Algebra Readiness Made Easy: Grade 3: An Essential Part of Every Math Curriculum (Best Practices in Action)

Grade 3

NEW YORK • TORONTO • LONDON • AUCKLAND • SYDNEY MEXICO CITY • NEW DELHI • HONG KONG • BUENOS AIRES Algebra Readiness Made Easy: Grade 3 © Greenes, Findell & Cavanagh, Scholastic Teaching Resources

Scholastic Inc. grants teachers permission to photocopy the activity sheets from this book for classroom use. No other part of this publication may be reproduced in whole or in part, or stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without written permission of the publisher. For information regarding permission, write to Scholastic Inc., 557 Broadway, New York, NY 10012. Editor: Maria L. Chang Cover design by Jason Robinson Interior design by Melinda Belter Illustrations by Teresa Anderko ISBN-13: 978-0-439-83930-3 ISBN-10: 0-439-83930-0 Copyright © 2008 by Carole Greenes, Carol Findell, and Mary Cavanagh All rights reserved. Printed in China. 1 2 3 4 5 6 7 8 9 10

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Algebra Readiness Made Easy: Grade 3 © Greenes, Findell & Cavanagh, Scholastic Teaching Resources

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INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

PROBLEM SETS Fox Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Hidden Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Mini Movers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Which Floor? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Balancing Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Favorite Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

PROBLEM-SOLVING TRANSPARENCY MASTER . . . . . . . . . . . . . 75

SOLVE IT TRANSPARENCY MASTERS . . . . . . . . . . . . . . . . . . . . 76

ANSWER KEY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 COLOR TRANSPARENCIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

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Table of Contents

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Introduction

Welcome to Algebra Readiness Made Easy! This book is designed to help you introduce children to problem-solving strategies and algebraic-reasoning techniques, to give them practice with major number concepts and skills, and to motivate them to write and talk about big ideas in mathematics. It also sets the stage for the formal study of algebra in the upper grades.

Algebra Standards The National Council of Teachers of Mathematics identifies algebra as one of the five major content areas of the mathematics curriculum to be studied by children in all grades (NCTM, 2000). The council emphasizes that early and regular experience with the key ideas of algebra helps students make the transition into the more formal study of algebra in late middle school or high school. This view is consistent with the general theory of learning—that understanding is enhanced when connections are made between what is new and what was previously studied. The key algebraic concepts developed in this book are: • representing quantitative relationships with numbers and symbols • replacing unknowns with their values • writing and solving equations • solving equations for the value of a variable • solving two equations with two unknowns • reasoning about proportional relationships • identifying and continuing patterns • exploring variables as representing varying quantities • identifying the functional relationship between two numbers

Building Key Math Skills NCTM also identifies problem solving as a key process skill and the teaching of strategies and methods of reasoning to solve problems as a major part of the mathematics curriculum for children of all ages. The problem-solving model first described in 1957 by renowned mathematician George Polya has been adopted by teachers and instructional developers nationwide and provides the framework for the problem-solving focus of this book. All the

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Algebra Readiness Made Easy: Grade 3 © Greenes, Findell & Cavanagh, Scholastic Teaching Resources

INTRODUCTION

problems contained here require children to interpret data displays—such as text, charts, diagrams, graphs, pictures, and tables—and answer questions about them. As they work on the problems, children learn and practice the following problem-solving strategies: • making lists or cases of possible solutions and testing those solutions • identifying, describing, and generalizing patterns • working backward • reasoning logically • reasoning proportionally The development of problem-solving strategies and algebraic concepts is linked to the development of number concepts and skills. As children solve the problems in this book, they’ll practice counting, computing, applying concepts of place value and number theory, and reasoning about the magnitudes of numbers. Throughout this book, we emphasize the language of mathematics. This language includes terminology (e.g., odd number, variable) as well as symbols (e.g., >, <). Children will see the language in the problems and illustrations and use the language in their discussions and written descriptions of their solution processes.

How to Use This Book Inside this book you’ll find six problem sets—each composed of nine problems featuring the same type of data display (e.g., diagrams, scales, and arrays of numbers)—that focus on one or more problem-solving strategies and algebraic concepts. Each set opens with an overview of the type of How old is each fox? problems/tasks in the set, the algebra and problemsolving focus, the number concepts or skills needed to solve the problems, the math language that is emphasized in the problems, and guiding questions to be used with the first two problems of the set to help children grasp the key concepts and strategies. The first two problems in each set are designed to be I’ll start with the fact that gives one fox’s age. discussed and solved in a whole-class setting. The first, “Solve the Problem,” introduces children to the type of Ima Thinker display and problem they will encounter in the rest of 1. Which fact did Ima use first? ____________________ 2. How old is Red? __________ the set. We suggest that you have children work on this 3. How old is Snooper? __________ first problem individually or in pairs before you engage 4. Who is the oldest? __________ in any formal instruction. Encourage children to wrestle with the problem and come up with some strategies they Name _____________________________________________ Date __________________

F O X F AC T S

S O L VE

T HE P R O BLE M

Use the facts. Figure out the ages.

Snooper

Red

Snowball

FA C T S :

Snooper is 4 years younger than Red.

B

Red is 5 years older than Snowball.

C

Snowball is 2 years old.

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

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Algebra Readiness Made Easy: Grade 3 © Greenes, Findell & Cavanagh, Scholastic Teaching Resources

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might use to solve it. Then gather children together and use the guiding questions provided to help them discover key mathematical relationships and understand the special vocabulary used in the problem. This whole-class discussion will enhance student understanding and success with the problem-solving strategies and algebraic concepts in each problem set. The second problem, “Make the Case,” comes as an overhead transparency and uses a multiple-choice format. Three different characters offer possible solutions to the problem. Children have to determine which character—Professor Pelly, Ms. Slide, and Dr. Tap—has the correct answer. Before they can identify the correct solution, children have to solve the problem themselves How old is each fox? and analyze each of the responses. Invite them to speculate about why the other two characters got the wrong answers. (Note: Although we offer a rationale for each wrong answer, other explanations are possible.) As children justify their choices in the “Make the Case” Roxy is 10 years problems, they gain greater experience using math Roxy is 12 years old. Screech is old. Screech is 2 years old. language. 2 years old. Howler is Howler is 10 years old. 10 years old. While working on these first two problems it is Ms. Slide important to encourage children to talk about their observations and hypotheses. This talk provides a window Roxy is 10 years old. Screech is into what children do and do not understand. Working 8 years old. Howler is on “Solve the Problem” and “Make the Case” should take Professor Pelly Dr. Tap 10 years old. Who is wearing a thinking cap? approximately one math period. The rest of the problems in each set are sequenced by difficulty. All problems feature a series of questions that involve analyses of the data display. In the first three or four problems of each set, problem-solving “guru” Ima Thinker provides hints about how to begin solving the problems. No hints are provided for the rest of the problems. If children have difficulty solving these latter problems, you might want to write “Ima” hints for each of them or ask children to develop hints before beginning to solve the problems. An answer key is provided at the back of the book. The problem sets are independent of one another and may be used in any order and incorporated into the regular mathematics curriculum at whatever point makes sense. We recommend that you work with each problem set in its entirety before moving on to the next one. Once you and your students work through the first two problems, you can assign problems 1 through 7 for children to do on their own or in pairs. You may wish to have them complete the problems during class or for homework. Name _____________________________________________ Date __________________ FO X FA C T S

MAK E THE C A SE

Use the facts. Figure out the ages.

Roxy

Screech

Howler

FA C T S : A

Roxy is 10 years older than Screech.

B

Screech is 8 years younger than Howler.

C

Howler is 10 years old.

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Algebra Readiness Made Easy: Grade 3 © Greenes, Findell & Cavanagh, Scholastic Teaching Resources

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

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INTRODUCTION

Using the Transparencies In addition to the reproducible problem sets, you’ll find ten overhead transparencies at the back of this book. (Black-line masters of all transparencies also appear in the book.) The first six transparencies are reproductions of the “Make the Case” problems, to help you in leading a whole-class disHow long is each fox? cussion of each problem. The remaining four transparencies are designed to be used together. Three of these transparencies feature six problems, one from each of the problem sets. Cut these three transparencies in half and overlay each probLook lem on the Problem-Solving Transparency. Then invite children to apply our three-step problem-solving process: SOLVE IT: FOX FACTS

PROBLEM-SOLVING TRANSPARENCY

SOLVE IT

The Twins

Roxy

Paint

FACTS:

1) Look: What is the problem? What information do you have? What information do you need? 2) Plan and Do: How will you solve the problem? What strategies will you use? What will you do first? What’s the next step? What comes after that?

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

1.

A

Roxy is 6 inches longer than each twin.

B

Each twin is 11 inches shorter than Paint.

C

Paint is 25 inches long.

Use the facts. Figure out the length of each fox.

What is the problem?

SOLVE IT: HIDDEN NUMBERS

What number is under each star? 2.

Plan and Do

What will you do first? How will you solve the problem?

+ 4 +

+ 8 = 30

The two stars are covering the same number.

3.

Answer and Check

How can you be sure your answer is correct?

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3) Answer and Check: What is the answer? How can you be sure that your answer is correct? These problem-solving transparencies encourage writing about mathematics and may be used at any time. They are particularly effective when used as culminating activities for the set of problems.

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References Cuevas, Gilbert, & Karol Yeatts. (2001). Navigating through algebra in grades 3–5. Reston, VA: National Council of Teachers of Mathematics. Greenes, Carole, & Carol Findell. (Eds.). (2005). Developing students’ algebraic reasoning abilities. (Vol. 3 in the NCSM Monograph Series). Boston, MA: Houghton Mifflin. Greenes, Carole, & Carol Findell. (2005). Groundworks: Algebraic thinking. Chicago: Wright Group/McGraw Hill. Moses, Barbara (Ed.). (1999). Algebraic thinking, grades K–12: Readings from NCTM’s school-based journals and other publications. Reston, VA: National Council of Teachers of Mathematics. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics. Polya, George. (1957). How to solve it. Princeton, NJ: Princeton University Press. Usiskin, Zalman. (1997). Doing algebra in grades K–4. Teaching Children Mathematics. 3(6), 346–356.

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Fox Facts

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Overview Children use clues to interpret mathematical relationships and work backward through the clues to answer the questions.

Algebra Represent quantitative relationships using numbers and symbols • Solve equations

Problem-Solving Strategies Work backward • Use logical reasoning

Related Math Skills Compute with whole numbers • Compare quantities

Math Language Age: older than, oldest, younger than • Length: inches, long, longer, shorter • Time: months, years • Weight: pounds, weighs more than, weighs less than

Introducing the Problem Set Make photocopies of “Solve the Problem: Fox Facts” (page 11) and distribute to children. Have children work in pairs, encouraging them to discuss strategies they might use to solve the problem. You may want to walk around and listen in on some of their discussions. After a few minutes, display the problem on the board (or on the overhead if you made a transparency) and use the following questions to guide a whole-class discussion on how to solve the problem: • What did Ima do first to figure out the ages? (She used Fact C, which gives Snowball’s age. Snowball is 2 years old.) • Whose age can you figure out next? (Red’s) • Why can’t you figure out Snooper’s age before figuring out Red’s age? (Fact A says that Snooper is 4 years younger than Red, so we have to figure out Red’s age before figuring out Snooper’s age.) Algebra Readiness Made Easy: Grade 3 © Greenes, Findell & Cavanagh, Scholastic Teaching Resources

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FOX FACTS

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Name _____________________________________________ Date __________________ F O X FA CT S

SO L V E TH E P R O BL E M

• How old is Red? (5 + 2, or 7 years old)

Use the facts. Figure out the ages.

• How old is Snooper? (7 – 4, or 3 years old) Work together as a class to answer the questions in “Solve the Problem: Fox Facts.”

Snooper

• Which fact did you use first to solve the problem? (Fact C) Why did you use that fact? (It gives Howler’s age of 10 years.)

Snowball

A

Snooper is 4 years younger than Red.

B

Red is 5 years older than Snowball.

C

Snowball is 2 years old.

I’ll start with the fact that gives one fox’s age.

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

• Who has the right answer? (Professor Pelly)

Red

FA C T S :

Math Chat With the Transparency Display the “Make the Case: Fox Facts” transparency on the overhead. Before children can decide which character is “wearing a thinking cap,” they need to figure out the answer to the problem. Encourage children to work in pairs to solve the problem. Then bring the class together for another whole-class discussion. Ask:

How old is each fox?

Ima Thinker 1. Which fact did Ima use first? ____________________ 2. How old is Red? __________ 3. How old is Snooper? __________ 4. Who is the oldest? __________ 11

Name _____________________________________________ Date __________________ FOX FA CTS

M AKE

• What did you do next? (Figure out Screech’s age. Fact B says that Screech is 8 years younger than Howler. So Screech is 10 – 8, or 2 years old.)

TH E CAS E

Use the facts. Figure out the ages.

• How can you figure out Roxy’s age? (Fact A says that Roxy is 10 years older than Screech. So Roxy is 2 + 10, or 12 years old.)

Roxy

Screech

Howler

FA C T S : A

Roxy is 10 years older than Screech.

B

Screech is 8 years younger than Howler.

C

Howler is 10 years old.

Roxy is 10 years

Roxy is 12 years old. Screech is 2 years old. Howler is 10 years old.

old. Screech is 2 years old. Howler is 10 years old.

Ms. Slide

• How do you think Dr. Tap got his answer? (Dr. Tap might have just used the numbers given in each clue as the ages of the foxes.)

Roxy is 10 years

Professor Pelly 12

10

old. Screech is 8 years old. Howler is 10 years old.

Dr. Tap

Who is wearing a thinking cap?

Algebra Readiness Made Easy: Grade 3 © Greenes, Findell & Cavanagh, Scholastic Teaching Resources

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

• How do you think Ms. Slide got her answer? (Ms. Slide probably used Facts B and C correctly. With Fact A, she used the number 10, but did not add it to Screech’s age.)

How old is each fox?

Name _____________________________________________ Date __________________ FOX FACTS

SOLVE THE PROBLEM

How old is each fox?

Use the facts. Figure out the ages.

Snooper

Red

Snowball

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

FACTS: A

Snooper is 4 years younger than Red.

B

Red is 5 years older than Snowball.

C

Snowball is 2 years old.

I’ll start with the fact that gives one fox’s age.

Ima Thinker 1. Which fact did Ima use first? ____________________ 2. How old is Red? __________ 3. How old is Snooper? __________ 4. Who is the oldest? __________ 11

Name _____________________________________________ Date __________________ FOX FACTS

MAKE THE CASE

How old is each fox?

Use the facts. Figure out the ages.

Roxy

Screech

Howler

FACTS: A

Roxy is 10 years older than Screech.

B

Screech is 8 years younger than Howler.

C

Howler is 10 years old.

Ms. Slide

Professor Pelly 12

Roxy is 10 years old. Screech is 8 years old. Howler is 10 years old.

Dr. Tap

Who is wearing a thinking cap?

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

Roxy is 10 years old. Screech is 2 years old. Howler is 10 years old.

Roxy is 12 years old. Screech is 2 years old. Howler is 10 years old.

Name _____________________________________________ Date __________________ FOX FACTS

PROBLEM

1

How old is each fox? Use the facts. Figure out the ages.

Skipper

The Twins

Daisy

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

FACTS: A

Skipper is 9 months older than the twins.

B

The twins are 2 months younger than Daisy.

C

Daisy is 5 months old.

I’ll start with the fact that gives one fox’s age.

Ima Thinker 1. Which fact did Ima use first? ____________________ 2. How old are the twins? __________ 3. How old is Skipper? __________ 4. How did you figure out Skipper’s age? _________________ ______________________________________________ 13

Name _____________________________________________ Date __________________ FOX FACTS

PROBLEM

2

How much does each fox weigh? Use the facts. Figure out how many pounds each fox weighs.

Red

Snowball

Screech

FACTS: A

Red weighs 2 pounds less than Snowball.

B

Snowball weighs 6 pounds more than Screech.

C

Screech weighs 8 pounds.

Ima Thinker 1. Which fact did Ima use first? ____________________ 2. How much does Snowball weigh? __________ 3. How much does Red weigh? __________ 4. How did you figure out Red’s weight? _________________ _____________________________________________ 14

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

I’ll start with the fact that gives one fox’s weight.

Name _____________________________________________ Date __________________ FOX FACTS

PROBLEM

3

How much does each fox weigh? Use the facts. Figure out how many pounds each fox weighs.

Paint

The Twins

Roxy

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

FACTS: A

Paint weighs 12 pounds more than one twin.

B

Each twin weighs 4 pounds less than Roxy.

C

Roxy weighs 7 pounds.

I’ll start with the fact that gives one fox’s weight.

Ima Thinker 1. Which fact did Ima use first? _______________________ 2. How much does each twin weigh? __________ 3. How much does Paint weigh? __________ 4. How did you figure out Paint’s weight? __________________ _______________________________________________

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Name _____________________________________________ Date __________________ FOX FACTS

PROBLEM

4

How much does each fox weigh? Use the facts. Figure out how many pounds each fox weighs.

Leap

Howler

Snooper

FACTS: A

Leap weighs 1 pound less than Howler.

B

Howler weighs 8 pounds more than Snooper.

C

Snooper weighs 2 pounds.

2. How much does Howler weigh? __________ 3. How much does Leap weigh? __________ 4. How did you figure out Leap’s weight? __________________ _______________________________________________

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Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

1. Which fact will you use first? ____________________

Name _____________________________________________ Date __________________ FOX FACTS

PROBLEM

5

How long is the body of each fox? Use the facts. Figure out the length of each fox without its tail.

Snooper

Snowball

Skipper

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

FACTS: A

Snooper’s body is 6 inches shorter than Snowball’s.

B

Snowball’s body is 2 inches longer than Skipper’s.

C

Skipper’s body is 18 inches long.

1. Which fact will you use first? ____________________ 2. How long is Snowball’s body? __________ 3. How long is Snooper’s body? __________ 4. How did you figure out the length of Snooper’s body? ________________________________________________

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Name _____________________________________________ Date __________________ FOX FACTS

PROBLEM

6

How long is the body of each fox? Use the facts. Figure out the length of each fox without its tail.

Leap

Daisy

Red

FACTS: A Leap’s body is 3 inches longer than Daisy’s. B Daisy’s body is 9 inches shorter than Red’s. C Red’s body is 25 inches long.

2. How long is Daisy’s body? __________ 3. How long is Leap’s body? __________ 4. How did you figure out the length of Leap’s body? _______________________________________________

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Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

1. Which fact will you use first? ____________________

Name _____________________________________________ Date __________________ FOX FACTS

PROBLEM

7

How long is the body of each fox?

Use the facts. Figure out the length of each fox without its tail.

Screech

Speedy

Howler

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

FACTS: A

Screech’s body is 9 inches shorter than Speedy’s.

B

Speedy’s body is 7 inches longer than Howler’s.

C

Howler’s body is 17 inches long.

1. Which fact will you use first? ____________________ 2. How long is Speedy’s body? __________ 3. How long is Screech’s body? __________ 4. How did you figure out the length of Screech’s body? ________________________________________________

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Hidden Numbers Overview

Given a number sentence with some addends “hidden” behind pictures, children figure out the number that each picture represents.

Algebra Replace symbols with their values • Understand that removing an addend changes the sum by the same amount • Solve equations for the value of a variable (unknown)

Problem-Solving Strategies Reason deductively

Related Math Skills Add and subtract with one- and two-digit numbers • Divide, basic facts

Math Language Same number • Sum

Introducing the Problem Set Make photocopies of “Solve the Problem: Hidden Numbers” (page 22) and distribute to children. Have children work in pairs, encouraging them to discuss strategies they might use to solve the problem. You may want to walk around and listen in on some of their discussions. After a few minutes, display the problem on the board (or on the overhead if you made a transparency) and use the following questions to guide a whole-class discussion on how to solve the problem: • What do you see in the picture? (An addition sentence showing two flags, the number 3, and the sum 13) • What do you need to do? (Figure out the number under each flag) • What will you do first? (Take away the 3 and change the sum.)

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Algebra Readiness Made Easy: Grade 3 © Greenes, Findell & Cavanagh, Scholastic Teaching Resources

HIDDEN NUMBERS

Name _____________________________________________ Date __________________ HIDDEN NUMBERS

SOLVE THE PROBLEM

What number is under each flag?

• What is the new sum? (13 – 3 = 10) • What does this new sum stand for? (The sum of the numbers under the two flags)

+

3

+

= 13

The two flags are covering the same number.

• How can you figure out the number under each flag? (If the sum of two flags is 10, then one flag is 5; 10 ÷ 2 = 5, or 5 + 5 = 10.)

The sum of the three numbers is 13.

First, I’ll take away the 3. Then, I ‘ll change the sum. Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

Work together as a class to answer the questions in “Solve the Problem: Hidden Numbers.”

Ima Thinker 1. Why did Ima take away the 3 first?

Math Chat With the Transparency Display the “Make the Case: Hidden Numbers” transparency on the overhead. Before children can decide which character is “wearing a thinking cap,” they need to figure out the answer to the problem. Encourage children to work in pairs to solve the problem. Then bring the class together for another whole-class discussion. Ask:

______________________________________________ 2. How did Ima change the sum? _______________________ ______________________________________________ 3. What number is under each flag? __________ 4. How did you figure out the answer to #3? ______________________________________________ 22

Name _____________________________________________ Date __________________ HIDDEN NUMBERS

MAKE THE CASE

• Who has the right answer? (Professor Pelly)

What number is under each bird?

• How did you figure it out? (16 – 2 = 14 and 14 ÷ 2 = 7)

+

• How do you think Ms. Slide got the answer of 8? (She probably divided 16 by 2 and got 8. She forgot to subtract the 2 first.)

2

= 16

The two birds are covering the same number. The sum of the three numbers is 16.

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

• How do you think Dr. Tap got the answer of 14? (He probably subtracted 2 from 16 and got 14. He forgot to divide the 14 by 2.)

+

It is 8. I’m sure of it.

This one is easy. It’s 7.

Ms. Slide

Professor Pelly

You are both wrong. It has to be 14.

Dr. Tap

Who is wearing a thinking cap?

Algebra Readiness Made Easy: Grade 3 © Greenes, Findell & Cavanagh, Scholastic Teaching Resources

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Name _____________________________________________ Date __________________ HIDDEN NUMBERS

SOLVE THE PROBLEM

What number is under each flag?

+

3

+

= 13

The two flags are covering the same number. The sum of the three numbers is 13.

First, I’ll take away the 3. Then, I’ll change the sum.

1. Why did Ima take away the 3 first? ______________________________________________ 2. How did Ima change the sum? _______________________ ______________________________________________ 3. What number is under each flag? __________ 4. How did you figure out the answer to #3? ______________________________________________ 22

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

Ima Thinker

Name _____________________________________________ Date __________________ HIDDEN NUMBERS

MAKE THE CASE

What number is under each bird?

+

+

2

= 16

The two birds are covering the same number.

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

The sum of the three numbers is 16.

It is 8. I’m sure of it.

This one is easy. It’s 7.

Ms. Slide

Professor Pelly

You are both wrong. It has to be 14.

Dr. Tap

Who is wearing a thinking cap?

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Name _____________________________________________ Date __________________ HIDDEN NUMBERS

PROBLEM

1

What number is under the smile?

+

4

5

+

= 15

The sum of the three numbers is 15.

First, I’ll take away the 4 and 5. Then, I’ll change the sum.

1. Why did Ima take away the 4 and 5 first? _______________________________________________ 2. How did Ima change the sum? _______________________________________________ 3. What number is under the smile? __________ 4. How did you figure out the answer to #3? _______________________________________________ 24

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

Ima Thinker

Name _____________________________________________ Date __________________ HIDDEN NUMBERS

PROBLEM

2

What number is under each dinosaur?

+

7

+

= 23

The two dinosaurs are covering the same number. The sum of the three numbers is 23.

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

First, I’ll take the 7 away. Then, I’ll change the sum.

Ima Thinker 1. Why did Ima take away the 7 first? _______________________________________________ 2. What number is under each dinosaur? _______ 3. How did you figure out the answer to #2? _______________________________________________ 4. Write the number on the dinosaur. What is the sum? _______

+2+3=? 25

Name _____________________________________________ Date __________________ HIDDEN NUMBERS

PROBLEM

3

What number is under each astronaut?

+

+

4

+

1

= 15

The two astronauts are covering the same number. The sum of the four numbers is 15.

First, I’ll take the 4 and 1 away. Then, I’ll change the sum.

1. Why did Ima take away the 4 and 1 first? _______________________________________________ 2. What number is under each astronaut? _______ 3. How did you figure out the answer to #2? _______________________________________________ 4. Write the number on the astronaut. What is the sum? __________ 26

+ 3 + 3 = ?

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

Ima Thinker

Name _____________________________________________ Date __________________ HIDDEN NUMBERS

PROBLEM

4

What number is under each koala?

+

3

+

+

5

= 14

The two koalas are covering the same number. The sum of the four numbers is 14.

1. What number is under each koala? __________

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

2. How did you figure out the answer to #1? _______________ _______________________________________________ 3. Write the number on each koala. What is the sum? ________

+

+ 2 + 3 = ?

4. Write the number on each koala. What is the missing number? __________

+

+ ? = 15

27

Name _____________________________________________ Date __________________ HIDDEN NUMBERS

PROBLEM

5

What number is under each turtle?

4

+

+

5

+

= 27

The two turtles are covering the same number. The sum of the four numbers is 27.

1. What number is under each turtle? __________ 2. How did you figure out the answer to #1?

3. Write the number on each turtle. What is the sum? _______

+

+ 1 + 6 = ?

4. Write the number on each turtle. What is the missing number? __________

+

28

+ ? = 22

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

_______________________________________________

Name _____________________________________________ Date __________________ HIDDEN NUMBERS

PROBLEM

6

What number is under each dolphin?

+

= 22

+ 10 +

The three dolphins are covering the same number. The sum of the four numbers is 22.

1. What number is under each dolphin? __________

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

2. How did you figure out the answer to #1? _______________________________________________ 3. Write the number on each dolphin. What is the sum? _______

+ 11 +

+ 12 = ?

4. Write the number on each dolphin. What is the missing number? __________

+

+ ? = 50

29

Name _____________________________________________ Date __________________ HIDDEN NUMBERS

PROBLEM

7

What number is under each zebra?

+

+ 1

+

= 31

The three zebras are covering the same number. The sum of the four numbers is 31.

1. What number is under each zebra? __________ 2. How did you figure out the answer to #1?

3. Write the number on each zebra. What is the sum? _________

+

+ 3 + 5 = ?

4. Write the number on each zebra. What is the missing number? _________

+

30

+ ? = 26

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

_______________________________________________

ALGE B

Mini Movers

3

SS NE

READ I RA

Overview Children examine two scales, each showing the total weight of different types of vehicles, then solve for the weight of each vehicle.

Algebra Solve two equations with two unknowns • Replace unknowns with their values

Problem-Solving Strategies Reason deductively • Test cases

Related Math Skills Add and subtract with one- and two-digit numbers • Multiply and divide, basic facts

Math Language Altogether • Pounds • Scale • Weigh • Weigh altogether

Introducing the Problem Set Make photocopies of “Solve the Problem: Mini Movers” (page 33) and distribute to children. Have children work in pairs, encouraging them to discuss strategies they might use to solve the problem. You may want to walk around and listen in on some of their discussions. After a few minutes, display the problem on the board (or on the overhead if you made a transparency) and use the following questions to guide a whole-class discussion on how to solve the problem: • Look at the two scales. What is on scale A? (2 toy dump trucks) • What is on scale B? (1 toy dump truck and 1 toy bus) • Which weight can you figure out first? (dump truck) • How will you figure it out? (On scale A, 2 dump trucks are 8 pounds, so one dump truck is half of 8, or 4 pounds.)

Algebra Readiness Made Easy: Grade 3 © Greenes, Findell & Cavanagh, Scholastic Teaching Resources

31

3

SS NE

ALGE B

MINI MOVERS

READ I RA

Name _____________________________________________ Date __________________ MINI MOVERS

SOLVE THE PROBLEM

How much do the toys weigh?

• How can knowing that a toy dump truck weighs 4 pounds help you figure out the weight of the toy bus? (If the dump truck is 4 pounds, and the total weight of the bus and dump truck is 11 pounds, then the bus must weigh 11 – 4, or 7 pounds.) Work together as a class to answer the questions in “Solve the Problem: Mini Movers.”

Display the “Make the Case: Mini Movers” transparency on the overhead. Before children can decide which character is “wearing a thinking cap,” they need to figure out the answer to the problem. Encourage children to work in pairs to solve the problem. Then bring the class together for another whole-class discussion. Ask:

11 pounds

SCALE A

SCALE B

I’ll start with Scale A. I can figure out how much one toy dump truck weighs.

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

Math Chat With the Transparency

8 pounds

Ima Thinker 1. One dump truck weighs _______ pounds. 2. The bus weighs _______ pounds. 3. The bus weighs _______ pounds more than the dump truck. 4. Two buses weigh _______ pounds.

33

• Who has the right answer? (Dr. Tap) Name _____________________________________________ Date __________________

• How did you figure it out? (Scale B shows that two vans are 18 pounds. So one van is 18 ÷ 2, or 9 pounds. Scale A shows the total weight as 21 pounds. Since the van is 9 pounds, subtract 9 from 21, or 21 – 9 = 12. The cement truck is12 pounds.)

MINI MOVERS

MAKE THE CASE

How much does the cement mixer weigh?

SCALE A

SCALE B

I believe the cement mixer weighs

The scale shows that it is 21 pounds.

• How do you think Ms. Slide got the answer of 3 pounds? (She might have subtracted 18 from 21 without realizing that 18 pounds is the total weight of two vans, not one.)

3 pounds.

Ms. Slide

Professor Pelly 34

You’re both wrong. The cement mixer weighs 12 pounds.

Dr. Tap

Who is wearing a thinking cap?

Algebra Readiness Made Easy: Grade 3 © Greenes, Findell & Cavanagh, Scholastic Teaching Resources

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

• How do you think Professor Pelly got the answer of 21 pounds? (He might have just looked at the number of pounds showing on scale A and not subtracted the weight of the van.)

32

18 pounds

21 pounds

Name _____________________________________________ Date __________________ MINI MOVERS

SOLVE

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

THE PROBLEM

How much do the toys weigh? 8 pounds

11 pounds

SCALE A

SCALE B

I’ll start with Scale A. I can figure out how much one toy dump truck weighs.

Ima Thinker 1. One dump truck weighs _______ pounds. 2. The bus weighs _______ pounds. 3. The bus weighs _______ pounds more than the dump truck. 4. Two buses weigh _______ pounds.

33

Name _____________________________________________ Date __________________ MINI MOVERS

MAKE THE CASE

How much does the cement mixer weigh? 18 pounds

21 pounds

SCALE A

SCALE B

The scale shows that it is 21 pounds.

Ms. Slide

Professor Pelly 34

You’re both wrong. The cement mixer weighs 12 pounds.

Dr. Tap

Who is wearing a thinking cap?

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

I believe the cement mixer weighs 3 pounds.

Name _____________________________________________ Date __________________ MINI MOVERS

PROBLEM

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

1

How much do the toys weigh? 9 pounds

3 pounds

SCALE A

SCALE B

I’ll start with Scale B. It shows the number of pounds for the sand truck.

Ima Thinker 1. The sand truck weighs _______ pounds. 2. The car weighs _______ pounds. 3. The car weighs _______ pounds more than the sand truck. 4. Two cars weigh _______ pounds.

35

Name _____________________________________________ Date __________________ MINI MOVERS

PROBLEM

2

How much do the toys weigh? 15 pounds

SCALE A

6 pounds

SCALE B

Ima Thinker 1. The pickup truck weighs _______ pounds. 2. The camper weighs _______ pounds. 3. The pickup truck weighs _______ pounds less than the camper. 4. Two campers weigh _______ pounds. 36

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

I’ll start with Scale B. It shows the number of pounds for the pickup truck.

Name _____________________________________________ Date __________________ MINI MOVERS

PROBLEM

3

How much do the toys weigh? 20 pounds

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

SCALE A

16 pounds

SCALE B

I’ll start with Scale A. I can figure out how much one school bus weighs.

Ima Thinker 1. One school bus weighs _______ pounds. 2. The flatbed truck weighs _______ pounds. 3. The school bus weighs _______ pounds more than the flatbed truck. 4. Two flatbed trucks and two school buses weigh _______ pounds all together. 37

Name _____________________________________________ Date __________________ MINI MOVERS

PROBLEM

4

How much do the toys weigh? 20 pounds

SCALE A

16 pounds

SCALE B

2. The tow truck weighs _______ pounds. 3. How did you figure out the answer to #2? _______________ _______________________________________________ 4. Two ambulances and two tow trucks weigh _______ pounds all together.

38

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

1. One ambulance weighs _______ pounds.

Name _____________________________________________ Date __________________ MINI MOVERS

PROBLEM

5

How much do the toys weigh? 27 pounds

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

SCALE A

14 pounds

SCALE B

1. One fire truck weighs _______ pounds. 2. The double-decker bus weighs _______ pounds. 3. How did you figure out the answer to #2? _______________ _______________________________________________ 4. Two fire trucks and two double-decker buses weigh _______ pounds all together.

39

Name _____________________________________________ Date __________________ MINI MOVERS

PROBLEM

6

How much do the toys weigh? 15 pounds

19 pounds

SCALE A

SCALE B

2. The oil truck weighs _______ pounds. 3. How did you figure out the answer to #2? _______________ _______________________________________________ 4. Three snow plows and three oil trucks weigh _______ pounds all together.

40

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

1. One snowplow weighs _______ pounds.

Name _____________________________________________ Date __________________ MINI MOVERS

PROBLEM

7

How much do the toys weigh? 30 pounds

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

SCALE A

36 pounds

SCALE B

1. One street sweeper weighs _______ pounds. 2. One motorcycle weighs _______ pounds. 3. How did you figure out the answer to #2? _______________ _______________________________________________ 4. Three street sweepers and three motorcycles weigh _______ pounds all together.

41

ALGE B

3

SS NE

READ I RA

Which Floor? Overview

Children use clues and reason logically to figure out the floor number that is represented by a letter.

Algebra Solve for values of unknowns • Replace letters with their values

Problem-Solving Strategies Make a list of possible solutions • Test possible solutions with clues • Use logical reasoning

Related Math Skills Add and subtract with one- and two-digit numbers • Compare numbers • Identify odd and even numbers • Identify tens and ones digits of two-digit numbers • Count by 2s, 3s, and 5s

Math Language Digit • Even number, odd number • Less than, greater than • Not . . .

Introducing the Problem Set Make photocopies of “Solve the Problem: Which Floor?” (page 44) and distribute to children. Have children work in pairs, encouraging them to discuss strategies they might use to solve the problem. You may want to walk around and listen in on some of their discussions. After a few minutes, display the problem on the board (or on the overhead if you made a transparency) and use the following questions to guide a whole-class discussion on how to solve the problem: • What numbers are on Ima’s list? (8, 7, 6, 5, 4, 3, 2, 1) • Why is 8 the greatest number on Ima’s list? (Clue 2 shows that when A is added to itself, the sum is less than 18. Since 8 + 8 = 16 [the greatest possible sum], 8 is the greatest number A can be.) 42

Algebra Readiness Made Easy: Grade 3 © Greenes, Findell & Cavanagh, Scholastic Teaching Resources

WHICH FLOOR?

Name _____________________________________________ Date __________________ WHICH FLOOR?

SOLVE THE PROBLEM

• Which clue do you think Ima will use next? (Either clue will help. Clue 1 eliminates 1, 3, 5, and 7, and Clue 3 eliminates 2, 4, and 6.)

What is the giraffe’s floor number?

The giraffe lives in this building.

A

stands for his floor number.

Use the clues to figure out

• What number is left? (8)

A.

CLUES: 1) You say A when you count by 2s, starting with 2.

• How can you check your answer? (Replace A with 8. Check 8 with each clue: Clue 1: You say 8 when you count by 2s. Clue 2: 8 + 8 = 16, and 16 is less than 18. Clue 3: 8 is greater than 6.)

2) 3)

A + A is less than 18. A is greater than 6.

I’ll start with

Work together as a class to answer the questions in “Solve the Problem: Which Floor?”

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

Clue 2. I’ll make a list of numbers for A .

1. What numbers are on Ima’s list?

Ima Thinker ___________________________ 2. Look at Clue 1. Which numbers can you cross off Ima’s list?

Math Chat With the Transparency

___________________________ 3. What is A ? __________

Display the “Make the Case: Which Floor?” transparency on the overhead. Before children can decide which character is “wearing a thinking cap,” they need to figure out the answer to the problem. Encourage children to work in pairs to solve the problem. Then bring the class together for another whole-class discussion. Ask:

4. How did you figure out the floor number? _______________ _______________________________________________ 44

Name _____________________________________________ Date __________________

THE CASE

• Who has the right answer? (Ms. Slide)

• How can you check Ms. Slide’s answer? (Replace B with 9 and check with each clue. Clue 1: 9 + 9 is greater than 12. Clue 2: 9 is less than 10. Clue 3: You say 9 when you count by 3s.) • How do you think Professor Pelly got the answer of 3? (3 fits Clues 2 and 3. Professor Pelly might have ignored Clue 1.)

What is the pig’s floor number?

The pig lives in this building.

B

stands for her floor number.

Use the clues to figure out

B.

CLUES: 1) 2) 3)

B + B is greater than 12. B is less than 10. You say B when you count by 3s, starting with 3.

B can only Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

• How did you figure it out? (From Clue 2, B can be 1, 2, 3, 4, 5, 6, 7, 8, or 9. Clue 1 says B + B is greater than 12, so when B is added to itself, the sum is greater than 12. Since 14 is the least possible sum, 7 is the least possible number that B can represent. That leaves 7, 8, and 9. From Clue 3, you say B when you count by 3s, so 7 and 8 are eliminated. That leaves B = 9.)

WHICH FLOOR?

MAKE

be 9.

Without a doubt, B is 3.

Ms. Slide

Honestly, B is 6.

Professor Pelly

Dr. Tap

Who is wearing a thinking cap?

45

• How do you think Dr. Tap got the answer of 6? (6 fits Clues 2 and 3. Dr. Tap might not have read Clue 1 correctly. If B is 6, then B + B is 12 and 12 is not greater than 12.) Algebra Readiness Made Easy: Grade 3 © Greenes, Findell & Cavanagh, Scholastic Teaching Resources

43

Name _____________________________________________ Date __________________ WHICH FLOOR?

SOLVE THE PROBLEM

What is the giraffe’s floor number?

The giraffe lives in this building.

A

stands for his floor number.

Use the clues to figure out

A.

CLUES: 1) You say A when you count by 2s, starting with 2. 2) 3)

A + A is less than 18. A is greater than 6.

1. What numbers are on Ima’s list?

Ima Thinker

___________________________ 2. Look at Clue 1. Which numbers can you cross off Ima’s list? ___________________________ 3. What is A ? __________ 4. How did you figure out the floor number? _______________ _______________________________________________ 44

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

I’ll start with Clue 2. I’ll make a list of numbers for A .

Name _____________________________________________ Date __________________ WHICH FLOOR?

MAKE THE CASE

What is the pig’s floor number?

The pig lives in this building.

B

stands for her floor number.

Use the clues to figure out

B.

CLUES: 1) 2) 3)

B + B is greater than 12. B is less than 10. You say B when you count by 3s, starting with 3.

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

B can only be 9.

Without a doubt, B is 3.

Ms. Slide

Honestly, B is 6.

Professor Pelly

Dr. Tap

Who is wearing a thinking cap?

45

Name _____________________________________________ Date __________________ WHICH FLOOR?

PROBLEM

1

What is the parrot’s floor number?

The parrot lives in this building.

C

stands for her floor number.

Use the clues to figure out

C.

CLUES: 1) 2) 3)

C is an odd number. C is greater than 1. C + C is less than 8.

1. What numbers are on Ima’s list? __________________________

Ima Thinker

2. Look at Clue 1. What number can you cross off Ima’s list? __________ 3. What is C ? __________ 4. How did you figure out the floor number? _______________ _______________________________________________ 46

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

I’ll start with Clue 3. I’ll make a list of numbers for C .

Name _____________________________________________ Date __________________ WHICH FLOOR?

PROBLEM

2

What is the rabbit’s floor number?

The rabbit lives in this building.

D

stands for his floor number.

Use the clues to figure out

D.

CLUES: 1) D is an odd number. 2) You say D when you count by 5s, starting with 5.

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

3)

D + D is less than 30.

I’ll start with Clue 3. I make a list of numbers for D .

1. What numbers are on Ima’s list?

Ima Thinker ___________________________ 2. Look at Clue 2. Which numbers can you cross off Ima’s list? ___________________________ 3. What is D ? __________ 4. How did you figure out the floor number? _______________ ______________________________________________ 47

Name _____________________________________________ Date __________________ WHICH FLOOR?

PROBLEM

3

What is the dog’s floor number?

The dog lives in this building.

E

stands for his floor number.

Use the clues to figure out

E.

CLUES: 1) 2) 3)

E is an even number. E + E is less than 24. E is a 2-digit number.

1. What numbers are on Ima’s list? __________________________

Ima Thinker

2. Look at Clue 3. Which numbers can you cross off of Ima’s list? __________________________ 3. What is E ? __________ 4. How did you figure out the floor number? _______________ _______________________________________________ 48

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

I’ll start with Clue 2. I’ll make a list of numbers for E .

Name _____________________________________________ Date __________________ WHICH FLOOR?

PROBLEM

4

What is the kangaroo’s floor number?

The kangaroo lives in this building.

F

stands for her floor number.

Use the clues to figure out

F.

CLUES: 1)

F is an odd number.

2) You say the number F when you count by 3s, starting with 3.

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

3)

F has a 1 in the tens place.

1. Which clue will you use first? __________ 2. When you use that clue, what numbers will be on your list? _______________________________________________ 3. What is F ? __________ 4. How did you figure out the kangaroo’s floor number? _______ _______________________________________________

49

Name _____________________________________________ Date __________________ WHICH FLOOR?

PROBLEM

5

What is the gecko’s floor number?

The gecko lives in this building.

G

stands for his floor number.

Use the clues to figure out

G.

CLUES: 1) You say G when you count by 2s, starting with 2. 2) 3)

G + G is less than 18. G is greater than 3 + 3.

2. When you use that clue, what numbers will be on your list? _______________________________________________ 3. What is G ? __________ 4. How did you figure out the gecko’s floor number? _______________________________________________ _______________________________________________

50

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

1. Which clue will you use first? __________

Name _____________________________________________ Date __________________ WHICH FLOOR?

PROBLEM

6

What is the penguin’s floor number?

The penguin lives in this building.

H

stands for her floor number.

Use the clues to figure out

H.

CLUES: 1) 2) 3) 4)

H is greater than 5 + 5. H is even. H is less than 19 – 4. You do not say H when you count

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

by 3s, starting with 3.

1. Which clue will you use first? __________ 2. When you use that clue, what numbers will be on your list? _______________________________________________ 3. What is H ? __________ 4. How did you figure out the penguin’s floor number? _______________________________________________ _______________________________________________

51

Name _____________________________________________ Date __________________ WHICH FLOOR?

PROBLEM

7

What is the monkey’s floor number?

The monkey lives in this building.

J

stands for his floor number.

Use the clues to figure out

J.

CLUES: 1) 2) 3) 4) 5)

J is greater than 13 – 8. J is less than 20 – 6. J is an odd number. J does not have 2 digits. J is not 3 + 4.

2. When you use that clue, what numbers will be on your list? _______________________________________________ 3. What is J ? __________ 4. How did you figure out the monkey’s floor number? _______________________________________________ _______________________________________________

52

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

1. Which clue will you use first? __________

ALGE B

Balancing Blocks

3

SS NE

READ I RA

Overview Shown two pan balances with blocks, children identify how many of one type of block will balance another set of blocks.

Algebra Understand that same blocks have the same value in a given problem • Substitute one set of blocks for a second set of equal weight in order to preserve balance • Reason about proportional relationships

Problem-Solving Strategies Reason deductively

Related Math Skills Add and multiply, basic facts • Recognize spheres, cylinders, cones, and cubes from their two-dimensional drawings

Math Language • Balance • Cone • Cube • Cylinder • Sphere • Weigh the same

Introducing the Problem Set Make photocopies of “Solve the Problem: Balancing Blocks” (page 55) and distribute to children. Have children work in pairs, encouraging them to discuss strategies they might use to solve the problem. You may want to walk around and listen in on some of their discussions. After a few minutes, display the problem on the board (or on the overhead if you made a transparency) and use the following questions to guide a whole-class discussion on how to solve the problem: • Look at the picture. What do you see? (Two pan balances with pans that show equal weights. The first pan balance shows that 3 spheres balance one cube. The second pan balance shows 2 cubes on one pan and a question mark on the other.)

Algebra Readiness Made Easy: Grade 3 © Greenes, Findell & Cavanagh, Scholastic Teaching Resources

53

3

SS NE

ALGE B

BALANCING BLOCKS

READ I RA

Name _____________________________________________ Date __________________ BALANCING BLOCKS

SOLVE THE PROBLEM

• Which weighs more—one sphere or one cube? (Cube) How do you know? (It takes 3 spheres to balance one cube, so the cube weighs three times as much as one sphere.) • What do you need to find out? (How many spheres will balance 2 cubes?)

• If one sphere weighs 2 pounds, how many pounds is one cube? (6 pounds) How do you know? (3 spheres balance one cube, and 2 + 2 + 2 = 6) Work together as a class to answer the questions in “Solve the Problem: Balancing Blocks.”

Blocks that are the same weigh the same.

I know how many spheres balance one cube. I’ll write that number on each cube. Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

• What did you do first? (Write 3 on each cube in the second pan balance.)

How many spheres will balance 2 cubes?

Ima Thinker 1. What number of spheres will Ima write on each cube? __________ 2. How many spheres balance 2 cubes? __________ 3. If one sphere weighs 2 pounds, how many pounds are 2 cubes? __________ 4. How did you figure out the answer to #3?_______________ _______________________________________________ 55

Math Chat With the Transparency Name _____________________________________________ Date __________________

Display the “Make the Case: Balancing Blocks” transparency on the overhead. Before children can decide which character is “wearing a thinking cap,” they need to figure out the answer to the problem. Encourage children to work in pairs to solve the problem, then bring the class together for another whole-class discussion. Ask:

BALANCING BLOCKS

MAKE THE CASE

How many cylinders will balance 6 cubes?

Blocks that are the same weigh the same.

• Who has the right answer? (Ms. Slide) That’s easy. It is 8.

Ms. Slide

• How do you think Professor Pelly got the answer of 8? (He might have added to find the total number of cubes on the two pan balances; 2 + 6 = 8.) • How do you think Dr. Tap got the answer of 12? (He might have multiplied the number of cubes in the two pan balances; 2 x 6 = 12.)

54

I know that it is 3.

Professor Pelly 56

You’re both wrong. I am sure it’s 12.

Dr. Tap

Who is wearing a thinking cap?

Algebra Readiness Made Easy: Grade 3 © Greenes, Findell & Cavanagh, Scholastic Teaching Resources

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

• How did you figure it out? (Two cubes balance one cylinder. Since 6 cubes is 3 times 2 cubes, it will take 3 x 1, or 3 cylinders to balance 6 cubes.)

Name _____________________________________________ Date __________________ BALANCING BLOCKS

SOLVE THE PROBLEM

How many spheres will balance 2 cubes?

Blocks that are the same weigh the same.

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

I know how many spheres balance one cube. I’ll write that number on each cube.

Ima Thinker 1. What number of spheres will Ima write on each cube? __________ 2. How many spheres balance 2 cubes? __________ 3. If one sphere weighs 2 pounds, how many pounds are 2 cubes? __________ 4. How did you figure out the answer to #3?_______________ _______________________________________________ 55

Name _____________________________________________ Date __________________ BALANCING BLOCKS

MAKE THE CASE

How many cylinders will balance 6 cubes?

Blocks that are the same weigh the same.

That’s easy. It is 8.

Ms. Slide

Professor Pelly 56

You’re both wrong. I am sure it’s 12.

Dr. Tap

Who is wearing a thinking cap?

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

I know that it is 3.

Name _____________________________________________ Date __________________ BALANCING BLOCKS

PROBLEM

1

How many cylinders will balance 2 spheres?

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

Blocks that are the same weigh the same.

I know how many cylinders balance one sphere. I’ll write that number on each sphere.

Ima Thinker 1. What number of cylinders will Ima write on each sphere? ________ 2. How many cylinders balance 2 spheres? __________ 3. If one cylinder weighs 3 pounds, how many pounds are 2 spheres? __________ 4. How did you figure out the answer to #3? ______________ _______________________________________________ 57

Name _____________________________________________ Date __________________ BALANCING BLOCKS

PROBLEM

2

How many cubes will balance 3 cylinders?

Blocks that are the same weigh the same.

Ima Thinker 1. What number of cubes will Ima write on each cylinder? ________ 2. How many cubes will balance 3 cylinders? __________ 3. If one cube weighs 5 pounds, how many pounds are 3 cylinders? __________ 4. How did you figure out the answer to #3? ______________ _______________________________________________ 58

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

I know how many cubes balance one cylinder. I’ll write that number on each cylinder.

Name _____________________________________________ Date __________________ BALANCING BLOCKS

PROBLEM

3

How many spheres will balance 3 cubes?

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

Blocks that are the same weigh the same.

I know how many spheres balance one cube. I’ll write that number on each cube.

Ima Thinker 1. What number of spheres will Ima write on each cube? __________ 2. How many spheres balance 3 cubes? __________ 3. If one sphere weighs 6 pounds, what is the weight of 3 cubes? __________ 4. How did you figure out the answer to #3? ______________ _______________________________________________ 59

Name _____________________________________________ Date __________________ PROBLEM

4

BALANCING BLOCKS

How many cubes will balance 4 cones?

Blocks that are the same weigh the same.

2. If one cube weighs 2 pounds, what is the weight of one cone? __________ 3. If one cube weighs 2 pounds, what is the weight of 8 cones? __________ 4. How did you figure out the answer to #3? ______________ _______________________________________________

60

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

1. How many cubes will balance 4 cones? __________

Name _____________________________________________ Date __________________ PROBLEM

5

BALANCING BLOCKS

How many cubes will balance 3 cones?

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

Blocks that are the same weigh the same.

1. How many cubes will balance 3 cones? __________ 2. If one cube weighs 4 pounds, what is the weight of one cone? __________ 3. If one cube weighs 4 pounds, what is the weight of 3 cones? __________ 4. How did you figure out the answer to #3? _______________________________________________

61

Name _____________________________________________ Date __________________ PROBLEM

6

BALANCING BLOCKS

How many cubes will balance 8 spheres?

Blocks that are the same weigh the same.

1. How many cubes will balance 8 spheres? __________

3. If one cube weighs 4 pounds, what is the weight of 8 spheres? __________ 4. How did you figure out the answer to #3? _______________ _______________________________________________

62

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

2. If one cube weighs 4 pounds, what is the weight of one sphere? __________

Name _____________________________________________ Date __________________ PROBLEM

7

BALANCING BLOCKS

How many spheres will balance 6 cones?

Blocks that are the same weigh the same.

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

1. How many spheres will balance 6 cones? __________ 2. If one sphere weighs 6 pounds, what is the weight of one cone? __________ 3. If one sphere weighs 6 pounds, what is the weight of 6 cones? __________ 4. How did you figure out the answer to #3? _______________ _______________________________________________

63

ALGE B

3

SS NE

READ I RA

Favorite Numbers Overview

Presented with an array of consecutive numbers, children identify the relationship between the last number in a row and the row number.

Algebra Identify patterns in rows of numbers and continue the pattern • Explore variables as representing varying quantities • Identify the functional relationship between the last number in a row and the row number

Problem-Solving Strategies Generalize patterns

Related Math Skills Add and subtract with whole numbers • Multiply and divide, basic facts • Apply inverse operations

Math Language Last number • Pattern • Row

Introducing the Problem Set Make photocopies of “Solve the Problem: Favorite Numbers” (page 66) and distribute to children. Have children work in pairs, encouraging them to discuss strategies they might use to solve the problem. You may want to walk around and listen in on some of their discussions. After a few minutes, display the problem on the board (or on the overhead if you made a transparency) and use the following questions to guide a wholeclass discussion on how to solve the problem: • What is the last number in Row 3? (6) in Row 4? (8) • What pattern does Ima see? (The last number in each row is equal to the row number plus the row number, or 2 times the row number.) 64

Algebra Readiness Made Easy: Grade 3 © Greenes, Findell & Cavanagh, Scholastic Teaching Resources

FAVORITE NUMBERS

Name _____________________________________________ Date __________________ F A VO RI TE NU M B ERS

SO L V E TH E P R O BL E M

What is my favorite number?

• What numbers are in Row 5? (1 through 10) My favorite number is the last number in Row 8.

• What is the last number in Row 5? (5 + 5 or 2 x 5 or 10)

Baxter Row Row Row Row

• What is the last number in Row 8? (8 + 8 or 2 x 8 or 16)

1: 2: 3: 4:

1 1 1 1

Kosmo 2 2 2 2

3 3 3

4 4 4

5 5

Nibble

6 6

7

8

This pattern of numbers continues.

Work together as a class to answer the questions in “Solve the Problem: Favorite Numbers.”

I see a pattern. Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

The last number in Row 1 is 1 + 1 or 2. The last number in Row 2 is 2 + 2 or 4. The last number in Row 3 is 3 + 3 or 6.

Math Chat With the Transparency Display the “Make the Case: Favorite Numbers” transparency on the overhead. Before children can decide which character is “wearing a thinking cap,” they need to figure out the answer to the problem. Encourage children to work in pairs to solve the problem. Then bring the class together for another whole-class discussion. Ask:

1. What numbers are in Row 8? _________________________________

Ima Thinker

2. Nibble’s favorite number is _______ . 3. Baxter’s favorite number is the last number in Row 10. Baxter’s favorite number is _______ . 4. Kosmo’s favorite number is 4 more than Baxter’s number. Kosmo’s favorite number is _______ . 66

Name _____________________________________________ Date __________________

• Who has the right answer? (Ms. Slide)

THE C AS E

• How did you figure it out? (The last number in a row is equal to the row number [R] plus 2. The last number in Row 10 is 10 + 2, or 12.)

Can you figure out my number? My favorite number is the last number in Row 10.

• How do you think Professor Pelly got the answer of 10? (He might have forgotten to add 2 to the row number.)

Ang

Row Row Row Row

1: 2: 3: 4:

1 1 1 1

2 2 2 2

3 3 3 3

4 4 4

5 5

6

This pattern of numbers continues.

That’s easy. Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

• How do you think Dr. Tap got the answer of 11? (He might have added 1 instead of 2 to the row number.)

F A VO RI TE NU M B ERS

MA KE

Ang’s favorite number is 12.

I’m sure that Ang’s favorite number is 10.

Ms. Slide

Ang’s favorite number has to be 11.

Professor Pelly

Dr. Tap

Who is wearing a thinking cap?

Algebra Readiness Made Easy: Grade 3 © Greenes, Findell & Cavanagh, Scholastic Teaching Resources

67

65

Name _____________________________________________ Date __________________ FAVORITE NUMBERS

SOLVE THE PROBLEM

What is my favorite number? My favorite number is the last number in Row 8. Baxter

Row Row Row Row

1: 2: 3: 4:

1 1 1 1

Kosmo 2 2 2 2

3 3 3

4 4 4

5 5

6 6

Nibble 7

8

This pattern of numbers continues.

1. What numbers are in Row 8? _________________________________

Ima Thinker

2. Nibble’s favorite number is _______ . 3. Baxter’s favorite number is the last number in Row 10. Baxter’s favorite number is _______ . 4. Kosmo’s favorite number is 4 more than Baxter’s number. Kosmo’s favorite number is _______ . 66

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

I see a pattern. The last number in Row 1 is 1 + 1 or 2. The last number in Row 2 is 2 + 2 or 4. The last number in Row 3 is 3 + 3 or 6.

Name _____________________________________________ Date __________________ FAVORITE NUMBERS

MAKE THE CASE

Can you figure out my number? My favorite number is the last number in Row 10.

Ang

Row Row Row Row

1: 2: 3: 4:

1 1 1 1

2 2 2 2

3 3 3 3

4 4 4

5 5

6

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

This pattern of numbers continues.

That’s easy. Ang’s favorite number is 12.

I’m sure that Ang’s favorite number is 10.

Ms. Slide

Ang’s favorite number has to be 11.

Professor Pelly

Dr. Tap

Who is wearing a thinking cap?

67

Name _____________________________________________ Date __________________ FAVORITE NUMBERS

PROBLEM

1

What is my favorite number? My favorite number is the last number in Row 10.

Thunder

Zula Row Row Row Row

1: 2: 3: 4:

1 1 1 1

2 2 2

3 3

4

Mamba

This pattern of numbers continues.

1. What numbers are in Row 10? _______________________________

Ima Thinker

2. Mamba’s favorite number is _______ . 3. Zula’s favorite number is the last number in Row 25. Zula’s favorite number is _______ . 4. Thunder’s favorite number is 10 less than Zula’s number. Thunder’s favorite number is _______ . 68

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

I see a pattern. The last number in Row 1 is 1. The last number in Row 2 is 2. The last number in Row 3 is 3.

Name _____________________________________________ Date __________________ FAVORITE NUMBERS

PROBLEM

2

What is my favorite number? My favorite number is the last number in Row 10. Echo

Spec Row Row Row Row

1: 2: 3: 4:

1 1 1 1

2 2 2 2

3 3 3

4 4

5

Hush

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

This pattern of numbers continues.

I see a pattern. The last number in Row 1 is 1 + 1 or 2. The last number in Row 2 is 2 + 1 or 3. The last number in Row 3 is 3 + 1 or 4.

1. What numbers are in Row 10? _______________________________

Ima Thinker

2. Hush’s favorite number is _______ . 3. Spec’s favorite number is the last number in Row 14. Spec’s favorite number is _______ . 4. Echo’s favorite number is 10 more than Spec’s number. Echo’s favorite number is _______ . 69

Name _____________________________________________ Date __________________ FAVORITE NUMBERS

PROBLEM

3

What is my favorite number? My favorite number is the last number in Row 7. Salty

Splash Row Row Row Row

1: 2: 3: 4:

1 1 1 1

2 2 2 2

3 3 3 3

4 4 4 4

5 5 5

6 6

Flipper 7

This pattern of numbers continues.

1. What numbers are in Row 7? __________________________________

Ima Thinker

2. Flipper’s favorite number is _______ . 3. Splash’s favorite number is the last number in Row 10. Splash’s favorite number is _______ . 4. Salty’s favorite number is 6 less than Splash’s number. Salty’s favorite number is _______ . 70

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

I see a pattern. The last number in Row 1 is 1 + 3 or 4. The last number in Row 2 is 2 + 3 or 5. The last number in Row 3 is 3 + 3 or 6.

Name _____________________________________________ Date __________________ FAVORITE NUMBERS

PROBLEM

4

What is my favorite number? My favorite number is the last number in Row 8.

Snuffle Row Row Row Row

1: 2: 3: 4:

1 1 1 1

Spice 2 2 2 2

3 3 3 3

4 4 4 4

5 5 5 5

6 6 6 6

7 7 7

8 8

9

Prairie

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

This pattern of numbers continues.

1. What numbers are in Row 8? ________________________ 2. Prairie’s favorite number is _______ . 3. Snuffle’s favorite number is the last number in Row 10. Snuffle’s favorite number is _______ . 4. Spice’s favorite number is 20 more than Snuffle’s number. Spice’s favorite number is _______ .

71

Name _____________________________________________ Date __________________ FAVORITE NUMBERS

PROBLEM

5

What is my favorite number?

Swizzle

My favorite number is the

Guda

Row Row Row Row

1: 2: 3: 4:

1 1 1 1

Hiss

2 2 2 2

3 3 3 3

4 4 4 4

last number in Row 7. 5 5 5 5

6 6 6

7 7

8

This pattern of numbers continues.

Lanka

2. Swizzle’s favorite number is the last number in Row 10. Swizzle’s favorite number is _______ . 3. Guda’s favorite number is 15. It is the last number in Row _______ . 4. Hiss’s favorite number is 20. It is the last number in Row _______ .

72

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

1. Lanka’s favorite number is _______ .

Name _____________________________________________ Date __________________ FAVORITE NUMBERS

PROBLEM

6

What is my favorite number? My favorite number is the last number in Row 9.

Zippy

Savanna Nomad Row Row Row Row

1: 2: 3: 4:

1 1 1 1

2 2 2 2

3 3 3 3

4 4 4

5 5 5

6 6

7 7

8

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

This pattern of numbers continues.

9 Stripe

1. Stripe’s favorite number is _______ . 2. Savanna’s favorite number is the last number in Row 10. Savanna’s favorite number is _______ . 3. Nomad’s favorite number is 15. It is the last number in Row _______ . 4. Zippy’s favorite number is 41. It is the last number in Row _______ .

73

Name _____________________________________________ Date __________________ FAVORITE NUMBERS

PROBLEM

7

What is my favorite number? My favorite number is the last number in Row 7. Rambo

Row Row Row Row

1: 2: 3: 4:

1 1 1 1

Beauty

Caboodle

Hulk 2 2 2 2

3 3 3 3

4 4 4

5 5 5

6 6 6

7 7

8 8

9 9

10

11

1. Beauty’s favorite number is _______ . 2. Rambo’s favorite number is the last number in Row 10. Rambo’s favorite number is _______ . 3. Caboodle’s favorite number is 60. It is the last number in Row _______ . 4. Hulk’s favorite number is 90. It is the last number in Row _______ .

74

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

This pattern of numbers continues.

12

PROBLEM-SOLVING TRANSPARENCY

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

SOLVE IT

1.

Look

2.

Plan and Do

3.

Answer and Check

What is the problem?

What will you do first? How will you solve the problem?

How can you be sure your answer is correct?

SOLVE IT: FOX FACTS

How long is each fox?

The Twins

Roxy

Paint

FACTS: A

Roxy is 6 inches longer than each twin.

B

Each twin is 11 inches shorter than Paint.

C

Paint is 25 inches long.

Use the facts. Figure out the length of each fox.

SOLVE IT: HIDDEN NUMBERS

What number is under each star?

+ 4 +

+ 8 = 30

The two stars are covering the same number.

76 Algebra Readiness Made Easy: Grade 3 © Greenes, Findell & Cavanagh, Scholastic Teaching Resources

SOLVE IT: MINI MOVERS

How many pounds is one toy jeep? 21 pounds

SCALE A

18 pounds

SCALE B

SOLVE IT: WHICH FLOOR?

What is the turtle’s floor number? The turtle lives in this building.

K stands for her floor number. Use the clues to figure out K . CLUES: 1) 2)

K + K + K is less than 30. You say K when you count by 3s, starting with 3.

3) 4)

K is an even number. K is greater than 9 – 6.

77 Algebra Readiness Made Easy: Grade 3 © Greenes, Findell & Cavanagh, Scholastic Teaching Resources

SOLVE IT: BALANCING BLOCKS

How many spheres will balance 9 cylinders?

Blocks that are the same weigh the same.

SOLVE IT: FAVORITE NUMBERS

Where is my favorite number? Bazooka’s favorite number is 26. It is the last number in Row _____. Row Row Row Row

1: 2: 3: 4:

1 1 1 1

2 2 2 2

3 3 3 3

4 4 4 4

5 5 5 5

6 6 6 6

7 7 7 7

8 8 8

9 9

10

This pattern of numbers continues.

78 Algebra Readiness Made Easy: Grade 3 © Greenes, Findell & Cavanagh, Scholastic Teaching Resources

ANSWER KEY Fox Facts (pages 11–19) Solve the Problem 1. Fact C 2. 7 years old 3. 3 years old 4. Red Make the Case Who is wearing a thinking cap? Professor Pelly Problem 1 1. Fact C 2. 3 months old 3. 12 months or 1 year old 4. Work backward. Fact C: Daisy is 5 months old. Fact B: The twins are each 5 – 2, or 3 months old. Fact A: Skipper is 3 + 9, or 12 months old. Problem 2 1. Fact C 2. 14 pounds 3. 12 pounds 4. Work backward. Fact C: Screech weighs 8 pounds. Fact B: Snowball weighs 8 + 6, or 14 pounds. Fact A: Red weighs 14 – 2, or 12 pounds. Problem 3 1. Fact C 2. 3 pounds 3. 15 pounds 4. Work backward. Fact C: Roxy weighs 7 pounds. Fact B: Each twin weighs 7 – 4, or 3 pounds. Fact A: Paint weighs 3 + 12, or 15 pounds. Problem 4 1. Fact C 2. 10 pounds 3. 9 pounds 4. Work backward. Fact C: Snooper weighs 2 pounds. Fact B: Howler weighs 2 + 8, or 10 pounds. Leap weighs 10 – 1, or 9 pounds. Problem 5 1. Fact C 2. 20 inches 3. 14 inches 4. Work backward. Fact C: Skipper’s body is 18 inches long. Fact B: Snowball’s body is 18 + 2, or 20 inches long. Fact A: Snooper’s body is 20 – 6, or 14 inches long. Problem 6 1. Fact C 2. 16 inches 3. 19 inches 4. Work backward. Fact C: Red’s body is 25 inches long. Fact B: Daisy’s body is 25 – 9, or 16 inches long. Fact A: Leap’s body is 16 + 3, or 19 inches long.

Problem 7 1. Fact C 2. 24 inches 3. 15 inches 4. Work backward. Fact C: Howler’s body is 17 inches long. Fact B: Speedy’s body is 17 + 7, or 24 inches long. Fact A: Screech’s body is 24 – 9, or 15 inches long. Solve It: Fox Facts 1. Look: Three facts are given about the body lengths of four foxes. To figure out the length of each fox, you have to use the facts. 2. Plan and Do: Work backward. Fact C: Paint’s body is 25 inches long. Fact B: Each twin’s body is 25 – 11, or 14 inches long. Fact A: Roxy’s body is 14 + 6, or 20 inches long. 3. Answer and Check: Roxy’s body is 20 inches long. Each twin’s body is 14 inches long. Paint’s body is 25 inches long. To check, test the lengths with the facts. Fact C: Paint is 25 inches long. Fact B: Each twin is 14 inches long and 25 – 11 = 14. Fact A: Roxy is 20 inches long and 14 + 6 = 20. Hidden Numbers (pages 22–30) Solve the Problem 1. Ima took the 3 away and subtracted it from the sum. The number left is the sum of the numbers under the 2 flags. 2. 13 – 3 = 10 3. 5 4. Take the 3 away and change the sum to 13 – 3, or 10. Then 2 flags are 10. That means that the number under each flag is 5; 5 + 5 = 10, or 10 ÷ 2 = 5. Make the Case Who is wearing a thinking cap? Professor Pelly Problem 1 1. Ima took the 4 and the 5 away and subtracted them from the sum. The number left is under one smile. 2. 15 – 4 – 5 = 6 3. 6 4. Take the 4 and the 5 away. Subtract 4 + 5, or 9 from the sum; 15 – 9 = 6. Problem 2 1. Ima took the 7 away and subtracted it from the sum. The number left is the sum of the numbers under the 2 dinosaurs. 2. 8 3. Take the 7 away and subtract 7 from the sum; 23 – 7 = 16. Since 2 dinosaurs are 16, one is 16 ÷ 2, or 8. 4. 13

Problem 3 1. Ima took away the 4 and the 1 and subtracted it from the sum. The number left is the sum of the two numbers under the astronauts. 2. 5 3. Take the 4 and the 1 away and subtract 4 + 1, or 5 from the sum; 15 – 5 = 10. Since 2 astronauts are 10, one is 10 ÷ 2, or 5. 4. 11 Problem 4 1. 3 2. Take the 3 and the 5 away and subtract 3 + 5, or 8 from the sum; 14 – 8 = 6. Since 2 koalas are 6, one is 6 ÷ 2, or 3. 3. 11 4. 9 Problem 5 1. 9 2. Take the 4 and the 5 away and subtract 4 + 5, or 9 from the sum; 27 – 9 = 18. Since 2 turtles are 18, one is 18 ÷ 2, or 9. 3. 25 4. 4 Problem 6 1. 4 2. Take the 10 away and subtract 10 from the sum; 22 – 10 = 12. Since 3 dolphins are 12, one is 12 ÷ 3, or 4. 3. 31 4. 42 Problem 7 1. 10 2. Take the 1 away and subtract 1 from the sum; 31 – 1 = 30. Since 3 zebras are 30, one is 30 ÷ 3, or 10. 3. 28 4. 6 Solve It: Hidden Numbers 1. Look: There is an addition sentence with 2 stars and 2 numbers. The sum is 30. The problem is to figure out the number under each star. 2. Plan and Do: Take the 4 and the 8 away and subtract 4 + 8, or 12 from the sum; 30 – 12 = 18. Since 2 stars are 18, each star is 18 ÷ 2, or 9. 3. Answer and Check: 9. To check, write 9 on each star and add the four numbers. The sum of 30 checks with the given sum. Mini Movers (pages 33–41) Solve the Problem 1. 4 3. 3 2. 7 4. 14 Make the Case Who is wearing a thinking cap? Dr. Tap

Problem 2 1. 6 3. 3 2. 9 4. 18 Problem 3 1. 10 3. 4 2. 6 4. 32 Problem 4 1. 8 2. 12 3. On scale B, 2 toy ambulances weigh 16 pounds, so one weighs 16 ÷ 2, or 8 pounds. On scale A, since one ambulance is 8 pounds, the toy tow truck is 20 – 8, or 12 pounds. 4. 40 Problem 5 1. 7 2. 20 3. On scale B, 2 toy fire trucks weigh 14 pounds, so one weighs 14 ÷ 2, or 7 pounds. On scale A, since the fire truck is 7 pounds, the toy double-decker bus is 27 – 7, or 20 pounds. 4. 54 Problem 6 1. 5 2. 4 3. On scale A, 3 snow plows are 15 pounds. One snow plow is 5 pounds. On scale B, since 3 snowplows are 15 pounds, the oil truck is 19 – 15, or 4 pounds. 4. 27 Problem 7 1. 10 2. 3 3. On scale A, 3 toy street sweepers are 30 pounds. On scale B, since 3 street sweepers are 30 pounds, the 2 toy motorcycles are 36 – 30, or 6 pounds, and one motorcycle is half of 6, or 3 pounds. 4. 39 Solve It: Mini Movers 1. Look: On scale A, 2 toy taxis and 3 toy jeeps are 21 pounds. On scale B, 3 toy taxis are 18 pounds. The problem is to figure out the weight of one toy jeep. 2. Plan and Do: On scale B, since 3 toy taxis are 18 pounds, one taxi is 18 ÷ 3, or 6 pounds. On scale A, the 2 toy taxis are 2 x 6, or 12 pounds. So, the 3 toy jeeps are 21 – 12, or 9 pounds, and each is 9 ÷ 3, or 3 pounds. 3. Answer and Check: A toy jeep is 3 pounds. To check, replace each toy jeep on the scales with 3 pounds and each taxi with 6 pounds, and figure out the sum of the weights on each scale. The sums should match the number of pounds shown by the scales.

Problem 1 1. 3 3. 3 2. 6 4. 12

Algebra Readiness Made Easy: Grade 3 © Greenes, Findell & Cavanagh, Scholastic Teaching Resources

79

Which Floor? (pages 44–52) Solve the Problem 1. 8, 7, 6, 5, 4, 3, 2, and 1 2. 1, 3, 5, and 7 3. A = 8 4. Clue 2: List the numbers: 8, 7, 6, 5, 4, 3, 2, and 1. Clue 1: Eliminate 1, 3, 5, and 7, leaving 2, 4, 6 and 8. Clue 3: Eliminate 2, 4, and 6, leaving 8. Make the Case Who is wearing a thinking cap? Ms. Slide *NOTE: Clues may be used in different orders to solve the problems. One order is shown for each problem. Problem 1 1. 3, 2, 1 2. 2 3. 3 4. Clue 3: List the numbers 3, 2, 1. Clue 2: Eliminate 1, leaving 2 and 3. Clue 1: Eliminate 2, leaving 3. Problem 2 1. 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1 2. Eliminate all numbers except for 10 and 5 3. 5 4. Clue 3: List the numbers 14, 13, . . . , 1. Clue 2: Eliminate all numbers except for 5 and 10. Clue 1: Eliminate 10, leaving 5. Problem 3 1. 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1 2. 1 through 9 3. 10 4. Clue 2: List the numbers 11, 10, . . . , 1. Clue 3: Eliminate 1 through 9. Clue 1: Eliminate 11, leaving 10. Problem 4 1. Clue 3 2. 10, 11, 12, 13, 14, 15 3. 15 4. Clue 3: List the numbers 10 through 15. Clue 2: Eliminate all numbers except for 12 and 15. Clue 1: Eliminate 12, leaving 15. Problem 5 1. Clue 2 2. 8, 7, 6, 5, 4, 3, 2, 1 3. 8 4. Clue 2: List the numbers 8, 7, . . . , 1. Clue 1: Eliminate all odd numbers, leaving 2, 4, 6, and 8. Clue 3: Eliminate 2, 4, and 6, leaving 8.

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Problem 6 1. Clue 3 2. 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1 3. 14 4. Clue 3: List the numbers 14, 13, . . . , 1. Clue 1: Eliminate 1 through 10, leaving 11, 12, 13, and 14. Clue 2: Eliminate 11 and 13, leaving 12 and 14. Clue 4: Eliminate 12, leaving 14. Problem 7 1. Clue 2 2. 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1 3. 9 4. Clue 2: List the numbers 13, 12, . . . , 1. Clue 1: Eliminate numbers 1 through 5, leaving 6 through 13. Clue 3: Eliminate even numbers, leaving 7, 9, 11, and 13. Clue 4: Eliminate 11 and 13, leaving 7 and 9. Clue 5: Eliminate 7, leaving 9. Solve It: Which Floor? 1. Look: K stands for the floor number. There are 4 clues to use to figure out the number. 2. Plan and Do: From Clue 1, K is 9, 8, . . . , 1. Clue 4 eliminates 1, 2, and 3, leaving 4 through 9. Clue 2 eliminates all numbers except for 6 and 9. Clue 3 eliminates 9, leaving 6. 3. Answer and Check: K = 6. To check, replace K in each clue with 6. Be sure the clues make sense. Clue 1: 6 + 6 + 6, or 18 is less than 30. Clue 2: You say 6 when you count by 3’s (3, 6, 9, . . .). Clue 3: 6 is an even number. Clue 4: 6 is greater than 9 – 6, or 3. Balancing Blocks (pages 55–63) Solve the Problem 1. 3 2. 6 spheres 3. 12 pounds 4. Three spheres balance one cube. If each sphere weighs 2 pounds, then 3 spheres weigh 3 x 2, or 6 pounds, and 1 cube weighs 6 pounds. Then 2 cubes weigh 2 x 6, or 12 pounds. Make the Case Who is wearing a thinking cap? Ms. Slide Problem 1 1. 2 2. 4 cylinders 3. 12 pounds 4. One cylinder is 3 pounds, so 2 cylinders are 2 x 3, or 6 pounds. That means that one sphere is 6 pounds. Then 2 spheres are 2 x 6, or 12 pounds. Problem 2 1. 3 2. 9 cubes 3. 45 pounds

4. One cube is 5 pounds, so 3 cubes are 3 x 5, or 15 pounds. That means that one cylinder is 15 pounds. Then 3 cylinders are 15 + 15 + 15, or 45 pounds. Problem 3 1. 2 2. 6 spheres 3. 36 pounds 4. One sphere is 6 pounds, so 2 spheres are 2 x 6, or 12 pounds. That means that one cube is 12 pounds. Then 3 cubes are 3 x 12, or 36 pounds. Problem 4 1. 12 cubes 2. 6 pounds 3. 48 pounds 4. One cube is 2 pounds, so 3 cubes are 3 x 2, or 6 pounds. That means that one cone is 6 pounds. Then 8 cones are 8 x 6, or 48 pounds. Problem 5 1. 12 cubes 2. 16 pounds 3. 48 pounds 4. One cube is 4 pounds, so 4 cubes are 4 x 4, or 16 pounds. That means that one cone is 16 pounds. Then 3 cones are 16 + 16 + 16, or 48 pounds. Problem 6 1. 4 cubes 2. 2 pounds 3. 16 pounds 4. One cube is 4 pounds, so 2 spheres are 4 pounds. That means that one sphere is 2 pounds. Then 8 spheres are 8 x 2, or 16 pounds. Problem 7 1. 2 spheres 2. 2 pounds 3. 12 pounds 4. One sphere is 6 pounds, so 3 cones are 6 pounds. Then 2 x 3, or 6 cones are 2 x 6, or 12 pounds. Solve It: Balancing Blocks 1. Look: On the first pan balance, one sphere balances 3 cylinders. On the second pan balance, one pan is empty and there are 9 cylinders in the other pan. The problem is to figure out the number of spheres that balance 9 cylinders. 2. Plan and Do: On the second pan balance, draw a ring around every set of 3 cylinders. Since one sphere balances 3 cylinders, 3 x 1, or 3 spheres will balance the 3 x 3, or 9 cylinders. 3. Answer and Check: 3 spheres. To check, assign numbers of pounds to the sphere and the cylinders that make sense in the first pan balance. It the cylinder is 1 pound, then a sphere is 3 pounds. In the second pan balance, the weight of the 9 cylinders

is 9 pounds. So 3 spheres weighing 3 x 3, or 9 pounds, will balance the cylinders. Favorite Numbers (pages 66–74) Solve the Problem 1. 1 through 16 3. 20 2. 16 4. 24 Make the Case Who is wearing a thinking cap? Ms. Slide Problem 1 1. 1 through 10 2. 10

3. 25 4. 15

Problem 2 1. 1 through 11 2. 11

3. 15 4. 25

Problem 3 1. 1 through 10 2. 10

3. 13 4. 7

Problem 4 1. 1 through 13 2. 13

3. 15 4. 35

Problem 5 1. 11 3. 11 2. 14 4. 16 Problem 6 1. 19 3. 7 2. 21 4. 20 Problem 7 1. 21 3. 20 2. 30 4. 30 Solve It: Favorite Numbers 1. Look: Rows of numbers starting with 1. The last number in each row is equal to 6 more than the row number (R), or R + 6. Bazooka’s favorite number is 26. The problem is to figure out the row with 26 as the last number. 2. Plan and Do: Figure out the row number with 26 as the last number. Since the last number in the row is 26, R + 6 = 26. So, R is 26 – 6, or 20. 3. Answer and Check: The answer is Row 20. To check, use a different method. The last number in Row 1 is 7. The difference between 7 and 26 is 19. Since each row has one more number than the row before it, the number 26 is in the 19th row below Row 1. Since 1 + 19 = 20, the number 26 is the last number in Row 20.

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How old is each fox? Use the facts. Figure out the ages.

Screech

Roxy

Howler

FACTS: A

Roxy is 10 years older than Screech.

B

Screech is 8 years younger than Howler.

C

Howler is 10 years old.

Roxy is 10 years old. Screech is 2 years old. Howler is 10 years old.

Roxy is 12 years old. Screech is 2 years old. Howler is 10 years old.

Ms. Slide

Professor Pelly

Roxy is 10 years old. Screech is 8 years old. Howler is 10 years old.

Dr. Tap

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What number is under each bird?

+

+

2

= 16

The two birds are covering the same number. The sum of the three numbers is 16.

It is 8. I’m sure of it.

This one is easy. It’s 7.

Ms. Slide

Professor Pelly

You are both wrong. It has to be 14.

Dr. Tap

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How much does the cement mixer weigh? 21 pounds

18 pounds

SCALE A

SCALE B

I believe the cement mixer weighs 3 pounds.

The scale shows that it is 21 pounds.

Ms. Slide

Professor Pelly

You’re both wrong. The cement mixer weighs 12 pounds.

Dr. Tap

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What is the pig’s floor number?

The pig lives in this building.

B stands for her floor number. Use the clues to figure out B . CLUES: 1) 2) 3)

B + B is greater than 12. B is less than 10. You say B when you count by 3s, starting with 3.

B can only

Without a doubt, B is 3.

be 9.

Ms. Slide

Honestly, B is 6.

Professor Pelly

Dr. Tap

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How many cylinders will balance 6 cubes?

Blocks that are the same weigh the same.

I know that it is 3.

That’s easy. It is 8.

Ms. Slide

Professor Pelly

You’re both wrong. I am sure it’s 12.

Dr. Tap

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Can you figure out my number? My favorite number is the last number in Row 10.

Row 1: Row 2:

1 1

2 2

3 3

4

Row 3: Row 4:

1 1

2 2

3 3

4 4

5 5

6

Ang

This pattern of numbers continues.

That’s easy. Ang’s favorite number is 12.

I’m sure that Ang’s favorite number is 10.

Ms. Slide

Professor Pelly

Ang’s favorite number has to be 11.

Dr. Tap

Who is wearing a thinking cap? Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

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PROBLEM - SOLVING TRANSPARENCY

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3

1.

Look

2.

Plan and Do

3.

Answer and Check

What is the problem?

What will you do first? How will you solve the problem?

How can you be sure your answer is correct?

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TEACHING TRANSPARENCY

SOLVE IT: FOX FACTS

How long is each fox?

Roxy

The Twins

Paint

FACTS: A

Roxy is 6 inches longer than each twin.

B

Each twin is 11 inches shorter than Paint.

C

Paint is 25 inches long.

Use the facts. Figure out the length of each fox.

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

SOLVE IT: HIDDEN NUMBERS

What number is under each star?

+ 4 +

+ 8 = 30

The two stars are covering the same number.

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SOLVE IT: MINI MOVERS

How many pounds is one toy jeep? 21 pounds

18 pounds

SCALE A

SCALE B

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

SOLVE IT: WHICH FLOOR?

What is the turtle’s floor number? The turtle lives in this building. K stands for her floor number. Use the clues to figure out K . CLUES: 1) 2)

K + K + K is less than 30. You say K when you count by 3s, starting with 3.

3) 4)

K is an even number. K is greater than 9 – 6.

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SOLVE IT: BALANCING BLOCKS

How many spheres will balance 9 cylinders?

Blocks that are the same weigh the same.

Algebra Readiness Made Easy: Gr. 3 © 2008 by Greenes, Findell & Cavanagh, Scholastic Teaching Resources

SOLVE IT: FAVORITE NUMBERS

Where is my favorite number? Bazooka’s favorite number is 26. It is the last number in Row _____. Row Row Row Row

1: 2: 3: 4:

1 1 1 1

2 2 2 2

3 3 3 3

4 4 4 4

5 5 5 5

6 6 6 6

7 7 7 7

8 8 8

9 9

10

This pattern of numbers continues.

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