of his, what fraction
18 Pieta was hiking between the alpine villages A and B. When she had travelled 59 of the distance between A and B, bad weather caused her to return to the half-way hut. a What fraction of the distance between A and B did Pieta walk when she returned to the hut? b On the following day, Pieta continued her hike. When Pieta reached B, what fraction of the distance between A and B had she travelled whilst hiking?
64
FRACTIONS (CHAPTER 4)
Unit 28
Ratio A ratio is used to compare two like quantities.
For example, if John is twice as heavy as Paul, we say that the ratio of John’s weight : Paul’s weight is 2 : 1, and we read this as 2 is to 1. ² ²
Notice that order is important. The ratio of Paul’s weight : John’s weight is 1 : 2. The ratio is written without units.
To write as a ratio we must convert both quantities to the same units. Examples: ²
10 seconds is to 1 minute = 10 seconds is to 60 seconds fas secondsg = 10 : 60
²
69 cents is to $3 = 69 cents is to 300 cents fas centsg = 69 : 300
To simplify a ratio we divide each number by their highest common factor. For example:
6 : 15 = 6 ¥ 3 : 15 ¥ 3 =2:5
Exercise 28 1 Express as a ratio without simplifying your answer a 23 cents is to 60 cents b 2 km is to 8 km d 7 years is to 14 years e 7 cm is to 10 cm
c f
3 kg is to 5 kg $1 is to $10
2 Express both quantities in the same units. Then write as a ratio without simplifying your answer: a 5 days is to a week b 3 months is to 1 year c 75 cm is to 2 m d 300 grams is to 1 kg e 60 cents is to $2:40 f $1:10 is to $2:20 3 Write in simplest form a 4:2 b f 5 : 10 g
2:6 15 : 25
c h
8:6 24 : 36
d i
4 We can write ‘10 cents for Charity for every dollars spent’ as the ratio money for Charity : money spent = 10 : 100 = 1 : 10.
Write as a ratio in simplest form a 4 litres of red paint are mixed with 1 litre of yellow paint. b There are 6 men for every 4 women on a committee. c There are 10 girls for every 2 boys in the choir. d George obtained 32 marks and Lisa obtained 36 marks in a test. e Kiri is 1 m tall and Ray is 1 m 20 cm tall. f The apple cost 60 cents and the chocolate bar cost $1:20 g The jug contains 50 mL of cordial and 1 litre of water.
9:3 60 : 100
e j
3 : 15 85 : 20
Remember to convert to the same units before writing as a ratio.
FRACTIONS (CHAPTER 4)
65
Examples: ² The ratio of black sheep to white sheep in a flock was 1 : 15. If there were 640 sheep in the flock, we can find how many of them were white, using this method:
² The ratio of footballers to hockey players in a class of boys was 8 : 3. If there were 6 hockey players, we can find how many boys played football in this way: footballers : hockey players =8:3 = 8£2 :3£2 f6 hockey playersg = 16 : 6
black sheep : white sheep = 1 : 15 [For every 1 part of the flock that is black, 15 parts of the flock are white.] The flock has 16 equal parts f1 + 15 = 16g 640 ¥ 16 = 40 so one part is 40 sheep:
) there were 16 boys who played football.
15 parts is 15 £ 40 = 600 (or 640 ¡ 40 = 600) ) there are 600 white sheep.
5 When dividing a quantity into these ratios, what is the total number of equal parts in the whole? a 1:2 b 1:3 c 2:3 d 5:6 e 20 : 1 f 1:1 g 4:3 h 100 : 1 i 3:7 j 9:4 6 $200 is divided in the ratio 5 : 3.
Find the larger share.
7 The ratio of girls to boys at Pony Club was 6 : 1. If 35 children attended Pony Club, how many were boys? 8 In a group of office workers, the ratio of coffee drinkers to tea drinkers is 3 : 2. If there were 25 office workers, how many drank coffee? 9 In a class of 24 children, the ratio of children who liked sailing to children who did not like sailing was 5 : 3. How many children liked sailing? 10 In a restaurant the ratio of people who ordered steak to people who ordered fish for their main course was 8 : 5. If 143 people dined at the restaurant, how many ordered fish? 11 A length of string was cut in two pieces in the ratio 4 : 3. If the longer piece was 72 cm, what was the length of the shorter piece? 12 A load was divided in the ratio 10 : 3 to be carted by a truck and a small van. If the weight of the smaller load was 420 kg, find the weight to be carried by the truck. 13 A supermarket sold 3 cans of dog food for every 2 cans of cat food. If 540 cans of dog food were sold in one week, how many cans of cat food were sold? 14 In his will, Samuel left money to be shared in the ratio 5 : 6 between Maria and Ivan. If Maria received $12 500, how much did Ivan receive?
66
FRACTIONS (CHAPTER 4)
Unit 29
Review of chapter 4
Review set 4A 1 Express the fraction two thirds a as a shaded region on a diagram c in symbols
b
on a number line
2 What fraction of the following diagrams is shaded? a
b
3 What fraction of the square is the shaded triangle?
4 The school canteen ordered 50 bottles of orange juice, 20 bottles of apple juice and 5 bottles of pineapple juice. a What fraction of the juice was apple? (Write your answer in lowest terms.) b What fraction of the juice was not pineapple? 5 Find
a
1 3
of 24
2 9
b
of 54
4 7
c
of 1 fortnight (answer in days)
6 If one quarter of my pocket money is $5, how much pocket money do I receive? 7 Write 3 34 as an improper fraction. 8 Copy and complete
2 3
a
=
¤ 15
b
45 54
9 Write as a whole number or a mixed number
= a
5 ¤ 15 5
b
16 3
10 Kristy rode her bicycle 2 km to school. Emma only had to ride one quarter of that distance. How far in metres did Emma ride? 11 When it is full, the petrol tank on my car holds 70 L. How many litres does it hold if it is 12 Find
a
1 4
+
3 8
13 Find
a
1¡
7 9
b
1 23
b
3 8
+
¡
2 9
1 4
14 A father and his young son went hiking. They divided the load to be carried in their back-packs in the ratio 9 : 2. a If the son carried 4 kg, what weight did his father carry? b What weight did they carry between them? 15 Jason said that he could eat 58 of a pizza and Sue said that she would like a Did they need to order more than 1 pizza? b What fraction of a pizza was left over?
1 2
of a pizza.
3 5
full?
FRACTIONS (CHAPTER 4)
67
Review set 4B 1 Write in words and symbols the fractions represented by the following: a (shaded part) b 2
3
2 24 sausages, 16 patties and 20 chops were barbequed. a What fraction of the pieces of meat were chops? (Answer in lowest terms.) b What fraction of the pieces of meat were not patties? (Answer in lowest terms.) 3 Copy and complete 4
a
Write
5 Find
a
4 9 1 8
a
3 25
=
¤ 100
b
25 75
=
with denominator 18. of $64
b
3 5
1 ¤
b
Write
3 5
with numerator 21.
of 65
6 The depth of water in a tank is 33 cm when the tank is 13 full. What would be the depth of water in the tank if it was full? 7 Two sevenths of the children in the class were absent with measles. If there were 28 children in the class, how many were at school? 8 Write as a whole number or a mixed number
a
17 11
b
26 2
9 Write 1 49 as an improper fraction. 10 Richard took one ninth of his alpacas to the Mount Pleasant Agricultural Show. If he took 8 alpacas to the Show, how many did he own? 11 Find
2 3
+
7 12
12 Joe and Sara were each given pieces of cake. Joe was given 14 of the cake and Sara was given 38 . a What fraction of the cake was eaten? b What fraction of the cake was left? 13 Find
a
2¡
3 5
b
1 38 ¡
3 4
14 The ratio of boys to girls in a class is 5 : 6. If there are 22 students in the class, how many are girls? 15 The ratio of red paint mixed with white paint to make a particular shade of pink is 1 : 4 by volume. a i If 20 litres of white paint is used, how much red paint must be added? ii How much pink paint is made? b If 40 litres of pink paint was made, how much red paint was used? 16 A business hired a truck to transport boxes of equipment. The total weight of the equipment was 2 tonnes, but the truck could only carry 58 of the boxes in one load. a What weight did the truck carry in the first load? Remember 1 tonne = 1000 kg. b If there were 72 boxes, how many did the truck carry in the first load?
68
DECIMALS
Unit 30
Representing decimals
We use fractions or decimal numbers to represent parts of a whole. In a decimal number, the decimal point separates the whole number part from the fraction part.
units
decimal point
1 10
0:1 means one tenth or
0:1
0:01 means one hundredth or
tenths
0:001 means one thousandth or
1 100 1 1000
CHAPTER 5
Look at the number 104:523 hundreds
tens
ones
decimal
tenths
hundredths
thousandths
(100s)
(10s)
(1s)
point
1 ( 10 s)
1 ( 100 s)
1 ( 1000 s)
1
0
4
:
5
2
3
The 0 in the tens place shows there are zero tens.
The number 104:523 is made up of one hundred, 4 ones, 5 tenths, 2 hundredths, and 3 thousandths. Some decimal numbers can be represented using grids. For example:
This grid contains 100 squares. 1 of the large square. So, each small square is 100 12 out of 100 squares are coloured. 12 The fraction coloured is 100 which, as a decimal is 0:12 PRINTABLE WORKSHEET
Exercise 30 1 The grid has 100 squares. If 5 squares out of 100 are coloured, this represents 0:05 or Colour the squares to represent the decimal number given. a b c d
0.23 (23 out of 100)
0.03
2 What decimal number is represented by i the coloured squares a
b
0.70
c
.
0.59
the uncoloured squares?
ii
5 100
d
DECIMALS (CHAPTER 5)
69
Grids can also represent tenths. 1 2 3 4 5 6 7 8 9 10
For example:
There are 10 hundredths in one tenth. In the figure, 10 squares out of 100 are coloured. 10 1 This is 100 or 10 fas a fractiong
0:10 or 0:1 fas a decimalg 10 hundredths = 1 tenth 3 Colour the squares to represent the decimal numbers given. a b c
0.4 0.9 4 What decimal number represents the coloured squares in i tenths ii hundredths? a
b
3 10
..... means
e
0:5 means ..... or ..... tenths
g
..... means
7 10
or three tenths or ..... tenths
6 Copy and complete the following a 0:01 means ..... or one hundredth 3 100
0.5
0.3
c
5 Copy and complete the following a 0:1 means ..... or one tenth c
d
d
2 10
b
0:2 means
or ..... tenths
d
0:4 means ..... or four tenths
f
0:6 means ..... or ..... tenths
h
..... means ..... or eight tenths
b
0:02 means
2 100
or ..... hundredths
c
..... means
or three hundredths
d
0:04 means ..... or four hundredths
e
0:05 means ..... or ..... hundredths
f
0:06 means ..... or ...... hundredths
g
..... means ..... or seven hundredths
h
..... means
i
0:09 means ..... or ..... hundredths
8 100
or ..... hundredths qA_p_
7
1
a If 1
represents a whole unit, what decimal is represented by i
1
b Write down the decimal number represented by: i
1
ii
qA_p_
iii
1
ii qA_p_
1
iii qA_p_
qA_p_
qA_p_
?
70
DECIMALS (CHAPTER 5)
Unit 31 Using a number line and the value of money Showing tenths Each division on this number 1 or 0:1 line represents 10
0.1 0
C
0.3
A 0.2 B 0.4
0.6
D
0.8
1
E
1.2
1.4
1.6
F
1.8
2
2.2
2.4
The decimal value of A is 0:1, B is 0:3, C is 0:8, D is 1:3, E is 1:9 and F is 2:3
Showing hundredths Each division on this number line is one hundredth or 0:01 of the whole. 0
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2
qA_p_ or qA_p:_p_
We do not usually write the number of every division on the number line.
qS_p_ or qS_p:_p_
A
2.4
2.5
B
C
2.6
D
On this number line A is 2:43, B is 2:51, C is 2:57 and D is 2:62
Exercise 31 1 Write down the value of the number at N on these number lines. a b 0
c
N
6
1
N
2
d
7
N
3
21
N
22
2 Copy the number lines given and mark these numbers on them. a A = 1:6,
B = 2:5, C = 2:9, D = 4:1
0
1
2
3
4
5
b E = 13:7, F = 14:2, G = 15:3, H = 16:5 13
14
15
16
17
3 Write down the value of the number at N on these number lines. N a b 0
0.2
0.1
0.3
N
c 1.6
0
0.1
0.2
3.4
3.5
0.3
N
d 1.8
1.7
N
1.9
3.6
3.7
4 Copy the number lines given and mark the following numbers on them. a A = 4:61, 4.6
B = 4:78, C = 4:83, D = 4:97 4.7
4.8
4.9
b E = 10:35, F = 10:46, G = 10:62, 10.3
5
a
c
10.4
Read the temperature on the thermometer shown. Read the length of Christina’s skirt from the tape measure.
5.0
H = 10:79
10.5
10.6
33
5.1
34
35
36
37
10.7
38
39
40
°C
10.8
b How much milk is in the jug? 1 litre 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
DECIMALS (CHAPTER 5)
The value of money $12:65 is read as “12 dollars and 65 cents”. The two digits after the decimal point show the cents. We would write $9:60 rather than $9:6 as $9:60 shows 9 dollars and 60 cents. $1 or 1 dollar equals 100 cents.
30 cents is zero dollars and 30 cents = $0:30 205 cents is 200 cents + 5 cents = $2:05
Examples: ²
²
0 dollars and 20 cents
² 10 dollars and 50 cents
7 dollars and 15 cents.
As decimals of a dollar the amounts are: $0:20
$10:50
$7:15
6 Write these amounts of money as decimals of one dollar. a b
c
d
e
f
7 Draw sketches of money to represent these amounts, using the smallest number of notes and coins.
You may use sketches like 20c , a
55 cents
b
$1:05
c
$5
, etc.
$3:10
d
$6:45
e
$14:00
8 Write each amount as dollars using a decimal point. a 36 cents b 5 cents d 99 cents e 4080 cents g twenty one dollars h two dollars ninety eight cents
c f i
$27:40
f
200 cents 710 cents sixty dollars five cents
9 Write each amount as dollars and cents. (3:8 dollars means $3:80) a 6:5 dollars b 14:1 dollars c 4:2 dollars d 2:7 dollars e 10:6 dollars f 0:9 dollars 10 Write these dollar amounts as cents. a $3:25 b $8:05 11 Write these amounts as dollars and cents. a 100 cents b 300 cents e 37 cents f 10 cents
c
$0:95
d
c g
$0:05
1004 cents 452 cents
e
d h
$152:45
2450 cents 10 000 cents
71
72
DECIMALS (CHAPTER 5)
Place value
Unit 32 We can write decimal numbers on a place value chart. number 503:082
hundreds 5
tens 0
ones 3
. :
tenths 0
hundredths 8
thousandths 2
In words, 503:082 is five hundred and three point zero eight two or five hundred and three and eighty two thousandths. In expanded decimal form,
503:082 = 500 + 3 + 0:08 + 0:002 0 8 2 + 100 + 1000 which is 500 + 3 + 10
There are no tenths shown. We must show that with a zero, 0.
Reminder: ²
If a decimal number is less than 1, we put a zero before the decimal point. 76 is written 0:76, not :76 So, 100
²
For whole numbers we do not include a decimal point. For example, twenty three is 23 and not 23:0
²
22:06 = 22 +
0 10
+
6 100 .
The 0 holds a place value and must not be left out.
Exercise 32 1 Copy and complete the place value table.
a b c d e f g h
number 3:6 50:6 231:4 26:52 285:21 688:02 60:862 100:05
hundreds
tens
ones
. . . . . . . . .
tenths
hundredths
PRINTABLE TEMPLATE
thousandths
2 Use your answers in question 1 to help you write these numbers in words. a 3:6 b 50:6 c 231:4 d 26:52 f 688:02 g 60:862 h 100:05 i 10:005 3 Write in expanded decimal form. For example, 2:31 = 2 + 0:3 + 0:01 a 3:01 b 0:68 c 54:361 e 603:2 f 106:4 g 30:402
e j
285:21 704:407
d h
50:004 100:101
d
4 10
4 Write these expanded numbers in decimal form. a
4+
1 10
+
3 100
b
3+
5 10
c
2+
2 100
+
3 1000
+
3 100
+
5 State the value of the digit 1 in the following a 16:5 b 0:01 c 167:432
d
3:15
e
0:471
6 State the value of the digit 5 in the following a 0:5 b 15:47 c 2:35
d
52:71
e
439:225
7 State the value of the digit 7 in the following a 0:73 b 21:607 c 7:032
d
6:372
e
173:62
6 1000
DECIMALS (CHAPTER 5)
Note:
fifty three and two tenths
c
six hundred and twenty three hundredths
42 1000
= 0:0 4 2
6
thousandths
b
hundredths
one and nine thousandths
tenths
a
dec. point
Number (in words)
Written numeral
1
:
0
0
9
1:009
5
3
:
2
0
0
:
2
tens units
hundreds
This table shows decimal numbers in words and numerals, and their place values.
3
600:23
That is, in 0:042 the value of the digit 4 is
2 thousandths
the value of the digit 2 is
4 hundredths 8 Use a c e g i
53:2
4 100
and
2 1000 .
a place value table to write the following in decimal form. seven tenths b three hundredths twenty one hundredths d thirty nine thousandths one hundred decimal five thousandths f seven and one tenth twenty and one hundredth h forty six and thirty one hundredths two hundreds and three tenths j five hundred and thirty four and two tenths
9 Write the following in decimal form a
4 10
b
3 10
c
7 10
d
15 100
e
24 100
f
8 100
g
209 1000
h
386 1000
i
27 1000
j
6 1000
k
2 1 10
l
13 2 100
m
6 7 100
n
56 2 1000
o
177 4 1000
To compare two or more decimal numbers we can write them under one another with the decimal points lining up. For example 16:347 As the 4 is larger, 16:426 > 16:347 16:426 10 Which is the larger number? a 22:463, 25:983 d 15:767, 15:788
b e
1:62, 1:75 6:999, 6:913
c f
0:487, 0:463 24:096, 24:09
11 Replace ¤ by =, > or < in a 0:83 ¤ 0:38 d 0:49 ¤ 0:94 g 0:325 ¤ 0:33
b e h
0:40 ¤ 0:4 0:213 ¤ 0:231 0:672 ¤ 0:627
c f i
0:084 ¤ 0:08 0:030 ¤ 0:03 0:9909 ¤ 0:999
c f
0:68, 0:74, 0:80, .... 9:8, 8:9, 8:0, ....
12 What is the next decimal number in these sequences? a 0:5, 1:0, 1:5, .... b 1:6, 1:9, 2:2, .... d 8:4, 8:2, 8:0, .... e 20:2, 18:8, 17:4, ....
13 Arrange these decimals in ascending order of size. That is, begin with the smallest. a $138:50, $130:85, $135:80, $108:35, $130:58, $103:85 b 4:803 km, 4:083 km, 4:38 km, 4:308 km, 4:83 km, 4:038 km c 10:89 sec, 10:908 sec, 10:98 sec, 10:098 sec, 10:089 sec, 10:809 sec 14 Arrange these decimals in descending order. That is, begin with the largest. a 0:25, 0:52, 0:205, 0:502 b 0:707, 0:770, 0:077 c 8:312, 8:123, 8:213, 8:321, 8:132, 8:231
73
74
DECIMALS (CHAPTER 5)
Unit 33
Adding and subtracting decimal numbers
When adding or subtracting two decimal numbers we write the numbers underneath each other and line up the decimal points. We add as normal and place the decimal point underneath the others in the answer. Examples of addition ² 15:71 + 4:93
²
105:27 + 9:361
15:71 + 4:93
105:270 + 9:361
20:64
114:631
1
1
²
49 + 3:25
fill in the zero
1
fill in with 2 zeros
49:00 + 3:25 1
52:25 DEMO
Exercise 33 1 Copy, fill in the zeros where needed and add these numbers: a 0:44 b 18:2 c + 1:3 + 4:8 + 2 Find a 0:2 + 0:7 e 2 + 0:005 i 4:37 + 18:59
b f j
1:3 + 2 2:6 + 2:8 1:04 + 0:777
c g k
3:9 14:26
d
4 + 6:9 0:2 + 23:65 26:531 + 18:402
d h l
142:1 + 9:87
0:04 + 0:007 6:36 + 0:04 123:841 + 22:976
3 Add these by writing in columns, lining up the decimal points: a 4:07, 3:95 and 2:64 b 11:02, 4:7 and 25:96 c 20, 2, 0:2 and 0:02 d 70:7, 7:07 and 0:707 4 Add a
d
3:93 6:71 + 4:01
b
76:410 93:321 + 6:214
c
607:125 46:204 + 9:005
25:461 4:325 + 19:113
e
4:210 54:314 + 9:123
f
12:020 9:061 + 14:115
5 Calculate: a $1:45 + $3:50 d 2:6 cm + 13:53 cm g 68 cents + $1:99 + $32 6
b e h
$4:93 + $20:05 29 cm + 6:38 cm $36 + $8:95 + 25 cents
c f i
$1:09 + $16 9:58 km + 17:5 km $26:40 + $2:55 + 5 cents
a At the shop I buy a jacket costing $128:50 and a pair of shoes costing $83:00. How much do I pay altogether? b I recently ran a 400 m race. I ran the first 100 m in 12:8 sec, the second in 13:4 sec, the third in 14:1 sec and the last 100 m in 10:9 sec. How long did it take me to run the whole 400 m? c The first hit of the golf ball went 251:5 m. My next shot went 83:8 m and my final putt went 4:8 m. How far did I hit the golf ball in total with my three shots? d I found three pieces of timber in the back yard and placed them end to end. If the pieces of timber measured 6:4 m, 3:85 m and 2:73 m, how far did they stretch when placed end to end? e When Jeremy went fishing on the weekend he was very successful. He managed to catch four trout weighing 6:4 kg, 3:27 kg, 5:83 kg and 6:28 kg. How much did his four trout weigh altogether?
EXTRA PRACTICE
DECIMALS (CHAPTER 5)
Examples of subtraction ²
DEMO
15 ¡ 4:93 9 1
15:00 ¡ 4:93 10:07
²
105:278 ¡ 9:36 9 4
fill in the zeros
1
105:278 ¡ 9:360 95:918
fill in the zero
7 Copy and fill in the zeros where needed and then subtract: a 4:63 b 2:04 c 16:7 ¡ 2:4 ¡ 0:7 ¡ 4:33 8 Find a d g j 9
0:7 ¡ 0:2 2:8 ¡ 2:6 2 ¡ 0:005 1:04 ¡ 0:777
b e h k
2 ¡ 1:3 6:36 ¡ 0:04 18:59 ¡ 4:37 26 ¡ 18:402
Find the difference between 16:3 and 5:82 What is 14:2 less than 83?
a c
b d
d
c f i l
48 ¡ 6:37
6:9 ¡ 3 23:65 ¡ 0:2 0:04 ¡ 0:007 123:841 ¡ 22:976
What is 2:4 less than 6:9? How much more is 8:4 than 8:301?
10 Subtract a 7:07 from 70:07 d 11:02 from 25:96
b e
0:02 from 2 $9:46 from $23:07
c f
3:95 from 4:07 $44:25 from $106:72
11 Find a d g j
b e h k
$6:92 ¡ $4:05 $60 ¡ $18:28 8:32 km ¡ 0:4 km $39:60 ¡ $4:85
c f i l
$35:08 ¡ $1:99 2:4 cm ¡ 1:6 cm 0:5 mL ¡ 0:355 mL $7:75 ¡ 80 cents
$2:65 ¡ $1:55 $420 ¡ $8:65 14:28 km ¡ 9:5 km $1:95 ¡ 65 cents
12 Find the missing digits in these subtractions to make them correct: a 25:24¤ b ¤6:¤2 ¡ 3:6¤3 ¡ 14:5¤ ¤¤:¤92 3¤:35
c
¤:¤¤¤ ¡ 1:6 0 8 2:7 1 7
13 Find the difference between a thirty point seven and twenty nine point four b fifty three point seven and forty two point zero eight c one hundred and fifteen point four five and thirty point two zero five d seven tenths and two hundredths e thirty nine dollars ten cents and twenty two dollars twenty five cents. 14
75
EXTRA PRACTICE
a Enis used to weigh 72:4 kg but after an illness she lost 9:7 kg. How much does she now weigh? b The temperature at 3 pm was 27:5o C but it had dropped 6:8o C by 6 pm. What was the temperature at 6 pm? c A truck weighed 3:27 tonnes at a police check point. The driver was told that this was overweight by 0:58 tonnes. How much should the truck have weighed to be legally correct? d A $32:60 sweat shirt is reduced by $9:35. How much does it now cost? e To join an elite police unit I have to be 185:4 cm tall. If I am now 163:9 cm tall, how much do I need to grow to join the unit? f The temperatures during a week were
Mon 32:3o
Tue 30:4o
Wed 30:2o
Thu 28:6o
What is the difference between the highest and lowest temperatures?
Fri 29:5o
Sat 30:6o
Sun 32:0o
76
DECIMALS (CHAPTER 5)
Unit 34
Multiplying & dividing by 10, 100, 1000
Rules for multiplying by 10, 100 and 1000 ² ² ² Examples:
When multiplying by 10 we move the decimal point one place to the right. When multiplying by 100 we move the decimal point two places to the right. When multiplying by 1000 we move the decimal point three places to the right. ²
0:14 £ 10 = 0:14 £ 10 = 1:4
²
70:8 £ 100 = 70:80 £ 100 = 7080
²
0:007 £ 1000 = 0:007 £ 1000 =7
Exercise 34
DEMO
1 Multiply the numbers a 2 £ 10 d 0:01 £ 10 g 6 £ 100 j 0:54 £ 100 m 7 £ 1000 p 0:38 £ 1000 2 Multiply the numbers to complete the table.
3 Complete the equation a 6 £ ¤ = 600 d 0:02 £ ¤ = 2
b e h k n q
6:3 £ 10 54 £ 10 9:2 £ 100 45 £ 100 6:2 £ 1000 6:75 £ 1000
c f i l o r
£10
£100
a b c d e
Number 0:009 0:12 0:5 4:6 19:07
b e
33 £ ¤ = 330 0:003 £ ¤ = 0:03
0:2 £ 10 60:6 £ 10 0:7 £ 100 70:4 £ 100 0:7 £ 1000 3:067 £ 1000
£1000
c f
3:4 £ ¤ = 34 5:64 £ ¤ = 5640
4 A bus ticket costs $2:25. Find the total cost of a 10 bus tickets b 100 bus tickets
c
1000 bus tickets
5 An icecream costs $1:80. Find the cost of a 10 icecreams b
c
1000 icecreams
6 A bicycle helmet costs $45:60. Find the cost of a 10 helmets b 100 helmets
c
1000 helmets
7 A pencil costs $0:30. Find the cost of a 10 pencils
c
1000 pencils
8 Find the cost of a 10 footballs at $26:30 each c 1000 ribbons at $1:50 each 9
b
100 icecreams
100 pencils b d
100 bottles of lemonade at $2:36 each 100 blocks of chocolate at $1:25 each
a Multiply 0:26 by 10, then multiply your answer by 10. b Multiply 0:26 by 100. c Compare your results in a and b. Copy and complete:
Multiplying by 10 and multiplying by 10 again is the same as multiplying by ..........
DECIMALS (CHAPTER 5)
Rules for dividing by 10, 100 and 1000 ² ² ²
When dividing by 10 we move the decimal point one place to the left. When dividing by 100 we move the decimal point two places to the left. When dividing by 1000 we move the decimal point three places to the left.
Examples: ²
0:8 ¥ 10
²
= 0:8 ¥ 10 = 0:08
²
= 06:1 ¥ 100 = 0:061
10 Divide the following numbers a 2 ¥ 10 d 0:01 ¥ 10 g 6 ¥ 100 j 50 ¥ 100 m 7 ¥ 1000 p 499 ¥ 1000 11 Divide the numbers to complete the table.
6:1 ¥ 100
b e h k n q
a b c d e
= 060:9 ¥ 1000 = 0:0609
6:3 ¥ 10 54:02 ¥ 10 9:2 ¥ 100 166 ¥ 100 6:2 ¥ 1000 701 ¥ 1000
Number 8 4:6 50 19:07 231:4
¥10
60:9 ¥ 1000
0:2 ¥ 10 606 ¥ 10 0:7 ¥ 100 300:7 ¥ 100 56:1 ¥ 1000 6854:9 ¥ 1000
c f i l o r
¥100
12 Write the divisor to complete the equation a 6 ¥ ¤ = 0:6 b 33 ¥ ¤ = 0:33 d 0:2 ¥ ¤ = 0:002 e 49 ¥ ¤ = 0:49
¥1000
DEMO
c f
3:4 ¥ ¤ = 0:34 634:1 ¥ ¤ = 0:6341
15 1000 exercise books cost $3200. Find the cost of a 100 books b 10 books
c
1 book
16 1000 tap washers cost $50. Find the cost of a 100 washers b 10 washers
c
1 washer
13 100 chocolate frogs cost $55. Find the cost of a 10 chocolate frogs b 1 chocolate frog 14 100 bottles of soft drink cost $125. Find the cost of a 10 bottles b 1 bottle
17 Replace ¤ with the correct symbol from >, <, =. a 8:2 £ 10 ¤ 820 ¥ 10 c 0:014 £ 100 ¤ 1:4 ¥ 10 e 26:8 ¥ 100 ¤ 2:68 £ 10 18
b d f
0:62 £ 100 ¤ 620 ¥ 100 75:61 ¥ 1000 ¤ 7:561 ¥ 100 0:07 £ 1000 ¤ 700 ¥ 10
a Divide 82 by 100, then divide your result by 10. b Divide 82 by 1000. c Compare your results in a and b. Copy and complete:
Dividing by 100 then dividing by 10 is the same as dividing by ......
77
78
DECIMALS (CHAPTER 5)
Unit 35
Conversions and percentage
Fractions to decimals Examples:
Method 1: Write the given fraction as a fraction with 10, 100 or 1000 as its denominator, then convert it to a decimal number.
3 5
²
7 20
²
=
3£2 5£2
=
7£5 20£5
=
6 10
=
35 100
= 0:6
= 0:35
Method 2:
Write the numerator as a decimal number and then divide numerator by denominator. ² To find
5 8
as a decimal, we need to find 5 ¥ 8. Write 5 as 5:0 with as many extra 0s as you need. Divide in the same way that you would divide if the decimal point was not there.
0:6 2 5 5 : 0 20 40
8
5 8
Line up the decimal points. So,
= 0:625
Exercise 35 1 Write as a fraction with a power of 10 in the denominator, then convert to a decimal. a g
1 2 7 20
b h
1 5 14 25
c i
1 4 3 8
2 5 3 50
d j
e k
3 4 9 20
4 5 49 500
f l
Copy and learn these conversions. 1 10 2 10 3 10
4 10 5 10 6 10
= 0:1 = 0:2 = 0:3
7 10 8 10 9 10
= 0:4 = 0:5 = 0:6
= 0:7 2 4
= 0:8
=
= 0:9
1 4 1 2 3 4
= 0:25 = 0:5 = 0:75
2 Use division to write the following fractions as decimal numbers. a
1 4
b
3 4
c
3 Use division to show that
a
1 3
1 8
= 0:333 333 ::::::
d
7 8
b
2 3
e
= 0:666 666 ::::::
Copy and learn these conversions. Eighths/Quarters 1 8 = 0:125 2 8
=
1 4
3 8
= 0:375
4 8
=
1 2
= 0:25
= 0:5
Eighths/Quarters 5 8 = 0:625 6 8
=
3 4
7 8
= 0:875
= 0:75
1 5
1 5 2 5 3 5
= 0:2
4 5
= 0:8
= 0:4 = 0:6
f
3 5
DECIMALS (CHAPTER 5)
Percentages Percentages are another way of writing fractions and decimals. Percent means ‘out of one hundred ’. One (unit) is one hundred percent, written as 1 = 100%. One percent, written 1%, is
1 100
or 0:01: So five percent, written 5%, is
5 100
or 0:05
Conversions 7 100 .
Percentage to Fraction
Divide by 100. For example, 7% =
Fraction to Percentage
Write the fraction with 100 in the denominator. Examples:
7 10
²
2 5
²
=
7£10 10£10
=
2£20 5£20
=
70 100
=
40 100
= 70%
= 40%
Percentage to Decimal
Divide by 100 by shifting the decimal point 2 places left. For example, 23% = 0:23 fshift the decimal point 2 places leftg
Decimal to Percentage
Multiply by 100 by shifting the decimal point 2 places right. For example, 0:20 = 020% = 20%
4 There are 100 squares in this grid. a What percentage are shaded? b What percentage are not shaded?
5 Write these percentages as fractions. a 1% b 50% c e 20% f 75% g i 9% j 29% k
Remember 1% = 0.01 so divide by 100 to convert to a fraction.
10% 80% 33%
d h l
25% 100% 78%
6 Write these fractions as percentages. a e
12 100 2 100
b f
8 100 37 100
c g
49 100 6 100
80 100 107 100
d h
7 Write these fractions with denominator 100 then convert to percentages. a e
3 10 3 5
b f
1 2 1 25
c g
4 5 11 20
d h
1 4 13 25
8 Write as decimal numbers. a 4% b e 100% f
10% 50%
c g
20% 75%
d h
25% 250%
9 Write as percentages. a 0:02 e 0:6
0:24 2:5
c g
0:75 3:2
d h
0:99 4
b f
79
80
DECIMALS (CHAPTER 5)
Unit 36
Multiplication and division
When multiplying a decimal number by a whole number, the number of digits after the decimal point in the question is equal to the number of digits after the decimal point in the answer. Examples: ²
5 £ 1:2 = 6:0 =6
fone figure after decimal point in question and in answerg
²
4 £ 0:06 = 0:24
ftwo figures after decimal point in question and in answerg
²
42:8 £ 25
4 £ 21 85 107
= 10 700:0 = 1070
2.8 25 40 60 0.0
Put the decimal point in the answer before leaving off unnecessary zeros.
fone figure after decimal point in the question and one figure after decimal point in answerg DEMO
Exercise 36 1 Find the value of a b
i i
3 £ 0:7 12 £ 0:2
ii ii
3 £ 0:07 12 £ 0:02
iii iii
3 £ 0:007 12 £ 0:002
Another method:
c
i
6 £ 0:11
ii
6 £ 0:011
iii
6 £ 0:0011
9 £ 2:7 is approximately 9 £ 3, so 9 £ 2:7 is approximately 27
Consider 9 £ 2:7
2 Find the value of a
i
6 £ 0:5
ii
6 £ 0:05
iii
6 £ 0:005
b
i
4 £ 2:5
ii
4 £ 0:25
iii
4 £ 0:025
c
i
5 £ 1:2
ii
5 £ 0:12
iii
5 £ 0:012
27 £ 9 243
so our answer is 24:3
3 Find the value of a
3:1 £7
b
0:11 £9
c
0:13 £2
f
2:3 £6
g
3:09 £4
h
0:011 £7
d
0:05 £11
e
0:007 £12
i
1:05 £8
j
2:002 £5
4 Find the value of a e 5
23 £ 0:41 16 £ 1:6 kg
b f
31 £ 0:56 17 £ $5:05
c g
17 £ 0:04 45 £ 4:2 m
d h
37 £ 0:014 25 £ $2:06
a At the local timber yard Jan bought five lengths of timber each 4:8 m long. What was the total length of timber bought? b If one cricket bat weighed 2:3 kg, how much would seven similar cricket bats weigh?
6 Find a d g j m
1% of $200 1% of $4:00 10% of 65 cm 25% of 100 litres 15% of 40 grams
b e h k n
1% of 500 people 10% of 26 litres 10% of $6:50 100% of $1000 20% of $2600
c f i l o
1% of 610 grams 10% of 7850 km 15% of $100 10% of 300 people 150% of 60 tonnes
Example: 25% of $30 = 0:25 £ $30 = $7:50
DECIMALS (CHAPTER 5)
81
Division by whole numbers ²
3¥4 )
²
0: 7 3 : 30
fadd as many zeros as you needg
5
20
3 ¥ 4 = 0:75
43:74 ¥ 9 )
4
9
4: 8 4 3 : 77
fdecimal point in answer goes above decimal point in questiong
6
54
43:74 ¥ 9 = 4:86
7 Find a 5:6 ¥ 7 d 37:54 ¥ 5 g 15:75 ¥ 5
b e h
3¥8 126:3 ¥ 3 14:88 ¥ 6
c f i
31:26 ¥ 2 1:632 ¥ 4 2:61 ¥ 9
8 Find a $4:86 ¥ 2 d $313:6 ¥ 7
b e
$45:75 ¥ 3 $98:72 ¥ 8
c f
$515:05 ¥ 5 $2:43 ¥ 9
Problem solving 9
a If six pens cost me $1:86, how much did each pen cost? b Eight identical pairs of shoes weigh a total of 2:72 kg. How much does each pair weigh? c Patrick cut a 3:25 m length of string into 5 identical length pieces. How long was each piece? d If I run 7 km per hour, how long will it take me to run 49:14 km?
10 A trip to our holiday house usually takes 4:8 hours. We attempt to have three stops along the way, equally spaced throughout the trip. After how many hours should we make the first stop? 11 Jason bought three bottles of orange juice costing $3:36 each. a What was the total cost? b How much did Jason pay if the total was rounded to the nearest 5 cents? c How much change from $20 did he receive? 12 It took Mary 13:8 seconds to run 100 m. If she ran at the same pace for the whole race, how long would it take her to run 1000 m? 13 Dawn is able to sew a skirt hem in 10:4 minutes. How long will it take her to sew nine skirt hems if she works at the same rate? 14 Jade bought a 4 metre roll of Christmas wrapping paper. If she used lengths of 1:2 m, 0:36 m and 0:75 metres to wrap presents, how much paper was left? 15
a If 1 litre = 1000 mL, write 1:25 litres as millilitres. b A bottle of soft drink contains 1:25 L and a can of softdrink contains 375 mL. Use your answer to a to help you find the difference in capacity of the two containers.
82
DECIMALS (CHAPTER 5)
Unit 37
Money and rounding decimals
Examples: ² Multiplication
² Estimating when dividing
² Division
$407:37 ¥ 19 + $400 ¥ 20 + $20
$38:75 £7 $271:25
8
7: 41 5 9 : 32 8
So $59:28 ¥ 8 = $7:41
Exercise 37 1 Find a d g j
$8 £ 4 $6:45 £ 8 $23:95 £ 9 $64:05 £ 7
b e h k
$1:70 £ 3 $3:25 £ 9 $36:05 £ 7 $49:95 £ 8
c f i l
$8:15 £ 6 $13:45 £ 5 $88:50 £ 4 $64:65 £ 6
2 Estimate a $19:80 ¥ 5 d $28:95 ¥ 3 g $201:85 ¥ 19 j $440 ¥ 37
b e h k
$2:98 ¥ 10 $91:50 ¥ 9 $880 ¥ 31 $482:40 ¥ 54
c f i l
$72:50 ¥ 7 $185:40 ¥ 11 $529:80 ¥ 48 $1048 ¥ 96
3 Find a d g j
$8 ¥ 4 $8:25 ¥ 3 $23:40 ¥ 9 $216:16 ¥ 8
b e h k
$80 ¥ 4 $36:09 ¥ 9 $99:75 ¥ 7 $599:40 ¥ 12
c f i l
$80:80 ¥ 4 $65:50 ¥ 5 $68:75 ¥ 11 $1781:50 ¥ 7
4 Find a $943:60 ¥ 2 d $905:50 ¥ 5 g $1125 ¥ 6
b e h
$750:80 ¥ 4 $872:10 ¥ 3 $1280:70 ¥ 9
c f
$922:50 ¥ 3 $395:50 ¥ 7
Remember that + means ‘is approximately equal to’.
Problem solving with money 5 a Find the cost of 2 kg of apples at $2:25 per kilogram. b How much would 3 kg of sausages at $4:45 per kilogram cost? c If one iceblock costs $0:75, how much will eight iceblocks cost me? d Sharon worked for 4 hours and was paid $8:60 per hour. How much did she earn? e Find the cost of 9 bus tickets at $2:35 each. f Find the cost of 8 books at $43:35 each. g When Eli filled his car with petrol he bought 50 L at $0:97/litre. How much did it cost? 6
a If 8 packets of crisps cost $20:40, find the cost of 1 packet. b If 4 kg of chicken cost $35:20, find the cost per kilogram of chicken. c If 7 boxes of chocolates cost $45:15, what was the cost of 1 box? d If I bought 4 £ 2 L of juice for $28:40, what was the cost per litre? e If 5 bunches of flowers cost $41:25, find the cost of 1 bunch. f If 4 iceblocks cost $1:80, find the cost of 1 iceblock. g If a dozen eggs cost $2:78, find the cost of half a dozen.
DECIMALS (CHAPTER 5)
83
Rounding decimal numbers Often we need an approximation of a decimal number instead of the actual number. For example: If the hot dog stand raised $393:85 at the school fete this would probably be reported as raising $394 or even as $390 to the nearest 10 dollars.
Rules for rounding ²
²
²
Rounding to the nearest whole number Look at the first decimal place. If the digit is 5, 6, 7, 8 or 9, round up. If the digit is 0, 1, 2, 3, or 4, round down. Rounding to the nearest one decimal place Look at the second decimal place. If the digit is 5, 6, 7, 8 or 9, round up. If the digit is 0, 1, 2, 3, or 4, round down. Rounding to the nearest two decimal places Look at the third decimal place. If the digit is 5, 6, 7, 8 or 9, round up. If the digit is 0, 1, 2, 3 or 4, round down.
Examples:
5:6
5:64
5:648
Rounding money When you pay cash, the amount you pay is rounded to the nearest 5 cents.
look at this then round
²
39:748 + 40 to the nearest whole number
look at this then round
²
39:748 + 39:7 to one decimal place
look at this then round
²
39:748 + 39:75 to two decimal places
For example, $41:82 $41:83 $41:86 $41:88
DEMO
would would would would
be be be be
$41:80 $41:85 $41:85 $41:90
7 Round to the nearest whole number. a 4:8 b 5:2 e 149:01 f 366:05 i 486:58 j 7999:14
c g k
6:73 409:96 4906:37
d h l
54:82 3684:2 4903:624
8 Round to the nearest one decimal place. a 46:12 b 5:09 e 49:99 f 721:119 i 100:55 j 105:63
c g k
12:882 22:25 48:9898
d h l
13:603 40:41 6:076 32
9 Round to the nearest two decimal places. a 5:8301 b 5:8031 e 6:8555 f 31:689 i 689:001 j 462:059
c g k
15:8003 45:022 6:011
d h l
4:9727 491:333 38:5678
10 How much cash would you expect to pay for these individual items? a an orange marked at $0:87 b a loaf of bread marked at $2:18 c a packet of cereal at $5:32 d a tub of icecream at $4:64 e a can of peaches at $1:85 f a packet of washing powder at $7:83 11 If you purchase all of the items in question 10 how much would you pay at the checkout if you pay cash?
84
DECIMALS (CHAPTER 5)
Unit 38
Review of chapter 5
Review set 5A 1 Write
31 1000
as a decimal number.
2 Give the place value of the digit 3 in 10:003 3
a Write ‘3 tenths 4 hundredths and 5 thousandths’ as a decimal number. b Express 21:027 in words. c Copy and complete: 51:746 = 50 + ::::: + 0:7 + ::::: + ::::: .
4 Place the following decimal numbers in order of size, from smallest to largest: 41:63, 4:163, 416:3, 4:613, 4:136, 46:13 5 Write the following amounts of money as decimals of one dollar: a b
6 Write each amount as dollars using a decimal point. a 50 cents b twenty four dollars and five cents 7 Write the fractions a
b
3 20
with 10 or 100 in the denominator
i 8 Write
2 5
2 25
ii
as decimal numbers
as a decimal number.
9 Write as a percentage
a 0:5
Find 10% of $200.
b 0:23 b
Find 25% of $400.
10
a
11
a Find the difference in length between 0:896 m and 3:207 m. b Find the product of 1000 and 6:925 c Find the quotient when 46:7 is divided by 100.
12
a Divide $627:55 equally between 7 children. b One hundred sheets of printing paper weigh 8 kilograms. How much does one sheet weigh? (Answer in grams: 8 kilograms is 8000 grams.) c Five people each paid $56:50 for a theatre ticket. Find the total paid for the tickets.
13 Anna decided to go to a movie. Her bus fare was $2:25 return, the movie ticket was $7:50, and she needed $1:55 for a drink and $1:85 for an icecream. a How much did it cost Anna? b How much change would she have from $20? c If four friends went to the movies with Anna, and they each spent the same amount as Anna, how much was spent by the five children?
DECIMALS (CHAPTER 5)
Review set 5B 1 Mark 1:35 on a number line: 2
7 1000
1
1.1
1.2
as a decimal number.
1.3
1.4
1.5
1.6
1.7
a
Write
c
Write ‘fifty four and fourteen thousandths’ as a decimal number.
b
1.8
1.9
2
2.1
Express 603:47 in written form.
3 Give the place value of the digit 2 in 1:327 4 Place the following decimal numbers in order of size, from largest to smallest. 5:432, 53:42, 5:234, 5:342, 534:2, 54:32 5 Write the following amounts of money as decimals of one dollar. a b
c
6 Write each amount as dollars using a decimal point. a 5 cents b forty five dollars and ten cents 7 Write the following fractions a
9
a c
b
with 10 or 100 in the denominator
i 8 Write
3 4
1 4
16 25
ii
as decimal numbers
as a decimal number.
Write 0:46 as a percentage. Write 16% as a decimal number.
b d
Write 1:55 as a percentage. Write 25% as a fraction in lowest terms.
10 Find 20% of $300. 11
a Subtract fifty nine decimal two five from eighty six decimal zero one. b Divide 0:02 by 10. c Multiply $62:31 by 5.
12 Kim was sending some Christmas presents to his brother overseas. The maximum weight allowed in a parcel was 10 kg. The total weight of the presents was 23:46 kilograms. a How many parcels did Kim need to send? b If the first two parcels weighed 9:42 kg and 9:05 kg, what weight remained? 13 Damien deposited amounts of $153:95 and $68:25 in a new bank account. How much more did he need to save to reach his target of $300:00?
85
86
MEASUREMENT
Length
Unit 39
In our everyday life we measure many things. For example, we measure lengths, volume (how much milk in a container) and mass (how heavy we are). We use the Metric System of units. ² ² ²
For measuring lengths we use metres, millimetres, centimetres and kilometres. For measuring mass we use grams, kilograms, milligrams and tonnes. (Chapter 8) For measuring capacity (amount of liquid in a container) we use litres.
Units of length To measure a line we usually use a ruler. Most rulers have marks showing millimetres (mm) and centimetres (cm) as shown here.
0
1 cm
2
3
5
6
metre centimetre millimetre kilometre
m cm mm km
4
CHAPTER 6
You may use either centimetres (cm) or millimetres (mm) to measure the length of something depending on how accurate you want to be. For longer distances we could use a tape measure or a trundle wheel. We measure in metres or kilometres.
Metric length units 1 centimetre = 10 millimetres 1 metre = 100 centimetres 1 metre = 1000 millimetres 1 kilometre = 1000 metres
Estimates These will help you visualise some of the units of length.
Shorthand for units:
1 mm
the thickness of a 5-cent coin
1 cm
the width of a finger nail
1m
half the height of a door
1 mm
1 cm
1m
1 km
twice around an AFL oval boundary
Exercise 39 1 When John drew these objects he left the units off. What should they be? a b c 10 ......
12 ......
54 .......
7
MEASUREMENT (CHAPTER 6)
d
e
87
f
156 ...... 5 ...... 84 ......
2 You can use these three measuring devices.
B
A
C
Which one should you use to measure a the length of a chalkboard b the width of an eraser c the length of a basketball court d the distance around the school oval e the distance jumped by an athlete f the length of a toothpick g the length of a speed boat h the length from the tee to the pin on a golf hole? 3 What units would be used to measure the lengths in the parts of question 2? 4 With your ruler, rule lines that are the following length a 5 mm b 30 mm c 72 mm f 10 cm g 3 cm h 24 cm
d i
240 mm 15 cm
e j
150 mm 20 cm
5 Look at the lines you have drawn in question 4. Are any the same length? If they are, write the length of the line in both millimetres and centimetres and see if you can tell how many millimetres there are in each centimetre. 6 Measure each of these lines in millimetres. a c 7 Measure each of these lines in centimetres. a c
b d
b d
8 To help you get to know how long a metre is, look at a metre ruler then estimate (guess) the length of the following a classroom length b classroom width c whiteboard length d friend’s height e soccer field length
ESTIMATING LENGTHS
9 Now check your estimates in question 8 by actually measuring them with a metre ruler. Write your answers down and see how close your estimate was.
88
MEASUREMENT (CHAPTER 6)
Converting length units and perimeter
Unit 40
We often need to convert between units of measure. For example, if I ran a 1500 metre event at sports day, how many kilometres is that? Builders usually measure window sizes in millimetres. How big in metres is a window that is 1530 mm long and 820 mm high? Conversion chart
´1000 km
´100 m
¸1000
2:5 m is
cm ¸100
2:5 £ 100 cm
²
´10
= 2:50 cm = 250 cm
mm
This means that to convert mm to cm we divide by 10.
¸10
To convert from big units to smaller units you multiply. To convert from small units to larger units you divide.
2:5 ¥ 1000 km
²
= 002:5 = 0:0025 km
Exercise 40 1 Convert to centimetres a 5m
b
2:6 m
c
0:2 m
d
0:08 m
2 Convert to millimetres a 5m
b
2:6 m
c
0:2 m
d
1:08 m
3 Convert to kilometres a 15 000 m
b
1500 m
c
600 m
d
6398 m
4 Convert to millimetres a 2:3 cm
b
26 cm
c
0:5 cm
d
6:85 cm
5 Convert to metres a 625 cm
b
82 cm
c
5 cm
d
0:2 cm
6 Convert to centimetres a 9 mm
b
96 mm
c
453:2 mm
7 Convert to metres a 546 mm
b
70 mm
c
5 mm
8 Convert to metres a 6 km
b
63 km
c
0:7 km
d
0:56 km
9 A length of timber measured 1563 mm. Write this length in metres. 10 Janine lived 0:65 km from her school. How far is that in metres? 11 Peter measured his long jump distance as 2:265 metres. Write the length of his jump in centimetres. 12 Jodie measured the length of a big black ant as 2:35 cm. Write the length in millimetres.
89
MEASUREMENT (CHAPTER 6)
Perimeter The perimeter of a shape is a measurement of the distance around the boundary of the shape. If the shape has straight sides, it is a polygon and the perimeter is found by adding the lengths of the sides. In the metric system we can measure perimeter in millimetres (mm), centimetres (cm), metres (m) or kilometres (km). Examples: ²
DEMO
18 cm
²
8 cm
10 m
8 cm
Perimeter = 10 + 9 + 15 m = 34 m
13 Find the perimeter of each of the following. a b 14 cm 27 mm
e
5 km
15 cm
h
5m
5 km
45 km
6.7 m 1.6 m
35 km
5.1 m
14 Determine the perimeter of these rectangles. a b 30 m
20 m
7 km
20 m 30 m
c
The same markings on the sides of a polygon show that the sides have the same length.
3 km 5 mm
d 52 cm
40 mm 17 cm
1.3 m 2.2 m
3.2 m
7m
15 mm 14 mm
4.6 m
5m
13 mm
12 mm
i
20 km
6m
6m
12 mm
15 mm
5 km
26 cm
6m
f
11 mm
4 km
13 cm
59 km 6 km
16 cm
19 cm
7m
44 km
35 km
15 mm
34 cm
6m
c 27 mm
23 cm
g
3 km
Perimeter = 8 + 5 + 3 + 4 + 6 km = 26 km
35 mm
28 cm
25 cm
6 km
5 km
9m
Perimeter = 18 + 8 + 18 + 8 cm = 52 cm
8 km
4 km
18 cm
d
²
15 m
90
MEASUREMENT (CHAPTER 6)
Perimeter continued
Unit 41
Perimeters are found by adding lengths of sides. Sometimes we must accurately measure these lengths in order to find the perimeter. Small marks on a figure (e.g., 3 a below) show equal lengths.
Special figures Sometimes we can use multiplication to find a perimeter. For example,
What is the perimeter of the regular octagon? As there are 8 equal sides the perimeter is 8 £ 7 cm = 56 cm. So we can use multiplication rather than addition.
7 cm
Exercise 41 1 Find the perimeter of these equilateral triangles. a b
c
8m
12 cm
21 km
2 Find the perimeter of these regular polygons with the dimensions as given. a b c
14 m
8 cm
d
e
16 mm
f
31 km
17 cm
23 cm
A regular polygon has all sides equal in length and all angles equal in size.
3 Find the length of one side of an equilateral triangle if the perimeter is a 36 cm b 60 mm c 93 km
d
372 mm
4 Find the length of one side of a regular pentagon if the perimeter is a 25 km b 45 cm c 70 mm
d
215 km
5 Find the length of one side of a regular octagon if the perimeter is a 32 cm b 56 mm c 96 cm
d
144 km
6 Find the perimeter of these by adding two adjacent lengths and doubling the result. a b c 9m
14 cm
18 m
26 cm
d
102 m
e
f 500 mm
68 km 92 km
68 m
1069 mm
155 cm 235 cm
MEASUREMENT (CHAPTER 6)
91
7 Use a ruler to help you find the perimeters of these figures. Measure to the nearest whole centimetre. a
b c
g
f
e i
d
j
k h
8 Choose one of these measurements A 69 cm B 150 m E 25 760 km F 44 mm I 70 m J 52 mm M 13 mm N 76 m
C G K
53 cm 150 million kms 11 000 km
D H L
7 km 47 cm 108 m
to best match the perimeters described in the following: a c e g i k m 9
distance around a standard envelope fencing on a suburban block of land the hole in the middle of a CD outside walls of a Pizza Hut orbit of Earth around the Sun around a bee’s thorax a CD cover
a The jogging track in a city park is shaped as shown. If I run one lap of the track, how far do I run?
b d f h j l
boundary of a small city circumference of a soccer ball coastline of Australia delete key on a computer keyboard distance around a baseball diamond circumference of the moon
56 m 120m
b Find the distance in kilometres that I would run in 5 laps.
80m
10 Della is asked to mark out a 60 metre long by 20 metre wide rectangular arena at Pony Club. If she walks around the arena to set out markers, how far does she walk? 11 Find the length of metal edging needed to surround a carpet which is 18 metres long and 12 metres wide. 12 A box of chocolates is in the shape of a hexagon. The sides of the box are 6 cm long. A ribbon is tied around the outside of the box and the bow takes an extra 20 cm of ribbon. How much ribbon is needed? 13 Kate is building a fence around her square housing block. Each side of the block is 42 metres. a Find the perimeter of the block. b If the gateway is 3 metres wide, find the length of fencing needed for the block. c If the fencing costs $12 per metre, find the cost of the fence. d If the gate costs $312, find the cost of enclosing the block.
92
MEASUREMENT (CHAPTER 6)
Area
Unit 42 Area is the size of a surface. It is the amount of surface inside a shape. Counting shapes of equal size gives us a way of measuring area. Area is measured in square units. For example,
this is a square cm, written cm2
this is a square mm, written mm2
Exercise 42 1 Obtain two packs of playing cards. Lay them on a desk in the class room to cover the desk with no gaps (except at their corners). Find the area of your desk in cards, e.g., area of desk = 60 cards. 2 Find the area of the classroom door (or the teacher’s desk top) using A4 sheets of paper. 3 Count the number of area units in each of the following pairs of shapes. Which shape has the larger area? a i ii
b
i
c
i
ii
ii
4 In a sentence or two explain why we use square units and not playing card units, A4 paper units, triangular units, etc. when measurng area. 5 Count the number of square units to find the area of each of these shapes. a b c
d
e
f
MEASUREMENT (CHAPTER 6)
g
Remember that units 2 or u 2 is the acceptable way of writing square units.
h
i
j
l
m
93
k
n
6 To find the area of each shape, count the number of square centimetres. Write cm2 after each number. a b c
7 Copy these shapes then rule lines to show square centimetres. Count and record the number of cm2 in each shape. a b c
For larger areas we use: 1 square metre (m2 ) is the area enclosed by a square of side length 1 m. 1 hectare (ha) is the area enclosed by a square of side length 100 m. 1 square kilometre (km2 ) is the area enclosed by a square of side length 1 km.
94
MEASUREMENT (CHAPTER 6)
Area of rectangles
Unit 43 6 units
Consider a rectangle 6 units long and 4 units wide. 4 units DEMO
Clearly the number of square units that this rectangle contains is 24 but 24 = 6 £ 4. This leads to the general rule:
length (l ) width (w)
Area of rectangle = length £ width (in square units)
Examples:
²
This rectangle
²
This rectangle
has area = length £ width = 7 cm £ 5 cm = 35 cm2
5 cm 7 cm
has area = length £ width = 12 m £ 3 m = 36 m2
3m 12 m
Exercise 43 1 Find the area of the rectangles. a
b
c
7m
3 cm
10 km
11 cm
9m
23 km
2 Determine the area of rectangles with these measurements. a 15 km by 4 km b 12 m by 8 m c 20 mm by 8 mm 3 Arrange the figures in order from smallest to largest area. a b 11 cm
d
73 m by 10 m
c 8 cm
4 cm 15 cm 8 cm
6 cm
4 Arrange the figures in question 3 in order from smallest to largest perimeter. 5 Arrange the following figures in ascending order of area. a b
9m
9m
c
15 m
8m 10 m
6 Arrange the figures in question 5 in order from smallest to largest perimeter. 7 Draw sketches of three different rectangles which have an area of 12 cm2 .
6m
MEASUREMENT (CHAPTER 6)
8 Here is a scale diagram of the floor plan of an apartment. a By measuring the outside walls, calculate its i length ii width
SHWR
KITCHEN
b Calculate for the outside walls i the perimeter ii the area c Calculate the area of i the kitchen ii the bathroom d In the bathroom, how much area has been allocated for the shower and bath combined?
ROBE
ROBE
LOUNGE
BEDROOM
Scale:
9 Use a calculator to help you complete the following table.
a b c d e f g h
Length 36 m 174 cm 28 km 25 cm 58 m
Width 13 m 86 cm 28 km
50 cm 48 mm
Area
250 cm2 1160 m2 2000 cm2 9600 mm2 1600 m2
1 cm º 1 m
Remember, a square is a special rectangle.
Perimeter
160 m
A shape with 2 dimensions has length and width (breadth).
Investigation You will need:
95
Area and perimeter are different An A4 page of cm2 graph paper, and coloured 1 cm squares.
What to do: 1
Take 3 coloured cm2 and arrange them as shown. The area of the squares is 3 square centimetres (3 cm2 ). The perimeter or distance around the square is 8 centimetres (8 cm).
2
Arrange the 3 square centimetres into other patterns making sure they all touch. For example: Each of the these arrangements has ² ² an area of 3 square centimetres (3 cm2 ). However the perimeter or distance around each arrangement varies. Area = 3 cm2 Area = 3 cm2 Perimeter = 12 cm Perimeter = 8 cm
3
Use this pattern method to find as many perimeters as possible using a 5 cm2 b 7 cm2 c 10 cm2
4
Determine the largest possible perimeter of arrangements using a 4 cm2 b 9 cm2 c 16 cm2
d
25 cm2
Determine and draw arrangements with the smallest possible perimeters using a 4 cm2 b 9 cm2 c 16 cm2
d
25 cm2
5
96
MEASUREMENT (CHAPTER 6)
Composite areas and problem solving
Unit 44
Composite shapes can be seen in real world examples where two or more rectangles (including squares) are present. For example, a lawn could surround a flower garden.
lawn flowers
4m
10 m
10 m 18 m
Discussion How could we find the area of the lawn shown in the diagram above? The area of this figure can be found in two different ways
8m 4m
3m 1m
5m
Method 1 (Area addition)
Method 2 (Area subtraction)
8m 4m 5m
3m
8m 3m 1m
3m
4m 1m 5m
Total area = area of rectangle + area of square =5£4 + 3£3 = 20 + 9 = 29 m2
3m
3m
Total area = area of large rectangle ¡ area of small rectangle =8£4 ¡ 3£1 = 32 ¡ 3 = 29 m2
Exercise 44 1 Find the area of these shapes using addition. a b
c
2m
2 km
3m 7m
3m
5 km 4m
9m
d
e
5m
2m 6m
f
2 cm
4 cm
8m
1m
6m 13 m
8m 12 m
MEASUREMENT (CHAPTER 6)
2 Find the area of these shapes using subtraction. a b
c
5m
2m
3 km
5m
1 km
4m
9m
3 km
d
97
2m
e 6m 4m
4m 12 m
10 m
10 m
18 m
Problem solving with areas 3 Find the area of a garden bed which is 15 metres long and 9 metres wide. 4 Find the area of a computer screen that is 36 cm long by 28 cm wide. 5 Find the area of a pane of glass to fit in a window frame that is 1:2 metres long and 0:8 metres wide. 6 A lawn is 10 m by 8 m and is surrounded by a path which is 1 m wide. a Find the area of the lawn. b Find the area of the lawn plus concrete.
lawn
concrete
c Find the area of the concrete. 7
a Find the area of a floor that is 12 metres long and 4 metres wide. b How many litre tins of paint would be needed to paint the floor if 1 litre of paint covers 16 m2 ?
8
a A bedroom wall is 5 metres long and 1:8 metres high. Find the area of the wall. b There is a window 1:2 metres long and 0:8 metres high in the wall. Find the area of wall to be painted.
9
a Find the area of a paddock that is a square with sides 400 metres. b If posts are placed every 4 metres in the fence, how many posts are needed when the paddock is fenced?
Square metres
Activity Make a square metre by using joined newspaper sheets. Use masking tape to join the sheets. What to do: 1 Find how many people can stand on the square metre. 2 Find how many copies of this book can lie flat on a square metre. Do not go over the edges. 3 Does a door of your classroom have an area which is a more than a square metre b less than two square metres? 4 Estimate how many square metres the chalkboard would be.
98
MEASUREMENT (CHAPTER 6)
Volume
Unit 45 A shape has 3 dimensions when it has length and width and height.
You have already used different units of area to measure the amount of surface inside a 2 dimensional shape. The volume of a solid is the amount of space it occupies. To measure 3 dimensional space it makes sense to use 3 dimensional units of measure.
As with units of area, the units used for the measurement of volume are related to the units used for the measurement of length. 1 cubic millimetre (mm3 ) is the volume of a cube with a side of length 1 mm. 1 cubic centimetre (cm3 ) is the volume of a cube with a side of length 1 cm. 1 cubic metre (m3 ) is the volume of a cube with a side of length 1 m.
Exercise 45 1 Suggest suitable units of volume for the following. a the earth removed for a backyard swimming pool The little 3 in mmC, cmC and mC shows that the shape has 3 dimensions, length, width and height.
b a child’s building block c a house brick d a granite boulder e a tablet prescribed by a doctor. 2 If each block is 1 cm3 , find the volume of a b
e
f
c
d
g
h
For 12 MA blocks it is possible to find four different shaped solids all with the same volume of 12 cubic units. These are
DEMO
Notice that standing them on different faces does not change the type. 3
Use MA blocks to find how many rectangular prisms can be constructed from the following numbers of blocks. ii Give the 3 dimensions and volume of each prism. a 16 b 18 c 8 d 20 f 25 g 64 h 36 i 100 i
e j
27 125
MEASUREMENT (CHAPTER 6)
For this figure
Notice that
DEMO
² ²
4
i ii a
it consists of 14 blocks so its volume is 14 units3 to complete it to make a rectangular prism, we would need 10 extra blocks. This prism would be 4 by 2 by 3 and so its volume = 4 £ 2 £ 3 = 24 units3 .
Find the volume in cubic units of the solids shown. Find the smallest number of cubic units that are needed to complete the solid as a rectangular prism. b c d
5 Find the volume of the following solids in the units as shown. a m3 b cm3 c mm3
d
u3
Rectangular prisms height
length
The volume of a box = length £ width £ height.
width
6 Find the volume of a
b
c 6 km
5 mm
4 cm
2 km
4 mm
2 cm
4 km
8 mm
d
7 cm
e
f 5 cm
3 mm
5 mm
10 mm
5 cm
6 cm
7 Draw this rectangular prism and find its volume a length 4 cm, width 3 cm and height 1 cm b length 10 m, width 2 m and height 2 m c length 5 mm, width 4 mm and height 4 mm 8 Find the volume of the cubes with sides a 3 cm b 5 mm
99
c
8m
100
MEASUREMENT (CHAPTER 6)
Capacity
Unit 46 Kilolitre (kL) and megalitre (ML) are used for large capacities, for example, the water used by your household in 1 year, or the capacity of a reservoir. The units of capacity we would use every day (for example, for a bottle of juice) are the litre (L) and millilitre (mL). The capacity of a container is the measure of the amount of liquid that it will hold.
1 m 3 holds 1 kL
1 L = 1000 mL 1 kL = 1000 L 1 ML = 1 000 000 L 1 mL = 1 cm3 1 L = 1000 cm3 1 kL = 1 000 000 cm3 = 1 m3
1m 1m
1m
The link between capacity units and volume units is
Did you know?
10 cm 10 cm
10 cm
If a cubic container has inside dimensions 10 cm by 10 cm by 10 cm, then it can contain exactly one litre of water and the water will weigh exactly one kilogram.
VIDEO CLIP
DEMO
Some devices to measure capacity
Conversion chart ´1000 ML
´1000
´1000
kL ¸1000
L ¸1000
mL ¸1000
Exercise 46 1 Write as millilitres a 8L
b
0:8 L
c
53 L
d
5:03 L
2 Write as litres a 3000 mL
b
8500 mL
c
600 mL
d
90 mL
3 Write as litres a 6 kL
b
66 kL
c
0:6 kL
d
6:06 kL
4 Write as kilolitres a 5000 L
b
500 L
c
6300 L
d
60 L
5 Which units of capacity would you use to measure the capacity of a a small carton of milk b a swimming pool
c
a thimble?
MEASUREMENT (CHAPTER 6)
For this open box
20 £ 16 £ 10 = 3200 If it was a solid its volume = 3200 cm3 . So, its capacity = 3200 mL which is (3200 ¥ 1000) L = 3:2 L
10 cm 16 cm 20 cm
6 Find the capacity, in millilitres, of the following box shaped containers. a b
c
15 cm
8 cm 10 cm 25 cm
101
10 cm
8 cm
30 cm
6 cm
7 Find the capacity, in litres, of the following box shaped containers. a b
c 5 cm
20 cm 20 cm
15 cm
10 cm
10 cm
15 cm 20 cm
20 cm 10 cm
8 A can of soft drink contains 375 mL. Find the number of litres of soft drink in a 10 cans b 100 cans 9 An aquarium is 60 cm long, 25 cm wide and 40 cm high. Find the capacity of the aquarium in litres.
40 cm
10 A rain water tank is 120 cm long, 40 cm wide and 100 cm high. Find the capacity of the tank in kilolitres. 11 A rainwater tank holds 5:6 kilolitres when full. If the tank is only contain?
60 cm 1 8
25 cm
full, how many litres of water does it
12 How many 500 mL bottles can be filled from an 8 litre container of sauce? 13 Kylie drank one quarter of a 2 litre bottle of water. How much did she drink (in millilitres)?
Activity
The capacity of a cubic centimetre is one millilitre You will need:
a lump of plasticine, a centicube, medicine glass, eye dropper, water.
What to do: 1 Carefully press the centicube into the lump of plasticine. Smooth the plasticine to the top of the cube so that it leaves a full impression. 2 Carefully remove the centicube from the plasticine to leave a cubic centimetre hole. 3 Repeat this 5 times. 4 Carefully fill a medicine glass to the 5 mL mark. You may need to use an eye dropper to make your measurement really accurate. 5 Carefully pour the water from the medicine glass into the 5 cubic centimetre holes. What do you notice? Give reasons why you needed to make 5 impressions.
102
MEASUREMENT (CHAPTER 6)
Review of chapter 6
Unit 47 Review set 6A 1 Measure each line. Give the length in i millimetres a b 2
centimetres.
ii
a Matthew ran 5000 metres. How many kilometres did he run? b Sam measured a length of timber as 426 mm. Write this length in metres.
3 Find the perimeter of each of the following a b
c 4 mm
15 cm
5 cm
12 cm
8 mm
13 cm
4 Suggest suitable units for measuring the following a the perimeter of your school grounds
b
the perimeter of a bus ticket
5 The perimeter of a regular hexagon is 24 cm. Find the length of one side. 6 Suggest suitable units for measuring the following a the area of a postage stamp
b
the area of Australia
7 Find the shaded area of each of the following if a square has length 1 unit. a b c
8 Find the area of each of the following a b
c 12 cm
12 m
8 mm
2 cm
9 A rectangular room is 45 square metres in area. If its length is 9 metres, find its width. 10 Find the volumes of the following solids by counting cubes. Each cube has volume 1 cubic centimetre. a b
11
c
a Determine the volume in cubic units of the shape shown. b Determine the least number of cubic units needed to complete the shape as a rectangular prism.
12 Suggest the best unit to measure the capacity of the following a a baby’s bottle b a swimming pool
c
the water in a lake
103
MEASUREMENT (CHAPTER 6)
Review set 6B 1 My suitcase is 65 centimetres long. Write this measurement in a millimetres 2 Find the perimeter of each of the following a b
b metres
c 34 km 51 km
3m
17 km
6m
8m
68 km
3 Suggest suitable units for measuring a the perimeter of your house block c the area of a tennis court
b d
the perimeter of a floor tile the area of an envelope
4 Find the shaded area of each of the following. (Each square has length 1 unit.) a b c
5 Find the area of each of the following a
b
3m
50 mm
8m
6 A square playing surface is 400 square metres in area. Find its dimensions. 7 Find the coloured area. 7m 4m 10 m
8 A rectangular farming block measures 550 metres long and 300 metres wide. a Find its perimeter. b Find the length of wire needed for a 3-strand wire fence around the perimeter. (Ignore gateways.) 9 Find the volumes of the following solids by counting cubes. Each cube has volume 1 cubic unit. a b
10
c
a Determine the volume in cubic units of the shape shown. b Determine the least number of cubic units needed to complete the shape as a cube.
11 Suggest the best unit of capacity for measuring the following a the milk carried by a milk tanker b a dose of medicine c a domestic aquarium
104
REVIEW OF CHAPTERS 4, 5 AND 6
TEST YOURSELF: Review of chapters 4, 5 and 6 1 What is the best unit to measure: a the capacity of a rainwater tank c the area of a key 2 Write as decimal numbers: 3 Find
a
1 4
of 36 eggs
a
59 100
b
3 7
b d
the length of a small insect the length of a paddock fence?
b
59 1000
of 28 apples.
4 Jane walked for 12 000 metres. What is this distance in a kilometres b centimetres? 5 What is the place value of the 9 in 6 If
1 3
a 45:96
b 3:319?
of my money is $21, how much have I?
7 Find the perimeter of: a
b
c 3m 3 cm
4m
8 Write in order of size from smallest to largest: a
4m
4 cm
7m
9 Write:
5m 4m
2 13 as an improper fraction
3:601, 3:106, 3:61, 3:16 17 4
b
as a mixed number.
10 A regular pentagon has perimeter 35 cm. Find the length of one side. 11 Write in decimal form: a eighty five cents
b
sixteen dollars and five cents
12 Copy and complete:
a
1 3
=
¤ 18
b
49 70
=
¤ 10
36 84
c
=
3 ¤
13 Find the shaded area of these figures if each small square has sides of length 1 unit. a b c
14 Copy and complete:
23:076 = 20 + 3 + = 20 + 3 +
¤ 10
+
¤ 100
+
¤ 1000
¤ 1000
= 23 + 0:07 + ¤ 15 A jug holds 4 L when full. How many litres does it hold when it is 16 Find the area of: a
b
3 8
full? c 6 cm
3 cm 4 cm 2 cm
8m 6 cm
10 cm
REVIEW OF CHAPTERS 4, 5 AND 6
17 Write as decimal numbers
1 5
a
b
7 20
c
18 After riding her horse for 3 km, Sally had ridden grandpa’s house?
1 5
105
3 50
of the way to her grandpa’s house. How far is it to her
19 A room has area 56 square metres and its length is 8 m. Find its width. 20 Write as a percentage: 21 Find:
1 3
a
+
1 6
a 0:2 b
1 3
¡
b
0:59
1 6
22 In the solids below each little cube has sides 1 cm. Find the volume of: a b c
23 Find:
a
10% of $600
24 Find:
a
1¡
6 11
b
25% of 200 litres
b
5 8
¡
1 4
25 A house yard is 30 m by 20 m. a Draw a sketch of the yard showing these dimensions. b Find the perimeter of the yard. c Find the area of the yard. 26 An amount of money was divided between Jon and Gordon in the ratio 3 : 4 with Jon getting the larger share. a If Jon got $800, how much did Gordon get? b What was the original amount of money? 27 Find the area of the shaded part.
7m
4m
10 m
15 m
28 A novel weighs 0:65 kg and 20 of them can be packed in a carton. a What is the total mass of novels to go in the carton? b If the carton weighs 1:2 kg what would be the total weight to be posted? c If the postage fee is 60 cents per kilogram, what will it cost to send the carton and its contents? 29 Helga ate
7 8
of a pizza and Peter ate
3 4
of another of the same size.
a What was the total amount of pizza consumed? b What fraction was left over? 30
a Divide $130:75 equally amongst 5 children. b 7 people each paid $23:75 for a ticket for a train trip. Find the total amount paid.
31 Clancy has 3 bottles of cola, 4 bottles of orange juice and 2 bottles of milk. Of the total, what fraction are: a orange juice b not milk?
106
LOCATION AND POSITION
Unit 48
Introduction to scales and grids
A scale shows the relationship between the size of the real object and its size on the plan. A scale drawing shows the object either smaller or larger than its real size, but keeping the proportions the same. These are house plans used by a builder or home buyer.
Sometimes our plans or diagrams are larger than the object they represent, for example, the plan for an electrical circuit, or a diagram showing a flea.
Scale 1 : 200
In order to use maps, plans and models we need to understand what the scale means. The scale on a map shows that the map has either been: ² reduced for large things or ² enlarged for small things. If we could enlarge the house plans 200 times they would be the actual house floor size.
CHAPTER 7
If a map has a scale of 1 cm represents 5 km then the real distance for a distance on the map of: ² 6 cm is
6 £ 5 km = 30 km
² 10:2 cm is
10:2 £ 5 km = 51 km
Note: The scale in the above example could also be represented as follows: 0 km
50 km
Exercise 48 1 A map gives a scale of 1 cm represents 20 km. Determine the actual length for a map length of a 10 cm b 5 cm c 4:5 cm 2 A map gives a scale of 2 cm represents 60 km. Find the real distance for a distance on the map of a 6 cm b 8 cm c 5 cm d 10:5 cm
This is a radar screen at the Adelaide Airport. Find the distance from the airport to plane
3
a A
A
B Adelaide 20 km 40 km 60 km 80 km 100 km 120 km
C
b
B
c C
107
LOCATION AND POSITION (CHAPTER 7)
4 If a map length is 1 cm compared with a real length of 3 km, give the scale in another way. 5 For the scale diagram of the dog alongside: a Measure the length of the picture from the tip of the nose to the point of the tail. Use the scale to find the actual length of the dog.
Scale: 1:30
b By measuring on the diagram, use the scale to find the height of the dog at the top of his head. 6 The scale of a model aeroplane is 1 cm represents 12 m. a Find the actual wingspan if the distance of the wingspan of the model is 5 cm. b Find the model length if the actual length of the plane is 60 m. 7 Using the scale given, calculate the distance from:
A
E
a A to B
F
J
C
b B to C c C to D d D to E e E to F f F to G G
g G to H
H
h H to I i I to J j J to K
D
I
B
Scale:
8
A
K 0m
25 m
50 m
B
this is a line on a scale diagram
Calculate the actual length of line AB if the scale used is a 1 cm represents 100 km b 1 cm represents 5 m d 5 cm represents 10 m e 1 12 cm represents 50 km 9 Ed built a doll’s house from a plan which had a scale of 1 : 20. Ed’s doll house scale diagram has a height of 40 mm, a width of 70 mm and windows which are 10 mm by 20 mm. What will be the actual a height of the doll’s house b width of the doll’s house c dimensions of the windows?
c f
1 2
cm represents 1 km 3 cm represents 4:5 m
108
LOCATION AND POSITION (CHAPTER 7)
Unit 49
Grids
On a grid, a letter and a number are used to direct us to a position.
Mitsubishi
Ferrari
6
Grids can be used to direct us to where an event or stall is at shows and exhibitions. Street directories use grids of this kind. This grid shows the position of various displays at a motor show.
Ford
Yamaha
Hino
5
Daewoo
4
Kia Holden
Honda
3
Mercedes
A vertical line goes
2
a horizontal line goes
1
Bedford Toyota
Kawasaki
B
A
C
D
E
F
Suzuki
G
H
We can see that: ² The Toyota display is at the intersection between the vertical line from F and the horizontal line from 1. So, its grid position is given as F1. ² The Daewoo display is at the intersection between the vertical line from B and the horizontal line from 4. So, its grid position is given as B4. Using horizontal and vertical steps we can find how to get from one place to another. ² To get from the Hino display to the Holden display we must move 3 steps to the right and 2 steps down. ² To get from the Suzuki display to the Yamaha display we must move 4 steps to the left and 4 steps up.
Exercise 49 1
6
Hot Dogs
4
3 2
Chinese Food
Icecream
5 Records
Plants
Pasta Dunking machine
Hamburgers
Toys Books
Soft Drink
1 A
The grid shows the position of various stalls at the local fair. Find: a the position of the following stalls i Hot Dogs ii Cakes iii Toys b using horizontal and vertical steps, how to get from the i Plants stall to the Thai food stall ii Dunking machine to the Noodles stall.
Thai Food
Noodles
B
C
D
Cakes E
F
G
H
2 The grid shows the homes of friends and the positions of other important places nearby. Find: a the location of the i tennis club ii pool iii video store iv theatre b using horizontal and vertical steps, how to get between the homes of i Mary and Amber ii Rob and John iii Sue and Peter.
8 7
Video Store
Rob
6
Theatre School
5 Mary
John Pool
2 1
Peter
Deli
4 3
Amber
Tennis Club
Playground
Oval A
B
C
D
E
Sue
F
G
H
Give the horizontal shift first, then the vertical shift.
LOCATION AND POSITION (CHAPTER 7)
3
8 7 6
Pine Fir
Fir
Maple
Wattle Oak
Pine Pine
1
A B C D E
4
Wattle
Elm Birch
4 2
Oak
Fern
Palm
5 3
Melaleuca
A
B
C D
E
F G H
I
F
I
G H
J
109
A gardener has the grid alongside for a field he has planted with trees. Use the grid to find: a the number of trees b the most common type of tree and the positions at which they are located c how to get from the elm to the palm d which type of tree is closest to the Melaleuca (using grid-steps only) e how to get from the birch to the fern (using grid-steps only). K
Colour the squares A8 F5 D6 A5 F11 H7 B4 I11 A4 F4 A9 J8 D5 L8 F6 B11 H6 D11 J6 F7 J5 L5
L M
1 2 3 4 5 6
7 8 9 10 11
on the grid. L10 J10 A6 E4 J9 D4 L6 A7 M11 D9 L9 H8 A10 I4 J7 A11 D10 F10 H11 H4 F9 L11
J4 D7 F8 E11 H5 J11 H9 D8 H10 L4 L7
PRINTABLE GRID
12 13 14
5 The local Primary School grounds have been laid out on a grid. a How many classrooms are there? b The year 1 class is in Classroom 4. Use horizontal and vertical steps to show how to get from their classroom to:
10 9 8 7
i the sandpit
6
ii the JP playground.
5
c Give the grid position that would be used at lunch time for play. d The principal is showing a new parent around the classrooms. Draw in colour the most efficient route to take.
Activity
11
4 3 2 1
Gym
Primary Playground
Classroom 5
Tennis Court
Classroom 1 Classroom 4
Languages Room
Classroom 2
Sandpit Classroom 3
Office
Music Room
Oval
J.P Playground
Garden
A B C D E
F
G H
I
J
K
Scale diagram of your bedroom
Measure the length and width of your bedroom and any large pieces of furniture in it, and the width of the door and windows. Take measurements so that you can position the door and the windows correctly on your diagram. Choose an appropriate scale and produce a labelled scale diagram of the floor.
110
LOCATION AND POSITION (CHAPTER 7)
Unit 50
Maps
Good maps will have a grid, a scale and show the direction of North. These maps help us to locate actual places. A scale is used to find actual distances between two places. The North arrow is used to find the direction of travel. Not all actual distances are measured as straight lines. On maps, some distances like paths, roads, rivers and coastlines are not straight lines. We can use a piece of cotton or string to help us be more accurate with lines which are not straight.
Exercise 50 1 These diagrams are drawn to scale. The scale is 1 cm represents 1 kilometre. a
A
B
Copy and complete: AB = ...... cm, so actual distance is ...... km. S
b
U W T
R
Copy and complete:
X
V
RS = ...... mm, VW = ...... mm,
ST = ...... mm, WX = ...... mm,
TU = ...... mm, UV = ...... mm, so total distance = ...... mm.
So, the actual distance = ...... km, c Carefully place a piece of string on the curve EF. E
F
Copy and complete: Total length = ...... cm, so actual distance = ...... km. 2 Measure the distances a and b to the nearest mm. Change the measurements to actual distances when the scale is 1 cm represents i
1 km
ii
10 km
a
iii
100 km
iv
50 km
b
3 The map alongside has a scale of 1 grid unit represents 2 km. Find: a the position of the i fishing centre ii lighthouse iii waterfall iv caravan park b the actual distance from the surf shop to the fishing centre c how far the ship is actually from the lighthouse d the shortest distance from the hill-top to the waterfall e the length of the jetty f the distance of the Waterfall from the Caravan Park.
15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
Lighthouse
Ship
Caravan Park
Hilltop
Jetty Surf shop
Waterfall
Fishing centre
A B C D E F G H I
J K L M N O P
LOCATION AND POSITION (CHAPTER 7)
4 The map alongside shows the countryside and some land marks. The scale is 1 grid unit represents 2 km. Find:
15 14 13
a the position of the i hotel
12 11 10 9 8 7 6 5 4 3 2 1
ii Post Office iii shearing shed b the actual distance from the bridge to the windmill c using a piece of string, the actual road distance i between the hotel and the nearest farm house
Windmill Hotel
Farm house
Bridge Post Office
Shearing shed
Dam
A B C D E F G H I
ii from the school to the Post Office. 5 The grid alongside shows the region around an inlet by the sea. Find:
School
111
13
Farm house
J K L M N O P Q R
Golf Club
12
Hotel
11
a the position of the i Whaling Station iii Golf Club v Jetty
Restaurant Hotel Bike Hire
ii iv vi
10 9
8 7
b If the scale is 1 grid unit represents 1 km, use a piece of string to find the actual distance along the road from the bike hire shop to the restaurant.
Jetty
6 5
Restaurant
Whaling Station
4 3 2
Bike Hire
1 A
6
B C D
E
F
G H
I
J
K
L M N
O
13 12 11 10
BRECON STREET
Mini Golf
Gondola
Ambulance Station
9 8
Kiwi Park
Primary School
Fire Station
ISLE STREET Rugby Fields
MAN STREET Carpark Berkels Burgers
Ski Hire
7
Medical Centre
Queenstown is in the South Island of New Zealand. It is a holiday resort. Using the street map alongside, answer the following questions. ii D6. a Find what is at: i B10
SHOTOVER STREET
THE MALL
4
1
A
b Give the location of the Ski Hire.
CHURCH STREET
c Give the location of the intersection of Isle Street and Camp Street.
DOC Police
2
Pizza Hut
Carpark
QUEENSTOWN BAY
3
BEECH STREET Council
5
Bike Hire
CAMP STREET
REES STREET
6
d B
C
D
E
F
G
H
represents the Post Office. What is its grid reference?
112
LOCATION AND POSITION (CHAPTER 7)
Unit 51
Direction
Although scales, grids, plans and maps help us locate information, it is much easier to find places when we know which direction to look.
Compass points
N W
E N
S
NW
The four main compass points are North, South, East and West as shown.
NE
W
They are called the cardinal points. Using these 4 directions which are 90o apart, we can divide each 90o into 45o s to make ‘half way’ directions: NE, SE, SW and NW.
E
SW
SE
So, for example, south-west (SW) is half way between south and west.
S
Exercise 51 1 Using the 8 main compass points, determine the direction from O of: a i J ii K iii L iv M v N b Determine the direction of: i M from L ii K from N
iii
O from M
L
K
W
E S
O M
N
J
N
2 A 2 dollar coin has been hidden on the school’s playing fields. For a location activity, 4 students have each been given a field compass, a metre trundle wheel and one of the following sets of directions. Each student must start from outside the classroom door. Directions must be followed in order from left to right. The direction codes are N = north, S = south, E = east and W = west. The distance code is m = metres.
Mike:
40 m N, 30 m W, 10 m N, 60 m E, 40 m S, 20 m E
Mary:
20 m N, 50 m W, 60 m N, 70 m E, 50 m S, 10 m E, 20 m N, 50 m E
Melissa:
30 m W, 30 m N, 30 m E, 30 m N, 30 m E, 10 m N, 30 m E
Martin:
60 m E, 40 m N, 20 m E, 40 m N, 80 m W, 20 m S
Scale: 1 grid unit is 10 m
N W
Classroom door
E S
Use different coloured pencils to draw the path each person followed. Then answer these questions: a Which student will be closest to the coin? b Using only 90o (right angle) turns, find GRID i which student walked the longest distance ii which student was the furthest from the coin. iii For each student, describe the shortest way to the coin from the point where their directions ended. PRINTABLE
c With the least number of turns, what is the shortest distance and direction from the classroom to the coin? d When the teacher walked in a straight line from the classroom door to the coin, how far did she walk?
LOCATION AND POSITION (CHAPTER 7)
3 The pattern alongside is a representation of a hang glider ’s flight path. The path is drawn to a ground distance scale of 1 grid unit represents 2 kilometres.
C
D
a Find the total ground distance of the flight.
c How far from the finish would the hang glider be if she landed 4 km SE of D?
B
N M
The circles are 10 km, 20 km, 30 km, 40 km and 50 km from their centre ². ² represents a point at the top of the hill. represents a road. Directions are given by the coloured compass points.
NW
O
D T
30 km SE 40 km NE 50 km W?
ii iv
NE R
J
Q
E
S
P
N
V
W
W
©A
F
I X
Starting from the top of the hill ², which point is i iii v
E
A (start/finish)
4 In the following diagram:
a
N
F
b Describe the flight path and find the distance travelled over the ground if the pilot first headed off in i an easterly direction (starting A - B) ii a north easterly direction (starting A - F).
113
C
H
G
¯
E
B O
Y
50 km NW 20 km SW
F
L
SW
Z N
b Determine the shortest distance i and direction by helicopter, of F from ² ii along the winding road from ²
to F
iii by helicopter from F via ~
to ²
SE
U
S Via is a Latin word which means ‘by way of ’.
iv around the circumference of the circle which has a 10 km radius.
5 Use the map of the Victorian Snow Fields shown alongside to answer the following questions.
Myrtleford
a How far by air is Harrietville from Mt Feathertop? 0
b Approximately how far by road is Myrtleford from Bright? c Locate a town west of Mt Beauty. d Locate a town south east of Mt Buffalo Chalet. e In what direction is Myrtleford from Bright? (Use the 8 main compass points.)
10
20 km
Bright Mt Beauty Mt Buffalo Chalet
Harrietville
Falls Creek Mt Feathertop
Mt Hotham
114
LOCATION AND POSITION (CHAPTER 7)
Unit 52
Plans
A plan is a 2-Dimensional representation of an object. It is usually drawn to scale on a plane (flat) surface.
a gym b pool
Classroom Block 1
Pool
c canteen d library e classroom block 1 f classroom block 2 2 This plan shows the front view of a small two storey holiday house. Use the scale given to find:
Library
Gym
Classroom Block 2
1 This plan shows some of the buildings in a school. Using the scale on the plan, find the actual size of the:
Canteen
Exercise 52
Scale: 1 cm represents 5 m Scale: 1 cm represents 1 m
a the maximum height of the house b the measurements of the smaller downstairs window c the maximum width of the house d the width between the walls of the second storey.
3 An area is to be paved as shown in the plan alongside. a Use the scale shown to find the area to be paved. b If the paving contractor charges $32 for each square metre of paving, find the total cost of the paving.
Scale: 1 cm represents 2 m Example of paving
4 This bathroom/toilet floor is to be tiled. a Calculate the area to be tiled excluding the basin and cupboards. b If the tiles cost $25 per square metre, what will be the cost of the tiles?
Scale: 1 cm º 1 m
LOCATION AND POSITION (CHAPTER 7)
5 The floor plan of a house is shown alongside. The measurements of each of the rooms are shown in metres. a Find the measurements of the dining room and lounge room combined. b If this area is to have a wooden floor which costs $50 per square metre, find the total cost of the wooden floor. c Find the cost of carpeting the second bedroom if the carpet costs $36 for each metre 2:7 m wide. d Make a key which explains the symbols used in this plan, for example, W.C. =
=
6 Using a scale of 1 cm represents 2 m, design a classroom that will fit:
² ² ² ² ² ²
14 desks which are 2 m by 1 m, leaving room for the 28 chairs 1 teacher’s desk which is 3 m by 1 m 4 book trolleys, each 1 m by 12 m PRINTABLE 1 computer and printer table, 3 m by 1 m GRID PAPER 1 white board, 4 m by 0:2 m 2 pin boards, 4 m by 0:2 m
The total area of the classroom floor is to be 80 m2 . Do not forget to include positions of doors and windows. If you have space left over, be imaginative with your classroom. entrance
7 Complete this plan of a maze. Keep your track 2 mm wide.
Include dead ends to confuse the user. If the scale is 1 unit ´ 1 m, find the shortest distance you could walk from the entrance to the pot of gold.
PRINTABLE MAZE
pot of gold
115
116
LOCATION AND POSITION (CHAPTER 7)
Unit 53
Coordinates
Graphs and maps are drawn on grids. Any point on them can be found by naming its coordinates. The first number is the x-coordinate and the second is the y-coordinate. 8
y
7 6 5
P 4 steps vertically
4 3 2 1
We write this in coordinate form as (5, 4).
5 steps horizontally 1
2
3
4
5
6
7
(7, 2) would represent the point which is 7 units horizontally and 2 units vertically from the origin.
x
8
Exercise 53 1
The horizontal coordinate is always named and located first.
In the graph alongside, the point P is 5 steps (units) horizontally and 4 steps (units) vertically from the origin.
DEMO 8
a Find the coordinates of the points:
7
i A ii B iii C iv D. b State the horizontal and vertical movements to go from:
A to C
i 2
7 6 5 4 3 2 1
B
6 5
C
4
A
3
B to D.
ii
y
2
D
1
y J
D
A H
C
B
E
I
1
2
3
4
5
6
7
x
8
a Write down the coordinates for each of the points shown.
G
b State the horizontal and vertical movements to go from:
F x 1 2 3 4 5 6 7 8
i A to B iv G to H
ii C to D v I to J
3 Draw a set of axes with scales and plot the following points. a P(4, 2) b Q(6, 3) c R(2, 7) d S(1, 8) e
iii E to F
T(5, 2)
f
U(3, 6)
4 The grid system shows a warehouse and six shops to which it delivers. a Write down in coordinate form the position of the warehouse and each of the shops. b If each square is 1 km by 1 km, find the total distance travelled if a truck delivers goods by going from the warehouse to shops A, B, C, D, E and F in order and then back to the warehouse along the grid lines.
y 7 shop C 6 shop D 5 shop F shop B 4 warehouse 3 2 shop E shop A 1 1 2 3 4 5 6 7 8
5
5 y 4 3 2 1 A
The grid alongside shows the vertices of the base of square ABCD. Draw a set of axes and use it to find the other 2 vertices. B
1 2 3 4 5 6 7 8
x
x
LOCATION AND POSITION (CHAPTER 7)
Activity
117
Hopping around a number plane Aim:
To draw and name a coordinate picture.
What to do: 1 Use an A4 sheet of 5 mm graph paper, pencil and ruler. 2 Number your horizontal axis (x) 0 to 30. 3 Number your vertical axis (y) 0 to 50. 4 Carefully follow the directions to locate and join the pairs of coordinates in the correct order.
The first number is always the horizontal or x-axis.
Begin with: (20, 16), (24, 19), lift pencil (21, 15), (25, 18), lift pencil (22, 14), (25, 16), lift pencil (22, 12), (25, 14), lift pencil (12, 45), (21, 45), (21, 43), (20, 44), (18, 44), (16, 43), (14, 44), (12, 44), (12, 45) Shade shape made by last nine pairs of coordinates. (18, 37), (22, 38), (22, 40), (23, 40), (23, 39), (25, 40), (25, 39), (23, 38), (25, 38) (25, 37), (23, 37), (23, 36), (22, 36), (22, 37), (18, 35), lift pencil (13, 35), (12, 31), (11, 28), (11, 27), (12, 26), (14, 24), (17, 24), lift pencil (22, 26), (26, 30), (27, 34), (24, 31), (22, 26), (21, 22), (22, 20), (20, 19) (18, 18), (16, 16), (11, 16), (11, 18), (18, 22), (12, 24), lift pencil (20, 45), (23, 46), (21, 47), (20, 47), (19, 46), (16, 46), (15, 47), (14, 49), (13, 47), (14, 46), (14, 45), lift pencil (15, 19), (13, 18), (13, 16), lift pencil (17, 19), (15, 18), (15, 16), lift pencil (14, 42), (15, 41), (16, 40), lift pencil (15, 41), (12, 40), (11, 40), (10, 41), (10, 43), (12, 45), lift pencil (12, 31), (11, 32), (9, 32), (6, 29), (9, 23), (5, 21), (2, 19), (2, 17), (8, 17), (9, 19), (12, 20), (12, 26), lift pencil (6, 20), (4, 19), (4, 17), lift pencil (8, 20), (6, 19), (6, 17), lift pencil (10, 43), (11, 42), (12, 43), lift pencil (13, 35), (9, 36), (9, 34), (8, 34), (8, 35), (7, 33), (6, 33) (7, 35), (5, 35), (5, 36), (7, 36), (5, 38), (6, 39), (8, 37), (8, 38), (9, 38), (9, 37), (13, 36) (14, 36), (14, 38), (13, 38), (11, 39), (11, 40), lift pencil (17, 24), (18, 28), (16, 31), (19, 33), (18, 35), lift pencil (24, 31), (19, 33), lift pencil (18, 37), (18, 39), (20, 41), (20, 44)
GAME
118
LOCATION AND POSITION (CHAPTER 7)
Unit 54
Review of chapter 7
Review set 7A 1
a Show the scale 1 cm represents 20 km in another way. b A map has a scale of 1 cm represents 20 km. What distance on the map would represent an actual distance of 170 km? 7
2 The animal pens at the local zoo have the positions given on the grid. Find: a the position of the pen for the i penguins ii giraffes iii kangaroos iv foxes b how to get from the i bird pen to the alligator pen ii llama pen to the ape pen iii monkey pen to the seal pen.
6
4 3 2
a
i ii iii
iv
Sydney to Brisbane Adelaide to Perth Melbourne to Adelaide then to Darwin Cairns to Alice Springs to Perth.
b Use the 8 compass directions provided to give approximate directions for the following. From Adelaide, what is the direction of: i ii iii iv
Mount Isa Hobart Canberra Rockhampton?
NW
N
Panda
Kangaroos
Penguins Deer
Monkey
Hyena
Elephant
Llama
Alligator
Giraffe B
C
D
E
F
G
H
NE
E S
Seals
Apes
Bears
1
W SW
Foxes
Birds
5
A
3 Using the scale on the map of Australia, find the distances from:
Tigers
Lion
Darwin
SE
Cairns
Alice Springs
Mt. Isa Rockhampton Brisbane
Kalgoorlie
Perth
Adelaide Canberra Sydney Melbourne scale: 1 cm represents 450 km Hobart
4 Refer to the Victorian Snow Fields map on page 113. a How far from Mt Feathertop is Mt Hotham? b How far by road is Harrietville from Bright? c Using the eight main compass points, in what direction is Mt Buffalo Chalet from Mt Hotham? d Give the direction of Mt Hotham from Mt Beauty. e Locate a town north west of Falls Creek. 5
a Draw a set of axes on graph paper and plot these points: A(1, 3), B(3, 7), C(4, 9). b State the horizontal and vertical movements to go from C to A. c Draw a straight line to pass through A and C. Give the coordinates of the point where the line cuts the vertical axis.
119
LOCATION AND POSITION (CHAPTER 7)
Review set 7B
KING WILLIAM ST
1 A house plan shows a scale of 1 : 250. On the plan the lounge room measures 2 cm £ 1:6 cm. a Calculate the actual dimensions of the lounge room. b Find its area. c Find the cost of tiling the lounge room if tiles cost State $38 per square metre. Bank
a Give the location of the centre of Victoria Square.
3
c Give the location of
5
State Bank Supreme Courts the Central Market the Town Hall St Francis Xaviers Cathedral.
i ii iii iv v
MORIALTA ST
4
SQUARE Tram terminus
St Francis Xaviers Cathedral
Law Courts
GOUGER ST
ANGAS ST
6
NELSON ST
F2.
Supreme Courts
CARRINGTON
A
B
C
D
E
3 Use the map of Central Australia below to answer these questions. a How far is Uluru from Mt Olga? b How far by road is Hermannsburg from Alice Springs? c How far by plane is Uluru from Alice Springs? d What feature is due south of Alice Springs? e What direction is Hermannsburg from Yulara? (Use the 8 main compass points.) CENTRAL AUSTRALIA
to Darwin
Pine Gap
Alice Springs
Hermannsburg Kings Canyon National Park
0
40
80 km
Yulara
Chambers Pillar Historical Reserve
Mt Olga Uluru (Ayers Rock)
to Coober Pedy
4 The grid shows the important places in a country town. a Write the position of each place in coordinate form. b State i ii iii
SA Police
WAKEFIELD ST
Hilton International Hotel
COGLIN ST
i B4 ii
GROTE ST
Central Market
b Find what is at
VICTORIA
Baptist Church
CHANCERY LANE
2
Pilgrim Uniting Church FLINDERS ST
St Aloysius College
2 Use the map of Adelaide city centre to answer these questions.
GPO
Town Hall
GAWLER PLACE
1
PIRIE
the horizontal and vertical steps to go from the Police Station to the Cemetery the School to the Town Hall the Post Office to the General Store.
8 7
y cemetery
school post office
6 5
garage
4
church
3 police 2 station general 1 store 1
2
town hall bank 3
4
5
6
x 7
8
F
120
SOLIDS AND MASS
Unit 55
Solids
A solid is any object which has length, width (breadth) and depth (height) and takes up space. We say the shape is 3-dimensional or has 3 dimensions.
height height
front face
width
width
length
length
A solid shape may be full (for example, a block of wood) or empty (for example a rubbish bin).
A solid may have flat or curved surfaces.
Prisms A prism is solid which has a constant shaped cross-section along its entire length. The side faces are always rectangles. These are other prisms.
CHAPTER 8
These are rectangular prisms.
SWISS Their cross-sections are
Exercise 55 1 By looking at the end shape of these prisms, choose the correct name from: square prism triangular prism rectangular prism hexagonal prism cylinder octagonal prism pentagonal prism a
b
c
d
e
f
2 The faces of a prism include the base shape and the other rectangular faces. For each shape in question 1, draw a sketch of all different faces. 3 Using matchsticks and plasticine make models of: a a prism which has faces which are all the same (identical). Name it! b a prism which has two triangles and three rectangles as faces. Name it!
SOLIDS AND MASS (CHAPTER 8)
121
Pyramids A pyramid is a solid with a polygon base. The other faces are triangles which come from the base to meet at a point called the vertex. 4 By looking at the shape of the bases of these pyramids, choose the correct name for each from the following list.
square-based pyramid pentagonal-based pyramid a
rectangular-based pyramid hexagonal-based pyramid
b
c
triangular-based pyramid
d
e
5 Draw the different shapes of the faces of each pyramid in question 4. 6 Use matchsticks and plasticine to make a a triangular-based pyramid
b
a square-based pyramid
7 Cuts through shapes are called cross-sections. Draw the cross-section shape shown in these figures: a b c
DEMO
d
e
f
8 For each solid in question 7 imagine the cross-section moving up and down on the figure. Does the shape remain the same or vary?
Cylinders, spheres and cones
cylinder
sphere
cone
Some shapes have curved surfaces. The pictures above show three common shapes. The tennis ball has the shape of a sphere while the ice-cream cone is roughly the shape of a cone. A cone can be thought of as a special case of a pyramid with a circular base.
122
SOLIDS AND MASS (CHAPTER 8)
Unit 56
Polyhedra
In this unit we will mainly consider those solids with flat surfaces which are polygons. A solid like this is a polyhedron. More than one polyhedron are polyhedra.
face
edges
Each flat surface is called a face. An edge is where two faces of a polyhedron meet. A vertex (plural vertices) is where three or more faces meet. It is a corner of the polyhedron.
vertex
vertices
For example, ²
B
A
C D F
G
E
H T
²
S P
R Q
The i ii iii
rectangular prism has 8 vertices A, B, C, D, E, F, G, H 12 edges AB, AD, AE, BC, BF, CD, CG, DH, EF, EH, FG, and GH 6 faces, rectangles ABCD, ABFE, BCGF, CDHG, ADHE and EFGH.
The i ii iii
pyramid has 5 vertices P, Q, R, S, T 8 edges, PQ, PS, PT, QR, QT, RS, RT, ST 5 faces, square PQRS and triangles PQT, PST, QRT and RST.
Exercise 56 1 For each of the following figures i name the shapes of the faces ii iv list the faces v a b
list the vertices name the figure B
L
M
N
I B A
C
E
G H
Z
X
Y W
V
f
D F
c
H R
e
list the edges
G
F
E
J
d
2
D
A
K
C
iii
G B
S P
F
Q
I
L
C D
A
H K
J
E
a For each figure in question 1, copy and complete the table below.
Number of vertices (V )
Number of faces (F )
Number of edges (E)
V +F ¡E
i ii iii iv v vi b What do you notice about the value of V + F ¡ E
for each figure?
c Make up another polyhedron of your choice to see if you get the same value for V + F ¡ E.
SOLIDS AND MASS (CHAPTER 8)
3 For each solid drawn below find i the number of vertices (V ) a
ii the number of faces (F ) b
4 Check your answers to question 3 by finding V + F ¡ E
iii c
in each case.
5 Find the name of each solid from the clues given: a All six faces are identical in shape. b It is a prism with two triangle and three rectangle faces. c It is a pyramid with four triangle faces and one square face. d It is a pyramid with four identical triangle faces. e This solid has a hexagonal base and all other faces are triangles. f This solid has one uniformly curved surface. g This solid has two flat faces and a curved surface. h This solid has one flat surface and a curved surface. 6 Example: This is a pyramid with 5 faces.
Draw a sketch of a a prism with 5 faces b a pyramid with 8 edges c a pyramid with 4 vertices d two prisms with 6 faces, 12 edges and 8 vertices e a prism with 8 faces, 18 edges and 12 vertices. 7 What is the difference between a triangular prism and a triangular pyramid? (Draw both and find the number of faces, edges and vertices for each solid.) 8 V +F ¡E =2 for polyhedra.
is known as Euler’s rule
Does Euler’s rule work for a a cylinder b a cone?
123
the number of edges (E)
124
SOLIDS AND MASS (CHAPTER 8)
Unit 57
Drawing solids
Drawing a solid on a sheet of paper can be difficult. Trying to represent something which has 3 dimensions (length, width and depth) on a plane which has only two dimensions (length and width but no depth) is not easy. The methods used to draw a line, figure or solid on a 2-dimensional plane are called projections or perspective drawings. By following the instructions in the exercise below, you should be able to draw several special solids.
Exercise 57 1 Drawing a rectangular prism or cube (box)
Step 1:
Draw a rectangle (or square) for the front of the box.
Step 2:
Draw equal lines back from the front face at the same angle.
Step 3:
Draw the final four lines to complete it. The unseen lines are shown with dots or dashes.
Draw 3 rectangular prisms of different shapes. 2 Drawing a cylinder
Step 1:
Draw an ellipse for the top face.
Step 2:
Draw the sides of the cylinder from the outer ends of the ellipse.
Step 3:
Complete the cylinder by drawing another ellipse for the base. Once again the unseen edge is dotted.
a Draw two cylinders of your own choosing using the method above. b Now draw one cylinder which is lying on its curved surface. Hint: Start with a horizontal line representing the flat surface. 3 Drawing a cone
Step 1:
Draw an ellipse to represent the base and mark its centre with a dot. The back edge should be dotted (or dashed).
Step 2:
From the dot, draw a vertical line to where the top point of the cone would be.
Step 3:
Join the slant edges to complete the cone.
a Draw 2 cones of your own choosing using the method given. b Draw 1 cone which rests on its curved surface.
SOLIDS AND MASS (CHAPTER 8)
125
4 Drawing a pyramid
Step 1:
Draw a parallelogram to represent the base.
Step 2:
To find the centre of the parallelogram, draw the diagonals and find their point of intersection. Draw a point above the centre to represent the apex (or top) of the pyramid.
Step 3:
Join each vertex of the base to the apex to complete the pyramid.
Step 4:
Looking at the picture of the pyramid above, not all edges can be seen at the one time. To show this we draw the invisible edges as dotted lines.
Note:
4
When the pyramid is viewed from above, all edges can be seen as shown here.
a Draw a square-based pyramid of your own choosing. b Draw a rectangular-based pyramid.
5 Follow these constructions and then name the shape you have drawn. Step 1: Draw an equilateral triangle. Step 2: Draw a second equilateral triangle the same size and shape but to the right and above the first. Step 3: Join the corresponding corners of the triangles. 6 Here is an oblique projection of a rectangular box, 2 units long by 1 unit wide by 1 unit high. (2 £ 1 £ 1 prism).
First measurement is width. Second measurement is breadth. Third measurement is height.
1 unit 2 units (eg. cm)
this line is drawn shorter than 1 unit
Draw an oblique projection for these rectangular prisms. a a 1 £ 1 £ 2 prism b a 1 £ 2 £ 1 prism c a 2 £ 2 £ 1 prism 7 Draw an oblique projection for the following a triangular prism b pentagonal prism
c
hexagonal prism
d
a 2 £ 1 £ 2 prism
126
SOLIDS AND MASS (CHAPTER 8)
Unit 58
Making solids from nets
A net is a shape or figure which is drawn in a plane and can be cut out and folded to form a solid.
For example, a cube has 6 faces, each of which is a square. So, the net for a cube must consist of 6 squares. However, not all arrangements of 6 squares in a plane can be folded to form a cube. Here are two possible arrangements: A
B
PRINTABLE TEMPLATE
A
can be folded into a cube by cutting along the solid lines and folding along the dotted lines.
B
cannot be folded to form a cube.
Try both of these for yourself. Nets can also be drawn for other rectangular and non-rectangular prisms as well as pyramids. Just as there is more than one net for a cube, other shapes can also have more than one net each. For example, the triangular prism could have nets
or
Exercise 58 1 Which solids can be formed from these nets? (Dotted lines are bend lines.) a b c
d
e
f
2 See if you can find all the different nets for making a cube. Step 1: Draw a ‘pattern’ of 6 squares which you think can be folded to form a cube. Step 2: Cut out and fold the pattern you have drawn to see if a cube can be formed. Keep a record of what you have tried and whether it worked. Step 3: Repeat the above until you have found all possible different patterns.
Patterns which are rotations or reflections of another cannot both be counted. For example,
and
cannot both be counted.
[Note: You should find at least 8 different patterns.]
SOLIDS AND MASS (CHAPTER 8)
127
3 Draw a possible net for these solids. a
b
c
e
f
1
2
1
d
1
3
2
[Hint: Before you try to draw a net, decide which shapes form the faces.] 4 A die is a solid in the shape of a cube. (Sometimes the corners are rounded.) The numbers 1 to 6 are arranged on the faces of the die so that all opposite faces have the same sum.
What number will be on the face opposite the face with the following number? a 4 b 5 c 1 5 Click on the icon to obtain the following nets which can be printed.
Make each solid from light card. a
hexagonal prism
b
octahedron
c
dodecahedron
PRINTABLE NETS
Activity
Nets as packages In groups: 1 List the products you know that have packaging made from “nets”, for example food and electrical goods. 2 List the advantages of using nets. 3 Design and label your own nets for removalist cartons, canned food or breakfast cereals.
128
SOLIDS AND MASS (CHAPTER 8)
Unit 59
Different views of objects
Objects which are three-dimensional can be represented by 2-dimensional drawings. A plan shows what an object looks like from above. An elevation is a view from the front or from the side.
Exercise 59 1 Match each elevation (side view) with its plan (top view). a b c
A
B
1L Milk
C
2 Draw a plan and elevation of a a chair b a TV set
c
3 Draw a plan and elevation(s) of these solids a b
c
d
D
a car
d
a golf ball
d
4 Draw an object which matches these plans and elevations.
top view (plan) a b
c
d
e
f
front elevation
side elevation
SOLIDS AND MASS (CHAPTER 8)
Activity
129
Constructing models In this section it is recommended that DIME solids and DIME cubes are used. Other solid construction materials could also be used.
It is easier to draw solids if you build them first. Construct these 3 dimensional solids using cubes. b
a
The shape in a appears to take 6 blocks to build. Could it take more? Explain! You will find it easier to draw the solids if you build them with cubes first. Example: Left end
The different views: the plan, elevation and end elevations of this object, are drawn alongside.
Right end
Front
VIDEO CLIP
Plan view 2 Left end
Right end
2 blocks deep
1
Plan view
1 block deep
Front
5
i ii
Use cubes to make these solids. Draw plan, elevation and end elevation views for the solids.
a
6
i ii
b
c
d
Use cubes to make these solids. From the views draw an oblique projection of the solid.
a
b L end
front 2
L end
R end
1
front 1
2
R end
1
2
plan
plan
c
d L end
front 1
1
1
1
plan
R end L end
front 2
2
plan
R end
130
SOLIDS AND MASS (CHAPTER 8)
Unit 60
Mass
Mass is measured in grams, milligrams, kilograms and tonnes. Mass is used to find how heavy an object is.
Mass measuring devices Three mass measuring devices or instruments are shown here. Modern chemical balances are electronic with digital displays. They are very accurate and measure to small parts of a gram.
The gram One gram is how heavy a 1 cm by 1 cm by 1 cm cube of water would be. We measure small masses using grams. Good estimates of one gram ² the mass of one paper clip
²
6 cm
Cut this off and crumple it into your hand. This mass is extremely close to one gram.
A4 sheet of 80 gsm photocopy paper
21 cm
The kilogram One kilogram is the mass of one litre of water. Good estimate of one kilogram Hold a full one litre carton of milk in your hand.
water inside container
This weighs slightly more than one kilogram.
1 cm
1 kilogram = 1000 grams 1 cm
The milligram
1 cm
Very small objects such as tablets have small masses. Milligrams are used to measure these masses. 1 milligram =
1 1000
of a gram 500 mg
200 mg
12 mg
2 mg
The tonne Very heavy objects such as motor cars, elephants and shipping containers have very large masses. 1 tonne = 1000 kilograms
Exercise 60 1 Which of these would be heavier than one kilogram? A a motor car B an apple D a large cat E a green pea
C F
a tea cup this book
SOLIDS AND MASS (CHAPTER 8)
131
2 These items have the units left off. Write them down as grams (g) or kilograms (kg). a b c d
20 .....
150 .....
40 .....
250 .....
3 State what unit of mass you think would be sensible to use when stating the mass of the following objects. a a cow b a drawing pin c a bag of potatoes d a CD e an elephant f a truck g a sheet of paper h a computer i a pair of scissors j the pencil you are writing with
Unit abbreviations g mg kg t
represents represents represents represents
If you are changing a larger unit into a smaller unit then you multiply.
grams milligrams kilograms tonnes
´1000 t
´1000 kg
´1000 g
¸1000
¸1000
mg ¸1000
If you are changing a smaller unit into a larger unit you divide.
To convert:
² 2 12 kg to g,
² 400 g to kg, 400 g = 400: ¥ 1000 kg So, 400 g = 0:4 kg
2 12 kg = 2:5 kg
2:5 kg = 2:500 £ 1000 g 2:5 kg = 2500 g
4 Find how many kilograms are in a 2000 g b 500 g
c
250 g
d
87 g
5 Find how many kilograms are in a 3t b 54 t
c
0:25 t
d
0:02 t
6 Find how many grams are in a 3 kg b
1:25 kg
c
0:8 kg
d
0:012 kg
7 Find how many tonnes are in a 7000 kg b 5526 kg
c
600 kg
d
50 kg
8 Find how many milligrams are in a 20 g b 2g
c
0:2 g
d
0:02 g
9 A packet contains 10 tablets, each weighing 500 mg. Find the total mass of the tablets, in grams. 10 A van can carry a maximum weight of 700 kg. Write this mass in tonnes. 11 Sliced ham costs $9:80 per kilogram. How much does it cost to buy 300 grams? 12 A large sheet of paper weighs 80 g. Find the mass of 1000 sheets. Answer in kilograms. 13 Mary and David are moving house. They have packed their household items into boxes. There are 53 boxes and they guess the boxes have an average mass of 12 kg. What mass are they moving? Is it more or less than half a tonne? 14 Hans bought these items at the supermarket: 350 g of bacon, 500 g of sausage, 340 g of cheese and 110 g of olives. Find the total mass of his purchases in kilograms.
132
SOLIDS AND MASS (CHAPTER 8)
Unit 61
Review of chapter 8
Review set 8A 1 Name the following special solids. a
2
For a b c d
b
O
the diagram given name the solid list the vertices name each face and give its shape list the edges.
C
D A
B
1
3 Draw a net for a rectangular prism shown:
1 3
4
a
Sketch a cylinder.
b
Draw a net for a cylinder.
Draw the solid alongside from different views showing the plan, elevation and end elevations.
5
6 From these views draw the oblique projection of the solid. L end
7 Give units suitable to measure the mass of a a pine cone
b
a semi-trailer
8 Five apples are approximately the same size. They weigh 1:2 kg. a Write 1:2 kg as grams. b Find the approximate weight of one apple in grams.
Review set 8B 1 Name the following special solids. a
2 For a b c d
the diagram given name the solid list the vertices name each face and give its shape list the edges.
b
D
A E B
3 Draw an oblique projection of a rectangular prism 2 units long by 2 units wide by 1 unit high.
C
front
F
R end
1 2 1 plan
SOLIDS AND MASS (CHAPTER 8)
4
a
Sketch a pyramid.
b
133
Draw a net for a pyramid.
5 Draw the solid alongside from different views showing the plan, elevation and end elevations. front
6 A block of chocolate weighs 250 grams. Find the mass in kilograms of a 10 blocks b 1000 blocks. 7 My Alaskan Malamute is a big strong dog. He weighs 54 kg, and I am told that he can pull a load which is 10 times his weight. a What weight can he pull? b Write your answer to a in tonnes.
Review set 8C 1 Name the following special solids. a
2
A B
C E
F
3
a
D
H G
Sketch a cone.
b
For a b c d
the diagram given name the solid list the vertices name each face and give its shape list the edges.
b
Draw a net for a cone.
4 Draw the solid alongside from different views showing the plan, elevation and end elevations. 5 From these views draw an oblique projection of the solid.
2 2 L end
6 Give units suitable to measure the mass of a a feather b 7
front
R end
1
plan
a biscuit
a Write 4 tonnes as kilograms. b If the total mass of luggage was 4 tonnes, how many people were travelling if they each took their maximum luggage allowance of 20 kg?
134
DATA COLLECTION AND REPRESENTATION
Unit 62
Organising data
Once data has been collected it needs to be organised into a table or graph. This must be of good standard so that information can be obtained from it. We organise data into:
Frequency tells how often a score occurs.
a dot plot where each dot represents one data value a frequency table with a tally column where each data value is represented by a stroke, j , and these © . are organised into bundles of 5 shown by jjjj ©
² ²
Tallies © , to complete a ‘bundle’ of 5. After you have put four j s, the fifth stroke is shown like this jjjj ©
j
represents
1
jj
represents
2
jjj
represents
3
represents
4
© jj © jjjj
represents
7
© © jjjj
represents
5
© jjj jjjj ©
represents
8
© j jjjj ©
represents
6
jjjj
and so on
Exercise 62 1 What frequencies are represented by the tally counts? © jjjj © jjjj © jjj a jjjj © b jjjj © © c
© jjjj © jjjj © j jjjj © © ©
d
© jjjj © jjjj © jjjj © jjjj © jj jjjj © © © © ©
Consider this data: The ages of netball players in a club are: 17 21 18 17 19 23 16 27 22 31 26 26 19 23 24 27 24 25 23 19 16 25 26 25 ² A dot plot of this data is:
Dot plot for netballer ages
32
30
31
28
29
26
27
24
25
22
23
20
21
18
19
16
17
CHAPTER 9
DEMO
We can often make comments about the data once it has been put in graphical form. For example, we could say ‘Most of the players are in the 23 to 27 age group. One player is much older and there is a reasonably large group of young players’. ² A frequency table of this data is:
Age 16 17 18 19 20 21
Tally jj jj j jjj j
Frequency 2 2 1 3 0 1
Age 22 23 24 25 26 27
Tally j jjj jj jjj jjj jj
Frequency 1 3 2 3 3 2
Age 28 29 30 31 32 33
Tally
j
Frequency 0 0 0 1 0 0
2 The weights of 15 soccer players were recorded to the nearest kg and the following data was obtained: 62 64 66 66 61 60 70 58 55 69 60 61 62 67 62 Construct a dot plot display of the data.
DATA COLLECTION AND REPRESENTATION (CHAPTER 9)
135
3 Construct a frequency table with headings ‘score’, ‘tally’, ‘frequency’ for the data. 7 8 8 5 6 8 5 4 9 7 7 7 11 6 8 5 7 9 10 3 8 9 6 5 4 8 8 8 10 2 6 6 7 6 6 5 8 9 8 8 a c
What is the frequency for a score of 8? What fraction of the scores are 7s?
How many scores are there? What fraction of the scores are less than 6?
b d
4 Convert the following dot plot into a frequency table.
number of students
Shoe sizes of year 6 students
Use headings of ‘shoe size’, ‘tally’, ‘frequency’. Why is this graph called a vertical dot plot?
8 6 4 2
size 5
5 Jodie noticed the colour of cars passing her in the street. She used a code to record the colours where R = red, B = blue, G = green, W = white and O = other colours. Jodie’s results were recorded for a sample of 50 cars.
6
7
8
9
10
11
12
13
Dot plot of car colour data number of cars
25 20
BGWWR OGWRW OOBBG OGRWR WWWGB BBGGW WWWOG WOBWW RWWRB BBBWR
15 10
a Complete a dot plot of the data using a grid like the one shown above.
5
b What fraction of the cars were painted i white ii red iii red or blue?
O
W
G
B
R
colour
6 Two year 6 classes did the same mathematics test out of 10 marks. Their results were:
8 7 6 9 10 9 7 9 8 5 9 8 7 7 7 9 4 8 7 9 9 6 7 7 3 8 6 6 6 7 5 6 10 2 5 7 8 8 6 4 5 5 7 6 6
Class 6A Class 6B
a Draw separate dot plots for each class. b State the highest and lowest mark for each class. c Which class performed better at the test? 7 The weights of 30 calves were obtained to the nearest kilogram 4 weeks after birth. The weights were:
58 50 60 60 55 59 60 64 58 66 62 63 68 53 67 59 53 50 50 54 53 57 57 57 67 57 68 51 59 69 a Construct a dot plot of the data. b What fraction of the calves weighed more than 60 kg? 8
For a b c d
number of times
Hockey goals scored each match
0
1
2
3
4
5
6
7
8
9
10 goals
the dot plot: Write down the smallest score. Write down the largest score. Which score occurs most frequently? What is the difference between the second to highest score and the second to lowest score?
136
DATA COLLECTION AND REPRESENTATION (CHAPTER 9)
Unit 63
Pictographs and strip graphs
Pictographs A pictograph is made up of identical pictures where each picture represents several of a particular object.
Country of Car Manufacture for Parents with Children at Mawson Primary School
The pictograph alongside uses the symbol , where represents 50 cars. This is called a key or legend.
Australia Japan Germany England Other
Numbers less than 50 are shown by using part-diagrams. For example, may represent 25 cars. The picture is chosen to show the object being talked about and is usually simple and eye-catching.
represents 50 cars
Discussion
Discussion
² What are the disadvantages of using pictographs to represent data (information)? Hint: Exactly how many Japanese cars are there in the Parent Car Pictograph shown above? ²
represents 50 cars is called the scale. What must be considered carefully when deciding what the scale should be for a pictograph?
Exercise 63 1 If a d
represents 10 oranges sold at the canteen, show how to represent 20 oranges sold b 50 oranges sold c 5 oranges sold 35 oranges sold e 12 oranges sold f 27 oranges sold
2 If a
represents 20 plums picked, estimate how many plums were picked when shown as b c d
3 If
represents 1000 football spectators, show how to represent
a 4 a b c d
1500 spectators
b
2300 spectators Population in Australian States
What information does the pictograph contain? What does represent? Which state has the highest population? Use the pictograph to estimate the population of i South Australia ii Victoria iii Western Australia
NSW Victoria Queensland South Australia Western Australia Tasmania
represents a million people
ACT Northern Teritory
5 In the Primary Schools’ Summer Sports Competition we noticed that 27 schools had tennis teams, 24 had cricket teams, 19 had golf teams and 17 had swimming teams.
Using
to represent 5 schools, draw a pictograph to represent the given information.
6 Kelly’s computers had these monthly computer sales: Jan Feb Mar Apr May Jun Jul Aug 68 21 28 32 50 72 81 19
Use
Sep 26
Oct 43
Nov 53
Dec 87
to represent 10 computers and draw a pictograph of the sales data.
DATA COLLECTION AND REPRESENTATION (CHAPTER 9)
137
It is important when drawing pictographs to include ² ² ²
a title, which tells us what the graph is showing (for example, bread sales at a bakery) axes, a vertical line (goes up) and a horizontal line (goes across) labels on the axes, for example Number of loaves baked on the vertical axis. Days of the week on the horizontal axis.
²
a key, for example
represents 100 loaves sold.
7 Draw pictographs for the data in these tables. Make sure you include a title, names on axes and an appropriate key. a Colour of cars in a park b Chocolates eaten by five children in a week
Yellow Blue White Red Green
Alex Claire Emmy Tess Ally
12 10 20 4 6
5 3 7 9 3
Strip graphs A strip graph is usually a horizontal rectangle (or strip) 100 mm or more long. It is divided into smaller rectangles, each representing an item being considered. For example, this eye colour data has the strip graph shown. Strip graph of eye colour data Grey
Brown
Blue
Eye colour Blue Brown Grey Hazel
Number 7 11 3 4
Hazel
8 The types of cars in the carpark on the school’s gala day were recorded as follows:
Ford 20
Holden 18
Toyota 13
Mitsubishi 9
Other 10
Total 70
You wish to display this information on a strip graph. a Explain why a strip of length 7 cm or 14 cm would be very suitable to represent this data. b If you chose to use a strip of length 14 cm, how long would the strip for Toyota be? c For the data, draw a strip graph of length 7 cm. 9 Categories of membership at the Maryborough Golf Club A
B
C
D
The above strip graph shows the ratios of people who are members of different categories of a golf club. Category A members can play golf 7 days a week. Category B players can only play on weekends. Category C members can play only on week days. Category D members are social members who are not allowed to play golf. There are 680 members in the club. Construct a table showing the number of members in each category.
People who share or have something which is common to them all can be grouped into the same category.
138
DATA COLLECTION AND REPRESENTATION (CHAPTER 9)
Unit 64
Column and bar graphs
Once data has been organised (usually by frequency table), the next step is to display it. We have already seen display using a pictograph and a strip graph. In this unit we display by column or bar graphs. These are two of the simplest ways to display data. A column graph displays data vertically.
A bar graph displays information horizontally.
The bars or columns are separated by spaces. By comparing the columns of this graph can you see that ² households with families are most likely to own one or more computers ² households with no children are least likely to have a computer
% of households with one or more computers
80 60 40
W
ith
0
To ta l ch ild re n No ch ild re n Ca pi ta lC ity Co un try
20
How our income is spent
Housing Food
Alongside is a bar graph which shows how income (money) is spent in Australian households.
Transportation Health Care
Notice that the largest portion of income is spent on housing and the least is spent on recreation.
Clothing Recreation 0
5
10
15
20
25 30 percentage
The most frequently occurring item is called the mode.
Exercise 64 1
a Draw a column graph for the data given in the frequency table which follows:
Eye colour frequency
Blue 7
Brown 11
Grey 3
Hazel 4
DEMO
b From the table or graph i write down the most frequently occurring eye colour
2 Protein is an important part of our diet and is found in many foods. The graph shows the percentage of protein present in 5 common foods. Find a which of the 5 foods is the best source of protein b the percentage of protein in i beef ii milk.
% protein
ii what percentage of eye-colours are blue? 10
Percentage protein in foods
8 6 4 2 0
Beef
Rice Eggs
Fish
Milk
DATA COLLECTION AND REPRESENTATION (CHAPTER 9)
3 The graph shows the percentage of sales made by a butcher for different meat types.
139
Butcher Sales Chicken
a What is the most popular meat sold by the butcher?
Pork
b What percentage of the sales were i pork ii beef?
Beef
c If the butcher sold 800 kg of meat during one day, find the weight of i pork sold ii beef sold. 4 The graph shows the melanoma rates for women (per 100 000 population) in five countries.
Lamb 0
10
20 %
30
40
NZ non-Maori Australia
a Which country shows the highest rate? b Which country shows the lowest rate?
Norway
c How many Australian women (out of each 100¡000 population) would you expect to have melanoma?
Melanoma skin cancer
Denmark Sweden
d Discuss why New Zealand women are marked ‘non-Maori’.
0
Just as with pictographs, column and bar graphs must have ² a title
5
10
15
² labelled axes
20
25 30 35 Number of people
² a scale
5 Draw a column graph of the number of students who learn to play different musical instruments at a certain primary school. Use a scale of 1 cm ´ 4 instruments.
Drums 8
Piano 16
Flute 6
Recorder 14
Violin 10
Cello 4
6 Draw a bar graph of the way a family spends its money in a week. Use a scale of 1 cm represents $20.
Food $120
Electricity $30
Clothes $40
7 This graph gives a comparison between the average number of days of stay in hospital for males and females in various age groups. a Find the average stay in hospital for i 1 to 14 year old males ii 65 to 74 year old females b Discussion: What are the general trends shown by the graph? [Hint: One trend is that as ages of males increase their average stay in hospital increases.]
Activity
Entertainment $30
Other $50
AVERAGE STAY IN HOSPITAL Average Length of Stay (days)
Mortgage $120
18 MALE FEMALE
14 10 6 2 <1
1-14 15-34 35-49 50-64 65-74 75+ AGE
Gathering graphs What to do: 1 Find four different column or bar graphs from newspapers or magazines. Cut them out and stick them in your work book. Explain what each graph is showing.
2 Construct a bar graph of your own involving members of your class or family. Collect your own data, for example, heights, shoe size, ages, pets.
140
DATA COLLECTION AND REPRESENTATION (CHAPTER 9)
Unit 65
Line graphs Line graphs are a set of points joined with straight lines.
They are used to show how things change over time (and so sometimes are called time graphs or time series). For example, Sophie took temperatures at 2-hourly intervals. Here is a table of her results. Time Temperature (o C)
6 am 19
8 am 21
10 am 24
12 noon 25
2 pm 26
4 pm 24
6 pm 20
We can graph this data. 26 24
Notice:
Temperatures for the day
The line on the bottom of the graph shows time in hours and is called the horizontal axis. The line on the left of the graph shows temperature in degrees and is called the vertical axis.
°C
22
20 18
5 pm
6 am 8 am 10 am 12 pm 2 pm 4 pm 6 pm time
From the line graph we can read off information such as: ² The highest temperature measured during the day was 26o C at 2 pm. ² The temperature increased from 6 am to 2 pm and decreased from 2 pm to 6 pm.
The line graph can sometimes be used to estimate “in between” values. For example, the temperature at 5 pm was approximately 22o C.
1 Managers of a retail store conduct a customer count to help them decide how to roster sales staff. The results are shown in the line graph alongside. a At what time was there the largest number of people in the store? b At what time was there the lowest number of people in the store? c Describe what happened in the store between 3 pm and 4 pm.
Number of shoppers in store
Exercise 65 Customer count
50 40 30
20
10 0 9am 11am
1pm
3pm
5pm
7pm
d Use the graph to estimate the number of people in the store at 6:30 pm. 2 The graph gives the monthly profits for Julia’s Shoe Shop. Profits are given in thousands of dollars. a What was the profit for May? b In what month did the largest profit occur? c What was the average monthly profit from January to July? d A large supermarket chain opened near Julia’s Shoe Shop causing a sharp drop in her sales. When did this occur? e What was her average monthly profit for the last 5 months of the year?
profit ($000’s)
5 4 3 2 1 0
J F M A M J J A S O N D months
time
3 The given graph is similar to those shown on television during the broadcast of a one-day cricket match. It shows how the runs are totalled during an innings. A wicket falls when a batsman gets out. Each dot represents the fall of a wicket. Use the graph to estimate a the score at the end of the innings b how many runs were scored before the fall of the first wicket c during which over the third wicket fell d what wicket fell in the 37th over e during which over the score reached 200.
total runs
DATA COLLECTION AND REPRESENTATION (CHAPTER 9)
141
Fall of wicket
250 200 150 100 50 0
overs 0
5 10 15 20 25 30 35 40 45
fYour answer will be approximate and may not agree exactly with the answers given in the back of the book.g
4 This graph shows the distance travelled starting at town O and finishing at town A, and the time taken to travel that distance. a
Distance travelled 600 500
i Find the coordinates of A.
300
iii Find the time taken to travel that distance.
200
b Find the distance travelled after 2 hours.
100
c Find the time taken to travel 300 km.
1
5 This graph shows the weight of a piglet in the first few weeks after it was born.
c Find the weight of the piglet at age i 1 week ii 5 weeks. d How old was the piglet when it weighed i 7 kg ii 14 kg?
b Estimate the height range for 9-year old girls. c What percentage of 17-year old girls would you expect to have heights between 150 and 175 cm? d Why were the tallest 5% and the shortest 5% of girls excluded from this graph?
3
4
5
6
time (hours)
Weight (kg)
16 14 12 10 8 6 4 2
b The piglet was born at age 0. What was its weight at birth?
a Estimate the height range for 11-year old girls.
2
Weight of a piglet
i What is shown on the horizontal axis? ii What is shown on the vertical axis?
6 The graph alongside shows the normal range of heights of girls in the age group 4 to 17 years for 90% of Australian girls. Their heights lie between the 5% and 95% lines.
A
400
ii Find the distance travelled between O and A.
a
distance (km)
age (weeks) 1 2 3 4 5 6 7
190 cm 180 170 160 150 140 130 120 110 100 90 80 4
5
Height of girls 4 - 17 years 95% 50% 5%
age 6
7
8
9 10 11 12 13 14 15 16 17
142
DATA COLLECTION AND REPRESENTATION (CHAPTER 9)
Unit 66
Pie charts and media graphs A pie chart is made up of a circle which is divided into sectors.
The pie chart below shows how Michael spent his $50 birthday present from his grandparents. Amount spent on book = 20% of $50 = 0.2 £ $50 = $10
Michael’s Purchases
CD 30% Tennis ball 25%
Magazine 25%
Sector angle is 20% of 360o = 72o . Book 20%
Book sector 20% spent on the purchase of a book.
Each sector shows one particular type of thing. The more things that are in that sector compared to the other sectors, the larger the sector angle will be.
Usually, percentages with names are found in the various sectors.
Exercise 66 1 The pie chart shows the results of a survey of 200 Year 6 students. All students were asked the question: “What is your favourite subject out of Science, Mathematics and English?” Use the pie chart to find
Maths 30%
Science 42%
English 28%
a the most popular subject b the least popular subject c the number of students whose favourite subject is Maths. 2 The pie chart alongside illustrates the proportion of water for various household uses. a For what purpose is the most water used? b For what purpose is the least amount of water used? c If the household used 500 kL of water during a particular period, estimate the quantity of water used in i the bathroom ii the laundry. 3 The pie chart alongside shows the percentages of women who wear certain sizes of clothing.
Cleaning 5%
Laundry 12%
Bathroom 40%
Other 10% Size 16 10%
a Find what size is most commonly worn. b 300 women attend a fashion parade. Estimate the number of them who would wear size 10 clothing.
Size 14 35%
Garden 43%
Size 10 15% Size 12 30%
DATA COLLECTION AND REPRESENTATION (CHAPTER 9)
Research
143
Media graphs From newspapers and magazines collect one of each of the following graph types: ² pie chart
² line graph
² column or bar graph.
What to do: 1 Briefly describe in your own words what information each graph contains. 2 State the main piece of information found from each graph. 3 Comment on how easy it was to interpret each graph. 4 Many graphs have misleading features. Information which gives the wrong idea or the wrong impression about something is called misleading information. For example, Steel Production - Graph A
30
28
'000s tonnes
'000s tonnes
30 26 24 22
Steel Production - Graph B
25 20
15 10 5
20 92
93
94 95 year
96
0
97
92
93
94 95 year
96
97
Graph A on the left has exaggerated the increase in production because the vertical scale does not start at zero. Graph B gives a much more accurate representation of the increase in production. Consider the following graphs: i
ii 270 260 250 240 230
B
A
milk production ('000s L)
sales ($)
Sales of two brands of washing powder
washing powder
10 0
A
B
Factory
70 65 60 55 50 45
Train Fares
94
95
Year
96
97
value ($millions)
iv cost (cents)
iii
20
Exports 4 3 2 1 0
1995
1996
1997 year
a Peter looked at graph i and said “Brand B is much better than brand A because it sells twice as much as A”. He has been misled. How has this been done? b Lyn looked at graph ii and said “Factory B produces far more milk than Factory A.” What has caused her to make this incorrect statement? c What are misleading features of graphs iii and iv?
144
DATA COLLECTION AND REPRESENTATION (CHAPTER 9)
Unit 67
Interpreting data
Once data has been organised and displayed we can often make observations and give interpretations of what we see.
frequency
² For example, look at this column graph: Mental arithmetic test marks
8 6 4 2 0
1
2
3
4
5
6
7
8
These facts can be seen from the graph. Can you find others? ² The highest mark was 10 and the lowest was 3. ² Most students obtained a mark of 7 or more. ² No student obtained a mark less than 3. ² There were 30 students who did the mental arithmetic test. ² Half the students obtained a mark of 8 or more.
9 10 mark
² This graph shows what type of food was bought by students at recess time. Each student was allowed to buy only one item of food.
80
frequency
60 40
Can you see that:
20
² ² ²
the most popular food bought was chips and that 60 0 were bought Bun Pasty Pie the least popular item was a bun and 25 were bought the total number of pies and pasties bought was 35 + 40 = 75 the total number of food items sold was 25 + 35 + 40 + 40 + 60 = 200
²
the fraction of students buying fruit was
²
40 200
type Fruit Chips
= 15 .
Exercise 67
Iced coffee
d If they bought only one drink, how many students bought a drink at lunch time?
Milk
c How many students bought i fruit juice ii iced coffee?
Choc milk
b What is the least popular drink?
70 60 50 40 30 20 10 0
Fruit juice
a What is the most popular drink?
Lunch time drinks frequency
Soft drink
1 The graph shows the type of drink bought by students at lunch time.
type
e How many students bought a drink containing milk? f If the school had 245 students attending, how many did not buy a drink? 2
Strip graph of favourite types of books read by year 7 students Adventure
Mystery
Science fiction
Fantasy
a Measure the length of each strip in millimetres. b What is the most popular book type? c What is the length of the whole strip? d What fraction of the surveyed students read Fantasy books as their favourites? e If the ‘Mystery’ strip represents 70 surveyed students, how many students in total were surveyed?
145
DATA COLLECTION AND REPRESENTATION (CHAPTER 9)
3 Tian is a keen fisherman and he has recorded his weekend fish catches over a 7-week period.
7 6 5 4 3 2 1
a In which week was Tian’s lowest catch? b What was Tian’s catch in week i
4
ii
5?
c What was Tian’s total catch over the 7-week period? d Did Tian’s weekly catch improve or get worse over the 7-week period? 4 Yearly profit and loss figures for a business are shown on the graph. a In what years was a profit made? b What was the result in i
1994
ii
1996?
c What was the overall profit during the 6-year period? d What was the overall pattern in profits over the 6-year period?
6 5 4 3 2 1 0 –1 –2
Profit/loss for “Sandra’s Shirts” (in thousands of dollars)
94
5 The daily newspapers regularly have graphs showing the value of Australian money compared with the other currencies. The graph alongside shows the value of the Australian dollar compared with the US dollar on a particular day. a
Weekend catches represents 2 fish
week
95
96
97
98
year
99
Currency values $Australian 42 36 30 24 18 12 6
i What is shown on the horizontal axis? ii What is shown on the vertical axis?
b Use the graph to find the value in $US of i $A18 ii $A36 iii $A9 c Find the value in Australian dollars of i $US20 ii $US50 iii $US65
$US
10 20 30 40 50 60 70
6 This graph shows the distance travelled as a jogger ran from his home to his Sports Club.
b
i What is shown on the horizontal axis? ii What is shown on the vertical axis? iii What does the end point of the graph show?
7 6 5 4 3 2 1
i How far from home was the Sports Club? ii How long did it take to reach the Sports Club?
c How far did the jogger travel in the first 10 min? d How long did it take the jogger to travel the first 3 km?
distance (km)
e What is missing from the graph?
10 20 30 40 50 60 70 80 time (minutes)
7 Sarah has two seafood shops, one in the city and one in the suburbs. For the city shop she uses no advertising. She started advertising the suburban shop at the beginning of the year. Her monthly profits are shown in the given back-toback bar graph.
Seafood shop profits
CITY SHOP
SUBURBAN SHOP
Jul
a What were the profits for each shop in February?
Jun May
b What was the total combined profit for April? c Find the total profit for each shop over the 7-month period. d What feature(s) of the graph indicate whether or not the advertising was effective?
Month
a
Apr Mar Feb Jan 8
7
6
5
4
3
2
1
0
0
1
2
3
Profits in 1000s of Dollars
4
5
6
7
8
146
DATA COLLECTION AND REPRESENTATION (CHAPTER 9)
Unit 68
Measuring the middle of a data set
One way of measuring the middle of set of data is to find its mean (or average). The mean (or average) is the total of the scores divided by the number of them. 36 3+4+6+7+8+8 = = 6: 6 6
²
For example, the mean of 3, 4, 6, 7, 8 and 8 is
²
If Michael caught 6, 4, 7, 8, 7 and 10 fish on the last six occasions he went fishing, his mean catch is 6 + 4 + 7 + 8 + 7 + 10 6 42 = 6 =7
total number of fish total number of occasions
Exercise 68 1 Find the mean (or average) of a 1, 2, 3, 4, 5, 6, 7, 8, 9 c 15, 18, 16, 18, 18, 17, 17
b d
5, 5, 5, 5, 5, 5, 5, 5, 5 3, 7, 0, 1, 0, 0, 5, 6, 0, 2
2 During the netball carnival Sara threw the following number of goals: 18, 13, 32, 27, 22, 17, 18, 17, 25 and 11. What was her mean score per match? 3 Two groups of students (A and B) do the same spelling test out of 10 marks. Group A: 9, 8, 9, 9, 10, 3, 6, 8, 6, 7, 9 Group B: 4, 7, 5, 7, 6, 9, 6, 8, 7, 6 a Calculate the mean of each group. b Is the following statement true or false? “Because of the unequal numbers of people doing the spelling test in each group it is unfair to compare their averages.” c Which group performed better at the test? 4 The given data shows the number of goals kicked by footballers in the local junior league teams. a Find the mean number of goals per game for each player. b Who won the trophy for the highest number of goals scored? c Discussion: Was the trophy winner in b really the best performer?
Name Michael Tao Mario Peter Sam
Goals 47 45 44 43 48
Games 10 9 8 10 12
5 At cricket Jon had scores of 23, 27, 32, 19, 35, 29, 32, 40, 33, whereas Jethro’s scores were 0, 2, 0, 5, 120, 138, 4, 1, 0. a Find the average for each batsman. b Discussion: In the final match the selectors chose Jon and dropped Jethro. Was this a fair decision? What do you think the selectors told Jethro as the reason for them selecting Jon ahead of him? 6 Find the mean weight of army recruits (in kg) given in the dot plot.
Weight of army recruits
60
65
70
75
80
85
90 weight (kg)
DATA COLLECTION AND REPRESENTATION (CHAPTER 9)
147
Using technology to graph data Many special computer programs are used to help us organise and graph data. STATISTICS PACKAGE
Haese & Harris statistical graphing package Click on the icon to enter the statistical graphing computer package. Alter the data in the table and see the effect on the graph. ² ²
Notice that labels on the axes can be changed and so can the graph’s heading. The type of graph can be changed by clicking on the icon to give the type that you want.
Put this data in the cells
Colour Frequency
white 38
red 27
blue 19
green 18
other 11
Experiment with the package and use it whenever possible.
Using Microsoft Excel Investigation
Using a spreadsheet for graphing data
Suppose you want to draw a frequency column graph of the car colour data: Colour Frequency
white 38
red 27
blue 19
green 18
The following steps using MS Excel: Step 1:
Start a new spreadsheet, type in the table as shown and then highlight the area shown.
Step 2:
Click on
Step 3:
Choose
You should get:
1 2 3 4 5 6 7
from the menu bar.
SPREADSHEET
other 11
A Colour white red blue green other
B Frequency 38 27 19 18 11
C
This is probably already highlighted. Click Finish
Frequency 40 35 30 25 20
Frequency
15 10 5 0
What to do:
white
red
blue
green
other
1 Gather statistics of your own or use data from questions in the previous exercise. Use the spreadsheet to draw an appropriate statistical graph of the data. 2 Find out how to adjust labels, scales, etc.
148
DATA COLLECTION AND REPRESENTATION (CHAPTER 9)
Unit 69
Review of chapter 9
Review set 9A 1 Draw a dot plot for the following ages of the children in the school music group: 13, 13, 12, 11, 13, 12, 9, 12, 10, 13, 11, 13, 10, 12, 12 a How many children are in the school music group? b What is the frequency of 13 year olds? c What fraction of the music group is less than 11 years old? 2 Susan received the following numbers of telephone calls at work each day for two weeks: 10, 5, 8, 11, 10, 9, 7, 15, 12, 4 Find the mean (average) number of telephone calls each day. 3 The display shows the ages of the mothers of children in a Year 6 class.
Ages of mothers of year 6 class
a What type of display is this? b How many ages are shown in the display? c What is the age of the oldest mother? 4
Score 9 10 11 12 13
5
Tally j jj jjjj © jj jjjj © © jjjj © Total
30
35
40
45
50
age
Copy and complete the frequency table for a sample. a What is the frequency of the score of 12? b How many numbers are in the sample? c Draw a column graph to display this data.
Frequency
How Year 6 students came to school Bike
Walk
Car
Bus
There are 21 students in the class.
a What type of graph is the one shown above? b Use your ruler to measure the length of each rectangle in the graph and so answer the following questions. i What fraction of the class rode bicycles to school? ii How many members of the class came to school by car?
6 a b c d
In the pictograph, what does
represent?
On which day were the milk sales i greatest ii least? How much milk was sold on i Thursday ii Friday? Find the average milk sale (to the nearest litre) per day.
Quick Mart Milk Sales
Monday Tuesday Wednesday Thursday Friday Saturday represents 10 litres sold
DATA COLLECTION AND REPRESENTATION (CHAPTER 9)
149
Review set 9B 1 Find the mean (average) of the set of scores: 13, 7, 0, 8, 11, 3 2
Favourite fruit for class 6J students Banana
Orange
Apple
Others
30 students in the class
a What type of graph is the one given above? b Use your ruler to measure the length of each rectangle in the graph and so answer these questions: i What fraction of the class have apples as their favourite fruit? ii How many members of the class say that bananas are their favourite fruit? Pumpkin harvest
year
3
99 98 97 96 95 94
In 1998 farmer Jones picked 6000 pumpkins. Find how many pumpkins he picked in each year from 1994 to 1999 and place your answers in a table.
frequency
11, 12, 11, 12, 12, 12, 13, 13, 11, 10, 11, 8, 9, 10, 10
6 a b c d 7
6 5 4 3 2 1 0
Mental arithmetic data
a b c d
4
5
6
7
8
9
10
What score occurred most frequently? How many students scored i 7 marks ii 5 marks? How many students did the test? What fraction of the students scored 8 or more?
What type of graph is given? What information is given by the graph? What is the frequency for the shoe size i 10 ii 12 12 ? What is the most common shoe size? Gold
$US 387 386 385 384 383
Shoe sizes at Cathy’s Shoes
frequency
5
frequency
4 Draw a dot plot for the ages in years of the children in the chess club:
7
8
9
10
11
12
13
14
size
The graph alongside shows the price of gold in $US on Monday to Friday in a particular week. a What does $US mean? b On what day was the price of gold i highest ii lowest? c Give the price of gold on Monday (to the nearest 10 cents).
382
8 Four hundred people whose houses had been burgled were asked what they were doing at the time of the burglary. The results are shown on the pie chart. Use the chart to find
other 6% at work 31%
a what most people were doing at the time of the burglary b the percentage of people who were at home at the time of the burglary.
at home 20% 10%
on holidays
10%
visiting 23%
shopping
150
TIME AND TEMPERATURE
Unit 70
Time lines
Time lines are simple graphs which display time, often in the form of dates underneath the line and key events which have happened (or ones we want to happen) above the line.
Exercise 70
2000
National Team Coach
Assistant Coach to National Team
Retired from competition
Knee operation
1995
1985
1990
Won gold medal
Won silver medal
Graduated from University National Team Member
First National Championship 1980
Started University
High School Athletics Champion 1975
1970
1965
Joined Athletics Club
Competed in District Sports Day
Moved home
Started school
Started walking
1960
Born
1 The following time line shows some of the important dates in Gavin’s life:
Use the time line to find a when Gavin i was born ii joined the athletics club iii started university iv won a gold medal b how long Gavin continued to compete after he was High School Athletics Champion c how old Gavin was when he i started school ii went to university
iii
was appointed National Team coach.
2000
Voice Recognition Computer
Game Boy
Sony Walkman Rubiks Cube 1980
Home video game
Electric toothbrush Skate board 1960
Long playing record
Aerosol spray can 1940
Pop up toaster
Hair dryer 1920
The zipper
Vacuum cleaner Razor blades 1900
1880
Coca Cola
Margarine Chewing gum 1860
Can opener
Safety match
Postage stamp 1840
1820
1800
Canned food
Sewing machine Lawn mower
2 The time line above shows the year in which the products first went on sale.
b Although the first canned food was available in 1811, how many years passed before a can opener was sold? c In which year was more than one product first sold? d For how many years before the 20th Century began had Coca-Cola been sold? e How many years after chewing gum was first sold did the electric toothbrush become available? f If the postage stamp is still being made 200 years after it was first sold, in what i year ii century will that be?
1800 1845 1863 1902
The first mass produced doll The first roller skates The first model train set Monopoly dice game
1943 1960 1965 1977
Slinky Etch-a-sketch GI Joe Smurfs
2000
1980
1960
1940
1920
1900
1880
1860
1840
1820
3 On a time line like the one below mark the following information: 1800
CHAPTER 10
a In which century were more products first sold?
TIME AND TEMPERATURE (CHAPTER 10)
151
4 Great times in history BC means ‘before Christ’. The year 2003 means 2003 years after the birth of Christ (AD). A 1947 A polaroid camera produces pictures in under one minute. B 1450 Gutenberg builds a printing press. Some people use CE (meaning Common C 700 BC Coins are used in Turkey for buying and selling goods. Era) instead of AD D 1890 An electric counting machine helps the American census. and BCE (meaning Before the Common E 999 First mechanical clock is invented by a monk. Era) instead of BC. F 810 First description of Arabic numerals. G 1636 An accurate pendulum clock is built. H 3000 BC An abacus, the first adding machine, is invented by Babylonians. I 1569 Mercator shows a new way of drawing maps. J 1938 The first ball point pen, a biro, is introduced. K 1642 A faster adding machine is designed by Pascal. L 2800 BC Egyptians devised the 12 month, 365 day calendar. M 1614 Scottish mathematician invents logarithmic tables. N 1946 The first electronic computer is demonstrated. O 100 Paper is invented in China. P 1500 The first watches are made.
Draw a 20 cm line. Divide the line into 5 equal lengths as shown in this representation. 3000
2000
1000
BC
0
AD
1000
2000
a Using the matching letter of the alphabet, arrange the above events in correct order on your number line. b Write the letters in order from back in time to the most recent. c How many years after paper was invented did the biro first appear? d How many years before Pascal’s adding machine was the Babylonian abacus used? e How long after the first mechanical clock was invented were the first watches made? 5 Create your own time line for these events, remembering that you must have equal time intervals and your information must be easily read.
The times of Harry Potter A B C D E F G H I J K L
1st January 15th January 19th January 23rd February 24th February 6th March 15th March 24th April 17th May 24th May 1st June 24th June
Lord Voldemort changed back to a weakling Bagman was overheard talking to three goblins Harry was almost caught by Snape Dobby stole gillyweed from Snape’s office Harry achieved the second Tri Wizard task Hermoine received threatening letters Bartemius Crouch escaped from Voldemort Harry talked with Sirius Black Bartemius Crouch met Harry Barty was killed by his son Harry dreamed about Voldemort and Barty Crouch Harry was given the prize money
Do not forget to include a heading and a scale on your time line.
152
TIME AND TEMPERATURE (CHAPTER 10)
Unit 71
Units of time
The most common units for measuring time are years, months, weeks, days, hours, minutes and seconds.
Relationship between time units
The number of days in the month varies: January February March April May June
July August September October November December
31 28 (29 in a leap year) 31 30 31 30
1 decade = 10 years 1 century = 100 years 1 millennium = 1000 years
1 week = 7 days 1 day = 24 hours 1 hour = 60 minutes 1 minute = 60 seconds
1 year = 12 months = 52 weeks = 365 days (or 366 in a leap year)
31 31 30 31 30 31
Thirty days has September, April, June and November. All the rest have thirty one, except February alone, which has but twenty eight days clear and twenty nine each leap year.
A leap year occurs if the year is divisible by 4 but not by 100 except if the year is divisible by 400. For example, 1996 and 2000 are leap years, but 1800 and 2100 are not leap years.
Exercise 71 1 How a d g j
many months in 3 years hours in 4 days years in 5 decades days from 1st May to 31st July?
b e h
days in 2 years minutes in 3 hours days in 2004 and 2005
c f i
weeks in 4 years seconds in 5 minutes centuries in 2000 years
Conversions ² To convert 3 weeks and 4 days into days we find 3 £ 7 + 4 f7 days = 1 weekg = 21 + 4 = 25 days
² To convert 3 12 hours to minutes we find
3 12 £ 60 = 3 £ 60 +
1 2
£ 60
= 180 £ 30 = 210 minutes
2 Convert into days a 5 weeks 6 days d 1 week 3 days g 26 weeks 1 day
b e h
10 weeks 2 days 8 weeks 3 days 52 weeks
c f
4 weeks 4 days 90 weeks 5 days
3 Convert into minutes a 4 hours 18 mins d 6 12 hours g 99 hours 10 mins
b e h
6 hours 54 mins 24 hours 58 mins 200 hours 25 minutes
c f
5 14 hours 72 hours 3 minutes
4 Convert into hours a 3 days 11 hours d 100 days 3 hours g 6 weeks 4 days
b e h
7 days 6 hours 5 days 19 hours 2 weeks 5 days 11 hours
c f
10 days 19 hours 8 12 days
TIME AND TEMPERATURE (CHAPTER 10)
5 Convert into seconds a 5 mins 22 seconds d 6 14 minutes g 5 hours 42 mins
b e h
10 minutes 47 seconds 7 mins 26 seconds 6 hours 32 mins 45 seconds
c f
153
3 12 minutes 2 hours 15 minutes
To place the times 1 hour 35 min, 1 14 hours, 94 minutes and 1:5 hours in order from shortest to longest time we must first convert each one to the same units. This will be minutes. 1 hour 35 mins = 60 minutes + 35 minutes 1 14 hours = 60 minutes + 15 minutes
³
60 4
= 95 minutes
´
= 75 minutes
= 15
94 minutes =
= 94 minutes
1:5 hours = 60 minutes + 30 minutes (60 £ 0:5 = 30) = 90 minutes So, the order is: 1 14 hours, 1:5 hours, 94 minutes, 1 hour 35 minutes. 6 Place these times in order from shortest to longest. a 3 12 days, 79 hours, 3 days 11 hours, 3 14 days b 187 seconds, 3 14 minutes, 3:5 minutes, 3 minutes 8 seconds c 29 days, 4 27 weeks, 4 weeks, 4 weeks 3 days 1 d 50 months, 4 13 years, 4 years 3 months, 4 12 years
e 11 years, 124 months, 1 decade, 9:5 years f 154 years, 1 12 centuries, 14 decades, 1 century and 6 decades g 53 weeks, 367 days, 1 year,
12 12 months
h 5 12 days, 145 hours, 5 days 11 hours, 7200 minutes i 4 hours 17 minutes, 16 000 seconds, 256 minutes, 4 14 hours j 5 weeks, 37 days, 5 weeks 20 hours, 886 hours 7 Use a calculator and the units of time tables to work out whether the following statements are true (T) or false (F). a 975 minutes > 16 hours b 39 years < 4 decades c 3742 seconds < 64 minutes d 640 years < 7 centuries 1 e 23 2 hours > 86 400 seconds f 1950 seconds = 32 12 minutes g 4123 years > 4 millennia h 37 years 8 months > 450 months i 57 942 days < 150 years j 3 days 7 hours 45 mins = 4895 mins k 9 weeks 4 days 13 hours > 1717 hours l 2 weeks 4 days 19 hours 27 mins = 27 087 mins m 14 weeks 3 days 2 hours 58 mins < 145 618 minutes
Activity
In your own time List five activities that you do each day. For example, eat breakfast. Time each activity to the nearest minute and record these figures. Find how long you would spend doing each activity during
a
one week
b
one month
c
one year
d
one decade.
154
TIME AND TEMPERATURE (CHAPTER 10)
Unit 72
A date with a calendar
Calendars help us keep track of the days, months and years. They help us keep appointments and remind us of important days in our lives, cultures and history.
Jan u
S ary M 200 T W 4 4 T F 11 5 6 1 Feb S 18 12 13 7 8 2 3 rua S ry 2 25 19 20 14 15 9 1 1 M T 0 0 26 2 W 27 21 22 16 17 8 3 T 28 9 2 4 3 15 1 29 30 24 22 16 1 0 11 5 12 3 7 M 1 2 1 S 3 2 13 ay 2 29 M 4 2 8 19 004 2 T 5 2 W 6 20 2 T 7 F 9 13 4 S 01 5 Jun 16 S 11 6 1 e 20 M 21 7 23 17 18 T 04 3 8 W 30 24 25 19 20 1 4 1 6 1 T 5 31 26 7 2 1 13 27 8 2 3 F S 28 22 20 14 1 9 1 4 29 5 1 0 5 21
26 / 8 / 04 day of month
last two digits of the year
month
In Australia a date is often shown using three numbers as shown above.
If Kiera will turn 21 on 15=9=08, then Kiera’s 21st birthday will be on the 15th day of September 2008.
Exercise 72 1 Write out in sentence form the meaning of the following. a Terri’s parents were married on 21=11=86. b Kahlia’s family migrated on 14=3=92. c Tim’s great grandfather will be 100 on 6=8=09. d Jenny bought the house on 3=10=98. e Man first stepped on the moon on 20=7=69. f Henry Ford’s first mass produced car was built on 12=8=08. 2 Write three examples of dates that a read the same backwards as forwards (like 29=4=92) b have all the same numbers (like 9=9=99) c read the same upside down (like 19=8=61) d have five consecutive digits (like 23=4=56) e total 100 like 29=12=59. 3
January 2004 S
M
T
4 11 18 25
5 12 19 26
6 13 20 27
S
M
T
2 9 16 23 30
3 10 17 24 31
4 11 18 25
W T 1 7 8 14 15 21 22 28 29
February 2004
F 2 9 16 23 30
S 3 10 17 24 31
S 1 8 15 22 29
M 2 9 16 23
W T
F
S
M
5 12 19 26
7 14 21 28
S 1 8 15 22 29
6 13 20 27
7 14 21 28
S
M
T
W T
3 10 17 24 31
4 11 18 25
5 12 19 26
6 13 20 27
May 2004 6 13 20 27
Use a c e
M
T
5 12 19 26
6 13 20 27
7 14 21 28
W 1 8 15 22 29
T 2 9 16 23 30
W 4 11 18 25
T 5 12 19 26
F 6 13 20 27
March 2004 S 7 14 21 28
S 7 14 21 28
M 1 8 15 22 29
S
M
T
4 11 18 25
5 12 19 26
6 13 20 27
S
M 1 8 15 22 29
June 2004
September 2004 S
T 3 10 17 24
F 3 10 17 24
S 4 11 18 25
T 1 8 15 22 29
W 2 9 16 23 30
T 3 10 17 24
F 4 11 18 25
7 14 21 28
W 3 10 17 24 31
T 4 11 18 25
April 2004
F 5 12 19 26
S 6 13 20 27
S
M
T
4 11 18 25
5 12 19 26
6 13 20 27
F 2 9 16 23 30
S 3 10 17 24 31
S 1 8 15 22 29
M 2 9 16 23 30
S 6 13 20 27
S
M
T
5 12 19 26
6 13 20 27
7 14 21 28
July 2004 S 5 12 19 26
October 2004 F 1 8 15 22 29
T 2 9 16 23 30
W T 1 7 8 14 15 21 22 28 29
the calendar above to write the day and date. six weeks after 18th January a fortnight before 10th October twenty eight days after 2nd February
7 14 21 28
b d f
T 2 9 16 23 30
W 3 10 17 24
T 4 11 18 25
F 2 9 16 23 30
S 3 10 17 24
August 2004
November 2004 S 2 9 16 23 30
W T 1 7 8 14 15 21 22 28 29
F 5 12 19 26
T 3 10 17 24 31
W 4 11 18 25
T 5 12 19 26
F 6 13 20 27
S 7 14 21 28
December 2004 W 1 8 15 22 29
T 2 9 16 23 30
F 3 10 17 24 31
S 4 11 18 25
five weeks before 29th March forty five days after 26th September two months before 3rd December
TIME AND TEMPERATURE (CHAPTER 10)
155
4 Each year is divided into four seasons. a Name the seasons and in which months they fall. b How many days are in each season? 5 Christmas day is on 25th December each year. If school begins on 28th January next year, how many days are between the two dates? 6 Students were handed a note on 29th August for a camp on 28th November. How many days did they have to wait for camp? 7 How many Mondays are there in January in the year 2004? 8 In the USA, dates are written with the month first then the day and last of all the year. For example 9=11=01 is actually the 11th September 2001. Convert these USA dates a 6=18=07 b 2=22=04 c 1=5=25 9
a If it is Monday tomorrow, what was it yesterday? b If yesterday was Tuesday, what will it be tomorrow? c If it was Thursday yesterday, what will it be the day after tomorrow? d If the day after tomorrow is Sunday, what was the day before yesterday? e If tomorrow is Friday, what will be the day a fortnight from today?
10 Using the six digit way of writing dates which of the following dates is not correct? a 31-12-98 b 31-10-31 c 29-02-99 d 11-11-11 e 18-12-05 f 29-02-96 g 31-06-99 h 31-04-02 i 31-07-98 j 31-09-00 k 25-13-01 l 31-11-13 11 Use a calculator and the calendar given in question 3 to answer the following. a How many minutes are there in i April ii summer iii all the months ending in ‘er’ iv all the Mondays v the whole year? b How many minutes between i the beginning of 1st of June and the end of 12th August ii noon on September 1st and midnight on 6th November iii 3 am October 8th and 7 pm on November 1st?
Activity
Calendar patterns What to do: 1 Collect a page from any old calendar. 2 Outline a square around any four numbers of the calendar. 3 Add the diagonals. What do you find? 4 Try any other square. What do you find? 5 Explain what is happening. 6 Try adding the diagonals of a 3 £ 3 square. Explain any patterns that you find. 7 What other patterns can you find?
MARCH M
T
6 7 13 14 20 21 27 28
W T F S S 1 2 3 4 5 8 9 10 11 12 15 16 17 18 19 22 23 24 25 26 29 30 31
AUGUST M 7 14 21 28
T W T F S S 1 2 3 4 5 6 8 9 10 11 12 13 15 16 17 18 19 20 22 23 24 25 26 27 29 30 31
156
TIME AND TEMPERATURE (CHAPTER 10)
Unit 73
Reading clocks and watches
Everyday can be broken up into units of time. We measure time with either 12 hour or 24 hour clocks. These times are read from analogue or digital clocks and watches.
12-Hour clocks Traditional analogue clocks give us 12-hour time. For example:
Digital clocks can also be 12-hour clocks which go through two cycles each day. They usually show a small a.m. or p.m. somewhere on their display. pm
5:00
a.m. means before the middle of the day p.m. means after the middle of the day
6
reads 5 o’clock and could be 5:00 am or 5:00 pm.
So, this time is 5:00 in the afternoon.
24-Hour clocks Most digital clocks today display 24 hour time. For a 12-hour clock, twenty past six or 6:20 could mean 6:20 am or 6:20 pm and we do not know which. A 24-hour clock overcomes this problem by displaying 06:20 for the morning (am) and 18:20 for the afternoon (pm). The 18:20 means 18 hours and 20 minutes since midnight, which is 6 hours 20 minutes since midday, that is, 6:20 pm.
Four digit notation 5:30 am appears as
9:20 pm appears as 21:20 and is written as 2120 hours.
5:30
and is written as 0530 hours.
Here are some examples which compare 12-hour time and 24-hour time. 12-hour time
12-hour time Digital display
24-hour time
24-hour time Digital display
midnight
0:00
0000 hours
0:00
6:58 am
6:58
0658 hours
6:58
12:00
1200 hours
12:00
10:16 pm
2216 hours
22:16
midday (noon) 10:16 pm
Exercise 73 1 Write as 24-hour time. a 5:27 am e noon
b f
9:55 am 6:48 pm
c g
midnight 7:46 pm
d h
12:08 am 11:59 pm
TIME AND TEMPERATURE (CHAPTER 10)
2 Write the following 24-hour times as 12-hour times. a 0400 hours b 0530 hours c e 2215 hours f 0735 hours g
1700 hours 2059 hours
1200 hours 1818 hours
d h
3 Write the following analogue times as 24-hour time. a b
c
morning
afternoon
d
am
e
f X
XI
XII
I
X
II
IX
III
VIII VII
VI
XI
XII
I
IX
IV
II III
VIII VII
V
pm
IV VI
V
morning
4 Convert the following to 24-hour time. a 14 past 8 am b
1 4
afternoon
past 8 pm
c
1 4
to 3 am
d
1 4
to 3 pm
e
10 to 7 am
f
18 to 6 pm
g
25 to 11 pm
h
half past 11 pm
i
13 past 4 am
j
3 minutes to midnight
k
11 minutes to noon
l
2 minutes past midnight
5 Find the missing times in these 12-hour watch patterns. a i ii iii
b
i
157
ii
iii
iv
v
iv
v
6 Find the missing times in these 24-hour clock patterns. a
i
b
i
c
i
11:45
22:35
ii
12:30
iii
13:15
iv
ii
11:35
iii
12:25
iv
13:15
v
iii
23:25
iv
23:50
v
ii
v
14 :45
7 Beginning at the time given, write the next 6 times at intervals of 20 minutes. a 10:40 am b 10 to 11 pm c 15:50 d quarter past 7 e 3:25 pm f 5 past 9 am g 22:40 h quarter to 8 8 Find the missing times in these patterns. a 11:05, 10:35, ....., 9:35, ....., ..... c 12:40, 12:15, ....., 11:25, ....., 10:35, ..... e
25 to 9, .....,
1 4
to 8, ....., 5 to 7,
1 2
past 6
b d
....., 8:15, 7:14, ....., 5:12, ....., 0110, 0040, ....., 2340, ....., 2240
f
....., quarter to two, ....., twenty five to one, midnight, .....
158
TIME AND TEMPERATURE (CHAPTER 10)
Unit 74
Clockwise direction and using a stopwatch
Clockwise direction The hands of a clock move in a particular direction. We call this clockwise direction. the second hand
the minute hand
the hour hand DEMO
All these hands move in a “clockwise” direction. In these real world examples the white wheel moves clockwise. DEMO
fan belt on a car The coloured wheel moved clockwise.
gears in a car The coloured wheel moves anticlockwise.
Exercise 74 1 Using the directions clockwise or anticlockwise, describe the direction of turn of the following. a b c
d
e
f A
B
2 Give the direction of turn for the following a opening the screw top of a soft drink bottle b the speedometer of an accelerating car c undoing a nut with a spanner d the tap to turn off the shower e turning the key to open a door f the blades of a fan when you are facing it. 3
a If the steering wheel of a car is turned left, which direction will the car move if it is going forward? b If the steering wheel of a car is turned right, which direction will the car move if it is reversing?
TIME AND TEMPERATURE (CHAPTER 10)
159
Using a stopwatch A stopwatch measures small fractions of time in seconds and parts of seconds.
Activity
Reaction time You will need: A digital watch which shows one hundredths of a second and a partner to work with. What to do: 1 2 3 4
06'35"49
1
1 100
Set the watch for 000 0000 00 100 . reads as 6 minutes 35.49 sec. Start the timer and stop it as close to 5 seconds as you can. Take turns with your partner to see whose reaction time is closest to 5 seconds. Record your times. Work out whose time was the closest.
Jason and Kelly try to stop a stopwatch at exactly 5 seconds. Jason stopped it at 00'05"19 and Kelly at 00'04"86 . Which one stopped it closer to 5 seconds? Jason:
000 0500 19 is 19 hundredths
Kelly:
000 0400 86 is 14 hundredths
³ ³
19 100 14 100
´ ´
past 5 seconds. before 5 seconds.
So, 000 0400 86 is closer to 5 seconds. 4 Which of these times is closer to 5 seconds? a i 000 0500 34 or ii 000 0500 38
b
i
000 0400 69 or ii 000 0500 28
5 Which of these times is closer to 10 seconds? a i 000 0900 34 or ii 000 1000 69
b
i
000 0900 37 or ii 000 1000 53
6 Which of these times is closer to 1 minute? a
i
000 5900 05
or ii
000 5900 50
b
i
010 0000 45
or ii
000 5800 85
c
i
010 0100 49
or ii
010 0000 99
d
i
010 0000 65
or ii
010 0000 09
Activity
Just a minute You will need: A stop watch or digital watch, coin, paper, pencil, 2 dice and a partner to work with. What to do: Complete the 30 second activity before you estimate for the 60 second activity.
Activity: Count the number of complete times. Write consecutive numbers starting with 1:
30 seconds Number of times Estimate Actual Difference
PRINTABLE TEMPLATE
60 seconds Number of times Estimate Actual Difference
Trace around a coin with a pencil. Roll a pair of sixes with 2 dice. 1 With which time was your estimate most accurate? Why was this so? 2 With which estimates were you most confident? Why was this so?
160
TIME AND TEMPERATURE (CHAPTER 10)
Unit 75
Timetables
Timetables, schedules, itineraries, guides, programmes and some charts tell us when events are to occur. Cruising Alaska
Given is a small timetable or itinerary of a yacht trip in Alaska. From such a table we can observe, for example, ² the order in which places will be visited ² what can be seen at some places.
6 days / 5 nights
Day 1
Arrive Ketchikan
Day 2
Misty Fjords excursion
Day 3
Cruise Ketchikan to Petersburg
Day 4
Le Conte Glacier
Day 5
Tracy Arm and Juneau
Day 6
Mendenhall Glacier, transfer to airport
Exercise 75 1 Given is a tide timetable for a particular day in 2000. a
When was the tide at its highest at Blight’s Bluff?
b
When was the tide at its lowest at Port Victory?
c
What is the difference in time between the morning low tide and the morning high tide at Power Point?
d
By what depth does the tide change at Turtle Beach in the afternoon? How much time is there between the two high tides at Eden Bay?
e
Blight’s Bluff Inner Harbour Turtle Beach Power Point Eden Bay Port Victory
Low tide 1.13 am 0:6 1:38 pm 0:4 4:37 am 1:7 7:01 pm 1:4 3:46 am 0:8 4:14 pm 0:3 4:20 am 0:7 4:45 pm 0:3 3:27 am 2:0 3:18 pm 1:6 1:27 am 0:5 2:20 pm 0:2
m m m m m m m m m m m m
High tide 7:16 am 2:6 m 7:43 pm 2:4 m 11:04 am 0:4 m 11:00 pm 0:5 m 9:29 am 2:6 m 10:16 pm 2:3 m 10:06 am 2:9 m 10:48 pm 2:6 m 9:24 am 0:2 m 9:42 pm 0:4 m 7:31 am 1:4 m 8:16 pm 1:0 m
2 Use the TV program to answer these questions.
Channel 4 05:30 06:30 07:00 08:00 08:30 09:30 11:00 11:30 12:00
Weather Watch Roger Robot Cartoon Collection Dazzlers Kids Korner Hot Hits Growing Gardens Football Flashbacks Spectator Sports
14:30 16:45 17:30 18:00 19:00 19:30 20:30 23:15 00:10
Movie Matinee Cartoon Capers Pick a Prize News and Weather Animal Antics North Park Saturday Special Sports Roundup Temporary close
a For how many minutes is News and Weather shown? b How much time was spent showing sport? c How much time passed between the end of Cartoon Collection and the start of Cartoon Capers? d Using the 12-hour clock, at what times did Saturday Special start and finish? e In minutes, how long did the movie matinee run? f Would a 180 minute video tape be long enough to record North Park and Saturday Special?
TIME AND TEMPERATURE (CHAPTER 10)
3 Alongside is a distance and time chart. The distances are in kilometres and the times shown in hours and minutes are the times taken driving sensibly in a car.
Mulagwa
Pineville
Wounded Knee
Crowtown
Timbuktoo
Dustbowl
Mosquito Valley
Bolero
Port Power
Falls Creek
Danger Crossing
Toffs Harbour
Camelia
Eyeone
distance Tango Mulagwa Pineville Wounded Knee Crowtown Timbuktoo Dustbowl Mosquito Valley Bolero Port Power Falls Creek Danger Crossing Toffs Harbour Camelia Eyeone
Tango
time
161
— 496 830 1102 575 1186 1327 536 706 477 952 922 868 130 519
6.55 — 1252 978 620 1062 1194 958 582 522 828 850 744 394 348
10.10 15.20 — 1337 755 1347 1479 300 1163 905 1316 1034 1380 858 1275
15.45 14.25 16.20 — 650 84 216 1446 402 800 170 303 234 1014 635
7.50 8.10 8.25 8.50 — 660 792 796 408 150 629 347 546 390 448
16.50 15.30 16.05 1.05 8.35 — 132 1456 486 810 254 313 318 1079 719
18.55 17.10 17.45 2.45 10.15 1.40 — 1588 618 942 386 445 450 1211 851
6.55 12.10 3.20 18.35 9.40 18.15 19.55 — 1141 809 1381 1143 1303 564 981
10.10 8.50 14.00 5.45 5.40 6.50 8.30 15.15 — 390 252 284 168 604 239
6.35 6.50 10.15 10.40 1.50 10.25 12.05 10.00 5.00 — 636 497 552 321 299
13.40 12.20 15.50 2.25 8.20 3.30 5.10 18.45 3.40 8.30 — 282 84 850 485
12.15 12.00 11.55 4.25 4.25 4.10 5.50 13.55 4.00 6.15 3.55 — 346 737 507
12.30 11.10 16.40 3.15 7.40 4.20 6.00 17.35 2.30 7.20 1.10 4.45 — 766 401
1.45 5.30 9.50 14.45 5.25 14.15 15.55 6.40 8.45 4.20 12.15 9.50 11.05 — 417
6.55 5.05 15.20 9.45 5.55 10.35 12.05 12.10 4.00 4.05 7.30 7.10 6.20 14.20 —
a How far is Toffs Harbour from Tango? b How long would it take to drive from Wounded Knee to Timbuktoo? c What is the nearest place to Bolero? d
i What is the longest time you would need to drive to travel between two places? ii Between which two places would that be?
e Which place is 15 hours 20 minutes from Eyeone? f If you left Mosquito Valley at 6 am, at what time would you expect to get to Crowtown? 4 This schedule of arrivals appears on a TV monitor at an international airport. a Convert each 24-hour arrival time to 12-hour time. b At what time is the Qantas flight from Bombay arriving? c BA009 arrives from ...... at ...... ? d If thunderstorms delay all arrivals by 4 hours what time will the i Singapore flight arrive ii Japan Airlines flight from Tokyo arrive? 5
Date Oct 19th Nov 2nd Nov 16th Nov 23rd
Time 2 4 6 2 4 6 2 4 6 2 4 6
pm pm pm pm pm pm pm pm pm pm pm pm
Teams 9 2 1 8 3 1 2 1 9 7 6 5
v v v v v v v v v v v v
4 8 5 5 4 9 5 6 8 1 1 6
1 3 6 2 7 6 3 7 3 8 3 9
v v v v v v v v v v v v
7 5 3 6 8 4 8 4 1 2 7 4
Flight QF100 BA009 SQ316 MH47 QF124 NZ54 QF301 JAL204
Arrivals From Los Angeles London Singapore Kuala Lumpur Bombay Melbourne Sydney Tokyo
Arr. Time 06:25 08:05 10:10 12:45 14:50 16:35 19:40 20:55
To the left is the draw for the local soccer teams for the first 4 weeks of the competition. a Which team has the most games at 2 pm each week? b How many teams are playing 5 games in the 4 weeks? c Which teams have a double header on November 16th? d Suppose you are playing in team 3. Record the dates and times you would need to attend your games.
162
TIME AND TEMPERATURE (CHAPTER 10)
Unit 76
Speed and temperature
Speed The speed of a moving object is a measure of how fast it is travelling. Speed is usually measured in kilometres per hour (km per hour or km/h or kmph) or in metres per second. A speed of 100 km per hour means that the car would travel 100 km in 1 hour. If a car is travelling at a steady speed of 90 kmph then ² in 1 hour it travels 90 km
² in 2 hours it travels 2 £ 90 km = 180 km
² in 20 minutes it travels 1 3 £ 90 km = 30 km
Exercise 76 1 A car travels at 80 km per hour. How far does it travel in a 3 hours b 1 12 hours c 30 minutes? 2 Sally rides her bike at a speed of 20 km per hour. How far will she travel in a 1 hour b 15 minutes
c
45 minutes?
3 A boat travels at 12 km per hour. How far will it travel in a 5 hours b 10 hours c 1 day
If Leon walks 18 km in 3 hours, he walks 6 km in 1 hour. So, his speed is 6 km per hour or 6 kmph. 4 Find the speed of a a person who walks 24 km in 4 hours b a car which travels 330 km in 3 hours c an aeroplane which travels 2800 km in 4 hours d a truck which travels 50 km in 30 minutes.
If a car travels at 90 kmph for a distance of 360 km it travels and
90 km in 1 hour 360 km in 4 hours
So, it would take 4 hours. 5
a A person jogs at 8 km/hour for a distance of 24 km. How long would it take? b An aeroplane travels at 800 kmph for a distance of 2400 km. How long would the flight take? c A motorcyclist travels at 100 kmph for 650 km. How long would the trip take?
6
a If you were travelling at 60 kmph, how far would you travel in 1 minute? b If you were travelling at 120 kmph, how far would you travel in 1 minute?
163
TIME AND TEMPERATURE (CHAPTER 10)
Temperature We measure temperature using thermometers. A thermometer contains a liquid which expands when the temperature increases and contracts when it cools. 7 At the moment the temperature is 25o C. What will be the temperature if it gets a 12o hotter b 15o cooler c 30o hotter d c
41
°C
40
39
38
F
37
36
400o C
°C
50
40
30
20
10
41
E
c
°C
100o C
0
50
the temperature of ice the temperature of a cold day the temperature of the sun
40
a c e
39
10 Match these temperatures with the statements A 10 000o C B 0o C C 8o C D
38
37
36
41
40
39
38
37
36
°C
9 What temperatures are shown on these medical thermometers? a b
°C
40
30
20
10
0
50
40
30
20
10
0
°C
8 What temperature is shown on these thermometers? a b
35o cooler?
45o C
the temperature of bath water the temperature of molten metal the temperature of boiling water
b d f
The maximum daily temperature is the highest temperature reached during the 24-hour day. The minimum daily temperature is the lowest temperature reached during the 24-hour day. 11 The table shows the temperatures (in o C) on the hour in Adelaide over a 24-hour period. a What was the temperature at Time Temp. Time Temp. i 4 am ii 7 pm? 12 midnight 11 1 pm 18 b What was the maximum temperature and when 1 am 10 2 pm 20 did it occur? 2 am 9 3 pm 18 c What was the minimum temperature and when 3 am 8 4 pm 16 did it occur? 4 am 7 5 pm 15 d What was the average temperature for 5 am 6 6 pm 14 6 am 5 7 pm 13 i the period from midnight to 6 am 7 am 6 8 pm 12 ii the temperature shown? 8 am 8 9 pm 12 e What time of the year would you expect these 9 am 13 10 pm 11 temperatures? 10 am 15 11 pm 11 f Draw a graph showing the movement in 11 am 16 12 mid10 night temperature during the 24-hour period. 12 noon 17
12 The following graph shows how the temperature changes during one day in Autumn at Smokey Bay. 30
°C
25 20 15 10 5 0 midnight
3 am
6 am
9 am
a What was the temperature at i
noon
5 pm
3 pm
ii
5 am
6 pm
9 pm
iii
midnight
noon?
b What was the maximum temperature and when did it happen? c What was the minimum temperature and when did it happen? d What was the average temperature from i
noon to 4 pm
ii
noon to 7 pm?
164
TIME AND TEMPERATURE (CHAPTER 10)
Unit 77
Review of chapter 10
Review set 10A
1970
1980
Opened chain of stores
Employed 50 staff
Opened second shop
Expanded educational section
Exported to UK and Canada
Exported to SE Asia
Moved to larger premises
Employed 20 staff
Commenced trading
1 The following line shows some important dates in the history of The Book Company.
1990
a Use the time line to find when The Book Company i started trading ii iii exported to SE Asia iv
2000
employed 20 staff expanded their educational section.
b How many years was it between when The Book Company i employed 20 staff and employed 50 staff ii opened a second shop and opened a chain of stores? 2
a Convert 7 weeks 2 days to days. b Place the following times in order from shortest to longest. 2 12 minutes, 2 minutes 25 seconds, 2:4 minutes, 148 seconds
3
a Convert into 24-hour time i 3:23 am
ii
1:02 pm
iii
8:36 pm
iii
18:52
b Convert the digital displays into 12-hour time i
2:30 PM
ii
2:06
4 The following schedule of arrivals has appeared on a TV monitor at Adelaide Airport. a b
c
Give the arrival time for the plane from Perth in 12hour time. Give the difference in time between the arrival of the two planes from i Melbourne ii Brisbane If the plane from Canberra is one hour and twenty minutes late, at what time (12-hour) will it arrive?
Flight QF571 QF497 QF499 QF517 QF789 QF808 QF523 QF663
ARRIVALS From Arrival time Sydney 12:18 Canberra 12:25 Melbourne 13:55 Brisbane 14:40 Melbourne 15:30 Perth 15:45 Darwin 16:20 Brisbane 16:55
5 The table shows the temperatures (in o C) on the hour in Port Augusta over the period 6 am to 6 pm. a What was the temperature at i 9 am ii 4 pm? Time Temp. (o C) Time Temp. (o C) b What was the maximum temperature? At what 6 am 17 1 pm 39 time did it occur? 7 am 19 2 pm 41 c What was the minimum temperature? At what 8 am 23 3 pm 38 time did it occur? 9 am 27 4 pm 37 10 am 31 5 pm 35 d At what period of the year are these temperatures 11 am 35 6 pm 34 likely to happen? noon 38 e What was the average temperature over the 12-hour period?
TIME AND TEMPERATURE (CHAPTER 10)
165
Review set 10B 1
a Convert 3 hours 18 minutes to minutes. b Place the following times in order from shortest to longest. 1 25 hours, 1 12 days, 1 day 3 hours, 1590 minutes
2
a Write the date 8=8=90 in words. b Use a calendar to write the date i four weeks after January 18
ii
30 days before Christmas day.
c If yesterday was Monday, what will the day be tomorrow? 3
a Write 30 minutes past midday in 24-hour time. b Write the following 24-hour times as 12-hour times. i 0200 hours ii 0020 hours
4
iii
2000 hours
a Fill in the missing time in the 12-hour clock pattern.
b Fill in the missing time in the 24-hour watch pattern.
15:20
16:50
17:35
c A 24-hour clock shows 19 : 45. Write this time in words. 5 Write the temperatures shown by these thermometers. a b °C 40
6
50
60
70
90
100
110
120
°C
c –10
0
10
20
°C
a If the temperature is 28o at 3 pm, but by 3 am it has fallen 30o , what is the temperature at 3 am? b The water in a kettle increases in temperature at the rate of 14o per minute. How long will it take to boil if the start temperature of the water is 16o ?
7 Use the television program guide given to answer the following questions. a Using the 12-hour clock, at what times did ‘Sport Stunts’ start and finish? b If I started watching television at 12:40 and watched for 53 minutes, i what program would be showing when I turned off the TV set? ii At what time did I turn off the television (use 12-hour time)? 8 A car travels at 90 kmph. a How far does it travel in
i
1 hour
ii
3 hours?
b How long will it take to travel 180 km? 9 Find the speed of a bus that travels 280 km in 4 hours.
11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30
Wednesday Festival of Films Cartoon Mania Car racing Top Dog Teen Hour Sport Stunts The Monsters My Cat called Felix
166
TIME AND TEMPERATURE (CHAPTER 10)
TEST YOURSELF : Review of chapters 7, 8, 9 and 10 1 Name these solids: a
2
b
c
a Convert 6 weeks 3 days to days. b Place these times in order from longest to shortest: 1 12 hours, 80 minutes, 1:4 hours, 1 hour 48 minutes c Write the date 14/7/2005 in words.
3
Using the pictograph shown:
Country of Car Manufacture for Parents with Children at Mawson Primary School Australia Japan Germany England Other
a
What does
b c
What information does the pictograph contain? Which was the most frequent country of origin of the cars? Estimate how many parents’ cars were Australian made.
d
represent?
represents 50 cars
4 A map has a scale of 1 cm represents 20 m. a What map length would represent an actual length of i 100 m ii 70 m? b What actual distance would be represented by a map distance of i 3 cm ii 7:5 cm? 5 For the diagram given a name the solid
A B C
b list the vertices c name each face and give its shape
D
d list the edges 6
E F
a Convert to 24-hour time i 5:20 am
ii
12 noon
iii
10:30 pm
ii
0:02
iii
22:05
b Convert to 12-hour time i
4:15
pm
7 Jacqui recorded the number of items of mail she received in the 20 working days before Christmas as: 7 5 4 2 5 5 6 8 9 3 6 7 7 4 2 7 4 8 5 3 a Draw a tally/frequency table to show this information. b Draw a dot plot to display this information.
8
8 The grid shows the homes of friends and the positions of other important places nearby. a Find the location of the i playground ii oval b Give i ii iii
7
Video Store
school
the direction of the tennis club from the playground the school from the tennis club Amber’s house from Mary’s house.
iv
deli
Peter
Deli Mary
John Pool
2 1
Theatre
School
4 3
N
Rob
6 5
iii
Amber
Tennis Club
Playground
Oval A
B
C
D
E
Sue
F
G
H
TIME AND TEMPERATURE (CHAPTER 10)
167
9 Find the mean (average of the scores): 5, 6, 8, 4, 12. 10
a At 6 am the temperature was 22o C and by 3 pm it was 39o C. What was the increase in temperature? b Find the temperature that is 15o C cooler than 24o C.
11
a A car travels at 60 kmph. How far does it travel in i 2 hours
ii
30 minutes?
b Find the speed of a car that travels 320 km in 4 hours. c A truck travels at 80 kmph. How long will it take to travel 400 km? 12 The grid alongside shows the vertices of the rectangle PQRS.
5
What are the coordinates of S? Give the horizontal and vertical steps to go from P to R.
3
a b
R
4 2 1
P 1
13
a The children in class 6A were asked to name their favourite pet. The results were displayed as shown. i ii iii iv
What type of graph is this? What was the most common favourite pet? How many more students preferred dogs to cats? What does the column marked ‘other’ mean?
b Draw a strip graph to represent the data. Use 1 cm represents 4 pets.
10 8
Q 2
3
4
5
6
7
Favourite pets frequency
6 4 2 0
pet cats
dogs
birds rabbits other
14 Draw plan, elevation and end elevation views for these solids. a b
15 Find how many kilograms are in
a 6840 g b
2:5 tonnes.
16 From these plans draw the oblique projection of the solid:
2 left end
front
right end
1
2
plan
17 The pie chart shows the survey of 100 year 6 students. All students were asked the question: “What is your favourite sport?” Use the pie chart to find a the most popular sport b the percentage of students whose favourite sports are netball or basketball c the number of students whose favourite sport is swimming.
basketball swimming 17% 14%
netball 19%
football 18% cricket 20%
soccer 12%
168
PATTERNS AND ALGEBRA
Unit 78
Number patterns (sequences)
To improve their fitness, Jacqui and Tim decide to run for 5 kilometres every fourth day in March starting on March 1st. So, they write down the dates on the calendar as: 1, 5, 9, 13, 17, 21, 25 and 29. This collection of numbers which has the pattern rule of ‘increasing by 4’ is called a number pattern or number sequence. ² If we start with 6, the number pattern found by increasing by 3 each time is 8, 8 + 3, 8 + 3 + 3, 8 + 3 + 3 + 3, .... etc.,
² If we start with 108, the number pattern found by decreasing by 7 each time is 108, 108 ¡ 7, 108 ¡ 7 ¡ 7, 108 ¡ 7 ¡ 7 ¡ 7, .... etc., This is the number sequence 108, 101, 94, 87, 80, 73.
This is the number sequence 8, 11, 14, 17, 20, 23.
Exercise 78 1 Write down the first 6 numbers in the pattern described by a starting at 3 and increasing by 2 each time b starting at 5 and increasing by 3 each time c starting at 11 and increasing by 7 each time d starting at 31 and decreasing by 2 each time e starting at 57 and decreasing by 5 each time f starting at 141 and decreasing by 3 each time.
Increasing means getting larger and decreasing means getting smaller.
2 Write down the next 3 lines of these number patterns a 1£5+1=6 b 1£6¡2 =4 2 £ 5 + 1 = 11 2 £ 6 ¡ 2 = 10 3 £ 5 + 1 = 16 3 £ 6 ¡ 2 = 16 4 £ 5 + 1 = 21 4 £ 6 ¡ 2 = 22
c
1 £ 7 + 3 = 10 2 £ 7 + 3 = 17 3 £ 7 + 3 = 24 4 £ 7 + 3 = 31
c
6£5¡5= 7£5¡5= 8£5¡5= .. .
CHAPTER 11
3 What would be the 20th member of the patterns for each question in 2? 4 Copy and complete: a 4£1¡1= 5£1¡1= 6£1¡1= .. .
10 £ 1 ¡ 1 =
b
4£3+2 = 5£3+2 = 6£3+2 = .. . 10 £ 3 + 2 =
12 £ 5 ¡ 5 =
5 This pattern: 6, 10, 14, 18, 22, etc can be described as starting with 6 and adding 4 each time. Describe in words the pattern rule for the sequence a 1, 3, 5, 7, 9, etc. b 8, 13, 18, 23, 28, etc. c 24, 22, 20, 18, 16, etc. d 81, 78, 75, 72, 69, etc. e 6, 12, 18, 24, 30, etc. f 3, 10, 17, 24, 31, etc.
PATTERNS AND ALGEBRA (CHAPTER 11)
6 Find a c e
the missing number ¢ in the following patterns. 1, 3, 5, ¢, 9, etc. b 29, 27, ¢, 23, 21, etc. d 4, 8, ¢, 16, 20, etc. f
2, 5, 8, ¢, 14, etc. 65, 62, ¢, 56, 53, etc. 3, 9, ¢, 21, 27, etc.
7 Write the next two numbers of the following patterns. Note: The patterns are multiples of one number or multiples of two numbers alternately. a 6, 12, 18, 24, 30, ...., .... b 4, 8, 12, 16, 20, ...., .... c 8, 16, 24, 32, ...., .... d 9, 18, 27, 36, ...., .... e 3, 4, 6, 8, 9, 12, 12, 16, ...., .... f 3, 7, 6, 14, 9, 21, 12, ...., .... g 4, 5, 8, 10, 12, 15, 16, ...., .... h 5, 6, 10, 12, 15, 18, 20, ...., .... 8 Look at this dot pattern.
member member member member
1 2 3 4
has has has has
1
1 dot 3 dots 6 dots 10 dots
3
6
10
The numbers 1, 3, 6, 10, .... are called triangular numbers as their dot representation is triangular.
a Draw the next three dot patterns. b From a what are the next three triangular numbers? c Without drawing them, what are the next two triangular numbers after those found in b? 9 This dot pattern shows square numbers.
1
4
9
16
a Draw the next two dot patterns. b From a, what are the next two square numbers? c The 3rd square number is 9. Is it true that the 10th square number is 10 £ 10? 10 Cube numbers can be shown using cubes. They are: , etc... 1
a Draw the 4th cube number representation.
8
b What is the 4th cube number? c Notice that
1=1£1£1 8=2£2£2 27 = 3 £ 3 £ 3 What is the value of cube number i 5
ii
23?
11 These cubes sit on a table and their bottom faces cannot be seen.
we can see 3 of its 6 faces
a Write down the number sequence of ‘seeable’ faces of these cubes. b If the 10th member was drawn, how many ‘seeable’ faces would there be?
27
169
170
PATTERNS AND ALGEBRA (CHAPTER 11)
Unit 79
Dot and matchstick patterns
Dot patterns We can make patterns with dots. For example,
is a dot pattern. 1st
2nd
3rd
4th
Notice that the number of dots in the diagrams of the pattern are: 2, 4, 6, 8, etc.
Diagram number 1st 2nd 3rd 4th
A table can be set up for each pattern (sequence). For the pattern above, the table could be:
Number of dots 2 4=2+2 6=4+2 8=6+2
We can see an increase of two dots from each number to the next one.
Exercise 79 1 Draw the next diagram in each of the following dot patterns. a b c
d
2 Using the diagrams in question 1, list the number of dots in each pattern for the first 5 members. 3 Look at the dot pattern. 1st
2nd
3rd
Look at the pattern now with the line partitions.
i.e., 3 + 1
2£3+1
3£3+1
a Draw the 6th member of the pattern with its partitions.
DEMO
b How many dots are there in the 6th member? c Predict how many dots there are in the 20th member. 4 Look at the dot pattern.
etc.
a As in question 3, show how to partition the pattern into 3 + 2, 2 £ 3 + 2, b How many dots are there in i the 6th member ii
the 15th member of the pattern?
5 For these patterns i draw the next two members ii partition the dots as in questions 3 and 4 so that a pattern is seen iii use the pattern to find the number of dots in the 30th member. a
b
3 £ 3 + 2 dots.
PATTERNS AND ALGEBRA (CHAPTER 11)
171
Matchstick patterns 6 For the following matchstick patterns i draw the next two members ii list the number of matches in each diagram up to the 6th diagram. a
b
c
d
e
f
g
h
7 Here are 3 different ways of representing the number sequence 3, 5, 7, 9, etc.
A
DEMO
B 3
5
...
... 7
3
9
a What are the next two members of i
C A
ii
3
B
5
7
5
9
7
9
...
iii C?
b Look at this partition for A. ...
number pattern ! 3 3 + 1 £ 2 3 + 2 £ 2 3 + 3 £ 2 Using diagrams, show how to partition B and C so that this number pattern is seen in them as well. 8 For the following matchstick patterns i draw the next two members ii show how to partition the members so that a number pattern like in question 7 is seen. iii Use the number pattern to find the number of matches needed for the 50th member. a
b
c
d
9 Draw two different sets of matchstick diagrams to represent the number sequence a 1, 3, 5, 7, etc. b 1, 4, 7, 10, .... c 2, 6, 10, 14, .... [Hint: In a 1 = 1, 3 = 1 + 2, 5 = 1 + 2 £ 2, 7 = 1 + 3 £ 2, ....]
172
PATTERNS AND ALGEBRA (CHAPTER 11)
Unit 80
Rules and problem solving
The members of number patterns are often found by ² ² ² ²
adding the same number to the previous member subtracting the same number from the previous member multiplying the same number by the previous member dividing the same number into the previous member. Previous means before. In the alphabet the 3 letters previous to H are E, F and G. The 2 letters previous to Q are O and P .
Rules in words ² For the number sequence 1, 4, 7, 10, .... +3
+3
+3
we see that 1 ! 4 ! 7 ! 10 So the rule is: “to get the next member, add 3 to the previous one”. ² For the number sequence 35, 31, 27, 23, .... ¡4
¡4
¡4
we see that 35 ! 31 ! 27 ! 23 So the rule is: “to get the next member, subtract 4 from the previous one”.
Exercise 80 1 In sentence form, write a rule to find the a 5, 8, 11, 14, .... b d 64, 32, 16, 8,.... e g 68, 59, 50, 41, .... h
next member of these number patterns: 1, 2, 4, 8, .... c 52, 49, 46, 43, .... 3, 10, 17, 24, .... f 2, 10, 50, 250, .... 20, 2, 0:2, 0:02, .... i 3, 3, 3, 3, ....
Finding missing numbers ² For 5, 12, 2, 26, 33, .... we notice that +7
+7
+7
+7
5 ! 12 ! 19 ! 26 ! 33, so the missing number is 19 and the rule is: “to get the next member, add 7 to the previous one”. ² For 5, 2, 20, 40, 80, .... we notice that £2
£2
£2
£2
5 ! 10 ! 20 ! 40 ! 80, so the missing number is 10 and the rule is: “to get the next member, multiply the previous one by 2”. 2 Find the missing number and a rule for finding other members of these patterns: a 3, 6, 9, 12, ¤, .... b 6, 12, 24, ¤, 96, .... c 53, 49, 45, ¤, 37, .... d 8, 13, ¤, 23, 28, .... e 48, 24, ¤, 6, 3, .... f 4, 12, ¤, 108, .... g 113, 102, 91, ¤, .... h 30, ¤, 0:3, 0:03, .... i 7, ¤, 63, 189, .... 3 Continue these patterns for the next three cases.
1+4=5 1+4+4=9 1 + 4 + 4 + 4 = 13 1 + 4 + 4 + 4 + 4 = 17
1+1£4=5 1+2£4=9 1 + 3 £ 4 = 13 1 + 4 £ 4 = 17
a What do you notice about the answers? b Which form is easier to write out? (Consider the 100th member of each.)
PATTERNS AND ALGEBRA (CHAPTER 11)
173
John starts selling refrigerators for a new business called COOLIT. His sales tar get is to sell 10 refrigerators in the first week. Every week after the first week he hopes to increase his sales by 3 over the previous week. Information like this can be put in a table. Week 1 2 3 4 5
Sales 10 = 10 10 + 3 = 13 13 + 3 = 16 16 + 3 = 19 19 + 3 = 22 Total 80
+3 +3 +3 +3
We can see that ²
the number sequence is 10, 13, 16, 19, 22
²
the total number sold in the next 5 weeks is 80.
4 John sells 13 refrigerators in the first week. He hopes to increase his sales by 5 over the previous week. a Draw up a table like the one above to show this information. b What will his total sales be over a 7 week period if he is able to make these sales? 5 When James weeded his garden plot, he decided to finish an area 2 m by 2 m each day. a Copy this diagram and mark on it each day’s work.
day 1 4m
b Write a number sequence to show how many square metres are done each day.
8m
c How long will it take him to finish?
6 Alicia plans to start her fitness program by walking 4 km on the first day and then walking 1 12 km more each day after that. Her program will take one week. a List the daily walking distances for Alicia. b How far does she intend to walk during the week? 7 Kiri is recovering from an accident to her arm. The doctor says that she must exercise it for 3 minutes on Monday and the time must be doubled on each day thereafter for one week. a List the daily exercise time for Kiri for the whole week. b How long must she exercise her arm on Sunday? c What is the total exercise time for the week? 8 How many cans are present in the 8th diagram of the sequence? a b
1st
2nd
3rd
9 Challenge Wei-lin earned $210 over a 6-day period. Each day she earned $7 more than on the previous day. How much did Wei-lin earn on the first day?
(Hint: Use trial and error.)
1st
2nd
3rd The “trial and error” method is sometimes called the “guess and check” method.
174
PATTERNS AND ALGEBRA (CHAPTER 11)
Unit 81
Graphing patterns and tables
We can draw graphs of number patterns. The graph of the number pattern 5, 7, 9, 11, 13, .... is drawn below. 15
This dot represents the value of the 4th member, which is 11.
11 10
PRINTABLE GRID PAPER
5 number 1
2
3
4
5
6
Exercise 81 1 List the number patterns for these graphs. a b 6 5 4 3 2 1 1
2
3
4
c
12 10 8 6 4 2
6 5 4 3 2 1 1
5 6 number
2
3
4
5 6 number
1
2
3
4
2 Which of the graphs in question 1 show increasing sequences (number patterns)? 3 On separate grid paper draw the graphs of the number sequences: a 2, 5, 8, 11, 14, 17 b 17, 13, 9, 5, 1 d 2, 4, 8, 16, 32 e 16, 8, 4, 2, 1
c f
3, 6, 12, 24, 48 1, 4, 9, 16, 25
4 Which of the graphs in question 3 have points lying in a straight line?
From a graph we can construct a table of values. For this graph 20
the table of values is
M
n M
15 10 n
2
3
4
5 Construct a table of values for a
b
24
40 K 30
16
20
T
8
2 10
3 15
1st row 2nd row
4 20
The first row always shows what is on the horizontal axis.
5 1
1 5
10 n
1 2 3 4 5 6 7
t
2
4
6
8
5 6 number
175
PATTERNS AND ALGEBRA (CHAPTER 11)
c
d 12 10 8 6 4 2
D
0
1
2 3
4
5
6
7
e 6 5 4 3 2 1
n
0
T
1
2 3
4
6 From these tables draw dot graphs of the information. a b a 1 2 3 4 5 b 7 14 21 28 35
5
6
7
a
0
1 1
a b
F
12 10 8 6 4 2
2 4
1
3 9
4 16
2 3
4
5
6
5 25
In a describe in words what is happening to the quantity b as a increases by 1.
Some members of this table are missing:
x y
1 5
2
3 7
4 8
this one is 5+1=6
5
6
this one is this one is 8+1=9 9+1=10 the y-values are increasing by 1
So the completed table is
x y
1 5
2 6
3 7
4 8
5 9
6 10
As the y-values are the x-values plus 4 we write y = x + 4. y = x + 4 is called the equation which connects x and y.
7 Complete these tables of values and find the equation connecting x and y-values. a b x 1 2 3 4 5 x 1 2 y 4 5 7 y 7 c
e
x y
1 10
2 9
3
4 7
5
d
x y
2 1
4
6 3
8
10 5
f
8 Jenny worked for 6 days, earning $150 a day. a On a grid like the one shown, plot the data. b Copy and complete:
d $
1
2
3
4
5
6
c Use the sequence to find how much Jenny would earn in i 10 days ii 30 days.
3 9
4
5 11
4 8
5
x y
1 2
2 4
3
x y
6 2
12
18 6
30 8
900 $ 800 700 600 500 400 300 200 100 0
2
4
6
day (d¡) 8
7
t
176
PATTERNS AND ALGEBRA (CHAPTER 11)
Unit 82
Using word formulae
² If an apricot pie costs $4 and you wish to buy several of them, then the formula for finding the total cost is: total cost is the number of pies bought multiplied by 4 dollars. ² If a taxi driver charges $3 for stopping (flag fall) and $1:50 for each kilometre travelled, then the formula for finding the total charge is: total charge is $3 plus the number of kilometres travelled multiplied by $1:50. ² A group of school students travelled by bus to see a play. If the cost of the bus hire is $2:50 for each student and the charge to see the play is $6:50 for each student, then the formula for finding the total cost is: total cost is the number of students travelling to the play by bus multiplied by $9:00. Formulae are used daily by many people. Often they do not realise that they are using them. Here are the first three members of a matchstick pattern The formula is the number of matches equals three times the number of squares plus one. ² We can check this word formula for these members. For 1 square, number of matches = 3 £ 1 + 1 = 4. X For 2 squares, number of matches = 3 £ 2 + 1 = 7. X For 3 squares, number of matches = 3 £ 3 + 1 = 10. X ² We can use the pattern to predict more complicated members. For 10 squares, number of matches = 3 £ 10 + 1 = 31. For 50 squares, number of matches = 3 £ 50 + 1 = 151.
Exercise 82 1 Here are the first three members of the matchstick pattern
.
The formula is the number of matches equals twice the number of triangles plus one. a Check the formula for the 3 members drawn above. b Use the formula to find the number of matches required to make the member with i 8 triangles ii 28 triangles. 2
are the first three members of a pattern of terrace houses. The formula is the number of matches equals five times the number of houses plus one. a Check the formula for the 3 members drawn above. b Use the formula to find the number of matches required to make the member with i 12 houses ii 58 houses.
3
are the first three members of a matchstick pattern. The formula is the number of matches equals seven times the member number plus one. a Check the formula for the three members above. b Use the formula to find the number of matches needed to make i the 10th member ii the 1000th member.
PATTERNS AND ALGEBRA (CHAPTER 11)
4 These are the first three members of a dot pattern
177
.
The formula is the number of black dots is double the number of coloured dots plus two. a Check this formula for the 3 members drawn above. b Find the number of black dots in the member with i 15 coloured dots ii 300 coloured dots. 5 For the pattern beginning
, a formula for the perimeter of each member is
given by perimeter equals twice the number of squares plus 2. Suppose each square is 1 cm by 1 cm. Use the formula to find the perimeter of the member consisting of a 6 squares b 30 squares c 300 squares. 6
(1)
(2)
(3)
is the beginning of a fencing pattern.
Diagram (3) consists of 4 posts (uprights) and 12 rails. The formula is number of rails equals four times the number of posts minus four. Find the number of rails needed to construct a fence with a 7 posts b 70 posts c 700 posts. 7 A taxi driver’s charge formula is the total charge is $3 plus the number of kilometres travelled multiplied by $1:50. Find the charge for a journey of a 1 km b 10 km c 17 km. 8 The cost of placing a small advertisement in the newspaper is given by the formula cost equals $7 plus $3 for each line of type. a Find the cost of placing an advert of i 5 lines ii 13 lines.
ACME TYRES The ACME tyre company is having an Auction at 11 am Friday, 6th May.
b How many complete lines could be placed if you have only $30? c How much would the illustrated advertisement cost? 9
1
5
17
53
is a number chain of length 4 using the formula
the next number is treble the previous one plus two. Draw number chains for a starting number is 2, length is 3, formula is the next number is twice the previous number plus five b starting number is 3, length is 4, formula is the next number is treble the previous number minus one c starting number is 4, length is 5, formula is the next number is half the previous one plus six. 10 Challenge 125
is a number chain with formula
the next number is twice the previous one plus three. What are the first 3 numbers in the chain?
Treble means multiply by 3.
178
PATTERNS AND ALGEBRA (CHAPTER 11)
Unit 83
Converting words to symbols
The word ‘is’ or ‘equals’ can be replaced by the symbol = , and letters can be used in place of some phrases to do with numbers of things. equals the number of boxes multiplied by |{z} $3 becomes For example, the formula |total {z cost} | {z }
=
C
|
{z
}
b
|
{z
£
}
3
C = b £ 3 dollars. Consider the word formula total profit is the number of bottles sold multiplied by $2. If T represents total profit and b represents the number of bottles then T = b £ 2 dollars. If 23 bottles are sold, then b = 23.
So, T
= 23 £ 2 dollars = 46 dollars.
Exercise 83 1 Consider the formula total cost equals the number of hammers multiplied by $7: a Write the formula in symbols. b Use the formula in a to find the cost of 17 hammers. 2 Consider the formula total profit is the number of cartons sold multiplied by $13. a Write the formula in symbolic form. b Use the formula in a to find the profit when 24 cartons are sold. 3 Jacko’s car uses 1 litre of petrol for every 14 kilometres travelled and he uses the formula litres needed equals kilometres travelled divided by 14 to work out how many litres are needed on a trip. a Write the formula in symbols. b Use the formula to determine the number of litres of petrol required for a 350 km trip. c If fuel costs $0:90 per litre, find the fuel cost for the trip.
The cost for a social club to go to the football is $12 per ticket and $150 for the hiring of the bus. ² If C is the total cost and m members go to the football then C = 12 £ m + 150 dollars. ² For 25 members on the bus, m = 25 and so C = 12 £ 25 + 150 = 450 dollars. ² Each of the 25 members pays
$450 = $18. 25
4 The total cost of an excursion is $8 per person and $200 for the bus hire. a If C represents the total cost for s people, write the formula using symbols. b Use the formula to find the total cost for a class of 29 students and their teacher. c Find the cost per person.
PATTERNS AND ALGEBRA (CHAPTER 11)
179
5 The total cost of putting on a ‘sausage sizzle’ is 50 cents per sausage cooked and $48 for hiring the barbeque. a Using C to represent total cost and s to represent number of sausages cooked, write the formula using symbols. b Use the formula to find the total cost of cooking 480 sausages. c If all sausages will be eaten find the cost at which each sausage should be sold in order to cover costs.
If tickets to the netball finals are $17 each then ²
the cost of 3 tickets = 3 £ $17
²
the cost of x tickets = x £ $17 this is a formula which is true for x = 1, 2, 3, 4, 5 ,....
6 There are 25 seats in each row in the football grandstand. How many seats are there in a 6 rows b r rows? 7 New books come in cartons of 18 books. How many books are there in a 5 cartons b x cartons? 8 The total cost of hiring a tennis court is $30 for the day. If each person pays an equal share, how much does each person pay if there are a 4 players b p players? 9 There are 32 biscuits in a family size packet. How many biscuits are there in a 5 packets b p packets?
Activity
Using a spreadsheet to produce patterns A spreadsheet is a computer program which consists of many cells in which words, numbers and formulae can be placed. An empty spreadsheet looks like
this is cell B2 this is cell C4 SPREADSHEET
Suppose you wish to use a spreadsheet to find the values of M in the formula M = 4 £ n + 13 for values of n from 1 to 30. Note that £ is done using ¤ on the keyboard.
Step 1:
In cell A1 type n and in cell B1 type M . The arrow keys enable you to move between cells.
Step 2:
In cell A2 type 1 and in A3 type the formula =A2 + 1 Select cell A3 and place the cursor on the bottom RH corner of the selection rectangle. When the cursor changes to a + sign press the LH button mouse and drag down to cell A31. This is called filling down. You should now have the numbers 1, 2, 3, 4, ... to 30 in the A-column.
Step 3:
In cell B2 type = 4*A2 + 13 and fill down to B31.
180
PATTERNS AND ALGEBRA (CHAPTER 11)
Unit 84
Algebraic expressions and equations
n are all examples of algebraic expressions 2 where x or n or any other chosen letter is used to represent an unknown number.
x + 2, 3 ¡ x, 5 £ n, and x + 6 = 11 is an equation.
If x is an unknown number, then
Do you know the difference between an expression and an equation?
Example
3 more than x is x + 3
f3 more than 8 is 8 + 3g
double x is 2 £ x
fdouble 8 is 2 £ 8g
5 less than x is x ¡ 5
f5 less than 8 is 8 ¡ 5g
Exercise 84 1 Yarna is 11 years old. How old will she be in a 3 years time b n years time
c
2 If y represents any unknown number, write expressions for a 4 more than the number b c twice the number d
x years time?
4 less than the number half the number.
3 In my pocket there are c coins. How many coins would be in my pocket if a I added two more b I took out 3 of them c I trebled the number d I halved the number?
I do not know how long my pen is and so I will say it is l cm long. How long is the part shown as in the following diagrams?
4
a
b
c
d
4 cm
6 cm
5 A pile of $1 coins is worth $n. a How many coins are in the pile? b How many coins are there in the pile if i iv
5 more are added ii 4 are removed iii the pile is trebled the pile is reduced to one third of its original size?
6 A pile of $2 coins is worth $d. a How many coins are in the pile? b How many coins are in the pile if i
6 are added
ii
8 are removed
iii
the pile is doubled?
7 x + 5 is an expression and x + 5 = 9 is an equation. a What does an equation have that an expression does not have? b Equation or expression? x i ii 7
x¡6=8
iii
3£x
iv
x =6 4
PATTERNS AND ALGEBRA (CHAPTER 11)
Substituting into an expression When we replace a letter by a number we call this substitution. For example, if we replace x by 5 then ²
x+2 =5+2 =7
²
x¡2 =5¡2 =3
²
x£2 =5£2 = 10
10 x
²
or
10 5 =2
=
8 If a = 8, what is the value of a e
a+4 16 + a
b f
a¡4 16 ¡ a
c g
a£4 16 £ a
d h
a¥4 16 ¥ a
9 If b = 3, find the value of a
b+7
b
11 ¡ b
c
b£8
d
b£b
e
12 b
f
13 + b
g
b+b
h
b¡b
Solving equations x + 7 = 11 is an equation, and as 4 + 7 = 11 then x = 4. x = 4 is called the solution of the equation. 10 Find a d g j m p
the number that replaces ¤ in each ¤+2=5 b 3+¤=8 e 13 + ¤ = 25 h ¤ + 23 = 32 k 15 ¥ ¤ = 3 n 2 = 10 4
q
of these algebraic sentences. 10 ¡ ¤ = 4 12 ¡ ¤ = 3 ¤ ¡ 5 = 10 ¤ ¡ 8 = 12 ¤¥6=5 2 =9 5
11 Find the number that would replace ¢ in these algebraic sentences. a ¢ + 7 = 13 b 13 ¡ ¢ = 6 d ¢ ¥ 3 = 11 e 3¥¢=3 g
56 =8 ¢
12 Solve for x: a 4+x=9 d 10 ¡ x = 4 g 3 £ x = 12
c f i l o
4 £ ¤ = 16 5 £ ¤ = 10 ¤ £ 3 = 12 ¤ £ 7 = 21 ¤ ¥ 3 = 10
r
2 =8 7
c f
¢+¢=8 ¢ £ 7 = 56
h
¢ + 7 = 52
i
¢ =9 6
b e h
6 + x = 11 12 ¡ x = 3 5 £ x = 20
c f i
12 + x = 15 15 ¡ x = 10 4 £ x = 24
j
15 =3 x
k
12 =4 x
l
x = 11 4
m
6 £ x = 42
n
8 £ x = 40
o
x £ 11 = 99
p
x =9 7
q
110 = 11 x
r
x =7 8
10 x = 10 ¥ x = 10 ¥ 5 =2
181
182
PATTERNS AND ALGEBRA (CHAPTER 11)
Unit 85
Graphing from a rule
Chad and Emily wash cars at car yards. They charge their customers using the rule C = (3 £ n) + 5 dollars where n is the number of cars washed. So, if they wash 7 cars, they earn C = (3 £ 7) + 5 = 21 + 5 = 26 dollars Using Chad and Emily’s formula we see that: When n = 1, C = (3 £ 1) + 5 = 8 When n = 2, C = (3 £ 2) + 5 = 11 When n = 3, C = (3 £ 3) + 5 = 14 When n = 4, C = (3 £ 4) + 5 = 17 When n = 5, C = (3 £ 5) + 5 = 20 From these values we obtain the graph shown.
20 18 16 14 12 10 8 6 4 2 0
Washing charges C
n=2, C=11 n=1, C=8
n 0
1
2
3
4
5
Exercise 85 1
2
a If P = (3 £ x) + 15, find P when x is i 4 ii 10
iii
23
iv
100
b If Z = (x £ 7) ¡ 11, find Z when x is i 2 ii 12
iii
40
iv
1000
a For the rule D = (2 £ n) + 3, find D when n = 1, n = 2, n = 3, n = 4, n = 5 and n = 6. b Copy and complete the table, using your answers in a.
n D
1
2
3
4
5
PRINTABLE GRIDS
D
6 n
c On grid paper, draw the graph of the values in the table. 3
a For the rule R = (3 £ n) ¡ 2,
find R when n = 1, n = 2, n = 3, n = 4, n = 5 and n = 6.
b Copy and complete the table, using your answers in a.
n R
1
2
3
4
5
R
6 n
c On grid paper, draw the graph of the values in the table. 4
a For the rule K = n £ n ¡ 1,
find K when n = 1, n = 2, n = 3, n = 4 and n = 5.
b Copy and complete the table, using your answers in a.
n K
1
2
3
4
K
5
c On grid paper, draw the graph of the values in the table.
n
PATTERNS AND ALGEBRA (CHAPTER 11)
Investigation
183
Water filling graphs
Click on the icon to see water entering various containers that you can construct.
DEMO
At the same time as water enters the vessel the graph of the height of water is plotted against the time taken. Experiment with different containers and try to predict what will happen to the graph when filling is slow, steady or fast.
depth of water
5 Draw a graph that could tell each of these stories.
b
distance from school
a You are taking a bath where you are checking the depth of water regularly. (Start with no water and finish with all water gone.)
time
You are riding on a bike or in a bus to school with your friends. Record the distance you are from school at regular time intervals. distance from home
time
c Take a trip to the shops and return home recording the distance you are from home at regular times.
Activity
time
Using algebra to describe the area of shapes If
has an area of p units and
has an area of b units then the area of the following
shape can be written in terms of p and b. For example, for the given shape, the total area is made up of 2p units and 1b unit. Therefore, the total area is Try these:
p+p+b =2£p+b = 2p + b in algebraic language
Use the knowledge you have learnt above to write the total area of each of the following shapes in terms of p and b.
a
b
c
d
e
f
g
h
i
Answers a p+b
b 2p
c
2b d 3p + b e 3b f 4p + 2b g 3p + 2b h
2p + 10b i 2p + 4b
184
PATTERNS AND ALGEBRA (CHAPTER 11)
Unit 86
Review of chapter 11
Review set 11 A 1 Write down the first 5 numbers in the pattern described by a starting at 2 and increasing by 7 each time b starting at 100 and decreasing by 7 each time. 2 Describe in words the pattern rule for the sequence 3, 8, 13, 18, 23, ..... 3
a Look at the pattern and draw the next two diagrams in the pattern. b List the number of dots in each pattern for the first 5 members. c Partition the members of the pattern to get a number pattern. d Predict the number of dots in the 8th pattern without drawing it.
4
a
i Give in sentence form a rule to find the next member of 23 ! 19 ! 15 ! 11: ii Find the next member.
b
i Find the rule in sentence form for 13, 19, 25, 2, 37. ii Find the missing number.
5 Sam is starting an exercise program. He exercises for 15 minutes each day for a week. Each week he will increase his daily exercise time by 5 minutes. a Write Sam’s daily exercise times for the first 5 weeks. b Write a rule to give Sam’s daily exercise time for any week. Use T to represent the exercise time and w to represent the number of weeks. c How many weeks from the start of Sam’s exercise program will it take him to reach 45 minutes per day? number
6 On grid paper, draw the graph of the number sequence 1, 3, 5, 7, 9. Do the points on this graph lie on a straight line?
member
7 Complete these tables: a x 1 2 3 y 4 6
4
5 8
b
6
x y
2 1
4 2
6
8
10 5
12
8 On a set of axes draw the graph of the points from 7 a. 9 If p = 8, find the value of a
p + 10
b
19 ¡ p
c
7£p
d
p 4
e
10 Solve these equations: a
y + 7 = 14
b
y ¡ 8 = 10
c
4 £ y = 20
d
p£p
y =3 4
11 The charges to travel on a ferry are $86 per bus plus $15 per passenger. a Find the charge if the bus carries i
2 passengers
ii
10 passengers
iii
30 passengers.
b If the bus carries 10 passengers, how much must each passenger be charged to cover the cost of the ferry crossing, including the charge for the bus?
PATTERNS AND ALGEBRA (CHAPTER 11)
185
Review set 11B 1 Write down the first 5 members in the pattern described by a starting at 5 and increasing by 8 each time b starting at 97 and decreasing by 8 each time 2 Describe in words the pattern rule for the sequence 43, 35, 27, 19, 11, ..... 3
a Look at the pattern and draw the next two diagrams.
b Construct a table with headings ‘Diagram number’ and ‘Number of matchsticks’. c List the number of matchsticks in each diagram up to the sixth diagram. d Predict the number of matchsticks in the 10th diagram without drawing it. 4
i Give in sentence form a rule to find the next member of 5 ! 10 ! 20 ! 40:
a
ii Find the next member. i Find the rule in sentence form for 81, 2, 9, 3, 1.
b
ii Find the missing term. 5 If n = 6, find the value of a
4+n
b
n¡5
c
6£n
d
n£n
e
6 Solve for x: a
3 + x = 12
b
10 ¡ x = 3
c
6 £ x = 24
d
12 n
18 =9 x
7 In his first Mathematics test Joseph scored 23 marks out of 50. He then improved his marks by 4 in each test. (All tests were out of 50 marks.) a Write the marks that Joseph scored in the first 4 tests. b Write a rule to find Joseph’s marks in his next tests if he continues to improve by 4 marks each test. Use M to represent the mark and t to represent the number of tests.
8
number
c How many marks would Joseph score in his seventh test?
A number sequence is shown on the graph alongside. a List the members of the sequence. b Is the sequence increasing or decreasing?
18 15 12 9 6 3 1
9
2
3
4
5
6
member
a Complete this table which is a pattern:
x y
4 7
6
8 11
12 15
b Graph the members of the table. c Describe the rule used in word form. 10 Boxes of books are being stacked on a pallet to be transported from Adelaide to Perth. The pallet weighs 30 kilograms and each box of books weighs 15:8 kilograms. a Find the weight of the load if i 5 boxes
ii
10 boxes
iii
40 boxes are sent.
b If the weight limit of a load is 200 kilograms, how many boxes can be sent on the pallet?
186
TRANSFORMATIONS
Unit 87
The language of transformations
Translation, reflection, rotation and enlargement are all transformations. TRANSLATION(1)
B
TRANSLATION (2)
TRANSLATION(2)
The word TRANSLATION has been shifted 3 units to the right and 2 unit upwards to (1). It has been shifted 4 units to the left and 3 units down to (2). ROTATION(1)
180°
A
Place a mirror along dotted line A and (1) is the reflection in A. What do you notice? Place a mirror along dotted line B and (2) is the reflection in B. Describe what you see.
ENLARGEMENT
(1)
ROTATION(2)
90°
REFLECTION
(1)
ROTATION(3)
ROTATION
270°
ENLARGEMENT ENLARGEMENT(2)
The word ROTATION has been rotated about ² clockwise through 90o to (1), through 180o to (2) and through 270o to (3).
The word ENLARGEMENT has been enlarged (made bigger) at (1) and reduced (made smaller) at (2).
Here are some more examples of translations, reflections, rotations and enlargements. a translation (or shift)
shift to a new position in a particular direction
CHAPTER 12
a rotation about O
a reflection
mirror line
an enlargement
O
The original shape is called the object. The shape which results from the transformation is called the image.
TRANSFORMATIONS (CHAPTER 12)
187
Congruent figures Two figures are congruent if they have exactly the same size and shape.
We are congruent.
If one figure is cut out and it can be placed exactly on top of the other, then these figures are congruent. The image and the object for a translation, rotation and reflection are congruent. The image and the object for an enlargement are not congruent because they are not the same size.
Exercise 87
1
2
3
4
5
6
Row 1
Row 2
Row 3
Row 4
Row 5
Use the fabric sample shown to answer the following questions. The rows and columns have been numbered to help you. 1 Start with row 1 and column 1 cat. a Give the row and column numbers for translations of this cat. b Give the row and column numbers for rotations of this cat. (Exclude rotations of 360o .) c Give the row number in column 1 for an enlargement of this cat. d Give the row numbers in column 3 for any single cat congruent to this cat. 2 Start with row 1 column 2 cat. a Give the row and column numbers for translations of this cat. b Discuss why row 4 column 2 is not a rotation of this cat. c Give the row and column numbers for rotations of this cat. (Exclude rotations of 360o .) d Give the row number in column 1 for a reflection of this cat. 3 Start with row 4 column 5 cat. a Give the row numbers for any cats congruent to this one that appear in column 2. b Give the row and column numbers of any cats that are a rotation of this cat. (Exclude rotations of 360o .) c Give the row and column numbers for any cats that are a reduction of this cat. 4 What is the transformation shown in the pair of cats? 5 Which two transformations are used to move the cats in a row 1 column 1 to row 1 column 5 b row 2 column 4 to row 5 column 4 c row 2 column 5 to row 5 column 5 d row 3 column 6 to row 2 column 6?
188
TRANSFORMATIONS (CHAPTER 12)
Unit 88
Tessellations
These are brick paving tessellations.
The first of these is a semi-regular tessellation as it uses two different shapes. The other two are pure (or true) tessellations.
Activity
Paving bricks What to do: , form at least two different tessellation patterns. The pic1 Using the “2 £ 1” rectangle ture of paving bricks below gives an example.
2 Repeat 1 using a “3 £ 1” rectangle.
To form a tessellation the shape may be translated, rotated or reflected (or maybe a combination of these).
Note:
From this basic shape
we can make
From this basic shape
we can make
Exercise 88 1 This tessellation is made from equilateral triangles.
By drawing the following shapes discover which ones produce pure tessellations. a d
a square a regular octagon
b e
2 Draw tessellations using these shapes. a b
a circle a parallelogram
c f
c
a regular hexagon a regular pentagon
TRANSFORMATIONS (CHAPTER 12)
3 Draw tessellations using these shapes. a b
189
c
4 Each of the three tile patterns is made by using tiles of the same size and shape.
Choose one of the above and make up a poster size (A3) coloured pattern suitable to be hung in the classroom. 5 What two different shapes make up these semi-regular tessellations? a b
Group discussion
In good shape
1 Research the shape of the cells in a beehive and explain why they are that shape. 2 Look at the shapes of paving blocks and explain what advantage some shapes have over others. When building walls, what are the advantages of rectangular bricks over square bricks? 3 Why are objects with an equilateral triangle shape easy to pack?
Activity
Computer tessellations This activity requires you to use a computer programme which enables us to tessellate. DEMO
What to do: 1 Pick a shape and learn how to translate, reflect and rotate it. 2 Create a tessellation on your screen. Colour your tessellation. 3 Print your final masterpiece.
190
TRANSFORMATIONS (CHAPTER 12)
Unit 89
Line symmetry
Line symmetry If a shape can be folded so that one half exactly matches the other half, then the shape has line symmetry. The fold line is the line of symmetry.
Note: If a mirror is placed along the line of symmetry, the reflection in the mirror will be exactly the same as the half of the figure “behind” the mirror.
Investigation You will need:
Finding lines of symmetry Mirror, tracing paper, ruler.
DEMO
What to do: 1 Examine the following figures and state the number of lines of symmetry each figure has. a b c
2 Copy the figures and use your ruler to draw the lines of symmetry on each figure. 3 Using the mirror and placing it along each of the lines of symmetry drawn in 2, show that the figure is reflected along that line.
Look at these shapes for lines of symmetry:
1 line of symmetry
no lines of symmetry
2 lines of symmetry.
Exercise 89 1 Copy these figures and draw the lines of symmetry a b
d
c
e
f
2 Using the digits
find all digits which have lines of symmetry. 3
a Copy these shapes and draw in all lines of symmetry. i
ii
iii
b Which of these figures has the most lines of symmetry?
iv
TRANSFORMATIONS (CHAPTER 12)
191
4 How many lines of symmetry do these patterns have? a b
These patterns are New Zealand Maori artwork.
Investigation
Making symmetrical shapes paper, scissors, pencil, ink/paint
You will need:
1 Take a piece of paper and fold it as shown alongside and then cut out a shape: 2 Open out the sheet of paper and observe the shapes that are revealed. 3 Record any observations about symmetry that you notice. 4 Try the following: a Fold the paper twice before cutting out your shape. b Fold the paper three times before cutting out your shape.
In each case record your observations about symmetry. 5 Place a blob of ink/paint in the centre of a rectangular sheet of paper. Fold the paper through the centre of the blob and press the two pieces together. Open the paper and again comment on the symmetry observed. ink
fold
fold line
ink blot
6
Try making other symmetrical patterns by folding a number of times and cutting out a shape. For example, how many folds would you need and what shape would you cut out to get the result shown? Note: Sometimes lines of symmetry are easier to see when the shape is drawn on a grid.
Martius Escher, who died in 1972 aged 74, created many clever and fascinating patterns, called tessellations. The above is one of his better known ones. Research the art of Escher in the library or on the internet.
192
TRANSFORMATIONS (CHAPTER 12)
Unit 90
Rotations and rotational symmetry
Rotations A rotation is a turning of a shape or figure about a point and through a given angle. We are all familiar with things that rotate.
DEMO
In the pictures, the hands of the clock rotate around the centre, and the wheels of the motor bike rotate around the axle. The point about which the hands of the clock, or the spokes of the wheel rotate, is called the centre of rotation. The angle through which the hands, or the spokes turn is called the angle of rotation. The globe of the world rotates about a line. The line is called the axis of rotation.
Activity
Using technology to rotate The purpose of this activity is to use a computer package to construct a shape that has rotational symmetry.
What to do: 1 Click on the icon. 2 From the menu, choose an angle to rotate through. 3 Make a simple design in the sector which appears. 4 Colour your sector and press
finish
ROTATING FIGURES
to see your creation.
Rotational symmetry
A full rotation does not mean that a shape has rotational symmetry. Every shape fits exactly onto itself after a rotation of 360°.
A shape has rotational symmetry if it can be fitted onto itself when it is turned through an angle of less than 360 o, i.e., less than one full turn. The centre of rotational symmetry is the point about which a shape can be rotated onto itself. If you cut out the shape, the centre of rotational symmetry is the point where you would put a pin so that the shape would spin.
O
The ‘windmill’ shown will fit onto itself every time it is turned through 90 o about O. O is the centre of rotational symmetry. (The design is symmetric about this point.)
Exercise 90 1 For the following shapes, find the centre of rotational symmetry. a b
c
TRANSFORMATIONS (CHAPTER 12)
d
e
f
g
h
i
193
(Hint: You may wish to trace the shape to help you answer this question.) 2 From the block alphabet alongside list the letters which become another letter of the alphabet with rotation: 3 Draw a an equilateral triangle
b
a square.
Examine each figure and locate the centre of rotation for each figure. is rotated clockwise about O through 120o it becomes
4 If
. O
O
If rotated through another 120o it becomes
and finally O
O
We say that an equilateral triangle has order of rotational symmetry 3 as it rotates onto itself about a 120o , 240o and 360o rotation. Find the order of rotational symmetry of: a b c d
Research Search through magazines, visit the internet, watch TV commercials, dance and drama, study sheet music and artwork for examples of line symmetry and rotational symmetry. Identify floor and tile patterns, wrapping paper, material, designs, gardens, business logos, and trademarks which are symmetrical. Where possible, make copies to share with your classmates.
For lots of wonderful transformations look into a kaleidoscope. Make rotations and look for the lines of symmetry.
194
TRANSFORMATIONS (CHAPTER 12)
Unit 91
Enlargements and reductions
We have all seen enlargements of photographs and slides being projected or looking through a microscope. Plans and maps are examples of reductions. Most photocopiers can make images either smaller or larger than the original. Alongside is a design which shows the use of enlargement (or reduction). Look at the figures in the grid below: A
B
C
D
DEMO
enlargement scale factor 2 enlargement scale factor 3
Notice that for an enlargement with scale factor 2, lengths have been doubled. With scale factor 3, lengths have been trebled. If shape B is reduced to shape A, lengths are halved, so the scale factor is 12 . If shape D is reduced to shape A, lengths are quartered, so the scale factor is 14 . Notice that the scale factor is
² ²
greater than 1 for an enlargement less than 1 for a reduction.
Exercise 91 1 In these diagrams, A has been enlarged to B. Find the scale factor. a b
A
B
A
c
B
A
B
2 In these diagrams, B is an enlargement or reduction of A. Find the scale factor. a b c
A
B
A
3 Copy the figures and enlarge with the given scale factor. a Scale factor 2 b Scale factor 12
A
B
c
Scale factor 3
B
TRANSFORMATIONS (CHAPTER 12) 1 2
d
Scale factor 2
e
Scale factor 3
f
Scale factor
g
Scale factor 2
h
Scale factor
1 2
i
Scale factor 3
Activity
195
Enlargements by grids You will need:
Paper, pencil, ruler
What to do: 1 Copy the picture alongside.
2 Draw a grid 5 mm by 5 mm over the top of the rabbit alongside as shown: 3 Draw a grid 10 mm by 10 mm alongside the grid already drawn.
PRINTABLE TEMPLATE
4 Copy the rabbit from the smaller grid onto the larger grid. Transfer points where the drawing crosses the existing grid lines to the corresponding points on the new grid. Join these points and finish the picture.
5 Use this method to change the size of 3 or 4 other pictures of your choosing. You may like to try making the picture smaller as well as larger, by making your new grid smaller than the original.
Activity
Using technology to enlarge and reduce What to do: 1
With your teacher visit the school’s photocopier with pictures and maps to be enlarged or reduced.
What feature makes it easy to enlarge or reduce? What is meant by a a 100% enlargement b a 200% enlargement
ENLARGER
c
a 50% enlargement?
2 Click on the icon to view an enlargement/reduction computer program. Click on any of the objects given in the menu. Then choose the scale factor. Then finish .
196
TRANSFORMATIONS (CHAPTER 12)
Unit 92
Review of chapter 12
Review set 12A 1 Draw tessellations of the following shapes. a
b
2 Copy the following figures and draw the lines of symmetry. a b
3 Copy the figure and find the centre of rotational symmetry. What is the order of rotational symmetry in this case?
In the diagram, A has been enlarged to B. Find the scale factor.
4 A B
5 a b c d 6
Copy the figure. Show the image of the figure for a reduction and scale factor 13 . Find the area of the original figure. Find the area of the reduced figure. What is the relationship between the two areas? Copy the figure and find the centre of rotational symmetry.
7 In the diagram A has been reduced to B. Find the scale factor.
8 A rotation of object A about O gives the image B. Through how many degrees has A been rotated?
A B
B O
A
TRANSFORMATIONS (CHAPTER 12)
Review set 12B 1 Draw a tessellation using the given shape.
2 Give two different tessellations of the given tile.
1 unit 2 units
3 Copy these figures and draw the lines of symmetry where possible. a parallelogram b equilateral triangle
4 Draw an equilateral triangle and find its centre of rotational symmetry. 5 Copy these figures and draw the lines of symmetry. a b
6 Copy the figure. Show the image of the figure for an enlargement with scale factor 2.
7 In the diagram, B is the image of A. a Has A been enlarged or reduced to B? b Give the scale factor. A B
8
a b c A
B
d C
9 Find the order of rotational symmetry of a b
Copy the figure for an enlargment with scale factor 3. Use a ruler to find the length of AC. Use a ruler to find the length of the image of AC. Use your answers to b and c to copy and complete: length of the image of AC = .... £ length of AC
c
d
197
198
CHANCE AND PROBABILITY
Unit 93
Describing chance
Words that are used to describe the chance of something happening in the future include: possible, impossible, likely, unlikely, maybe, certain, uncertain, no chance, good chance, highly probable, probable, improbable, doubtful, often, little chance, rarely.
Exercise 93 1 Describe using a word or phrase, the chance of the following happening. a There will be a public holiday on January the 1st. b You will die by accidental drowning.
a ‘6’
c Your birthday next year will fall on a week day. d Next year there will be 30 days in February. e You will eat a meal between 5 pm and 10 pm tonight. f You will roll a ‘6’ next time with a die. g You could run from Adelaide to Melbourne in one day. h The next child born at a particular hospital will be a girl. i You will have a shower tomorrow. 2
2
3
1
3
2
3
2
A spinner is constructed using 10 equal sectors of a circle as shown. The result of a ‘spin’ is where the arrow stops. At the moment the spinner is on ‘1 grey’. On either side of it are the sectors ‘3 white’ and ‘3 blue’. Copy and complete these chance statements.
3
1
3
a It is ...... for the spinner to stop on a 4. b There is an ...... chance that the spinner will stop on a grey sector. c It is ...... that the spinner will stop on a number. d It is ...... that the spinner will stop on a white sector. e It is ....... that the spinner will stop on a number greater than one. 3 Which of these is possible? a One day you will buy a house. b One day you will dig a hole to the centre of the Earth. c One day you will marry. d You will eat a meal tomorrow. 4 Write two things which are unlikely to happen to you. Why?
CHAPTER 13
5 When a coin is tossed there is a fifty-fifty chance that it could fall heads or fall tails. Give two other examples of when a fifty-fifty chance could occur. 6 Explain why these events are impossible. a a cat could give birth to puppies b the Moon will collide with the Earth c the Sun will set in the East d you will swim from Sydney to Los Angeles
CHANCE AND PROBABILITY (CHAPTER 13)
199
Example: A dam contains 80 adult fish of which 10 are trout. One fish is caught. We can say that it is unlikely that the fish caught is a trout as only 10 in 80 or 1 in 8 fish in the dam are trout. 7 Below is a chance line. From the listed words and phrases below, find replacements for a, b, c, d, e and f. a
c
b
d
e
impossible
more likely than not
f certain
equal chance, almost impossible, a little less than equal chance, highly likely, unlikely, very unlikely 8 A bag contains 20 marbles, of which 19 are grey and one is blue. A marble is randomly chosen from the bag. a How likely is the marble to be blue? b How likely is the marble to be grey?
To randomly choose means that each object has the same chance of being selected.
c How likely is the marble to be red? d How likely is the marble to be blue or grey? e True or false? “There is a 1 in 20 chance that the marble is blue.”
9
A can contains 7 black, 3 white, 4 blue and 6 grey disks. One disk is randomly chosen from the can. a How likely is it that the disk is i black, blue or grey ii white iii black or white? b True or false? “There is a 1 in 5 chance that the disk is blue.”
10 Describe using either certain, possible or impossible. a When tossing a coin it falls ‘tails’ uppermost.
a die
b When tossing a coin 8 times it falls ‘heads’ on every occasion. c When rolling a 6-sided die, a 6 results. d When rolling a 6-sided die, an 8 results. e When rolling a die, a result between 0 and 10 occurs.
a pair of dice
11 Using the words from question 7, copy and complete: a It is ..... for the spinner to stop on a 4. b There is an ...... chance that the spinner will stop on a grey sector. c It is ..... that the spinner will stop on a number.
3
2
d It is ...... that the spinner will stop on a white sector. e It is ..... that the spinner will stop on a number greater than one.
1
200
CHANCE AND PROBABILITY (CHAPTER 13)
All possible results
Unit 94 When tossing a 20-cent coin there are two possible results, a ‘head’ or a ‘tail’. The ‘head’ side of a coin usually shows the head of a Monarch, Prime Minister or President. If we let H represent a ‘head’ and T represent a ‘tail’ we could indicate the possible results by ² ²
listing them illustrating them on a tree diagram.
a head
a tail
Tossing one coin The possible results are:
²
fH, Tg when listing them or
H
²
T
using a tree diagram.
When tossing a coin, if each outcome has the same chance of occurring we would expect heads : tails = 50% : 50%.
Exercise 94 1 Consider the children possible in families of one child (according to sex). a List the possible children in words. b List the possible children in shorthand notation. c Draw a tree diagram showing the possible children. 2
a List the possible results when a single die is rolled. b Draw a tree diagram of the possible results.
1 st
3 If B represents ‘a boy’ and G represents ‘a girl’ a draw a tree diagram to represent the children possible in twochild families. Use labels of ‘first child’ and ‘second child’ on your diagram. b List the children possible in two-child families using Bs and Gs.
2 nd
B
G
Investigation
Tossing one coin
What to do: 1 Toss a coin 10 times. Repeat this a second time, a third time, etc. for 6 times. 2 Are the results for each set of 10 tosses, 5 heads and 5 tails? 3 Are the results from one toss to the next predictable? 4 Click on the icon to toss one coin 1000 times. What are the results? Repeat the 1000 tosses simulation several times. Are the results predictable? Now repeat several times the simulation of one coin tossing 10 000 times.
TOSSING ONE COIN
Number of heads
Number of tails
First 10 Next 10 Next 10 Next 10 Next 10 Next 10
5 True or false? ² “Over a large number of tosses, individual results are unpredictable, but approximately and 12 are tails.” ² “The larger the number of tosses the closer we are to getting 50% heads, 50% tails.”
1 2
are heads
CHANCE AND PROBABILITY (CHAPTER 13)
Investigation
201
Tossing two coins
What to do:
Possible result two heads head & tail two tails Total
1 In pairs toss two coins 100 times and record your results on a table like the one given. 2 Add into your results the results of 4 other pairings. 3 Explain why there are about twice as many ‘head and tail’ results as ‘two heads’ or ‘two tails’.
Frequency
100
COIN TOSSING SIMULATION
4 Use the coin tossing simulation to toss two coins 10¡000 times and repeat this 4 more times, each time recording results.
From the investigation you should have discovered that we get about 25% ‘two heads’, 50% ‘a head and a tail’
and 25% ‘two tails’.
1 2
1 4
Tally
1 4
This is because we have 4 possible results, 2 of which appear the same.
The shorthand way of showing this is the listing f |{z} HH 1 4
H
H T
T H
HT, TH, | {z } 1 2
TT g. |{z} 1 4
is the tree diagram which shows all four possibilities.
T
4 O represent an ‘odd’ result and E represent an ‘even’ result when a die is rolled. The die is rolled twice. a Draw a tree diagram of possible results. b List the possible results. 5
win lose
win lose win
a b
Explain what the diagram alongside could represent. List the 4 possible results.
lose H
6 A coin is tossed and a die is rolled. Copy and complete:
5
a List the possible results (H5 is one of them). b How many results are possible?
7 A restaurant has the following menu. Amy and Jon eat at the restaurant and both choose from each course on the menu. a Draw a tree diagram showing all possible choices for the 3-course meal.
2
coin
die
Entree
Main course
Sweets
Pasta Prawn cocktail Fish
Steak Venison Lamb
Cheesecake Ice cream
b How many different choices are possible? c Amy chose pasta for her entree. How many choices does she have for the other two courses? d Jon chose icecream for his sweets. How many choices did he have for the rest of his meal?
202
CHANCE AND PROBABILITY (CHAPTER 13)
Unit 95
Probability
The chance of something happening is its probability. We assign numbers to all probabilities. These numbers are between 0 and 1 (including 0 and 1). If something cannot happen its probability is 0. If it is certain to occur we use 1. Decimals, fractions and percentages are all used when giving probabilities. All sector angles are 120o , so each letter P, Q or R is equally likely to occur at a spin.
For this spinner
P
R
Q ² the chance of spinning a P is
1 3
or 33 13 %
² the chance of spinning P or Q is
2 3
or
² the chance of spinning a T is 0 or 0% fas no T existsg
66 23 %
² the chance of spinning P, Q or R is 1 or 100%
Exercise 95 1 Convert to fractions a a chance of 1 in 4 d a chance of 3 in 5
b e
a chance of 1 in 7 a chance of 2 in 11
a chance of 2 in 3 a chance of 8 in 11
c f
2 A die is rolled. a List the possible results. b What is the probability of getting ii
B
A
C
3
a6
D
i
a 2 or a 3
iii
a7
iv
a result less than 7?
A spinner has equal sector angles of 90o . Show on a number line the chances of getting a aB b an E c a B or a D d an A, B, C or D.
4 A small lottery has 100 tickets and you buy 3 of them. What is the probability that you will a win first prize b not win first prize? 5 A hat contains 3 black, 1 white and 4 blue cards and one of these cards is randomly selected. What is the chance that it is a black b white c blue d purple e black or white f black, white or blue? 6 On a number line like this one mark the probability of these things happening:
0
0.5
1
a I will obtain a head or a tail next time I toss a coin.
d I will get a head next time I toss a coin. f I will get a 3 next time I spin this spinner.
4
e 7 £ 8 = 63
3
c The sun will rise tomorrow.
2
b I will grow antlers by 6:00 pm.
1
CHANCE AND PROBABILITY (CHAPTER 13)
203
7 The probability that Joe will be late for school is 6%. What is the probability he will not be late? 8 The probability that the computer program will fail is 14 . What is the probability that it will not fail? 9 This table shows the pets owned by students in a class. a If a boy is randomly chosen from the class what is the probability we will own i a cat ii a dog iii a fish? b Repeat question a but this time for a girl. c Repeat question a but this form for any student.
Pets cats dogs fish Total
Boys 5 9 2 16
Girls 4 10 0 14
Total 9 19 2 30
10 There are 18 socks in a drawer (6 red, 4 blue, 1 white, the rest black). In the dark one sock is pulled out. What is the probability that it is a red b blue c white d black e green f blue or black g white, red or green h black, blue or red.
Game
Dicing with danger This is a short game for two to four players. The players take it in turns to roll a die three times. Counters are used to show the position of each player on the game tree. The winner is the person who has the highest score after 10 games. 1 to 4
5 or 6
Roll 1 1 to 4
5 or 6
1 to 4
5 or 6
Roll 2 1 to 4
5 or 6
1 to 4
5 or 6
1 to 4
5 or 6
1 to 4
5 or 6
Roll 3 win 2
Example:
lose 2
win 2
lose 3
lose 1
win 3
win 4
win 5
Edward and Jane played the game once and the results were: Edward 5, Jane 2, Edward 4, Jane 3, Edward 5, Jane 5. The result is: Edward wins 3 points, Jane loses 2 points.
A score card example:
Game 1 2 3 .. .
10
Edward game progress +3 +3 ¡2 +1 +2 +3
Jane game progress ¡2 ¡2 +5 +3 +4 +7
SCORE CARD TEMPLATE
204
CHANCE AND PROBABILITY (CHAPTER 13)
Unit 96
Review of chapter 13
Review set 13A 1 Use words such as certain, highly likely, equal chance to describe the chance of the following happening. a It will be cold in winter in Tasmania. b Tomorrow’s maximum (highest) temperature will be 60o C. c When you are an adult you will be taller than you are now. d If you tossed a coin it would land heads. 2 An eight-sided die has the numbers 1 to 8 marked on its faces. If the die is rolled, show on a number line the chance of a 1 b 1 or 2 c 9 d an even number.
43
65
87
3
21
4
1 2 5 8
A spinner is constructed using 8 equal sectors of the circle as shown. The result of a spin is where the arrow stops. The spinner is shown on 1 white. Segments 2, 5 and 8 are grey. Copy and complete the following chance statements. a The spinner is ..... to stop on a number from 1 to 8. b It is ..... that the spinner will stop on white or grey. c There is an ..... that the spinner will stop on an odd number. d It is ..... that the spinner will stop on 9.
a You are given the playing cards shown. The cards are well shuffled. 2 3 One card is chosen at random. i List the possible colour outcomes for the card. ii List the possible outcomes according to the number on the card. iii List the possible outcomes according to the suit, for example, diamonds.
2
3
2
b The first card is replaced. The cards are shuffled. Another card is chosen at random. Draw a tree diagram for the possible outcomes for choosing the two cards according to the number on them. 5 A bag contains 5 disks. Three are blue, one is yellow and one is pink. One disk is taken out of the bag and its colour is noted. a What is the chance that the disk is yellow? b Find the chance that the disk is blue. c What is the chance that the disk is not pink?
The disk is put back into the bag and the bag is shaken. Another disk is taken at random from the bag. d List the possible outcomes for the colours of the two disks. Write B for blue, Y for yellow and P for pink. e Draw a tree diagram to show this information. 6 A board is divided into 8 equal parts as shown. A marker is thrown on to the board. Show on a number line the chance of the marker landing on a A b A or B c X d any letter of the alphabet e one of the first 4 letters of the alphabet 7
A C E G
B D F H
A survey showed that six students in a class of 25 were lefthanded. a Estimate the probability that a randomly chosen student in the school is left-handed. Write your answer as a percentage. b Estimate the number of students in a school of 225 students who are left-handed.
205
CHANCE AND PROBABILITY (CHAPTER 13)
Review set 13B 1 Use the words impossible, possible, or certain to describe the following. a An isosceles triangle has two equal sides. b All angles of a triangle are acute. c All angles of a triangle are obtuse.
A circular spinner has 16 equal segments. One segment is coloured grey, 8 are white and the remainder are blue. Copy the following chance line and mark your replacement words for the questions on it.
2
impossible
a c 3
The spinner is ...... to stop on grey. The spinner is ...... to stop on blue.
a You are given the playing cards shown alongside.
equal chance
b d
A
certain
The spinner is ...... to stop on white or blue. The spinner is ...... to stop on white.
2
2
3
3
4
4
They are shuffled and one card is chosen at random. i List the possible colour outcomes for the card. ii List the possible outcomes according to the numbers on the cards (an ace is 1). iii List the possible outcomes according to the suit. b The first card is replaced. The cards are shuffled and a second card is drawn at random. i Draw a tree diagram to show the possible colour outcomes for the two cards. ii List the possible colour outcomes for the two cards. 4 One card is chosen at random from the cards shown in question 3. Find the probability that the card a is an ace b is blue c has an odd number d is not a four e is a club 5 A survey showed that 60% of households have the daily paper delivered. a What is the probability that a household chosen at random i
has the paper delivered
ii
does not have the paper delivered?
b If there were 8000 households in a town, how many would you expect to have the paper delivered? 6 It was noticed that 3 students in the school choir wore glasses. There were 50 students in the choir. a Find the probability that a choir member i wore glasses ii did not wear glasses. b Estimate the probability that a randomly selected student from the school wears glasses. c Estimate the number of students in a school of 1000 students who wear glasses. d Could the sample ‘students in the choir’ be a biased sample?
206
REVIEW OF CHAPTERS 11, 12 AND 13
TEST YOURSELF: Review of chapters 11, 12 and 13 1 Write down the first 6 numbers in the pattern which: a starts at 2 and increases by 5 each time b starts at 33 and decreases by 2 each time 2 Draw tessellations of these shapes: a
b
3 Choose from the words: impossible, certain, equal chance, likely, unlikely, little chance, good chance, to describe the chance of these events happening. a You will grow to be 3 metres tall. b You will get a head if you toss a coin. c You could stand on one leg all day. d Tomorrow will not be Sunday. e You will live to 90 years of age. f There will be 31 days in October this year. 4 Describe the pattern rule for the sequence 1, 6, 11, 16, 21, .... in words. 5
a Look at the pattern and draw the next two diagrams in the pattern. b List the number of dots in each pattern for the next 3 members. c Partition the members of the pattern to show a number pattern. d Predict the number of dots in the 8th pattern without drawing it.
6 Copy these figures and draw lines of symmetry where possible. a b
7 A six-sided die is rolled. Show on a number line the chance of getting: a a6 b a number less than 7 c an odd number 8
a Find the rule in sentence form for 7, 13, 19, ¤, 31. b Find the missing number.
9 Copy the figure and find the centre of rotational symmetry. What is the order of rotational symmetry in this case?
c
REVIEW OF CHAPTERS 11, 12 AND 13
207
For the spinner shown, use chance words to fill in the missing word:
10
a b c
The spinner is ..... to stop on white or blue. The spinner has ..... of stopping on blue. It is ..... for the spinner to stop on red.
11 In the diagram, give the scale factor when: a A is enlarged to B b A is reduced to C c C is enlarged to B
A B
C
12 James is feeding a litter of puppies. In the first week he feeds them 7 cans of food. Each week following he increases the number of cans by 2 as the puppies grow. a Write down the number of cans of food James feeds his puppies for the first 4 weeks. b Write a rule to give the number of cans n of food James feeds his puppies in week W . c How many cans of food will James feed in week 8?
3
13 You are given the playing cards shown. The cards are well shuffled. One card is chosen at random.
4
4
4
5
i List the possible colour outcomes for the card. ii List the possible outcomes according to the number on the card. iii List the possible outcomes according to the suit. b Find the probability that the card is: i a3 ii a 4 iii not a spade iv black a
c The first card is replaced. The cards are shuffled and another card is chosen at random. Draw a tree diagram for the possible outcomes for choosing two cards according to the number on them. 14 Complete these tables: a x 1 2 3 y 17 14 11
4
b
5 5
15 If x = 2, find the value of: a x+7 b x¡2 16
c a b
X
16 ¥ x
x y
2 8
4 12
6
d
4£x
8 20
10
e
x£x
Draw an enlargement of the figure with scale factor 3. Copy and complete: On the enlargement the length of XY is ..... times the length of XY on the starting diagram.
Y
17 Solve these equations: a
a + 9 = 16
b
a ¡ 9 = 16
c
3 £ a = 15
d
16 =2 a
208
ANSWERS
n five hundred and five thousand o five hundred thousand five hundred p fifty thousand and fifty
Exercise 1 1 a 12 b 20 c 51 d 16 e 31 f 83 g 124 h 1257 2 a XVIII b XXXIV c CCLXXIX d CMII e MXLVI f MMDLI 3 a 88 = LXXXVIII b 19 = XIX c MMIV 4 a i CCCLIV swords ii DCCVIII swords b MCCXCIV denarii c CDXLV metres 5 a 13 b 26 c 204 d 3240 e 723 f 5259 6 a
b
d
c
e
f
12 a d f h
40 + 4 = 44 b 11 + 6 = 17 c 200 ¡ 3 = 197 80 ¡ 8 = 72 e 6000 ¡ 18 = 5982 3000 ¡ 200 = 2800 g 11 000 + 50 = 11 050 509 000 + 38 = 509 038
13 a b c d
i 5403 ii five thousand four hundred and three i 50 043 ii fifty thousand and forty three i 504 003 ii five hundred and four thousand and three i 54 300 ii fifty four thousand three hundred
Exercise 3 1372 b 2408
1 a
Exercise 2
2 a
1 a 7 b 70 c 7 d 700 e 70 f 7000 g 700 h 7000 i 7 j 70 000 k 7000 l 70 000 b
2 a 5 b 500 c 50 d 500 000 3 a 20, 4, 800 b 2000, 4, 800 c 2, 400, 80 000 d 200 000, 40 000, 8000 4 a 579 b 98 320 c 774 411 d 678, 687, 768, 786, 867, 876 5 a 16, 20, 26, 60, 62 b 18, 26, 29, 64, 67, 85 c 7, 70, 700, 707, 770 d two thousand and eight, 2080, two thousand eight hundred, 2808 6 a 32, 21, 20, 17, 16 b 95, 77, 64, 49, 36, 28 c 880, 808, 800, 80, 8 d 2606, two thousand six hundred, 2060, two thousand and six 7 a e h k m p
7<9 b 9>7 c 2+2=4 d 3¡1=9¡7 6 + 1 > 5 f 7 ¡ 3 = 9 ¡ 5 g 16 > 5 5 < 16 ¡ 9 i 12 = 24 ¥ 2 j 11 £ 2 = 44 ¥ 2 15 ¡ 9 = 2 £ 3 l 7 + 13 = 5 £ 4 118¡17 = 98+3 n 7900 < 9700 o 7900 > 7090 25 £ 4 > 99 q 99 < 25 £ 4 r 345 678 < 345 687
8 a 49 b 704 c 386 d 2634 e 60 583 f 93 008 g 46 375 h 908 023 i 270 308 9 a c d e f g h
4 £ 100 + 8 £ 10 + 6 b 3 £ 100 + 4 £ 10 2 £ 1000 + 4 £ 100 + 3 £ 10 + 8 4 £ 1000 + 8 £ 10 + 3 2 £ 10 000 + 4 £ 1000 + 5 £ 100 + 6 £ 10 + 9 3 £ 10 000 + 9 £ 1000 + 8 £ 100 + 4 4 £ 100 000 + 3 £ 100 + 8 2 £ 100 000 + 5 £ 10 000 + 4 £ 1000 + 3 £ 100 + 7 £ 10 + 2
10 a 36 b 70 c 30 d 18 e 900 f 9000 g 520 h 502 i 6014 j 6440 k 14 004 l 40 040 m 15 869 n 95 311 o 708 198 11
sixty six b six hundred and sixty seven hundred and fifteen eight hundred and eighty eight four thousand three hundred and eighty nine six thousand and ten g ninety thousand thirty eight thousand seven hundred fifteen thousand and forty forty four thousand four hundred and forty four four hundred and eight thousand eight hundred and four l two hundred and forty six thousand three hundred and fifty seven m fifty thousand five hundred a c d e f h i j k
3 a 60 798 b 307 203 c 36 280 d 97 075 e 270 682 f 3 092 000 4 a 599 999 b 20 050 c 3 001 d e 4900 f 27 990 5 a
38 000
987 654 b 102 345
6 885 309 eight hundred and eighty five thousand three hundred and nine 7 1 089 999 9
a 432 917 b 380 946
Ten thous.
10 a b 11
8 1023 1032 1203
1
1230 2013 1302 2031 1320 2103
2130 2301 2310
3012 3120 3021 3201 3102 3210
Thous.
Hundreds
Tens
Units
7
3 2
7 3
5 5
a $87:65 b $240:75 c $569:08 d $154:40 e $983:07 f $176:10
12 a B i sum of pairs is ninety nine thousand six hundred and seventy seven ii difference is twenty seven b B i sum of pairs is twenty nine thousand four hundred and ninety eight ii difference is five thousand nine hundred and forty sum of A columns = 61 604, sum of B columns = 67 571 13 a 7, 2, 6, 0, 5, 9 b 89 541 c 976 520 d e 6
205 679
14 a 69, 207, 621, 1863, 5589, 16 767, 50 301, 150 903, 452 709 b 9 times c 452 709 d 2 times
Exercise 4 1 a 20 b 70 c 70 d 100 e 350 f 560 g 410 h 600 i 3020 j 2860 k 3090 l 8890 m 2900 n 10 000 o 30 910 p 49 900 2 a 100 b 700 c 600 d 900 e 300 f 1000 g 2100 h 3900 i 1000 j 13 500 k 99 200 l 10 100 3 a 1000 b 1000 c 1000 d 5000 e 8000 f 7000 g 10 000 h 9000 i 13 000 j 8000 k 246 000 l 499 000 4 a 20 000 b 50 000 c 50 000 d f 50 000 g 90 000 h 100 000
80 000 e 90 000
209
ANSWERS
5 a 200 000 b 300 000 c 700 000 d 700 000 e 100 000 f 500 000 g 300 000 h 100 000
c
3
6 a $190 b $19 000 c 380 km d $800 e 10 m f 9 kL g 29 000 h $270 000 i 500 000
6
7 a $0:50 b $2:75 c $1:85 d $1:85 e $34:00 f $25:05 g $16:75 h $5:00 i $13:00 j $102:25 k $430:85 l $93:90 8 a $84:70 b $31:65 c $10:00
4
5 7
a
9 a $4:00 b $9:00 c $4:00 d $11:00 e $8:00 f $19:00 g $20:00 h $39:00 i $40:00 j $61:00
b
11 18
10 a $6 b $19 c $14 d $37 e $60 f $140 g $84 h $848 i $1027
19 17
12
15
0
b
2
4
5
7
10
13
14
15
16
0
30
40
50
60
70
12
16
20
24
28
8
9
10
18
19
20
0
4
8
0
f
25
0
75 100 125
15
14
13 15
90 100 32
36
200
16
11
17
40
8 a
e
13
18
12
19
d
16
17
13
14
c
18
12
c
1 a
11
19
14
16
Exercise 5
7
2
8
250
6
4
1
500 1000 1500 2000 2500 3000 3500 4000
2 a 3 + 6 + 9 = 18 b 11 + 9 ¡ 13 = 7 c 20 + 30 ¡ 5 £ 10 = 0 d 250 + 400 ¡ 350 = 300 e 70 ¡ 40 + 20 = 50 f 17 ¡ 3 £ 4 ¡ 5 + 4 = 4
2
b
9
7
8
7
2
4
3
1
5
6
c
10
7
6
2
10
3
4
8
3
5
9
10
1
3 a 0
10
9 + 8 ¡ 6 = 11
5
20
9
b 0
10
2 + 4 + 8 ¡ 2 = 12
9
20
7 3 9
c 0
100
200
40 + 70 + 90 ¡ 50 = 150
2 6 7 2
d 0
100
200
55 + 60 + 75 ¡ 40 = 150
e 0
3 £ 9 ¡ 8 = 19
10
20
30
5 6 3 8 4
8 9 6 1
4 4 9 6 7
8 6
4 5
6 2
4 1
2 5
4 2
7
9 3 0 6
4 0 9 8
5 3 8 0 4
3
8 5
Review set 1A
f 0
10
20
4 £ 6 + 5 = 29
4 26 + 4 £ 3 = 38 m
5
2
3
7
6 a
4
8
b
2
5
6
1
1 a
30
CX
3 a
200 b 2000 4 21, 102, 112, 121, 201, 211
5 a
30 459 b 9£1000+4£100+7 6 11 236 7 53 072
8 a
7790 > 7709 b 30 £ 5 = 200 ¡ 50 c 63 < 6 £ 12
9 4011 + 306 = 4317 10 a 7050 b 19 999
11 9
8 b 54 2 a XXIII b
10
11
Thousands a b
2
2
Hundreds 4 5
Tens 6 4
Units 0 0
12 a 1 dollar 75 cents b 25 dollars 45 cents c 6500 $s
7 3
6 5
7 4
4
13 a 60 000 b 2 kg c $2:00
5 3
14 a 2049 b 642 387 15 162 000 cents
6
16
0
2
4 5 6
17 4 £ 4 ¡ 6 = 10
8
10
12
15
210
ANSWERS
Review set 1B 1 a 19 b 35 2 a XI b XLIII 3 a 9000 b 90 000 4 8425, 8254, 4825, 4258, 582, 558 6 95 406 6 a 97 320 b 3 £ 10 000 + 7 £ 1000 + 2 £ 10 + 9
8 a 8000 < 80 000 b 0 ¥ 9 = 9 £ 0 c 1194 < 1419 4m d
10 a $23 b $240 11
P
7 a
$4:90
1724 b
a
i ]KLM ii 70o iii acute i ]DEF ii 140o iii obtuse i ]RST ii 165o iii obtuse i reflex ]UVW ii 225o iii reflex
6 a ]RST, ]SRT, ]RTS b ]IJK, ]JKM, ]IMK, ]JIM, ]KIM, ]IKM, ]JIK, ]IKJ
7 a fifty one thousand six hundred and two b 50 610 c 863 + 794 = 1657 9 a 270 b 53 000 c
4 a b c d
12 a $23:16 b $45:95 13 60 dollars 5 cents 14
3
4
5
6
7
8
9
10
11 12 13 14
8 a
4 £ 3¡8 = 4
5
10
b
1 a
Q
M
d
S
R
Y
P 3 a False b False c True d True
e False f True
4 a pentagon b AB, BC, CD, DE, AE c AC, AD, BD, BE, CE 5 a AB, BA b KL, LK c
2 a
b
c
d
scalene d isosceles
b
c
d
e
f
4 a
triangles are isosceles
b
triangles are (in general) scalene
5 a True - all sides equal in length and opposite sides parallel b True - all angles 90o , opposite sides equal in length and parallel c True - opposite sides are parallel d True - opposite sides are parallel
9 a True b False c False d True not all straight sides c lines cross over
a all sides are not equal b c all angles are not equal
3 a parallelogram b rectangle c kite d rhombus e parallelogram f square
XY, XZ, YZ, YX, ZX, ZY
7 a
11
equilateral c
N
2 a AB and CD b B c BC d AB and CD
10 a not closed b
isosceles b
Q P
c
True
Exercise 9
15
Exercise 7 1 a
False b True c
9 AB ? BC AB ? AD BC ? CD AD ? CD
16 0
S
Q
15 a 13 ¡ 4 ¡ 5 = 4 b 3 £ 30 + 40 = 130
C
B
M
L
607 903
R
b
all angles are not equal
6 a scalene b isosceles c scalene d isosceles e equilateral f isosceles
Exercise 10 1 a 40o b 24o c 100o d 80o e 33o f 62o g 74o h 31o i 19o j 140o k 41o l 90o
12 a pentagon b hexagon c heptagon d octagon e nonagon f decagon
2 60o
Exercise 8
3 a 76o
b 15o
c 45o
d 51o
e 60o
f 45o
o
1 a
b
4 a 60 b ¢ABC is equilateral c AB = 5 cm, BC = 5 cm
c
5 a d
e
¢ABC is isosceles b ]ACB = ]ABC = 65o
6 a ¢ABC is right angled isosceles b ]ACB = 45o c BC = 5:7 cm
f
Exercise 11 1 a 58o
2 a obtuse b reflex c acute d right angle e straight angle f acute g obtuse h right angle 3 a
Q R
S
b
2 a
Y
c 96o
b 100o
c
d 86o
e 90o
f 50o
90o
Review set 2A
c
X Z
105o
b 102o
M
L
1 a
K
2 a
AB is parallel to CD b o
30
b
o
155
Q c
]AQP d right angle
ANSWERS
3 a isosceles b scalene c isosceles d equilateral
B
4 a
b
Q
A P c
10 a b d f h 11
5 a 32o
6 a
b 68o
b
60°
c
d
A
1 + 2 + 3 + 4 + 5 + 6 + 7 + 89 1 + 23 + 45 + 67 + 89 c 1234 + 5 + 6789 12 345 + 6789 e 123 + 456 + 78 + 9 1 + 2 + 3 + 4 + 56 + 789 g 123 + 45 + 6 + 789 1 + 234 + 56 + 789
a 1234 + 567 ¡ 89 b 123 456 ¡ 789 c 12 + 345 ¡ 67 + 89 d 1234 ¡ 567 + 89
Exercise 14
60° 60°
211
B
1 a f j o
630 b 6300 c 63 000 d 630 000 e 2380 23 800 g 238 000 h 2 380 000 i 3700 370 000 k 370 l 37 000 m 504 000 n 5040 5 040 000 p 50 400
2 a f k o s
9600 b 304 000 c 6320 d 840 000 e 80 000 340 000 g 597 200 h 30 900 i 260 j 8300 24 300 l 368 000 m $780 n $59 700 $864 000 p $170 000 q 480 cents r 35 200 km 842 000 L t 43 800 cm u 58 000 g
4 a 15 b 85 c 29 d 380 e 11 f 50 g 50 h 34 i 56 j 56 k 5600 l 560 m 680 n 22 o 840 p 10
C 7 a parallelogram b PQ and SR or PS and QR c PQ = SR or PS = QR 8 110o
5 a 90 b 79 c 45 d 6 e 507 f 6500 g 307 h 910 i $64 j $28 k $92 l $100 m 94 km n 680 g o 69 cents p 240 L q $27:05 r $38:47
Review set 2B
6 a ¥10 b £100 c £10 d £10 e ¥100 f ¥100 g £1000 h ¥1000 i ¥1000
1 a parallelogram b AB and DC or AD and BC c AB = DC or AD = BC d A e ]ACD 2 145o 4 a 91o
3 a kite b isosceles c 60o
d equilateral
7 a 300 cents b 3000 cents c 30 000 cents d 300 000 cents e 49 200 cents
A
8 a 2000 m b 5000 m c 50 000 m d e 7 500 000 m
b 58o
5 a AB ? XY
b
200 000 m
Exercise 15 O c
B
d
1 a
168 b 160 c 4992 d 5526 e 2092 f 17 227
2 a 1680 b 16 170 c 5580 d 18 480 e 15 440 f 15 400 g 26 100 h 147 240 3 a 600 b 6000 c 660 d 1020 e 1800 f 18 000 g 1980 h 810
6 a rectangle b AD = BC or AB = DC c AD and BC or AB and DC 7 65o
4 a 40 b 400 c 4000 d 40 000 e 21 f 210 g 2100 h 210 000 i 66 j 660 k 6600 l 6600
8 a rectangle, square b parallelogram, rectangle, square, rhombus c square, rhombus
5 a 180 b 350 c 320 d 3000 e 1900 f 6000 g 1700 h 23 000 i 48 000 j 64 000 k 8000 l 540 000 6 a
130 b 567 c 1520 d 168 hours
2 a 519 b 9582 c 4458 d 58 242 e 10 196 f 98 737
7 a g m s
152 868 364 7056
3 a 58 b 75 c 99 d 146 e 236 f 255 g 173 h 295 i 463
8 a 1176 b 4214 c 5628 d 3510 e 1539 f 3132 g 3066 h 1221
4 a 42 b 33 c 17 d 498 e 102 f 124 g 74 h 2853 i 6654
9 a 63 200 b 36 630 c 36 068 d 38 368 e 14 694 f 28 980 g 21 402 h 21 645
5 a 13 b 25 214 c 2889
Exercise 16
6 a 45 b 891 c 1368 d 58 023 e 84 f 56
1 a 3 b 30 c 300 d 3000 e 8 f 80 g 800 h 80 000 i 126 j 59 k 152 l 69 m 321 n 795 o 1047 p 24 314
Exercise 13 1 a 81 b 255 c 1562 d 749 e 11 456 f 11 506
7 a 1226 b 1210 c 42 934 d 110 418 e 4222 f 55 768 g 3178 h 3013 i 185 j 36 843 k 899 901 8 a 15 b 38 c 860 d 42 e 325 f 182 g 289 9 a 15 b 38 c $860 d 325 e 144 f 289 km g 57 h 2945 km i $877 j 1110
b h n t
544 c 4560 d 918 e 741 f 1218 4158 i 1323 j 1849 k 2400 l 810 3362 o 1012 p 3234 q 2407 r 2438 1064 u 2844 v 754 w 1377 x 1936
2 a 31 r 1 b 11 r 2 c 4 r 7 d 8 r 1 e 174 r 1 f 132 r 3 g 206 r 1 h 91 r 4 i 309 r 4 j 293 r 5 k 429 r 2 l 330 r 7 3 a 4 b 9 c 9 d 24 e 34 f 59 g 97 h 73 i 27 r 15 j 79 r 25 k 141 r 1 l 121 r 51
212
ANSWERS
4 a 12 b $132 c 435 dollars d 49 weeks e 365 cartons f 91 50-cent pieces
8 a Yes b Yes c No d No e g Yes h No i Yes j Yes
5 a 8 b 12 c 8 d 5 e 12 f 12 g 20 h 12 i 12 j 5 k 20 l 8 m 5 n 8 o 20 p 8 q 8 r 8 s 12 t 8
9 a 2, 3 b 5 c none d 2, 3, 5 e 3, 5 f none g 3 h 2 i 3 j 3 k 2 l none m 3 n none o 3
6 a $18 400 b i 1151 tickets ii $9081 c i $42 324 ii $80 272
10 36 or 63
7 224 sheep 8 1620 stamps 9 215 sheep 10 $720 14 276 papers, $27:60 15 9805
2 a 0 c 20 e 41 g 90 i 5567 k 53 3 a 160 b 210 c 120 d 300 e 450 f 270 g 360 h 720 i 350 j 360 k 450 l 210 4 a 369 b 546 c 212 d 828 e 162 f 552 g 410 h 114 5 a 1500 b 1800 c 3000 d g 14 000 h 20 000
1600 e 5600 f 1600
6 a 1926 b 2082 c 5656 d g 2610 h 4606
3008 e 2982 f 3764
7 a 1500 b 4800 c 2400 d 8000 e 20 000 f 27 000 g 49 000 h 72 000 i 480 000 j 180 000 k 240 000 l 210 000 8 a 500 cars b 200 drinks 9 a 800 b 1800 c 6400 d 20 000 e 9000 f 14 000 g 45 000 h 450 000
e 11
1134 15 484
2604
b f
c
126 240
49 320 g
d
299
429
a 20 hours b $12 000 c 20 helpers d 600 students
Exercise 18 1 a c e g h j l
1 £ 12, 2 £ 6, 3 £ 4 b 1 £ 15, 3 £ 5 1 £ 18, 2 £ 9, 3 £ 6 d 1 £ 20, 2 £ 10, 4 £ 5 1 £ 22, 2 £ 11 f 1 £ 24, 2 £ 12, 3 £ 8, 4 £ 6 1 £ 28, 2 £ 14, 4 £ 7 1 £ 30, 2 £ 15, 3 £ 10, 5 £ 6 i 1 £ 9, 3 £ 3 1 £ 25, 5 £ 5 k 1 £ 36, 2 £ 18, 3 £ 12, 4 £ 9, 6 £ 6 1 £ 48, 2 £ 24, 3 £ 16, 4 £ 12, 6 £ 8
2 a f i l
1, 3 b 1, 2, 4 c 1, 5 d 1, 2, 4, 8 e 1, 3, 9 1, 2, 3, 4, 6, 12 g 1, 13 h 1, 2, 4, 8, 16 1, 2, 3, 6, 9, 18 j 1, 7, 49 k 1, 3, 7, 9, 21, 63 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
3 a Even b Odd c Odd d Even e Even f Odd g Even h Even i Odd j Odd k Even l Odd 4 a d g j
0
0 e 0 f 0 g 0
Exercise 19
Exercise 17 1 a 120 b 700 c 500 d 200 e 2000 f 400 g 8000 h 3000 i 10 000 j 60 000 k 80 000 l 10 000
10 a
a 18 b 18 c 14 d 100 e 99 f 99 g h 83 i 99
11
12 a 0 b 0 c undefined d h 0 i undefined
$540 12 14 rows 13 A by 9 runs
11
Yes f No
22 = 2 £ 11 12 = 2 £ 6 56 = 8 £ 7 45 = 15 £ 3
b 32 = 8 £ 4 c e 28 = 14 £ 2 f h 48 = 4 £ 12 i k 121 = 11 £ 11
5 a 1, 2, 3, 4, 6, 12 b d 6
16 = 4 £ 4 50 = 10 £ 5 45 = 9 £ 5 l 48 = 3 £ 16
1, 2, 3, 6, 9, 18 c 1, 2, 3, 6
6 a 1, 2, 4 b 1, 3, 9 c 1, 7 d 1, 2, 4 e 1, 3, 9 f 1, 3, 9 g 1, 2, 4, 8 h 1, 2, 3, 6 7 a 5 b 8 c
6 d 5 e 2 f 9 g 7 h 8
8 £ 5 = 40, 80 £ 5 = 400, 800 £ 5 = 4000, 880 £ 5 = 4400 b 3 £ 6 = 18, 30 £ 6 = 180, 300 £ 6 = 1800, 330 £ 6 = 1980 c 9 £ 7 = 63, 9 £ 70 = 630, 9 £ 700 = 6300, 9 £ 770 = 6930 d 3 £ 20 = 60, 4 £ 20 = 80, 6 £ 20 = 120, 30 £ 20 = 600
1 a
2 a b c d e f
3, 6, 9, 12, 15, 18, 21, 24, 27, 30 6, 12, 18, 24, 30, 36, 42, 48, 54, 60 7, 14, 21, 28, 35, 42, 49, 56, 63, 70 35, 40, 45, 50, 55, 60, 65, 70, 75, 80 160, 168, 176, 184, 192, 200, 208, 216, 224 244, 248, 252, 256, 260, 264, 268, 272, 276
3 a c d e f
50, 60, 70, 80, 90, 100 b 54, 63, 72, 81, 90, 99 39, 42, 45, 48, 51, 54 1000, 1100, 1200, 1300, 1400, 1500 210, 240, 270, 300, 330, 360 60, 75, 90, 105, 120, 135
4 a 2, 4, 6, 8, 10, 12 b 10, 20, 30, 40, 50, 60 c 20, 40, 60, 80, 100, 120 d 8, 16, 24, 32, 40, 48 e 11, 22, 33, 44, 55, 66 5 a 42, 45, 48, 51 b 48, 56, 64, 72 c 45, 54, 63, 72 d 48, 60, 72, 84 e 50, 100, 150, 200 6 a
12 b 30 c 28 d 18 e 24 f 20 g 30 h 24
7 a 36, 48 b 45 c 48 d 35 e 48 f 36, 42, 48 g 36 h 40 8 a
72 b 80 c 90 d 72 9 a False b False
10 6 nights, i.e., on the following Sunday 11
720 seconds
12 a 4 ! 24 ! 16 b 40 ! 240 ! 232 c 7 ! 42 ! 34 d 12 ! 72 ! 64 e 100 ! 600 ! 592 13 a 10 ! 40 ! 46 b 15 ! 60 ! 66 c 25 ! 100 ! 106 d 200 ! 800 ! 806 e 0!0!6 14 a 10 ! 5 ! 30 b 2 ! 1 ! 6 c 200 ! 100 ! 600 d 20 ! 10 ! 60 e 50 ! 25 ! 150 15 a 21 ! 3 ! 6 b 70 ! 10 ! 20 c 91 ! 13 ! 26 d 700 ! 100 ! 200 e 147 ! 21 ! 42 16 a 9 ! 3 ! 12 ! 21 b 27 ! 9 ! 36 ! 45 c 36 ! 12 ! 48 ! 57 d 90 ! 30 ! 120 ! 129 e 333 ! 111 ! 444 ! 453 17 a +6, +9, 29, 38 b ¡3, £7, 7, 49 c £4, +1, 44, 45 d £6, ¡3, 120, 117 e ¥5, +7, 12, 19 f ¥3, £10, 11, 110
ANSWERS
Review set 3A
Exercise 21
1 a 319 b 1086 2 a 340 b 93 c $79 000
1 a
1 4
b
4 7
c
3 6
d
3 8
e
7 12
3 a 5628 b 21 990 c 3800
2 a
1 4
b
1 2
c
3 4
d
2 3
e
1 4
3 a
i
ii
b i
ii
4 a 1050 b 2375 c 35 5 a 396 b 2400 6 679 450 ends in 0 7 1, 2, 3, 4, 6, 8, 12, 24
f
2 3
8 27, 30, 33, 36, 39 9 a 9 b 24 10 a 53 525 b 17 339 c $800 11
$335 200
12 a 2 ! 12 ! 0 b 10 ! 60 ! 48 c 11 ! 66 ! 54 d 30 ! 180 ! 168 13 a 60 000 b 64 000 c 63 730 14 a 500 b 12 000 c $400 d
$1200
Review set 3B 1 a 279 b 6809 2 a 4163 b 1292 c 109 d 5830 3 a 71 000 b 951 c
$23 4 36 300 b 833
5 a 134 b 50 c 7
4
Symbol
Words
Num.
Den.
a
2 5
two fifths
2
5
b
3 8
three eighths
3
8
c
5 6
five sixths
5
6
d
2 10
two tenths
2
10
6 a 4 ! 12 ! 14 b 8 ! 24 ! 26 c 12 ! 36 ! 38 d 20 ! 60 ! 62
a
7 35 000 8 65 731 ends in an odd number
c
9 1, 2, 4, 8, 16 10 a 33 b 60 11
0 0
$105 984
1
2 5
b
0
3 8
d 5 6
1
0
1
2 10
12 250 grams 13 a 7 ! 2 ! 6 b 27 ! 22 ! 66
5 a
yes b no
14 6 days 15 579, 404, 267, 837, 296
6 a
i
ii
iii
b i
ii
iii
c i
ii
iii
c no d yes e no f no
Test yourself (chapters 1, 2 and 3) 1 a i 17 ii 159 b i XVII ii XXXIX iii CXLVIII 2 a 300 b 30 000 3 47, 74, 457, 475, 547, 574, 745, 754 4 a 506 491 b 6 £ 100 000 + 9 £ 10 000 + 3 £ 100 + 7 5 a True b True c False 6 a 6000 b 5800 c 5770 d
$23:85
7 a
i
ii
7 a 1246 b 567 083 8 a 6300 cents b 6365 cents 9 a 2035 b 2937 10 a $3:25 b $14:80 c $36 984:25 11
$60 12 $38:20 iii 90o
b 180o
14 a equilateral b scalene c
isosceles
13 a i
45o
ii 45o
15 a x = 50 b 16
x = 110 c x = 60
a
b
c
d
17 a False b False 18 40o
b
c
iii
iv
i
ii
iii
iv
i
ii
iii
iv
equal parts are shaded 3 = 12 4 16
equal parts are shaded 3 = 12 4 16
19 a 696 b 1109 20 a 670 b $27600 c 370 21 a 351 b 3354 22 a 123 b 208 c 368 23 414 24
a 630 b 6300 c 63 000 d 630 000
25 1, 2, 3, 4, 6, 9, 12, 18, 36 26 12, 16, 20, 24, 28 27 36 28 a 36 578 people b
1294 people c $1 097 340
1
equal parts are shaded 3 = 12 4 16
213
214
ANSWERS
6 a
1 2
b
i
2 5
j
7 a
1 3
b
2 5
7 a 5 b 4 c 4 d 4 e 5 f 7 g 65 g h 60 cents i 15 min
8 a
1 4
b
1 12
8 4 games
9 a
3 5
b
1 4
Exercise 22 1 a 2
8 11
b
5 13
3
4 9
4
13 36
5 20
c i 50 250
ii
48 60
5
3 20 435 1000
6
9 16 apples 10 13 cars 11 $40 12 20 plants
13 a i 90o ii 180o 30 1 b i 360 = 12 ii
iii 270o 60 = 16 iii 360
240 360
=
1 a
Exercise 23 36 m e 12 weeks
2 34 days 3 $51 4 64 jars 5 $1000 6 a 12 apples b $570 c 16 days d 400 g e 1600 m f 90 marks 7 120 cows 8 2 kg 9 96 matches 10 $1030 a
1 , 2
b
1 2 , , 3 3
c
1 2 3 4 , , , , 5 5 5 5
1, 1 12 , 2, 2 12 , 3, 3 12 , 4 1, 1 13 , 1 23 , 2, 2 13 , 2 23 , 3 1, 1 15 , 1 25 , 1 35 , 1 45 , 2
1 2 3 4 5 , , , , , 6 6 6 6 6
12 a
2 6
b 13 a b
1 3 , 3 6
1 16 ,
1,
1 36 ,
1 46 ,
1 56 ,
1 13 ,
1 36
2 , 3
1 26
=
1 2 3 4 5 6 7 , , , , , , , 8 8 8 8 8 8 8
1,
1 18 ,
1 28 ,
2 8
3 , 4
1 28
=
=
1 4 , 4 8
=
1 4 , 2 6
1 26 ,
=
1 6 , 2 8
=
=
1 38 ,
1 14 ,
=
2
=
1 12 ,
1 46
1 48 ,
1 58 ,
1 68 ,
1 12 ,
1 68
1 48
=
1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 12 12 12 12 12 12 12 12 12 12
14 a b
2 12
= 16 ,
10 12
=
3 12
= 14 ,
4 12
= 13 ,
6 12
= 12 ,
8 12
1 a 2 a
2 6
3 a
2 5
c
3 7
=
8 16
=
1 3
= =
3£4 7£4
3 2
g
22 80
i
=
3 15
k
77 11
4 a
8 12
5 a
ii
1 2
b
2£8 5£8
e
b
=
16 40
=
12 28
3£12 2£12
=
3¥3 15¥3
= =
b
= =
77¥11 11¥11 9 12
3 6
=
3 8
= 23 ,
9 12
1 5
=
6 1
18 12
j 7 1
36 12
d
b i
iii
=
=
=7 l
=
=
= 6£3 1£3
18¥6 12¥6
10 12
2 5
c
2 5
d
3 4
e
1 3
f
1 2
1 3
c
2 3
c
8 9
d
2 3
d
11 18
e
=
=
18 3
=
=
1 78 ,
=
1 34
= 34 ,
2
3 3
10 10
c 1=
=
e
7 12
f
=9 7 12
6 3
1 12
3 2
b
1 34
7 4
c
2 23
8 3
d
2 14
9 4
e
2 15
11 5
3 a
3 2
b
i
7 2
j
1 12
4 3
c
5 3
13 4
k
11 3
b 2 12
7 4
d
9 4
e
18 5
l
c 1 13
e 2=
12 6
9 2
m 2 23
d
7 3
f
n
e 5 23
g
12 5
h
14 3
o
16 3
f 1 14
14 5
g 3 14
1 h 5 14 i 1 35 j 2 15 k 6 25 l 1 16 m 3 56 n 2 38 o 3 10 5 5 1 12 pizzas 6
3 34 apples
Exercise 26 1 a i
3 5
1 2
b
1 12
1 13
j
2 5
c 1 d 1 12
k
+
6 7
e
1 15
l
1 12
m
+
3 a
1 4
+
1 8
c
1 2
+
5 12
d
3 4
+
3 8
e
1 10
+
2 5
=
f
3 14
+
1 2
=
4 a
2 3
2 8
=
1 8
=
3 8
+
5 12
=
3 8
=
9 8
1 10
+
4 10
=
5 10
=
1 2
3 14
+
7 14
=
10 14
=
5 7
6 12
=
6 8
=
3 4
b
+ +
3 10
c
2 3
b
+
3 8
=
7 8
+
1 2
2 5
=
4 10
=
9 10 1 4
+
+
=
3 12
=
7 12
1 6
=
4 6
5 10
+
1 3 4 12
+
+
1 6
=
5 6
= 1 18
d
9 10
5 18
i
7 10
j
1 2
1 k 1 12
5 a
1 14
b
7 9
c
4 5
d
h 1 38
=
4 8
h 1 13
11 12
h
4 g 1 15
+
3 8
So,
=
n 3 o 1 2
So,
=
5 9
f 1 g
So,
=
+ 9 1
5 6
f
d 2=
3 2
81¥9 9¥9
9 20
h
Mixed numb. Improp. fraction
c
9 16
=
3 10
g
1 2
e
49 70
27¥3 48¥3
81 9
f
a
b
15 40
7£7 10£7
27 48
h
2 3
=
1 4
12 16
=
3£5 8£5
f 6=
11 40
c
4 6
=
7 10
d
=
c c
b
36 24
=
22¥2 80¥2
=
3 4
e
Diagram
2 a 4 16
1 3
12 4
f 3=
4 a
1 23
1
5 6
1 4
d
b 1=
Exercise 24 1 2
1 4
5 12
4 4
1=
2
15 27 children 16 2 12 hours 17 15 goals
11
c
Exercise 25
2 3
14 a i 11 ii 33 b i 19 ii 57 c i 61 g ii 183 g d i $6 ii $18 e 400 m f 22 days
1 a $45 b 770 g c 300 bags d f 4 h 10 mins
1 3
1 i 1 10
13 24
5 6
e l
f
g
3 4
20 27
1 e 1 21
7 j 1 16
7 8
k
f 1 18 7 1 24
l 1 14
1 p 3
ANSWERS 3 8
6 a
1 2
b
i 1 12
7 8
c
1 j 1 10
7 7 a 1 10
2 k 1 15
b 2 23
8 a 1 89
1 2
d
9
g 1 38
h 1 38
2 e 4 25
1 8
b
Exercise 30 1 a
d 1 25 28
7 8
a
7 8
f
3 l 1 10
7 c 1 10
b 3 14
5 9
e
f 2 19 20 1 2
10 a
b
1 2
b
Exercise 27 1 a
3 8
b
i
1 2
j
3 a 1 12 5
1 38
2 9
3 4
buckets 6 9 10
10 Yes,
d 1 78
c 2 14
n
e 1 58 7 12
2 9
g
1 6
4 21
o
10 21
p
1 8
1 3
x
7 24
f
6 25
u
d 1 13
kg 7
full 11
5 9
m
7 100
t
1 10
e
9 16
l
c 1 38
b 2 12
3 8
d
11 100
s
3 4
b
1 4
k
r
3 8
2 a
1 9
c
1 3
11 16
q
7 12
11 50
v
w
f 1 12 3 4
4
g 2 78
1 2
h
1 12
kg
8 20 offices 9 39 km 3 8
$200 12
1 5
16 a i 1 18
18 a
c 30o
1 12
b
0.70
15 a 4 14 kg b 1 3
ii $12 b i
0.03 d
8 9
h
13 a $1860 b $20 460 14 a
0.23 c
1 6
ii
2 a i 0:32 ii 0:68 b i 0:20 ii 0:80 c i 0:32 ii 0:68 d i 0:42 ii 0:58
1 34 kg
iii $20 17
0.59
8 9
3 a
b
1 19
b
Exercise 28 1 a 23 : 60 b 2 : 8 c 3 : 5 d 7 : 14 e 7 : 10 f 1 : 10 2 a 5 : 7 b 3 : 12 c 75 : 200 d 300 : 1000 e 60 : 240 f 110 : 220 3 a 2:1 b 1:3 c 4:3 d 3:1 e 1:5 f 1 : 2 g 3 : 5 h 2 : 3 i 3 : 5 j 17 : 4
0.4 c
0.9 d
4 a 4:1 b 3:2 c 5:1 d 8:9 e 5:6 f 1 : 2 g 1 : 20 5 a 3 b 4 c i 10 j 13 6 $125 7
5 d 11 e 21 f 2 g 7 h 101
5 boys 8 15 coffee drinkers 9 15 children
10 55 people 11
4 a i 0:5 ii 0:50 b i 0:3 ii 0:30 c i 0:4 ii 0:40 d i 0:2 ii 0:20
14 $15 000
Review set 4A 1 a
b
2 a i
5 9
ii 1 16
0
3
1 16
4
11
2 3
=
10 15
45 54 5 8
b
42 L 12 a
=
5 6
15 4
$20 7
13 a
2 9
10 500 m 1 8
b
22 kg 15 a Yes, 1 18
14 a 18 kg b
14 15
b
9 3 b 5 13 1 89
b
4 15
a
2 3
c
1
two thirds
5 a 8 b 12 c 8 days 6 8 a
7 8
b
Review set 4B 1 a two fifths, 2 a
1 3
4 a
4 9
b =
2 5
11 15
8 18
7 b two and seven twelfths, 2 12
3 a 3 5
b
11
12 a
5 8
3 25
21 35 6 1 11 3 8
=
7 20 children 8 a 1 14
0.5
54 cm 12 1400 kg 13 360 cans
b
=
12 100
b
25 75
1 3
5 a $8 b 39 6 99 cm b 13 9 13 a
1 25
13 9
b
14 12 girls 15 a i 5 L ii 25 L b 16 1250 kg b 45 boxes
=
10 72 alpacas 5 8
8L
5 a b c d e f g h
1 or one tenth 0:1 means 10 2 or two tenths 0:2 means 10 3 0:3 means 10 or three tenths 4 or four tenths 0:4 means 10 5 0:5 means 10 or five tenths 6 or six tenths 0:6 means 10 7 0:7 means 10 or seven tenths 8 or eight tenths 0:8 means 10
6 a b c d e f g h i
1 or one hundredth 0:01 means 100 2 or two hundredths 0:02 means 100 3 or three hundredths 0:03 means 100 4 or four hundredths 0:04 means 100 5 0:05 means 100 or five hundredths 6 or six hundredths 0:06 means 100 7 0:07 means 100 or seven hundredths 8 or eight hundredths 0:08 means 100 9 or nine hundredths 0:09 means 100
7 a i 0:1 ii 0:01 iii 0:001 b i 1:223 ii 1:024 iii 0:206
0.3
215
216
ANSWERS
Exercise 31 1 a N is 0:7 b N is 2:3 c N is 6:8 d N is 21:4 A
2 a
0
B C
1
b
2
E
F
13
D
3
4
G
14
5 H
15
16
17
3 a N is 0:15 b N is 0:24 c N is 1:77 d N is 3:59 4 a
A
B
4.6
E
b 10.3
D 4.9
F 10.4
o
C
4.8
4.7
5.0
5.1
G 10.5
10.7
3:01 = 3 + 0:01 b 0:68 = 0:6 + 0:08 54:361 = 50 + 4 + 0:3 + 0:06 + 0:001 50:004 = 50 + 0:004 e 603:2 = 600 + 3 + 0:2 106:4 = 100 + 6 + 0:4 g 30:402 = 30 + 0:4 + 0:002 100:101 = 100 + 0:1 + 0:001
4 a
4:13 b 3:5 c 2:023 d 0:436
5 a
10 b
6 a
5 10
b 5 units c
7 a
7 10
b
8
H
10.6
3 a c d f h
b 0:3 L c 75:5 cm
5 a 38:4
6 a $0:35 b $10:20 c $25:55 d $13:35 e $104:40 f $85:25 7 a
b 50 ¢
d
5¢
5¢
e
$5 20 ¢
c $1
$10 $2
5¢
$1
f
$1 20 ¢
$2
$5
$20
$2
$2
10 ¢
9 a
20 ¢
20 ¢
7 1000
1
0
2 5
2 4 0 3
1 10
c 100 d
hun tens ones
a b c d e f g h i j
10.8
1 100
0 7 0 6 0 4
5 100
e
tenths
7 0 2 0 0 1 0 3 3 2
5 1000
d 50 e 7 100
c 7 units d : : : : : : : : : : :
1 1000
e 70
1 ths 100
1 ths 1000
3 1 3 0
9 5
1 1
Number
0:7 0:03 0:21 0:039 100:005 7:1 20:01 46:31 200:3 534:2
0:4 b 0:3 c 0:7 d 0:15 e 0:24 f 0:08 g 0:209 h 0:386 i 0:027 j 0:006 k 1:2 l 2:13 m 7:06 n 2:056 o 4:177
8 a $0:36 b $0:05 c $2:00 d $0:99 e $40:80 f $7:10 g $21:00 h $2:98 i $60:05
10 a 25:983 b 1:75 c 0:487 e 15:788 e 6:999 f 24:096
9 a 6 dollars 50 cents b 14 dollars 10 cents c 4 dollars 20 cents d 2 dollars 70 cents e 10 dollars 60 cents f 90 cents
11
10 a 325 cents b 805 cents c e 15 245 cents
12 a 2:0 b
11
95 cents d 5 cents
a $1:00 b $3:00 c $10:04 d $24:50 e $0:37 f $0:10 g $4:52 h $100:00
Exercise 32 1 a b c d e f g h
number
hun. tens ones
3:6 50:6 231:4 26:52 285:21 688:02 60:862 100:05
5 3 2 8 8 6 0
2 2 6 1
3 0 1 6 5 8 0 0
.
1 ths 10
: : : : : : : :
6 6 4 5 2 0 8 0
1 ths 100
1 ths 1000
a 0:83 > 0:38 b 0:40 = 0:4 c 0:084 > 0:08 d 0:49 < 0:94 e 0:213 < 0:231 f 0:030 = 0:03 g 0:325 < 0:33 h 0:672 > 0:627 i 0:9909 < 0:999 2:5 c 0:86 d 7:8 e 16:0 f 7:1
13 a $103:85, $108:35, $130:58, $130:85, $135:80, $138:50 b 4:038 km, 4:083 km, 4:308 km, 4:38 km, 4:803 km, 4:83 km c 10:089 sec, 10:098 sec, 10:809 sec, 10:89 sec, 10:908 sec, 10:98 sec 14 a 0:52, 0:502, 0:25, 0:205 b 0:770, 0:707, 0:077 c 8:321, 8:312, 8:231, 8:213, 8:132, 8:123
Exercise 33 2 1 2 6 5
1 a
1:74 b 23 c 18:16 d
151:97
2 a 0:9 b 3:3 c 10:9 d 0:047 e 2:005 f 5:4 g 23:85 h 6:4 i 22:96 j 1:817 k 44:933 l 146:817 2
2 a three decimal six or three and six tenths b fifty decimal six or fifty and six tenths c two hundred and thirty one decimal four or two hundred and thirty one and four tenths d twenty six decimal five two or twenty six and fifty two hundredths e two hundred and eighty five decimal two one or two hundred and eighty five and twenty one hundredths f six hundred and eighty eight decimal zero two or six hundred and eighty eight and two hundredths g sixty decimal eight six two or sixty and eight hundred and sixty two thousandths h one hundred decimal zero five or one hundred, and five hundredths i ten decimal zero zero five or ten and five thousandths j seven hundred and four decimal four zero seven or seven hundred and four and four hundred and seven thousandths
3 a
10:66 b 41:68 c 22:22 d 78:477
4 a 14:65 b 175:945 c 662:334 d 48:899 e f 35:196
67:647
5 a $4:95 b $24:98 c $17:09 d 16:13 cm e 35:38 cm f 27:08 km g $34:67 h $45:20 i $29:00 6 a $211:50 b 51:2 sec c 340:1 m d 12:98 m e 21:78 kg 7 a
2:23 b 1:34 c 12:37 d 41:63
8 a 0:5 b 0:7 c 3:9 d 0:2 e 6:32 f 23:45 g 1:995 h 14:22 i 0:033 j 0:263 k 7:598 l 100:865 9 a
10:48 b 4:5 c 68:8 d 0:099
10 a 63 b 1:98 c 0:12 d 14:94 e $13:61 f $62:47 11
a $1:10 b $2:87 c $33:09 d $411:35 e $41:72 f 0:8 cm g 4:78 km h 7:92 km i 0:145 mL j $1:30 k $34:75 l $6:95
ANSWERS
12 a
25:245 ¡ 3:653 21:592
46:92 ¡ 14:57 32:35
b
4:325 ¡ 1:608 2:717
c
5 a e
13 a 1:3 b 11:62 c 85:245 d 0:68 e $16:85 14 a 62:7 kg b 20:7o C c e 21:5 cm f 3:7o C
2:69 tonnes d $23:25
1 a 20 b 63 c 2 d 0:1 e 540 f 606 g 600 h 920 i 70 j 54 k 4500 l 7040 m 7000 n 6200 o 700 p 380 q 6750 r 3067 a b c d e
£10 0:09 1:2 5 46 190:7
Number 0:009 0:12 0:5 4:6 19:07
£100 0:9 12 50 460 1907
£1000 9 120 500 4600 19 070
9 a 2:6, 26 b 26 c 0:26 £ 10 £ 10 = 0:26 £ 100 = 26 Multiplying by 10 and multiplying by 10 again is the same as multiplying by 100. 0:2 b 0:63 c 0:02 d 0:001 e 5:402 f 60:6 0:06 h 0:092 i 0:007 j 0:5 k 1:66 l 3:007 0:007 n 0:0062 o 0:0561 p 0:499 q 0:701 6:8549
12 a 10 b
¥100 0:08 0:046 0:5 0:1907 2:314
18 a 0:82, 0:082 b 0:082 c 82 ¥ 100 ¥ 10 = 82 ¥ 1000 = 0:082 Dividing by 100 then dividing by 10 is the same as dividing by 1000.
e
75 100
25 100
= 0:75 f
8 10
h
56 100
= 0:56 i
375 1000
= 0:375 j
k
45 100
= 0:45 l
98 1000
= 0:098
2 a 0:25 b
0:75 c
4 a 43% b 57%
= 0:8 g
=1
= 30% b = 25% e = 55% h
50 100 60 100 52 100
= 50% c = 60% f
80 100 4 100
= 80% = 4%
= 52%
8 a 0:04 b 0:1 c 0:2 d 0:25 e 1 f 0:5 g 0:75 h 2:5
1 iii 0:1
= 0:25 d
35 100
4 10
= 0:4
= 0:35 6 100
5 a
24 m b 16:1 kg
6 a $2 b 5 people c 6:1 g d $0:04 e 2:6 L f 785 km g 6:5 cm h $0:65 i $15 j 25 L k $1000 l 30 people m 6 g n $520 o 90 t 7 a 0:8 b 0:375 c 15:63 d 7:508 e 42:1 f 0:408 g 3:15 h 2:48 i 0:29
$0:31 b 0:34 kg c 0:65 m d 7:02 hours a $10:08 b $10:10 c $9:90
15 a 1250 mL b 875 mL
1 a $32 b $5:10 c $48:90 d $51:60 e $29:25 f $67:25 g $215:55 h $252:35 i $354 j $448:35 k $399:60 l $387:90 2 a $4 b $0:30 c $10 d $10 e $10 f $20 g $10 h $30 i $10 j $10 k $10 l $10 3 a $2 b $20 c $20:20 d $2:75 e $4:01 f $13:10 g $2:60 h $14:25 i $6:25 j $27:02 k $49:95 l $254:50 4 a $471:80 b $187:70 c $307:50 d $181:10 e $290:70 f $56:50 g $187:50 h $142:30
Exercise 35 = 0:2 c
h
1 4
Exercise 37
$0:50 c $0:05
2 10
= 100 100
12 138 sec 13 93:6 min 14 1:69 m
100 c 10 d 100 e 100 f 1000
= 0:5 b
g
30 100 25 100 55 100
10 1:2 hours 11
17 a 8:2 £ 10 = 820 ¥ 10 b 0:62 £ 100 > 620 ¥ 100 c 0:014£100 > 1:4¥10 d 75:61¥1000 = 7:561¥100 e 26:8 ¥ 100 < 2:68 £ 100 f 0:07 £ 1000 = 700 ¥ 10
5 10
d
9 a
13 a $5:50 b $0:55 14 a $12:50 b $1:25
1 a
l
25 100
8 a $2:43 b $15:25 c $103:01 d $44:80 e $12:34 f $0:27
¥1000 0:008 0:0046 0:05 0:019 07 0:2314
15 a $320 b $32 c $3:20 16 a $5 b
g
1 d 10 80 4 = 100 5 78 39 = 100 50
=
4 a 9:43 b 17:36 c 0:68 d 0:518 e 25:6 kg f $85:85 g 189 m h $51:50
8 a $263 b $236 c $1500 d $125
a b c d e
k
= 34 33 100
3 a 21:7 b 0:99 c 0:26 d 0:55 e 0:084 f 13:8 g 12:36 h 0:077 i 8:4 j 10:01
7 a $3 b $30 c $300
¥10 0:8 0:46 5 1:907 23:14
10 100
c
2 a i 3 ii 0:3 iii 0:03 b i 10 ii c i 6 ii 0:6 iii 0:06
$180 c $1800
Number 8 4:6 50 19:07 231:4
f 29 100
1 2 75 100
=
1 a i 2:1 ii 0:21 iii 0:021 b i 2:4 ii 0:24 iii 0:024 c i 0:66 ii 0:066 iii 0:0066
6 a $456 b $4560 c $45 600
11
j
50 100
Exercise 36
4 a $22:50 b $225 c $2250
10 a g m r
=
1 5
9 a 2% b 24% c 75% d 99% e 60% f 250% g 320% h 400%
3 a 100 b 10 c 10 d 100 e 10 f 1000 5 a $18 b
b
6 a 12% b 8% c 49% d 80% e 2% f 37% g 6% h 107% 7 a
Exercise 34
2
i
1 100 20 100 9 100
217
= 0:06
0:125 d 0:875 e 0:2 f 0:6
5 a $4:50 b $13:35 c $6 d $34:40 e $21:15 f $346:80 g $48:50 6 a $2:55 b $8:80 c $6:45 d $3:55 e $8:25 f $0:45 g $1:39 7 a 5 b 5 c 7 d 55 e 149 f 366 g 410 h 3684 i 487 j 7999 k 4906 l 4904 8 a 46:1 b 5:1 c 12:9 d 13:6 e 50:0 f 721:1 g 22:3 h 40:4 i 100:6 j 105:6 k 49:0 l 6:1
218
ANSWERS
9 a 5:83 b 5:80 c 15:80 d 4:97 e 6:86 f 31:69 g 45:02 h 491:33 i 689:00 j 462:06 k 6:01 l 38:57 10 a $0:85 b $2:20 c $5:30 d $4:65 e $1:85 f $7:85 11
4 4:136, 4:163, 4:613, 41:63, 46:13, 416:3 5 a $0:55 b $15:35 6 a $0:50 b $24:05 15 100
ii 0:4 b i
0:15 8
ii
0:08
9 a 50% b 23% 10 a $20 b $100 11
14 a 100 m b
20 km c 90 mm d 138 cm
24 m b 36 cm c 63 km
2 a 32 cm b 84 m c 112 mm d e 170 cm f 372 km
3 1000
3 a 0:345 b twenty one and twenty seven thousandths c 51:746 = 50 + 1 + 0:7 + 0:04 + 0:006
4 10
13 a 90 cm b 104 mm c 138 km d 123 cm e 25 km f 92 mm g 48 m h 100 km i 24:7 m
1 a
Review set 5A
7 a i
226:5 cm 12 23:5 mm
Exercise 41
$22:70
1 0:031 2
9 1:563 m 10 650 m 11
3 a
12 cm b 20 mm c 31 km d 124 mm
4 a
5 km b 9 cm c 14 mm d 43 km
5 a
4 cm b 7 mm c 12 cm d 18 km
6 a 54 m b 80 cm f 780 cm
c 340 m d 320 km e
3138 mm
7 a 41 cm b 32 cm c 29 cm d 16 cm e 9 cm f 14 cm g 12 cm h 14 cm i 15 cm j 20 cm k 11 cm
a 2:311 m b 6925 c 0:467
12 a $89:65 b 80 g
184 cm
c $282:50
13 a $13:15 b $6:85 c $65:75
8 a H b D c B d A e F f E g N h J i G j L k M l K m C n I
Review set 5B
9 a
1
12 56 cm 13 a 168 m b 165 m c $1980 d $2292
1.35 1
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
2
2.1
2 a 0:007 b six hundred and three decimal four seven c 54:014 3
2 100
4 534:2, 54:32, 53:42, 5:432, 5:342, 5:234
5 a $0:55 b $3:40 c $28:05 6 a $0:05 b $45:10 7 a i
75 100
ii 0:75 b
9 a 46% b 155% c 11
i
64 100
ii 0:64 8 0:25
0:16 d
1 4
10 $60
a 26:76 b 0:002 c $311:55
12 a 3 parcels b
4:99 kg 13 $77:80
Exercise 39 1 a cm b
f m
2 a B b A c B d C e B f A g B h C 3 a m b mm c m d m e m f cm g m h m 5 30 mm and 3 cm are the same length 240 mm and 24 cm are the same length 150 mm and 15 cm are the same length 10 mm = 1 cm 6 a 50 mm b
36 mm c 32 mm d 62 mm
7 a 4 cm b 6 cm c 5 cm d 2:5 cm
3 a i 14 triangles ii 15 triangles, ii is larger b i 32 rectangles ii 32 rectangles, equal area c i 13 hexagons ii 12 hexagons, i is larger Square units are the best measure of area as they fit together evenly with no gaps and are the easiest for counting to find the total area of a shape. 5 a 5 square units b 12 square units c 34 square units d 40 square units e 35 square units f 128 square units g 36 square units h 34 square units i 30 square units j 36 square units k 58 square units l 45 square units m 44 square units n 44 square units 4
7 a
12 cm2 2
6 cm
b 12 cm2 2
c 13 cm2 c 9 cm2
b 8 cm
Exercise 43 1 a
63 m2
b 33 cm2
c 230 km2
2 a
60 km2
b 96 m2
c 160 mm2
3 a
66 cm2
b 60 cm2
c 64 cm2
4 a
34 cm b 38 cm
c 32 cm
) order is c, a, b
5 a
81 m2
c 90 m2
) order is b, a, c
6 a
36 m b 36 m c 42 m ) order is a, b, c or b, a, c
b 80 m2
d 730 m2 )
order is b, c, a
7
Exercise 40 1 a 500 cm b 260 cm c 20 cm d 8 cm 3 a 15 km b 1:5 km
c 0:6 km d 6:398 km
260 mm c 5 mm d 68:5 mm
2 cm
3 cm
2 a 5000 mm b 2600 mm c 200 mm d 1080 mm 4 a 23 mm b
60 m
Exercise 42
6 a mm c mm d cm e cm
456 m b 2:28 km 10 160 m 11
6 cm 4 cm 1 cm
5 a 6:25 m b 0:82 m c 0:05 m d 0:002 m 6 a 0:9 cm b
9:6 cm c 45:32 cm
7 a 0:546 m b 0:07 m c 0:005 m 8 a 6000 m b
63 000 m c
700 m d 560 m
12 cm 8 a i 10 m ii 6:5 m b i 33 m ii 65 m2 c i 6 m2 ii 11:2 m2 d 4 m2
ANSWERS
9 a b c d e f g h
Length
Width
Area
Perimeter
36 m 174 cm 28 km 25 cm 58 m 40 cm 200 mm 40 m
13 m 86 cm 28 km 10 cm 20 m 50 cm 48 mm 40 m
468 m2 14 964 cm2 784 km2 250 cm2 1160 m2 2000 cm2 9600 mm2 1600 m2
98 m 520 cm 112 km 70 cm 156 m 180 cm 496 mm 160 m
4 a i 18 u3 c i 30 u3 26 m3
5 a
219
ii 6 u3 b i 34 u3 ii 6 u3 ii 18 u3 d i 35 u3 ii 40 u3 b 24 cm3
c 35 mm3
6 a 160 mm3 b 48 km3 e 125 cm3 f 180 cm3 7 a
d 40 u3
c 56 cm3
1 cm
d 150 mm3
volume = 12 cm 3
3 cm 4 cm b
2m
Exercise 44 2
2
c 16 m
2
2
1 a 69 m b 24 km f 222 m2 2
2 a 15 km 2
3 135 m
b 39 m
1008 cm
4 2
6 a 80 m
2
7 a 48 m
2
2
d 34 m
2
c 18 m
d 120 m
2
5 0:96 m
2
c 40 m2 2
b 3 tins 8 a 9 m
e 140 m
4 mm b 8:04 m
1 a m
b cm 3
2 a 2 cm f 6 cm3
3
cm
c 3
b 2 cm g 4 cm3
d m 3
c 3 cm h 9 cm3
3
e mm
3
d 4 cm3
3 a 1 £ 1 £ 16 1£2£8 1£4£4 2£2£4
total is 4 volume is always 16 cm3
b 1 £ 1 £ 18 1£2£9 1£3£6 2£3£3
total is 4 volume is always 18 cm3
c 1£1£8 1£2£4 2£2£2
4 mm
5 mm 8 a
3
volume = 80 mm 3
2
Exercise 45 3
volume = 40 m 3
c
400 posts
b
2m
10 m
2
2
b 120 m
9 a 160 000 m2
e 116 cm
2
e
6 cm3
total is 3 volume is always 8 cm3
d 1 £ 1 £ 20 1 £ 2 £ 10 1£4£5 2£2£5
total is 4 volume is always 20 cm3
e 1 £ 1 £ 27 1£3£9 3£3£3
total is 3 volume is always 27 cm3
f 1 £ 1 £ 25 1£5£5
total is 2 volume is always 25 cm3 total is 7 volume is always 64 cm3
g 1 £ 1 £ 64 1 £ 2 £ 32 1 £ 4 £ 16 1£8£8
2 £ 2 £ 16 2£4£8 4£4£4
h 1 £ 1 £ 36 1 £ 2 £ 18 1 £ 3 £ 12 1£4£9
1£6£6 2£2£9 2£3£6 3£3£4
i 1 £ 1 £ 100 1 £ 2 £ 50 1 £ 4 £ 25 1 £ 5 £ 20
1 £ 10 £ 10 2 £ 2 £ 25 2 £ 5 £ 10 4£5£5
j 1 £ 1 £ 125 1 £ 5 £ 25 5£5£5
total is 3 volume is always 125 cm3
total is 8 volume is always 36 cm3 total is 8 volume is always 100 cm3
27 cm3
b 125 mm3
c 512 m2
Exercise 46 1 a
8000 mL b 800 mL c 53 000 mL d 5030 mL
2 a
3 L b 8:5 L c 0:6 L d 0:09 L
3 a
6000 L b 66 000 L c
4 a
5 kL b 0:5 kL c 6:3 kL d 0:06 kL
5 a
mL b kL c mL
6 a
2000 mL b 720 mL c 4500 mL
7 a
4 L b 1 L c 3 L 8 a 3:75 L b 37:5 L
9 60 L 10 0:48 kL 11
600 L d
6060 L
700 L 12 16 bottles
13 500 mL
Review set 6A 1 a
i 40 mm ii 4 cm b
2 a
5 km b 0:426 m
3 a
60 cm b 32 mm c 30 cm
4 a
metres b cm
5 4 cm 6 a cm2 7 a
7 u2
b 6 u2
8 a
144 m2 3
10 a 5 cm 11
a 12 u3
b km2 c 14 u2
b 24 cm2 3
6 u3
c 48 mm2
9
5m
c 24 cm3
b 6 cm b
i 28 mm ii 2:8 cm
12 a mL b kL
c ML
Review set 6B 1 a
650 mm b 0:65 m
2 a
22 m b 30 m c 170 km
3 a
m b cm c m2
4 a
8 u2
5 a
24 m2 2
11
c 8 u2
b 2500 mm2
6 20 m by 20 m
8 a 1700 m b 5100 m
7 54 m 9 a
b 7 u2
d cm2
3
7u
b 6 u3
c 27 u3
a kL b mL c L
10 a 8 u3
b 19 u3
220
ANSWERS
Test yourself (chapters 4, 5 and 6) 2
1 a kL b mm c
cm
9 10
4 a 12 km b 1 200 000 cm 5 a
12 a
1 3
11
49 70
b
=
36 84
c
1:5 L 16 a 64 m2
3 7
=
c 11 units2
0 7 14 23:076 = 20 + 3 + 10 + 100 + 76 = 20 + 3 + 1000 = 23 + 0:07 + 0:006
15
4 14
b
$16:05
7 10
b 18 units2
13 a 9 units
7 3
a
a $0:85 b
6 18 2
=
9 1000
22 m b 10 cm c 32 m
8 3:106, 3:16, 3:601, 3:61 9 10 7 cm
b
6 1000
5
b 18 cm2
c 104 cm2
a b
17 a 0:2 b 0:35 c 0:06 18 15 km 19 7 m 1 2
20 a 20% b 59% 21 a 2
22 a 5 cm
2
b 24 cm
c
5 classrooms d i 5 steps right, 3 steps down ii 2 steps right, 7 steps down F1 or I10
2
24 cm
c
23 a $60 b 50 litres 24 a
5 11
30 m
25 a
1 6
b
3 8
b b c
J K L M
1 2 3 4 5 6 7 8 9 10 11 12 13 14
d m
0:059 3 a 9 eggs b 12 apples
2 a 0:59 b 6 $63 7 a
A B C D E F G H I
4
100 m 600 m2
11 classroom 5 10 9 classroom 1 classroom 4 8 7 classroom 2 6 5 classroom 3 4 3 2 office 1 A B C D E F G H I
20 m
J K
Exercise 50 26 a $600 b $1400 27 122 m2 28 a 13 kg b 29 a 31 a
1 58 4 9
14:2 kg c $8:50
pizzas b b
3 8
30 a $26:15 b $166:25
7 9
Exercise 48 1 a 200 km b
100 km c
90 km
2 a 180 km b
240 km c
150 km d 315 km
3 a 80 km b 60 km c 4
110 km
0 3 km or 1 : 300 000 or 1 cm represents 3 km
5 a 90 cm b 66 cm 6 a
60 m b 5 cm
7 a 37:5 m b 40:5 m c 27:5 m d 32:5 m e 32:5 m f 21:5 m g 25 m h 35 m i 45 m j 40:5 m
1 a AB = 9:4 cm, so actual distance is 9:4 km b RS = 26 mm, ST = 13 mm, TU = 17 mm, UV = 23 mm, VW = 14 mm, WX = 10 mm so total distance = 103 mm So, the actual distance = 10:3 km c Total length = 15:5 cm, so actual distance = 15:5 km 2 a 5:2 cm i 5:2 km ii 52 km iii 520 km iv 260 km b 6:3 cm i 6:3 km ii 63 km iii 630 km iv 315 km i E3 ii N15 iii N7 iv region covered by B11 to C11 and B12 to C12 b 8 km c 24 km d 10 km e 3 km f 8:2 km
3 a
4 a i J10 ii O5 iii H2 b c i + 18:4 km ii + 33 km
14:5 km
5 a i A4 ii H6 iii D12 iv J12 v F6 vi L1 b + 6:5 km 6 a i ambulance station ii bike hire b B7 c E9 d between F3 and F4
Exercise 51
8 a 1500 km b 75 m c 30 km d 30 m e 500 km f 22:5 m
1 a i south ii NW iii NE iv SW v SE b i SW ii NW iii NE
9 a 80 cm b 140 cm c 20 cm by 40 cm
2 a Melissa b i Mary ii Mike and Martin iii Mike: 10 m E 60 m N, Mary 20 m W 2 m N, Melissa is at the coin, Martin 60 m E 10 m N c 60 m E 70 m N d 92 m
Exercise 49 1 a i A6 ii G1 iii E3 b i 7 steps right, 2 steps upwards ii 3 steps right, 4 steps upwards 2 a i D7 ii C2 iii A7 iv H6 b i 5 steps right, 5 steps upwards ii 5 steps right, 3 steps downwards iii 3 steps left, 3 steps upwards 3 a 15 trees b Pine, B8, D1, G2 c 3 steps left, 1 step upwards d Pine e 3 steps right, 2 steps upwards
3 a + 97 km b i 20 km E, 28:4 km NW, 12 km E, 11:2 km SE, 8 km W, 17:2 km SW ii 17:2 km NE, 8 km E, 11:2 km NW, 12 km W, 28:4 km SE, 20 km W c + 23 km 4 a i Z ii D iii P iv B v ~ b i 50 km SW ii + 70 km iii + 88 km iv + 63 km
ANSWERS
5 a 6 12 km b 30 km
c Bright d Harrietville e NW
5 a
Exercise 52 1 a 35 m £ 15 m b 25 m £ 10 m c 15 m £ 10 m d 10 m £ 10 m e 45 m £ 10 m f 30 m £ 10 m 2 a 6 m b 2:5 m £ 1 m c 9 m d 4 m 3 a 88 m2
b $2816
4 a 19 m2
b $475
8 7 6 5 4 3 2 1
b 3 units left, 6 units down c (0, 1)
C
y
221
B
A x 1 2 3 4 5 6 7 8
Review 7B7B Reviewsetset
5 a 4:5 m £ 6:3 m b $1417:50 c $122:40 d W.C. = toilet, = doorway; = window; SHR = shower; WIR = walk in robe; VAN = vanity; WM = washing machine; T = tumble dryer; L = laundry sink; = sink; ST = stove; F = fridge; P = pantry
2 a C3 b i Hilton International Hotel ii Baptist Church c i E1 ii C5 iii A4 and A5 iv C1 and D1 v D4
Exercise 53
4 a
1 a i (6, 3) ii (4, 6) iii (1, 4) iv (7, 1) b i 5 units left, 1 unit upwards ii 3 units right, 5 units downwards 2 a A is (3, 6), B is (6, 4), C is (1, 2), D is (7, 7), E is (4, 1), F is (6, 0), G is (8, 3), H is (2, 4), I is (7, 2), J is (1, 7) b i 3 units to the right then 2 units down ii 6 units to the right then 5 units up iii 2 units to the right then 1 unit down iv 6 units to the left then 1 unit up v 6 units to the left then 5 units up 3
8 7 6 5 4 3 2 1
y S
1 a
5 m £ 4 m b 20 m2
c $760
3 a 36 km b 140 km c 364 km d Chambers Pillar Historical Reserve e NE Police station (0, 3), School (1, 7), Garage (2, 4), General store (3, 1), Post office (4, 6), Bank (6, 0), Church (6, 5), Town hall (7, 2), Cemetery (7, 7) b i 7 units to the right and up 4 units ii 6 units to the right and down 5 units iii 1 unit to the left and down 5 units
Exercise Exercise5555 1 a triangular prism b square prism c hexagonal prism d pentagonal prism e rectangular prism f octagonal prism 2 a b
R
U
c P
T
Q
1 2 3 4 5 6 7 8
d x
e 4 a Warehouse (3, 3), Shop A (6, 1), Shop B (7, 5), Shop C (5, 7), Shop D (4, 5), Shop E (1, 1), Shop F (1, 4) b 30 km 5
5
f
C (4, 4), D (1, 4)
y
4
D
C
3 a
A
B
4 a triangular-based pyramid b pentagonal-based pyramid c square-based pyramid d rectangular-based pyramid e hexagonal-based pyramid
3 2 1 1
2
3 4 5 6 7
8
x
Review set 7A
b
1 a 1 : 2 000 000 b 8:5 cm 2 a i H5 ii B1 iii A3 iv E6 b i 5 units right and 4 units down ii 2 units right and 3 units up iii 4 units right and 3 units up 3 a i 700 km ii 2100 km iii 3200 km iv 3300 km b i North ii SE iii East iv NE 4 a 12 12 km b 27 km c d S e Myrtleford
5 a
NW
c
d
e
cube b triangular prism
222
ANSWERS
7 a
b
c
d
e
f
triangular prism 5 faces, 9 edges 6 vertices
7
triangular-based pyramid 4 faces, 6 edges 4 vertices
8 a remains the same b remains the same c remains the same d varies e varies f remains the same
8 a
Exercise 56 2 triangles, 3 rectangles ii I, J, K, L, M, N IJ, IK, JK, JM, KL, IN, LN, LM, MN KIJ, JKLM, IJMN, IKLN, LMN triangular prism 6 rectangles ii A, B, C, D, E, F, G, H AB, BC, CD, DA, EF, FG, GH, HE, AE, BF, CG, DH ABCD, EFGH, ABFE, BCGF, CDHG, ADHE rectangular prism 4 triangles, 1 square ii V, W, X, Y, Z VW, WX, XY, YV, ZV, ZW, ZX, ZY VWXY, ZYV, ZYX, ZWX, ZVW square-based pyramid 6 squares ii A, B, C, D, E, F, G, H AB, BC, CD, DA, EF, FG, GH, HE, AE, BF, CG, DH ABCD, EFGH, ABFE, BCGF, CDHG, ADHE cube 4 triangles ii P, Q, R, S PQ, PS, QS, RP, RQ, RS iv PQR, PRS, PQS, RSQ tetrahedron 2 hexagons, 6 rectangles A, B, C, D, E, F, G, H, I, J, K, L AB, BC, CD, DE, EF, FA, GH, HI, IJ, JK, KL, LG, AL, BG, CH, DI, EJ, FK iv ABCDEF, GHIJKL, ABGL, BCHG, CDIH, DEJI, EFKJ, FALK v hexagonal prism
1 a i iii iv v b i iii iv v c i iii iv v d i iii iv v e i iii v f i ii iii
2 a i ii iii iv v vi
No. of vertices (V ) 6 8 5 8 4 12
No. of faces (F ) 5 6 5 6 4 8
No. of edges (E) 9 12 8 12 6 18
V +F ¡E 2 2 2 2 2 2
V + F ¡ E = 0 + 3 ¡ 2 = 1 6= 2, ) no, it does not work for a cylinder.
b V + F ¡ E = 1 + 2 ¡ 1 = 2, ) yes, it does work for a cone.
Exercise 57 1
2 a
b
3 a
b
4 a
b
5 triangular prism
6 a
b
2 units 2 units 1 unit 1 unit
b V +F ¡E =2 3 a i 10 ii 7 iii 15 b c i 9 ii 9 iii 16
i 8 ii 6 iii 12
c
d
1 unit
4 V + F ¡ E = 2 in each case 5 a cube b triangular prism c square-based pyramid d triangular-based pyramid (tetrahedron) e hexagonal-based pyramid f sphere g cylinder h cone 6 a
b
d
1 unit
1 unit
c
2 units 7 a
2 units
2 units 2 units b
1 unit
c
e
or
Exercise 58 1 a triangular-based pyramid b square-based pyramid c rectangular prism d hexagonal-based pyramid e cone f cylinder
ANSWERS
2
4 a
b
c
d
e
f
5 a ii
Left end
Front
1
3 a 1
1
2
Left end
b ii
Right end
Front
Right end
1 2
1
b
1
1
Plan view
Plan view
1
c ii
Left end
Front
d ii
Right end
Left end
Front
Right end
1 1
1
2
1
Plan view
c
2
3 units
6 a
b
c
d
f
1 unit
Exercise 60
2 units
1 A, D 2 a
150 g b 20 kg c 250 g d 40 kg
3 a kg b mg c kg d g e t f t g g h kg i g j g 4 a 3 b 2 c
6
4 a
2 kg b 0:5 kg c 0:25 kg d 0:087 kg
Exercise 59
5 a 3000 kg b 54 000 kg c 250 kg d 20 kg
1 a B b D c A d C
6 a
3000 g b
7 a
7 t b 5:526 t c 0:6 t d 0:05 t
2 a
b
1250 g c 800 g d
12 g
8 a 20 000 mg b 2000 mg c 200 mg d 20 mg
plan
plan
elevation
elevation
11
13 636 kg; more,
1 2
1 a
d
elevation
t = 500 kg 14
plan
elevation
2 a square-based pyramid b O, A, B, C, D c OAB, OBC, OCD, OAD are triangles, ABCD is square d AB, BC, CD, AD, OA, OB, OC, OD 3
b
3 3
plan
elevation
c
plan
elevation
1
elevation
d
plan
1:3 kg
cube b triangular prism
3 3 a
$2:94 12 80 kg
Review set 8A
c
plan
9 5 g 10 0:7 t
plan
elevations
3 3
4 a
1
Plan view
d
e
223
b
224 5
ANSWERS Left end
Front
2
1
2
1
Right end
6
3
Score 2 3 4 5 6 7 8 9 10 11
Plan view
7 a grams b tonnes 8 a 1200 g b
240 g
Review set 8B 1 a rectangular prism b tetrahedron 2 a triangular prism b A, B, C, D, E, F c ABC and DEF are triangles, ABED, ACFD and BCFE are rectangles d AB, AC, BC, DE, DF, EF, AD, BE, CF 3
4
1 unit
4 a
5
Shoe size 5 5 12 6 6 12 7 7 12 8 8 12 9 9 12 10 10 12 11 11 12 12 12 12 13
2 units
2 units
b
Left end
Front
2
1
Right end
2
Plan view
6 a 2:5 kg b 250 kg
7 a 540 kg b 0:54 t
Review set 8C 1 a pentagonal prism b cylinder
3 a
4
5 a
Front
Right end
10 5
b i
9 25
4000 kg b 200 people
G
R colour
B
Weight of soccer players
frequency
ii
7 50
iii
9 25
Class 6A
0 1 2 3 4 5 6 7 8 9 10
Number of students
Exercise 62 1 a 9 b 13 c 16 d 27
55
W
O
Plan view
2
Frequency 1 0 2 1 3 6 8 7 6 4 2 3 2 1 2 1 1
5
1
6 a mg b g 7 a
j jjj jj j
65
70 weight (kg)
Score
Class 6B
0 1 2 3 4 5 6 7 8 9 10
60
9 40
15
Number of students
2
jj j jjj © © jjjj © © jjjj © © jjjj © © jjjj jjjj jj jjj jj j jj j j
d
Dot plot of Car Colour Data
6 a 1
Tally j
a 11 b 40 3 c 20
20
b
Left end
Frequency 1 1 2 5 7 6 11 4 2 1 40
It is called a vertical dot plot as dots are used to represent each student wearing a particular shoe size and the dots are arranged vertically.
frequency
2 a rectangular prism b A, B, C, D, E, F, G, H c ABFE and CDHG are squares, ABCD, EFGH, BCGF and AEHD are rectangles d AB, AD, AE, BC, BF, CD, CG, DH, EF, EH, FG, HG.
Tally j j jj © © jjjj © jj © jjjj © j © jjjj © © © j © jjjj jjjj jjjj jj j Total
b 6A: highest 10, lowest 4 6B: highest 10, lowest 2
Score
c 6A performed better
225
ANSWERS
7 a
frequency
50
Weight of calves
55
60
b
65
3 10
The total is 70, so if the length is 7 cm = 70 mm, each 1 is represented by 1 mm. If 14 cm = 140 mm is used, each 1 is represented by 2 mm. b 26 mm c Strip graph of car types
8 a
70 weight (kg)
8 a 1 goal b 10 goals c 6 goals d 5 goals
Exercise 63
Category A B C D
9
1 a
b
c
d
e
f
2 a 40 b 10 c 25 d 68
4 a Populations of each Australian State or Territory b a million people c NSW d i 1:5 million ii 4:7 million iii 1:9 million
1 a
represents 5 schools
5
sport
Tennis Cricket Golf Swimming
Eye colour
12 10 8 6 4 2 0
Blue
Brown
Grey
Hazel eye colour
2 a
Eggs b i 7% ii 8%
3 a
Lamb b i 20% ii 30% c i 160 kg ii 240 kg
4 a
NZ (non-Maori) b Denmark and Sweden c 25
5
frequency
Number of students playing instruments
represents 10 computers
16 12 8 4 0
Drums
Piano
Flute
Recorder
Violin
Cello
instrument
Money spent in a week
6
item
sales
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
MITSUOTHER BISHI
Number of members 170 230 190 90
brown ii 28%
b i
Kelly’s Computers monthly sales
6
TOYOTA
Exercise 64
b
frequency
3 a
HOLDEN
FORD
Other
Education Clothes Electricity
Colours of cars in a park
colour
7 a
Yellow Blue White Red Green
Food Mortgage
0 represents 5 cars number of cars Chocolates eaten in a week
child
b
Alex Claire
represents 3 chocolates
Emmy
7 a
20
40
i 3 days ii 11 days
60
80
100 120 money spent ($)
Exercise 65 1 a 11 am b 8 pm c a rapid increase from 13 to 39 people d
15 people
2 a
$3000 b March c $4000 d August e $1300
3 a
225 runs b 32 runs c 19th d 7th e
4 a
i A(6, 450) ii 450 km iii 6 h b 150 km
44th c 4h
5 a i age (weeks) ii weight (kg) b 4 kg c i 5 kg ii 13 kg d i 2 weeks ii 6 weeks
Tess Ally chocolates eaten
6 a 130 - 155 cm b 120 - 142 cm c about 90% d to eliminate extremes, i.e., abnormally short or tall
226
ANSWERS
c
Exercise 66
8
4
2 a the garden b cleaning c i 200 kL ii 60 kL
2
30 women
0
Review set 9B 1 7 2 a a strip graph b i Year 1999 1998 1997 1996 1995 1994
3
US dollars b
8 a
at work b 20%
i Tuesday ii Thursday c $386:30
Smurfs J
4 a 13
3000
2 9:1 calls
1000
2000
b HLCOFEBPIMGKDJNA c 1838 years d 4642 years e 501 years The times of Harry Potter
5 A
BC
DE F G
H
I JK
L
1st Jul
a 7 b 19
F E B PI G D A M
O
1st Jun
Frequency 1 2 4 7 5 19
C
1st May
Tally j jj jjjj © jj © jjjj © © jjjj Total
BC 0 AD
1000
H L
16 c 50 years
1st Jan
3 a dot plot b
2000
N
1 5
1st Apr
12
1st Mar
11
K
1st Feb
10
2000
mass produced doll
14 years e 89 years
1980
2 a 20th b 44 years c 1901 d f i 2040 ii 21st
Age
9
4 7
1 a i 1960 ii 1973 iii 1979 iv 1992 b 18 years c i 7 years old ii 19 years old iii 41 years old
1800
Numbers of students
13
Exercise 70
School music group
Score 9 10 11 12 13
12
Age 8 b i 5 students ii 0 students c 21 students d
7 a
3
a 15 children b 5 c
4
11
6 a vertical dot plot b frequency of shoe sizes sold at “Cathy’s Shoes” c i 4 ii 2 d 9 12
Review set 9A 1
10
Etch-a-sketch GI Joe
6 75:55 kg
5 a
9
1960
5 a Jon 30, Jethro 30 b probably that Jon is more consistent
8
Slinky
4 a Michael 4:7, Tao 5:0, Mario 5:5, Peter 4:3, Sam 4:0 b Sam c No, he simply played more games. Mario performed best (highest average).
Chess club members
1940
3 a Group A: 7:6 Group B: 6:5 b not true, this is the whole point of finding averages c Group A
4
1920
17 d 2:4 2 20 goals
ii 12 students
Monopoly dice game
1 a 5 b 5 c
1 5
Number of pumpkins 6500 6000 4500 3500 3000 3000
1900
Exercise 68
ii 6 students
frequency
7 a City $4000, Sub $2000 b $8000 c City $26 000, Sub $28 000 d Steady increase in length of bars for suburban shop shows the advertising was effective.
1 7
6 a 10 L sold b i Friday ii Wednesday c i 30 L ii 55 L d 38 L
5 a i US dollars ii Australian dollars b i US$30 ii US$60 iii US$15 c i A$12 ii A$30 iii A$39 6 a i time (mins) ii distance (km) iii when he reached the Sports Club b i 6 km ii 80 minutes c 2 km d 20 minutes e a heading
score
13
1880
4 a 1995, 1997, 1998, 1999 b i a $2000 loss ii no profit (“break even”) c $15 000 d generally increasing
12
model train set
3 a week 2 b i 8 fish ii 9 fish c 56 fish d improved
a strip graph b i
1860
2 a Adventure (45 mm), Mystery (35 mm), Science fiction (30 mm), Fantasy (20 mm) 2 b Adventure c 130 mm d 13 e 260 students
5 a
11
roller skates
i 50 ii 10 d 200
10
1840
Exercise 67 1 a soft drink b iced coffee c e 90 f 45
9
1820
3 a size 14 b
frequency
6
1 a Science b English c 60 students
ANSWERS
Exercise 71 1 a 36 months b 730 days c 208 weeks d 96 hours e 180 minutes f 300 seconds g 50 years h 731 days i 20 centuries j 92 days 2 a 41 days b 72 days c 32 days d 10 days e 59 days f 635 days g 183 days h 364 days 3 a 258 min b 414 min c 315 min d 390 min e 1498 min f 4323 min g 5950 min h 12 025 min 4 a 83 hrs b 174 hrs c 259 hrs d 2403 hrs e 139 hrs f 204 hrs g 1104 hrs h 467 hrs 5 a 322 secs b 647 secs c 210 secs d 375 secs e 446 secs f 8100 secs g 20 520 secs h 23 565 secs 6 a b c d e f g h i j
3 14
3 12
days, 79 hrs, 3 days 11 hrs, days 187 secs, 3 min 8 secs, 3 14 mins, 3:5 mins 4 weeks, 29 days, 4 27 weeks, 4 wks 3 days 1 4 12 yrs, 50 mths, 4 yrs 3 mths, 4 13 yrs 9:5 yrs, 1 decade, 124 months, 11 yrs 14 decades, 1 12 centuries, 154 yrs, 1 century and 6 decades 1 year, 367 days, 53 weeks, 12 12 months 7200 min, 5 days 11 hrs, 5 12 days, 145 hours 4 14 hrs, 256 min, 4 hrs 17 min, 16 000 secs 5 wks, 5 wks 20 hrs, 886 hrs, 37 days
7 a T b T c T d T e F f T g T h T i F j F k F l T m F
3 a 0730 hours b 1511 hours c 1155 hours d 2045 hours e 0345 hours f 2020 hours 4 a d g j
0815 hours 1445 hours 2235 hours 2357 hours
5 a
iv 12:35 b ii 11:00 iv 1:30
6 a
iv 14:00 b i 10:45 v 14:05 c ii 23:00 v 00:15
7 a
11:00 am, 11:20 am, 11:40 am, 12:00 noon, 12:20 pm, 12:40 pm 10 past 11 pm, half past 11 pm, 10 to 12 pm, 10 past 12 am, half past 12 am, 10 to 1 am 16:10, 16:30, 16:50, 17:10, 17:30, 17:50 25 to 8, 5 to 8, quarter past 8, 25 to 9, 5 to 9, quarter past 9 3:45 pm, 4:05 pm, 4:25 pm, 4:45 pm, 5:05 pm, 5:25 pm 25 past 9 am, quarter to 10 am, 5 past 10 am, 25 past 10 am, quarter to 11 am, 5 past 11 am 23:00, 23:20, 23:40, 00:00, 00:20, 00:40 5 past 8, 25 past 8, quarter to 9, 25 past 9, 25 past 9, quarter to 10
b c d e f g h 8 a b c d e f
Exercise 72 1 a Terri’s parents were married on the 21st day of November 1986. b Kahlia’s family migrated on the 14th day of March 1992. c Tim’s great grandfather will be one hundred on the 6th day of August 2009. d Jenny bought the house on the 3rd day of October 1998. e Man first stepped on the moon on the 20th day of July 1969. f Henry Ford’s first mass produced car was built on the 12th day of August 1908. 3 a Sunday 29th February b Monday 23rd February c Sunday 26th September d Wednesday 10th November e Monday 1st March f Sunday 3rd October 4 a Autumn - March, April, May Winter - June, July, August Spring - September, October, November Summer - December, January, February b Autumn has 92 days; Winter has 92 days; Spring has 91 days; Summer has 90 days (91 in a leap year) 5 33 days 6 91 days 7 4 Mondays 8 a 18th June 1907 or 2007 b 22nd February 1904 or 2004 c 5th January 1925 or 2025 9 a Saturday b Thursday c Sunday d Wednesday e Thursday 10 c, g, h, j, k, l 11
a i 43 200 min ii 131 040 min iii 175 680 min iv 74 880 min v 527 040 min b i 105 120 min ii 95 760 min iii 35 520 min
Exercise 73 1 a 0527 hours b 0955 hours c 0000 hours d 0008 hours e 1200 hours f 1848 hours g 1946 hours h 2359 hours 2 a 4:00 am b 5:30 am c 5:00 pm d 12:00 noon e 10:15 pm f 7:35 am g 8:59 pm h 6:18 pm
227
b e h k
2015 hours 0650 hours 2330 hours 1149 hours
c f i l
0245 hours 1742 hours 0413 hours 0002 hours
11:05, 10:35, 10:05, 9:35, 9:05, 8:35 9:16, 8:15, 7:14, 6:13, 5:12, 4:11 12:40, 12:15, 11:50, 11:25, 11:00, 10:35, 10:10 0110, 0040, 0010, 2340, 2310, 2240 25 to 9, 10 past 8, 14 to 8, 20 past 7, 5 to 7, 12 past 6 20 past 2, quarter to 2, 10 past 1, twenty five to one, midnight, twenty five past 11
Exercise 74 1 a clockwise b clockwise c anticlockwise d anticlockwise e clockwise f clockwise 2 a anticlockwise b clockwise c can be either but usually anticlockwise d clockwise e can be either f clockwise 3 a
anticlockwise b
4 a
i b ii 5 a i b ii
anticlockwise
6 a
ii b i c ii
d ii
Exercise 75 1 a 7:16 am b 2:20 pm c e 12 hours 18 minutes
5 hours 46 minutes d 2 m
2 a 60 mins b 3 hours 55 minutes c 8 hours 45 minutes d started at 8:30 pm, ended at 11:15 pm e 135 min f no 3 a 868 km b 1 hour 5 minutes c Toffs Harbour d i 19 hours 55 mins ii Mosquito Valley and Dustbowl e Pineville f 3:40 pm 6:25 am, 8:05 am, 10:10 am, 12:45 pm, 2:50 pm, 4:35 pm, 7:40 pm, 8:55 pm b 14:50 c London at 08:05 d i 14:10 ii 00:55 the next day
4 a
5 a teams 2 and 8 b 3 teams - teams 4, 5 and 7 c teams 1, 3 and 8 d Oct 19th 4 pm, Oct 19th 6 pm, Nov 2nd 4 pm, Nov 16th 2 pm, Nov 16th 6 pm, Nov 23rd 4 pm
Exercise 76 1 a
240 km b
120 km c 40 km
2 a
20 km b 5 km c 15 km
3 a
60 km b 120 km c 288 km
4 a 6 kmph b 110 kmph c 700 kmph d 100 kmph 5 a
3 h b 3 h c 6 h 30 min 6 a 1 km b 2 km
228
ANSWERS
7 a 37o C b 10o C c 55o C d ¡10o C
7 a
no. of items 2 3 4 5 6 7 8 9
8 a 20o C b 35o C c 41o C 9 a 38:4o C b 39:3o C c 40:1o C 10 a B b 11
F c C d E e A f D
a i 7o C ii 13o C b 20o C at 2 pm c 5o C at 6 am d i 8o C ii 12:04o C e winter f Adelaide’s temperatures
20 temp. (°C) 10 time
Items of mail
12
m id n 2: ight 00 4: AM 00 6: AM 00 8: AM 00 10 AM :0 0 A 12 M no 2: on 00 4: PM 00 6: PM 00 8: PM 00 10 PM :0 12 0 m PM id ni gh t
0
frequency 2 2 3 4 2 4 2 1
frequency
b
tally jj jj jjj jjjj jj jjjj jj j
3
2 12 a i 20o C ii 9o C iii 22o C b 26o C at 2 pm c 9o C at 5 am d i 24o C ii 22o C
Review set 10A
3 a i 0323 hours ii 1302 hours iii 2036 hours b i 2:30 pm ii 2:06 am iii 6:52 pm 4 a 3:45 pm b i 1 h 35 min ii 2 h 15 min c 1:45 pm
a i
120 km ii 30 km b 80 kmph c 5 h
12 a (2,5) b 3 units right and 4 units upwards 13 a i vertical column graph ii a dog iii 3 students iv any other pets (not cats, dogs, birds or rabbits)
birds
b
1 1 a 198 min b 25 hours, 1 12 days, 1590 min, 1 day 3 hours
14 a
2
2 a 8th day of August in 1990 b i February 15th ii November 25th c Wednesday ii 12:20 am iii 8 pm
4 a 3:15 b 16:05 c The time is 45 minutes past 7 o’clock in the evening (or 15 minutes to 8 o’clock). a ¡2o
rabbits
1 cm º 4 pets
Review set 10B
1
plan b
left end right end
1
plan 15 a b
elevation
1 1
b 6 min
7 a 1:30 pm start, 2 pm finish b i Sports Stunts ii 1:33 pm
1
other
dogs
cats
5 a i 27o C ii 37o C b 41o C at 2 pm c 17o C at 6 am d summer e 31:85o C
105o C c ¡5o C 6
6 7 8 9 number of items
8 a i D1 ii A1 iii F5 iv B4 b i North ii south east iii north east 11
2 a 51 days b 2:4 mins, 2 mins 25 secs, 148 secs, 2 12 mins
5 a 60o C b
5
9 7 10 a 17o C b 9o C
1 a i 1968 ii 1970 iii 1978 iv 1984 b i 25 years ii 8 years
3 a 1230 h b i 2 am
4
6:84 kg 2500 kg
elevation
left end
right end
16
8 a i 90 km ii 270 km b 2 h 9 70 kmph
Test yourself (chapters 7, 8, 9 and 10) 1 a rectangular prism b tetrahedron c pentagonal pyramid 2 a 45 days b 80 min, 1:4 h, 1 12 h, 1 h 48 min c the fourteenth day of July, year two thousand and five 3 a 50 cars b the country of manufacture of parents’ cars for Mawson Primary School children c Australia d 35 4 a i 5 cm ii 3:5 cm b i 60 m ii 150 m 5 a triangular prism b A, B, C, D, E, F c ABC and DEF are triangles, ABED, BCFE and ACFD are rectangles d AB, AC, BC, DE, DF, EF, AD, BE, CF 6 a i 0520 h ii 1200 h iii 2230 h b i 4:15 pm ii 12:02 am iii 10:05 pm
17 a cricket b 33% c 17 students
Exercise 78 1 a 3, 5, 7, 9, 11, 13 b 5, 8, 11, 14, 17, 20 c 11, 18, 25, 32, 39, 46 d 31, 29, 27, 25, 23, 21 e 57, 52, 47, 42, 37, 32 f 141, 138, 135, 132, 129, 126 2 a 5 £ 5 + 1 = 26 b 5 £ 6 ¡ 2 = 28 6 £ 6 ¡ 2 = 34 6 £ 5 + 1 = 31 7 £ 6 ¡ 2 = 40 7 £ 5 + 1 = 36
c 5 £ 7 + 3 = 38 6 £ 7 + 3 = 45 7 £ 7 + 3 = 52
3 a 20 £ 5 + 1 = 101 b 20 £ 6 ¡ 2 = 118 c 20 £ 7 + 3 = 143 4 a 4 £ 1 ¡ 1 = 3 b 4 £ 3 + 2 = 14 c 6 £ 5 ¡ 5 = 25 7 £ 5 ¡ 5 = 30 5 £ 3 + 2 = 17 5£1¡1=4 8 £ 5 ¡ 5 = 35 6 £ 3 + 2 = 20 6£1¡1=5 9 £ 5 ¡ 5 = 40 7 £ 3 + 2 = 23 7£1¡1=6 10 £ 5 ¡ 5 = 45 8 £ 3 + 2 = 26 8£1¡1=7 11 £ 5 ¡ 5 = 50 9 £ 3 + 2 = 29 9£1¡1=8 12 £ 5 ¡ 5 = 55 10 £ 3 + 2 = 32 10 £ 1 ¡ 1 = 9
ANSWERS
5 a b c d e f
starts at 1 and adds 2 each time starts at 8 and adds 5 each time starts at 24 and takes 2 each time starts at 81 and takes 3 each time starts at 6 and adds 6 each time starts at 3 and adds 7 each time
e i ii 4, 7, 10, 13, 16, 19 f i
6 a ¢ = 7 b ¢ = 11 c ¢ = 25 d ¢ = 59 e ¢ = 12 f ¢ = 15 7 a 36, 42 b 24, 28 c 40, 48 d 45, 54 e 15, 20 f 28, 15 g 20, 20 h 24, 25
ii 2, 5, 8, 11, 14, 17 g i ii 5, 9, 13, 17, 21, 25 h i
8 a ii 7, 10, 13, 16, 19, 22 ,
,
7 a
i
b 15, 21, 28 c 36, 45
ii iii
9 a b , b 25, 36 c yes
8 a
10 a
64 i 125 ii 12 167
b c
i ii iii 101 matches
b i 11
a 3, 5, 7, 9 b 21 faces
ii
Exercise 79
iii 152 matches
1 a
b
c
d
c i ii iii 150 matches
2 a 1, 3, 5, 7, 9 b 4, 6, 8, 10, 12 c 3, 5, 7, 9, 11 d 3, 6, 10, 15, 21 3 a
b 6 £ 3 + 1 = 19 c 20 £ 3 + 1 = 61
4 a
b i 6 £ 3 + 2 = 20 ii 15 £ 3 + 2 = 47
d i ii iii 103 matches 9 a
5 a i, ii
,
iii 63 dots
or b or
iii 92 dots
b i, ii ,
c or
6 a i ii
2, 4, 6, 8, 10, 12
b i ii
1, 3, 5, 7, 9, 11
c i ii
4, 8, 12, 16, 20, 24
d i ii
3, 7, 11, 15, 19, 23
Exercise 80 1 a b c d e f g h i
to get the next term add 3 to the previous one to get the next term multiply the previous one by 2 to get the next term take 3 from the previous one to get the next term divide the previous one by 2 to get the next term add 7 to the previous one to get the next term multiply the previous one by 5 to get the next term take 9 from the previous one to get the next term divide the previous one by 10 the next term is the same as the previous one
2 a
¤ = 15. To get the next add 3 to the previous one.
229
230
ANSWERS
¤ = 48. To get the next multiply the previous one by 2. ¤ = 41. To get the next take 4 from the previous one. ¤ = 18. To get the next add 5 to the previous one. ¤ = 12. To get the next divide the previous one by 2. ¤ = 36. To get the next multiply the previous one by 3. ¤ = 80. To get the next take 11 from the previous number. h ¤ = 3. To get the next divide the previous one by 10. i ¤ = 21. To get the next multiply the previous one by 3. b c d e f g
3
1 + 5 £ 4 = 21 1 + 6 £ 4 = 25 1 + 7 £ 4 = 29
1 + 4 + 4 + 4 + 4 + 4 = 21 1 + 4 + 4 + 4 + 4 + 4 + 4 = 25 1 + 4 + 4 + 4 + 4 + 4 + 4 + 4 = 29
a They are the same for each pattern. b 2nd form, i.e., 1 + 100 £ 4 = 401 is easier than 1 + 4 + 4 + 4 + :::: + 4 = 401
|
4 a
Week 1 2 3 4 5
{z
5 a
b 196 sales
n T
1 4
day 4
day 6
1 4
2 5
3 6
4 7
5 8
6 9
d
a T
1 0
2 1
3 2
4 3
5 4
6 5
e
t F
1 3
2 5
3 7
4 9
5 11
6 a
35 30 25 20 15 10 5
b
day 8
b 192 min c 381 min
8 a 18 cans b 36 cans 9 $17:50
Exercise 81 1 a 2, 3, 4, 5, 6 b 3, 5, 7, 9, 11 c 6, 5, 4, 3, 2, 1 b
15
15 10
5
5 1 2
3 4
5 6 member
1 2
d
50
3 4
5 6 member
40 30
40
10
20 10 1 2
3 4
1
2
3
4
5 6 member
1
2
3
4
5 6 member
5 6 member
f
20 15
30 20
10
10
5 1
2
3
4
5 6 member
3
5 30
4
6 35
7 40
7 6
5
6
a
6
a
25
2
3
4
5
y = x+3
x y
1 4
2 5
3 6
4 7
b
x y
1 7
2 8
3 9
4 10
c
x y
1 10
d
x y
1 2
2 4
3 6
4 8
5 10
y=2£x
e
x y
2 1
4 2
6 3
8 4
10 5
y=
1 2
£x
f
x y
6 2
12 4
y=
1 3
£x
8 a
$ 900 800 700 600 500 400 300 200 100
20
30
4 25
7 a
20
10
2
1
2 a and b increase 20
3 20
20 15 10 5
6 a 4, 5 12 , 7, 8 12 , 10, 11 12 , 13 b 59 12 km
e
2 15
As a increases by 1, b increases by 7.
8m b 4, 8, 12, 16, 20, 24, 28, 32 c 8 days 7 a 3, 6, 12, 24, 48, 96, 192
6 24
n D
day 7 4m
day 2
5 20
c
+5 +5 +5 +5
day 5
4 16
1 10
1
day 3
day 1
c
3 12
t K
5 a
3 a
2 8
b
}
100 times
Sales 13 = 13 13 + 5 = 18 18 + 5 = 23 23 + 5 = 28 28 + 5 = 33 Total 115
4 graphs a and b
2 9
1
3 8
5 8
4 7
18 6
2
5 11
y =x+6
5 6
y = 11 ¡ x (or x + y = 11)
24 8
3
30 10
4
5
6
d
ANSWERS
b
d $
1 2 3 4 5 6 150 300 450 600 750 900
c i $1500 ii $4500
Exercise 82 1 a
2 a
3 a
member
member
member
triangles
formula
1
2£1+1=3 X
2
2£2+1=5 X
3
2£3+1=7 X
b i 17 ii 57
b i 61 ii 291
y+4 b y¡4 c 2£y
2 a 3 a
c+2 b
(2 £ l) cm b (l ¡ 4) cm
5 a
n b i n + 5 ii n ¡ 4 iii n £ 3 iv n ¥ 3
6 a
d ¥ 2 b i (d ¥ 2) + 6 ii (d ¥ 2) ¡ 8 iii d
12 b 4 c 32 d 2 e 24 f 8 g 128 h 2
9 a
10 b 8 c 24 d
1
5£1+1=6 X
2
5 £ 2 + 1 = 11 X
3
5 £ 3 + 1 = 16 X
houses
formula
1
7£1+1=8 X
2
7 £ 2 + 1 = 15 X
b
3
7 £ 3 + 1 = 22 X
c
formula 2£1+2=4 X 2£2+2=6 X 2£3+2=8 X
a ¢ = 6 b ¢ = 7 c ¢ = 4 d ¢ = 33 e ¢ = 1 f ¢ = 8 g ¢ = 7 h ¢ = 45 i ¢ = 54
11
12 a f k o
x=5 b x=5 c x=3 d x=6 e x=9 x=5 g x=4 h x=4 i x=6 j x=5 x = 3 l x = 44 m x = 7 n x = 5 x = 9 p x = 63 q x = 10 r x = 56
Exercise 85 1 a i 27 ii 45 iii 84 iv 315 b i 3 ii 73 iii 269 iv 6989
b i 32 ii 602
5 a 14 cm b 62 cm c 602 cm 6 a 24 b 276 c 2796
D = 5, D = 7, D = 9, D = 11, D = 13, D = 15 n D 20
c 10
2 ! 9 ! 23
b
0
3 a
c
3 ! 8 ! 23 ! 68
3 a b c 4 a
L = k ¥ 14 L = litres needed 25 litres k = kilometres travelled $22:50 C = 8 £ s + 200 dollars b $440 c $14:67
8 a $30 ¥ 4 = $7:50 b $30 ¥ p 9 a 160 biscuits b 32 £ p biscuits
1 1
n R 20
0
4 a b
b
3
5
6
7
2 4
3 7
4 10
5 13
6 16
4
5
6
R
n 0
1
2
3
7
K = 0, K = 3, K = 8, K = 15, K = 24 n K 30
1 0
2 3
3 8
4 15
5 24
1
2
3
4
5
K
20 10 0
6 a 25 £ 6 = 150 seats b 25 £ r seats 18 £ x books
4
n 0
5
5 a C = 0:50 £ s + 48 dollars b $288 c $0:60 7 a 18 £ 5 = 90 books b
6 15
5 a
n 0
6
7
depth water
P = total profit c = number of cartons
5 13
R = 1, R = 4, R = 7, R = 10, R = 13, R = 16
Exercise 83
2 a P = c £ 13 dollars b $312
2
4 11
10
13 ! 29 ! 61 ! 125
C = total cost, h = number of hammers
1
3 9
D
15
4 ! 8 ! 10 ! 11 ! 11 12
1 a C = h £ 7 dollars b $119
2 7
5
7 a $4:50 b $18:00 c $28:50
9 a
1 5
10
b
8 a i $22 ii $46 b 7 lines c $19
9 e 4 f 16 g 6 h 0
10 a 3 b 6 c 4 d 5 e 9 f 2 g 12 h 15 i 4 j 9 k 20 l 3 m 5 n 30 o 30 p 40 q 45 r 56
formula
black 4 6 8
c (l + 6) cm d (l ¥ 2) cm
8 a
15
coloured 1 2 3
c¡3 c c£3 d c¥2
7 a An equation has an ‘=’ sign but an expression does not. b i expression ii equation iii expression iv equation
2 a
4 a
d y¥2
4 a
houses
b i 71 ii 7001
Exercise 84 1 a 14 years b (11 + n) years c (11 + x) years
231
time
232
distance from school
ANSWERS
b
i To get the next member multiply the previous member by 2. ii 80 b i To get the next member divide the previous one by 3. ii ¤ = 27
4 a (bus)
distance from home
time
c
5 a
10 b 1 c 36 d
6 a
x=9 b x=7 c x=4 d x=2
7 a
23, 27, 31, 35 b
8 a
18, 15, 12, 9, 6, 3 b decreasing
9 a time
b
Review set 11A 1 a 2, 9, 16, 23, 30 b 100, 93, 86, 79, 72 2 start with 3 and add 5 each time b 2, 5, 8, 11, 14 d
3 a, c
5
1
3
2
5
4
6
member
10
8
12 15
4
6
8
20 y 15
0
x 0
2
10
12
14
c To get the next member add 1 to the previous member. 10 a i 109 kg ii 188 kg iii 662 kg b 10 boxes
Exercise 87
x y
1 4
2 5
3 6
4 7
5 8
x y
2 1
4 2
6 3
8 4
10 5
12 6
9
a d a c a
y
8
10
6 4
11
2
x 1 2
3 4
5 6
2 a Row 1, col 5; Row 5, col 1; Row 5, col 4 b The tail is on the wrong side. c Row 3, col 3; Row 3, col 6; Row 4, col 5
d Row 2
3 a Row 1 b Row 1, col 2; Row 1, col 5; Row 5, col 1; Row 5, col 4 c Row 2, col 3; Row 2, col 6; Row 4, col 1; Row 4, col 4
Yes, points lie on a straight line.
b
10 13
1 a
10
7 a
8 11
Row 1, col 4; Row 3, col 2; Row 3, col 5; Row 5, col 3; Row 5, col 6 b Row 2, col 3; Row 2, col 6; Row 4, col 1; Row 4, col 4 c Row 5 d Row 2, Row 5
number
6
6 9
5
4 a i To get the next member take 4 from the previous one. ii 7
5 a 15, 20, 25, 30, 35 b T = 5 £ w + 10 minutes c 7 weeks
4 7
M = 4 £ t + 19 c 47
10
23
b i To get the next member add 6 to the previous one. ii ¤ = 31
x y
36 e 2
6 9
4 a reflection
18 b 11 c 56 2 e 64 y = 7 b y = 18 y = 5 d y = 12 i $116 ii $236 iii $536 b $23:60
5 a translation and enlargement b translation and reflection c translation and reflection d translation and reduction
Exercise 88 1 a 2 a
yes b no
c yes d no e yes f no b
Review set 11B 1 a 5, 13, 21, 29, 37 b 97, 89, 81, 73, 65
c
2 Start with 43 and subtract 8 each time. 3 a
b
Diagram number 1 2 3 4 5
Number of matches 6 11 16 21 26
c 6, 11, 16, 21, 26, 31 d 51 matches
3 a
b
c 5 a an octagon and a square b a regular hexagon and an equilateral triangle
ANSWERS
Exercise 89
3 a
1 a
b
c
d
e
f
b
c
2 3 a i
ii d
iii
e
iv
b ii 4 a 1 (vertical) b 1 (horizontal)
f
g
Exercise 90 1 a
b
O
c
O O centre is O
h
centre is O
d
i
centre is O
e
f centre centre
centre
g
Does not have rotational symmetry
centre
h
centre
i
Review set 12A 2
becomes itself
becomes itself
becomes itself
becomes
becomes
becomes itself
becomes itself
becomes itself
1 a
b
2 a
b
3
order of rotational symmetry is 2
becomes itself 3 a
here
b 4 2
centre
centre
4 a 4 b 2 c
6 d infinite
1 a 2 b 3 c
1 12
2 a 2 b
1 6
c
3 4
b c
d reduced area = 6
Exercise 91
5 a
1 9
centre
original area 7
1 2
8 180o
18 u2 2 u2
233
234
ANSWERS
10 a possible b possible (but highly unlikely) c possible d impossible e certain
Review set 12B 1
a impossible b a little less than equal c certain d very unlikely e very likely
11
Exercise 94 1 a
boy, girl b fB, Gg c
2 a
f1, 2, 3, 4, 5, 6g b
2
B G 1 2 3 4 5 6
3 a not possible b 3 a
G 5 a
4 a
b
b fBB, BG, GB, GGg
2nd child B G B G
1st child B
here
4
b fOO, OE, EO, EEg
2nd roll O E
1st roll O
O E
E 6
The results for A when A plays B say at two sets of tennis. b fwin win, win lose, lose win, lose loseg
5 a
Coin
6 7 a reduced b 8 a
1 5
b c
1 2 3 4 5 6 1 2 3 4 5 6
H
+ 11:2 mm + 33:5 mm
T
A B C d length of the image of AC = 3 £ length of AC 9 a 5 b 2 c
Die
2 d infinite
a fH1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6g b 12 7 a
Entree
Main
P
S V L
PC
S V L
F
S V L
Exercise 93 1 a certain b highly unlikely c likely d impossible e highly likely f unlikely g impossible h even chance i highly likely 2 a impossible b even chance c certain d very unlikely e very likely 3 a possible b impossible c possible d possible Two examples are: A spinner marked ‘A’ on one half and ‘B’ on the other. A hat with the equal numbers of two different coloured objects, e.g., 2 red and 2 blue discs. 6 a A cat can only give birth to kittens. b The moon is constantly orbitting the Earth so can not collide with it. c The sun rises in the East and sets in the West so can not set in the East. d It is phyically impossible to swim such a distance.
5
7 a almost impossible b very unlikely c unlikely d a little less than even chance e even chance f highly likely 8 a very unlikely b very likely c impossible d certain e true 9 a i very likely ii very unlikely iii an even chance b true as 4 in 20 is the same as 1 in 5
b 18 c 6 d 9
Dessert C I C I C I C I C I C I C I C I C I
Exercise 95 1 a
1 4
1 7
b
c
2 3
d
2 a f1, 2, 3, 4, 5, 6g b i 3 b
a
0
Qr_
4 a
3 100
5 a
3 8
1 6
e ii
2 11 1 3
f
8 11
iii 0 iv 1
c
d
Qw_
1
97 100
b b
3 5
1 8
c
1 2
d 0 e
6 b, e
f
d
0
0.25
0.5
1 2
f 1
a, c 1
235
ANSWERS 3 4
7 94% 8 9 a i
5 16
c i
3 10 1 3
10 a
b
ii
9 16
ii
19 30
2 9
b
c
iii
1 8
iii
1 15
1 18
b 7 18
d
2 7
i
5 7
ii
e 0 f
11 18
iii 0 7 18
g
h
17 18
Review set 13A 1 a highly likely b d even chance 2
3 a impossible b equal chance c little chance d unlikely e unlikely f certain
impossible c highly likely
4 The pattern starts at 1 and increases by 5 each time.
c
a
b
d
0
Qi_
Qr_
Qw_
5 a
1
3 a certain b likely c even chance d impossible 4 a
5 a
fblack, blueg f2, 3g fspades, clubs, diamondsg
i ii iii
1 5
b
3 5
2nd 2 3 2 3
2 3
4 5
c
1st
b
e
,
6 a
b
Y
a Qi_
0
b Qr_
e Qw_
c
a
7
0 Qy_ impossible
d
9 a
1
c
b
Qw_ equal chance
1 certain
To get the next number, add 6 to the previous one. b 25
8 a c
,
d 17 dots
B Y P B Y P B Y P
B
P
6
c
2nd
1st
d fBB, BY, BP, YB, YP, YY, PB, PY, PPg
,
b 9, 11, 13
b 3
centre
7 a 24% b 54
Review set 13B 1 a certain b possible c impossible 2
impossible
certain
a
d
c
very unlikely a little less equal than equal chance chance
b
very likely
3 a i fblue, blackg ii f1, 2, 3, 4g iii fdiamonds, clubs, hearts, spadesg b i
10 a very likely b 11
blue
a 2 b
13 a i
c 4
1 5
ii
3 5
iii
3 5
iv
3 5
c
1 7
b
5 a i
3 5
6 a i
3 50
3 7
ii ii
c 2 5 47 50
3 7
5 7
d
e
2 7
4
a
x y
1 17
2 14
3 11
4 8
5 5
b
x y
2 8
4 12
6 16
8 20
10 24
X
Y
4800 houseboats
b b
3 50
c 60 students
d It could be a biased sample. Children who wear glasses may be less likely to play vigorous sport and more likely to choose a quieter activity like the choir.
3 4 5 3 4 5 3 4 5
3
5
14
g
2nd
1st
ii fblue blue, blue black, black blue, black blackg 4 a
c 21 cans
fblue, blackg ii f3, 4, 5g iii fÄ, |, ¨,
b i
blue black
black
equal chance c impossible
12 a 7, 9, 11, 13 b N = 5 + 2 £ w
2nd card blue black
1st card
1 2
15
a 9 b 0 c 8 d 8 e 4
16 a
Test yourself (chapters 11, 12 and 13) 1 a 2, 7, 12, 17, 22, 27 b 33, 31, 29, 27, 25, 23 2 a
b On the enlargement the length of XY is 3 times the length of XY on the starting diagram. 17 a a = 7 b a = 25 c a = 5 d a = 8
236
CORRELATION CHART: R-7 SACSA MATHEMATICS TEACHING RESOURCE
Correlation Chart: R-7 SACSA Mathematics Teaching Resource
Knowledge, skills and understandings
Core Skills Mathematics 6
Mathematics for Year 6 (second edition)
Chapter
Unit
Chapter
Unit
9
A, B
STRAND: Exploring, analysing and modelling data Data collection and representation •conduct surveys to collect data •utilise tally system •present data graphically, e.g., frequency table •construct graphs on grid paper (e.g., pictographs, bar graphs, composite bar graphs, column graphs, line graphs) •construct tables and graphs using graphing software •label graphs with titles, axes, key and scales •interpret graphs, including pie graphs, from various sources •calculate the mean (average) of a set of data Chance and probability •describe the likelihood of events in everyday situations using appropriate everyday language (e.g., likely, unlikely, possible, probable, certain, impossible) •order the terms from impossible to certain •describe the likelihood of events in everyday situations using appropriate mathematical terminology (e.g., 50:50 chance, 1 in 4 chance, no chance, equal chance) •utilise graphic organisers (e.g., tree diagrams) to develop lists of possible outcomes •predict and record possible outcomes of an event •use data to order chance events from least likely to most likely •explain the differences between predicted results and actual results of an experiment (e.g., coin tossing) •use samples to make predictions about a larger population from which the sample comes (e.g., coin tossing – predict the result from a sample of 100 tosses)
9
62
9
B
9 9 9 9 9
62, 63, 64, 65, 66 64 62, 63, 64, 65, 66 67 68
9 9 9 9 9
C D C C, D, E F
13 13
93 93
13 13
A, B, C C, D
13
95
13
C
13 13 13
94 94, 95 93, 94, 95
13 13 13
B B C, D, E
13
94
13
B
13
94
13
B, E, F
6 6 6 6
39 39 40 40
6 6 6 6
A A B B
6 6 6
40, 41 41 44
6 6 6
C, D, E D F
6 6
40, 41 42, 43, 44
6 6
C, D, E G, H, I
6
43
6
I
6
44
6
J
6
44
6
K
STRAND: Measurement Length, perimeter and area •select and use the appropriate device and unit to measure lengths or distances •measure and record lengths or distances, including kilometres •convert between units of length (e.g., mm to cm, cm to m, m to km) •calculate lengths or distances using decimals to three decimal places •estimate length and perimeter with a reasonable degree of accuracy and confirm by measuring them accurately •compare perimeters of different shapes •construct a square metre using a variety of lengths and widths •understand and show that perimeter of shapes can be the same regardless of length of sides •estimate and record areas in square metres and square centimetres (cm²) •explain that the area of squares and rectangles can be found by multiplying the length by the breadth: A=L´W or A=L´B •calculate the area of irregular shapes composed of square and rectangular sections •apply knowledge of length, perimeter and area through practical problem solving activities
237
CORRELATION CHART: R-7 SACSA MATHEMATICS TEACHING RESOURCE
Core Skills Mathematics 6
Mathematics for Year 6 (second edition)
Chapter Volume and capacity •understand the concept of kilolitre (i.e., 1000 litres = 1 kilolitre) 6 •use the abbreviation for millilitres (mL), litres (L) and kilolitres (kL) 6 •construct 3-D objects using cubic centimetre blocks and measures volume by counting the number of blocks 6 •use the abbreviations for cubic centimetres (cm³) and cubic metres (m³) 6 •estimate the volume of rectangular prisms using cubic centimetres 6 •explain that the volume of rectangular prisms can be found by multiplying the length by the width by the height: V=L´W´H 6 •select and use the appropriate device and unit to measure capacity 6 •calculate capacity using millilitres and litres to 3 decimal places 6
Mass •estimate the mass of familiar objects •select and use the appropriate device and unit to measure mass •compare the mass of different objects •use the abbreviations for milligrams (mg), grams (g), tonnes (t) and kilograms (kg) •convert between milligrams, kilograms, grams and tonnes to 3 decimal places •apply the knowledge of mass to practical problem solving situations
Unit 46 46
Chapter 6 6
Unit M M
45 45 45
6 6 6
L L L
45 46 46
6 6 6
N M N
8 8 8
60 60 60
8 8 8
F F F
8 8 8
60 60 60
8 8 8
F F F
Time •use a stopwatch to accurately time events to hundredths of seconds •tell the time using analogue, 24-hour and digital clocks •convert between analogue, 24-hour and digital time •convert from one time unit to another (e.g., ‘How many seconds are there in one hour?’) •calculate the duration of an event using starting and finishing times •use a calendar as a planning tool •read a simple timetable •understand terminology such as AD, BC, CE, BCE •read and construct a timeline, including AD and BC
10 10 10
74 73 73
10 10 10
D D D
10 10 10 10 10 10
71 75 72 75 70 70
10 10 10 10 10 10
B, F B, F C F A A
Temperature •determine and record temperature variations •estimate and read maximum and minimum temperatures •calculate and interpret average temperatures
10 10 10
76 76 76
10 10 10
H I I
1 1 1 1 1 1 1 1 3
1 4 2 3 2 4 2 2 13
1 1 1 1 1 1 1 1 3
A B, C D H D F D D A, F
3
13
3
A, C
3
15
3
D
3
15
3
D
STRAND: Number Whole numbers •recognise the existence of different number systems (e.g., Greek, Roman, Hindu-Arabic) •provide examples of the use of number in everyday life •read, write and record number to one million, using numerals and words •explain place value of digits in number to 1¡000¡000 •write numbers to 1¡000¡000 in expanded form •round to the nearest 10, 100, 1000, 10¡000 and 100¡000 •place numbers in descending and ascending order •compare numbers and use symbols (e.g., =, ¹, < and >) •explain mental strategies used to solve and subtraction problems •choose appropriately between mental, written and calculator methods for addition and subtraction problems •use rounding and a mental strategy to multiply a 2 digit number by a 2 digit number to obtain an approximate answer (e.g., 67´53»70´50=3500) •multiply a 2 digit number by a 2 digit number using the extended form (long multiplication)
238
CORRELATION CHART: R-7 SACSA MATHEMATICS TEACHING RESOURCE
Core Skills Mathematics 6
Mathematics for Year 6 (second edition)
•divide a number with 3 or more digits by multiples of 10 (including remainders) •select and use appropriate operations to solve contextual word problems
Chapter 3 3
Fractions, decimals, percentages, ratios and rates •provide examples of the use of decimals in everyday life •explain the place value of tenths, hundredths and thousandths •read and write decimals to thousandths, in both numerals and words •write decimals in expanded form (e.g., 1.25=1u+2t+5h or 1+0.2+0.05) •round to the nearest whole number, tenth or hundredth •compare and order decimals (descending and ascending) •use symbols (e.g., =, ¹, < and >) to compare decimals •add or subtract decimal numbers that have a different number of decimal places •multiply and divide tenths, hundredths and thousandths by a single digit to terminating numbers •multiply and divide decimal numbers, including money, by 10, 100 and 1000 •multiply and divide decimal numbers, including money, by single digit numbers in everyday contexts (e.g., cost of 3 computer games at $29.95 each, cost of 1 iceblock if a pack of 8 costs $3.90) •continue, create and describe patterns involving fractions (e.g., Qr_ , Qw_ , Er_ , 1) •convert fractions to lowest terms •convert improper fractions to mixed numbers by division •convert mixed numbers to improper fractions •add and subtract simple fractions by changing one denominator (e.g., We_ + Qy_ ) •demonstrate understanding of addition and subtraction of fractions through everyday problem solving (e.g., ‘I ate half a pie and my friend ate two-thirds of a pie. How many pies did we need? How much pie is left over?’) •convert simple decimals to fractions (e.g., 0.125= qA_pS_pG_p_ = Qi_ ; 0.25= qS_pG_p_ = Qr_\\) •convert fractions to decimals (e.g., Er_ = qJ_pG_p_ = 0.75) •explain the use of percentages in everyday life •express everyday percentages as fractions and decimals (e.g., 10%, 20%, 25%, 50%, 75%, 100%) •express simple fractions and decimals as percentages (e.g., 50%= qG_p:_p_ = Qw_\\)
Unit 16 16
Chapter 3 3
Unit E F
5 5 5 5 5 5 5
30 32 32 32 37 32 32
5 5 5 5 5 5 5
B, D D D D P D D
5
33
5
F, G
5 5
36 34
5 5
L H
5 4 4 4 4
36, 37 23 24 25 25
5 4 4 4 4
L, M, N, O F H I I
4
26, 27
4
J, K
4
27
4
L
5 5 5
35 35 35
5 5 5
J I E, J
5
35
5
K
5
35
5
K
11 11 11 11
78, 79 78, 79 80 80
11 11 11 11
A, B, C A, B, C D D
11 11
81 80
11 11
E D
11
84
11
I
2 2
7, 8 7
2 2
A, D D
STRAND: Pattern and algebraic reasoning Algebra •build a simple numerical or geometric pattern using materials (e.g., matchstick patterns) •complete the pattern for a numerical or geometric series (e.g., 2, 4, 8, 16) •calculate the value of a missing number in a series of values •explain how the answers in a series of values are determined •determine and record a rule, in words, to describe the pattern presented in a table •apply a rule to a table to calculate the missing values •calculate the value of missing number in a number sentence (e.g., 7´D=42, what is the value of D ?)
STRAND: Spatial sense and geometric reasoning Lines & Angles •use symbols for ‘is parallel to’ (¡k¡) and ‘is perpendicular to’ (?) •identify and draw perpendicular lines
239
CORRELATION CHART: R-7 SACSA MATHEMATICS TEACHING RESOURCE
Core Skills Mathematics 6
Mathematics for Year 6 (second edition)
Chapter
Unit
Chapter
Unit
2
7
2
A
2
7, 8
2
A, C
2 2
8 8
2 2
C C
2 2
8 8
2 2
C C
2 2
8 10
2 2
C F
2 12 12
11 87 87
2 12 12
G A A
8
59
8
B, D
8 8 8 8
55, 56 59 57, 59 56
8 8 8 8
A B, D B, C, D A
8 8
56 56
8 8
A A
2 2
9 9
2 2
E E
12 12
90 90
12 12
D D
12 12
90 88
12 12
D B
12
91
12
E
12
91
12
E
7
49, 53
7
B, D
7
50, 52
7
C, E
7
48
7
A
7 7
51 50, 52
7 7
D C, E
•name and label lines, rays and line segments (e.g., AB, AB, AB) •use common conventions to indicate right angles, equal angles and parallel lines •classify and identify angles as right, acute, obtuse, reflex, straight or a revolution •construct, label and name angles using angle ABC notation •estimate and measure angles in degrees using a protractor and geometry software •construct an angle of a given size using a protractor •apply your understanding of angles to spatial sense and geometric reasoning activities (e.g., movement of the hands of a clock) •prove and use the fact that the sum of the interior angles of a triangle is 180° •prove and use the fact that the sum of the interior angles of a quadrilateral is 360° •understand the meaning of the term congruence •recognise congruence in lines, shapes and solids 2-D and 3 -D shape •construct a model of a simple 3-D shape from drawings of different views •use the terminology in describing 3-D objects including base, edge, surface, vertex and face •visualise and sketch simple solids from different views •construct a model of a simple solid from an isometric drawing •identify and name the properties of rectangular prisms and triangular prisms •identify and name the properties of square-based and triangular-based pyramids •use the formal names for prisms and identify pyramids •compare and describe the side and angle properties of isosceles, equilateral and scalene triangles •identify isosceles, scalene and equilateral triangles Transformation •rotate shapes clockwise/anticlockwise •identify and name shapes that have rotational symmetry •use both pen and paper, and geometry software to construct a shape that has rotational symmetry •recognise tessellations in the everyday environment (e.g., weaving) •make enlargements and reductions of 2-D shapes, pictures and maps using pen and paper or using geometry software •discuss similarities and differences of the same object or scene represented in different sizes (e.g., drawings enlarged on photocopier, drawings or pictures using geometry software) Location and position •use a coordinate grid to make simple 2-D shapes •read and interpret maps, plans, scale drawings and diagrams which have been drawn to scale •read and write scales in words and through diagrams (e.g., 1 cm represents 5 km; 1:500¡000) •use a magnetic compass to find North and hence the direction associated with the other three major compass points •identify and record familiar routes, locations
240
INDEX
INDEX
12-hour time 24-hour time acute angle analogue angle of rotation approximation area bar graph capacity cardinal points chance clockwise column graph congruent coordinate cross-section denominator difference digital divisibility rules dot pattern dot plot elevation enlargement equation equilateral triangle equivalent fractions Euler’s rule even number expanded form expression factor fraction frequency table gram grid hectare improper fraction intersecting lines isosceles triangle line line graph line of symmetry line segment litre lowest common multiple lowest terms MA blocks matchstick pattern mean metric system mixed number mode net
156 156 22 156 192 40 92 138 100 112 198 158 138 187 116 121 50 32 156 43 170 134 128 186, 194 180 24 56 123 42 72 180 42 50 134 130 108 93 58 20 24 20 140 190 20 100 44 57 12 171 146 86 58 138 126
number line number pattern numerator obtuse angle odd number parallel lines percentage perimeter perpendicular lines pictograph pie chart place value plan plan view point polygon polyhedron prism probability product projection pyramid quadrilateral quotient random ratio ray reflection reflex angle regular polygon remainder revolution right angle Roman numerals rotation rotational symmetry rounding off scale drawing scale factor scalene triangle solid solution of equation speed square unit straight angle strip graph substitution sum tessellation time conversions time line time series translation tree diagram triangular numbers undefined vertex volume
16, 50, 70 168 50 22 42 20 79 89 22 136 142 10, 72 114 128 20 21 122 120 202 36 124 121 24 38 199 64 20 186 22 21 38 22 22 8 186, 192 192 14, 40, 83 106 194 24 120 181 162 94 22 137 181 32 188 152 150 140 186 200 169 43 122 98
