California Math Triumphs VOL 2A FRACTIONS (CALIFORNIA MATH TRIUMPHS VOL 2A)

Authors Basich Whitney • Brown • Dawson • Gonsalves • Silbey • Vielhaber

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Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without prior permission of the publisher. Send all inquiries to: Glencoe/McGraw-Hill 8787 Orion Place Columbus, OH 43240-4027 ISBN: 978-0-07-878205-3 MHID: 0-07-878205-8 Printed in the United States of America. 1 2 3 4 5 6 7 8 9 10 055/027 16 15 14 13 12 11 10 09 08 07

California Math Triumphs Volume 2A

California Math Triumphs Volume 1 Place Value and Basic Number Skills 1A Chapter 1 Counting 1A Chapter 2 Place Value 1A Chapter 3 Addition and Subtraction 1B Chapter 4 Multiplication 1B Chapter 5 Division 1B Chapter 6 Integers Volume 2 Fractions and Decimals 2A Chapter 1 Parts of a Whole 2A Chapter 2 Equivalence of Fractions 2B Chapter 3 Operations with Fractions 2B Chapter 4 Positive and Negative Fractions and Decimals

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Volume 3 Ratios, Rates, and Percents 3A Chapter 1 Ratios and Rates 3A Chapter 2 Percents, Fractions, and Decimals 3B Chapter 3 Using Percents 3B Chapter 4 Rates and Proportional Reasoning Volume 4 The Core Processes of Mathematics 4A Chapter 1 Operations and Equality 4A Chapter 2 Math Fundamentals 4B Chapter 3 Math Expressions 4B Chapter 4 Linear Equations 4B Chapter 5 Inequalities Volume 5 Functions and Equations 5A Chapter 1 Patterns and Relationships 5A Chapter 2 Graphing 5B Chapter 3 Proportional Relationships 5B Chapter 4 The Relationship Between Graphs and Functions Volume 6 Measurement 6A Chapter 1 How Measurements Are Made 6A Chapter 2 Length and Area in the Real World 6B Chapter 3 Exact Measures in Geometry 6B Chapter 4 Angles and Circles iii

Authors and Consultants AUTHORS

Frances Basich Whitney

Kathleen M. Brown

Dixie Dawson

Project Director, Mathematics K–12 Santa Cruz County Office of Education Capitola, California

Math Curriculum Staff Developer Washington Middle School Long Beach, California

Math Curriculum Leader Long Beach Unified Long Beach, California

Philip Gonsalves

Robyn Silbey

Kathy Vielhaber

Mathematics Coordinator Alameda County Office of Education Hayward, California

Math Specialist Montgomery County Public Schools Gaithersburg, Maryland

Mathematics Consultant St. Louis, Missouri

Viken Hovsepian Professor of Mathematics Rio Hondo College Whittier, California

Dinah Zike Educational Consultant, Dinah-Might Activities, Inc. San Antonio, Texas

CONSULTANTS Assessment Donna M. Kopenski, Ed.D. Math Coordinator K–5 City Heights Educational Collaborative San Diego, California

Instructional Planning and Support

ELL Support and Vocabulary

Beatrice Luchin

ReLeah Cossett Lent

Mathematics Consultant League City, Texas

Author/Educational Consultant Alford, Florida

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CONTRIBUTING AUTHORS

California Advisory Board CALIFORNIA ADVISORY BOARD

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Glencoe wishes to thank the following professionals for their invaluable feedback during the development of the program. They reviewed the table of contents, the prototype of the Student Study Guide, the prototype of the Teacher Wraparound Edition, and the professional development plan.

Linda Anderson

Cheryl L. Avalos

Bonnie Awes

Kathleen M. Brown

4th/5th Grade Teacher Oliveira Elementary School, Fremont, California

Mathematics Consultant Retired Teacher Hacienda Heights, California

Teacher, 6th Grade Math Monroe Clark Middle School San Diego, California

Math Curriculum Staff Developer Washington Middle School Long Beach, California

Carol Cronk

Audrey M. Day

Jill Fetters

Grant A. Fraser, Ph.D.

Mathematics Program Specialist San Bernardino City Unified School District San Bernardino, California

Classroom Teacher Rosa Parks Elementary School San Diego, California

Math Teacher Tevis Jr. High School Bakersfield, California

Professor of Mathematics California State University, Los Angeles Los Angeles, California

Eric Kimmel

Donna M. Kopenski, Ed.D.

Michael A. Pease

Chuck Podhorsky, Ph.D.

Mathematics Department Chair Frontier High School Bakersfield, California

Math Coordinator K–5 City Heights Educational Collaborative San Diego, California

Instructional Math Coach Aspire Public Schools Oakland, California

Math Director City Heights Educational Collaborative San Diego, California

Arthur K. Wayman, Ph.D.

Frances Basich Whitney

Mario Borrayo

Melissa Bray

Professor Emeritus California State University, Long Beach Long Beach, California

Project Director, Mathematics K–12 Santa Cruz County Office of Education Capitola, CA

Teacher Rosa Parks Elementary San Diego, California

K–8 Math Resource Teacher Modesto City Schools Modesto, California

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California Reviewers CALIFORNIA REVIEWERS Each California Reviewer reviewed at least two chapters of the Student Study Guides, providing feedback and suggestions for improving the effectiveness of the mathematics instruction. Melody McGuire

Math Teacher California College Preparatory Academy Oakland, California

6th and 7th Grade Math Teacher McKinleyville Middle School McKinleyville, California

Eppie Leamy Chung

Monica S. Patterson

Teacher Modesto City Schools Modesto, California

Educator Aspire Public Schools Modesto, California

Judy Descoteaux

Rechelle Pearlman

Mathematics Teacher Thornton Junior High School Fremont, California

4th Grade Teacher Wanda Hirsch Elementary School Tracy, California

Paul J. Fogarty

Armida Picon

Mathematics Lead Aspire Public Schools Modesto, California

5th Grade Teacher Mineral King School Visalia, California

Lisa Majarian

Anthony J. Solina

Classroom Teacher Cottonwood Creek Elementary Visalia, California

Lead Educator Aspire Public Schools Stockton, California

vi

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Bobbi Anne Barnowsky

Volume 2A Fractions and Decimals Chapter

Parts of a Whole

1

1-1 Parts of a Whole and Parts of a Set ................................4. 2NS4.0, 4NS1.5

1-2 Recognize, Name, and Compare Unit Fractions ........11 2NS4.1

Progress Check.................................................................18 1-3 Representing Fractions....................................................19 2NS4.3, 4NS1.7

Assessment Study Guide .....................................................................26 Chapter Test .....................................................................28

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Standards Practice ...................................................30

Bixby Creek Bridge on Highway 1, south of Carmel

Chapters 1 and 2 are contained in Volume 2A. Chapters 3 and 4 are contained in Volume 2B.

Standards Addressed in This Chapter 2NS4.0 Students understand that fractions and decimals may refer to parts of a set and parts of a whole. 2NS4.1

Recognize, name, and 1 1 compare unit fractions from ___ to __. 12 2 2NS4.3 Know that when all fractional parts are included, such as fourfourths, the result is equal to the whole and to one. 4NS1.5 Explain different interpretations of fractions, for example, parts of a whole, parts of a set, and division of whole numbers by whole numbers; explain equivalence of fractions (see Standard 4.0). 4NS1.7 Write the fraction represented by a drawing of parts of a figure; represent a given fraction by using drawings; and relate a fraction to a simple decimal on a number line.

vii Roy Ooms/Masterfile

Contents Chapter

Equivalence of Fractions

2

Standards Addressed in This Chapter 2-1 Equivalent Fractions and Equivalent Forms of One ..............................................34 2NS4.3, 3NS3.1, 4NS1.5

2-2 Mixed Numbers and Improper Fractions....................41 2NS4.3, 4NS1.5, 5NS1.5

Progress Check 1 .............................................................50 2-3 Least Common Denominator and Greatest Common Factors ......................................51 4NS1.5

2-4 Compare and Order Fractions...................................... 59 3NS3.1, 6NS1.1

Progress Check 2 .............................................................68 2-5 Simplify Fractions ...........................................................69 3NS3.1, 4NS1.5

Assessment

Chapter Test .....................................................................82 Standards Practice ...................................................84 Alabama Hills, Owens Valley

viii Daryl Benson/Masterfile

3NS3.1 Compare fractions represented by drawings or concrete materials to show equivalency and to add and subtract 1 simple fractions in context (e.g., __ of a 2 2 pizza is the same amount as __ of another 4 3 pizza that is the same size; show that __ is 8 1 larger than __). 4 4NS1.5 Explain different interpretations of fractions, for example, parts of a whole, parts of a set, and division of whole numbers by whole numbers; explain the equivalence of fractions (see Standard 4.0). 5NS1.5 Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers. 6NS1.1 Compare and order positive and negative fractions, decimals, and mixed numbers and place them on a number line.

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Study Guide .....................................................................77

2NS4.3 Know that when all fractional parts are included, such as fourfourths, the result is equal to the whole and to one.

Contents Chapter

Operations with Fractions

3

3-1 Add Fractions with Like Denominators .......................4 3NS3.2, 6NS2.1

3-2 Subtract Fractions with Like Denominators ...............11 3NS3.2, 6NS2.1

Progress Check 1 .............................................................18 3-3 Multiply Fractions ...........................................................19 5NS2.0, 5NS2.5, 6NS2.1

3-4 Divide Fractions ............................................................. 25 5NS2.5, 6NS2.1

Progress Check 2 .............................................................32 3-5 Add Fractions with Unlike Denominators ..................33 3NS3.2, 5NS2.0, 6NS2.1

3-6 Subtract Fractions with Unlike Denominators ...........39 3NS3.2, 5NS2.0, 6NS2.1

Chapters 1 and 2 are contained in Volume 2A. Chapters 3 and 4 are contained in Volume 2B.

Standards Addressed in This Chapter 3NS3.2 Add and subtract simple 1 3 fractions (e.g., determine that __ + __ is the 8 8 1 same as __). 2 5NS2.0 Students perform calculations and solve problems involving addition, subtraction, and simple multiplication and division of fractions and decimals. 5NS2.5 Compute and perform simple multiplication and division of fractions and apply these procedures to solving problems. 6NS2.1 Solve problems involving addition, subtraction, multiplication, and division of positive fractions and explain why a particular operation was used for a given situation.

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Progress Check 3 .............................................................45 Assessment Study Guide .....................................................................46 Chapter Test .....................................................................50

San Diego Harbor

Standards Practice ...................................................52

ix Jeremy Jeremy Woodhouse/Masterfile Woodhouse/Masterfile

Contents Chapter

Positive and Negative Fractions and Decimals

4

4-1 Introduction to Decimals ...............................................56 3NS3.4, 4NS1.6, 4NS1.7

4-2 Decimals and Money ......................................................63 2NS5.1, 2NS5.2

Progress Check 1 .............................................................72 4-3 Compare and Order Decimals ......................................73 5NS1.5, 6NS1.1

4-4 Compare and Order Fractions and Decimals ............ 81 5NS1.5, 6NS1.1, 4NS1.7

Progress Check 2 .............................................................88 4-5 Add Decimals ..................................................................89 4NS2.0, 5NS2.0, 5NS2.1, 7NS1.2

4-6 Subtract Decimals........................................................... 97 4NS2.0, 5NS2.0, 5NS2.1, 7NS1.2

Progress Check 3 ...........................................................104 5NS2.0, 5NS2.1, 7NS1.2

4-8 Divide Decimals ........................................................... 113 5NS2.0, 5NS2.1, 7NS1.2

Progress Check 4 ...........................................................120 4-9 Operations with Positive and Negative Numbers ............................................... 121 4NS1.8, 5NS2.1, 6NS2.3, 7NS1.2

Assessment Study Guide ...................................................................128 Chapter Test ...................................................................134 Standards Practice .................................................136 Antelope Valley

x Daryl Benson/Masterfile

3NS3.4 Know and understand that fractions and decimals are two different representations of the 1 same concept (e.g., 50 cents is __ of a dollar, 2 3 __ 75 cents is of a dollar). 4 4NS1.6 Write tenths and hundredths in decimal and fraction notations and know the fraction and decimal equivalents for halves and fourths 3 1 2 (e.g., _ = 0.5 or 0.50; __ = 1_ = 1.75). 4 2 4 4NS1.7 Write the fraction represented by a drawing of parts of a figure; represent a given fraction by using drawings; and relate a fraction to a simple decimal on a number line. 2NS5.1 Solve problems using combinations of coins and bills. 2NS5.2 Know and use the decimal notation and the dollar and cent symbols for money. 5NS1.5 Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers. 6NS1.1 Compare and order positive and negative fractions, decimals, and mixed numbers and place them on a number line. 4NS2.0 Students extend their use and understanding of whole numbers to the addition and subtraction of simple decimals. 5NS2.0 Students perform calculations and solve problems involving addition, subtraction, and simple multiplication and division of fractions and decimals. 7NS1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to whole-number powers. 5NS2.1 Add, subtract, multiply, and divide with decimals; add with negative integers; subtract positive integers from negative integers; and verify the reasonableness of the results. 4NS1.8 Use concepts of negative numbers (e.g., on a number line, in counting, in temperature, in “owing”). 6NS2.3 Solve addition, subtraction, multiplication, and division problems, including those arising in concrete situations, that use positive and negative integers and combinations of these operations.

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

4-7 Multiply Decimals.........................................................105

Standards Addressed in This Chapter

R E G N E V A SC HUNT Let’s Get Started Use the Scavenger Hunt below to learn where things are located in each chapter.

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

1 What is the title of Chapter 1? 2

What is the Key Concept of Lesson 2-1?

3

What are the four steps of problem solving?

4

What are the vocabulary words for Lesson 2-5?

5

How many Examples are presented in Lesson 1-3?

6

What are the California Standards covered in Lesson 2-4?

7

List two ways in which fractions are represented on page 23?

8

What do you think is the purpose of the Standard Practice on pages 30–31?

9

On what pages will you find the Study Guide for Chapter 1?

10

In Chapter 2, find the logo and Internet address that tells you where you can take the Online Readiness Quiz.

1

Chapter

1

Parts of a Whole We use fractions every day. Any time we want to describe parts of a whole or parts of a set, we can use fractions. For example, three out of eight 3 players, or __ of the players, are on the red practice squad. 8

Copyright © by The McGraw-Hill Companies, Inc.

2

Chapter 1 Parts of a Whole

Larry Dale Gordon/Getty Images

STEP

STEP

1 Quiz

2 Preview

Are you ready for Chapter 1? Take the online readiness quiz at ca.mathtriumphs.com to find out. Get ready for Chapter 1. Review these skills and compare them with what you’ll learn in this chapter.

What You Know

What You Will Learn

You know that half of one dollar is 50 cents. Half of 50 cents is a quarter, or 25 cents.

Lessons 3-1 and 3-2

1 1 2

1 4

You know that when you have a pizza and share it among 8 people that you cut it into 8 equal-sized pieces.

Copyright © by The McGraw-Hill Companies, Inc.

Each person gets an equal-sized piece.

1 2

1 4

1 4

1 4

Lessons 3-2 and 3-3 The pizza is divided into 8 equal parts. The entire pizza can be 8 represented by the fraction __. 8 1 __ Each person gets of the pizza. 8 If one person ate 4 pieces, he would 4 , or __ 1 , of the pizza. have eaten __ 8 2

1 1 1 piece = __ 4 pieces = __ 2 8 1 of 1 of the pizza is greater than __ __ 2 8 the pizza.

3 (bkgd)Larry Dale Gordon/Getty Images, (tl)Michael Houghton/StudiOhio, United States coin images from the United States Mint, (bl br)Burke/Triolo Productions/FoodPix/Jupiter Images

Lesson

1-1 Parts of a Whole and Parts of a Set KEY Concept In a fraction , the number above the fraction bar is called the numerator . The number below the fraction bar is called the denominator .

A fraction names part of a whole . The flag of France is divided into three equal parts: red, white, 1 of the whole flag. and blue. Each color of the flag represents __ 3 number of red parts 1 ______________________ = __ number of colors in flag 3

VOCABULARY fraction a number that represents part of a whole or part of a set 1 1 1 3 Examples: __, __, __, __ 2 3 4 4 numerator the number above the bar in a fraction that tells how many equal parts are being used 5 ← numerator __ 6 denominator the number below the bar in a fraction that tells how many equal parts are in the whole or the set 5 __ 6 ← denominator whole the entire amount or object

A fraction can also name part of a set. In a chess set that contains 32 pieces, 16 of the pieces are 16 pawns. Among all the chess pieces, ___ are pawns. 32 number of pawns 16 _____________________ ___ = 32 number of pieces in all

When using a whole shape, parts are regions of equal size inside the whole. When using sets, all of the items together make up the entire set. 4

Chapter 1 Parts of a Whole

(t)Matthias Kulka/zefa/Corbis, (b)Comstock Images/Alamy

Copyright © by The McGraw-Hill Companies, Inc.

Notice that the “whole” is the area of all of the flag. Each 1 , represents an equal area of the flag. color, or __ 3

2NS4.0 Students understand that fractions and decimals may refer to parts of a set and parts of a whole. 4NS1.5 Explain different interpretations of fractions, for example, parts of a whole, parts of a set, and division of whole numbers by whole numbers; explain equivalence of fractions.

Example 1 Write a fraction that represents the shaded region of the rectangle.

1. Ten equal parts make up the whole. This number is the denominator. 2. Three parts are shaded. This number is the numerator. 3. Write the fraction, numerator over denominator. 3 ___ 10

YOUR TURN! Write a fraction that represents the shaded region of the circle. 1. How many equal parts make up the whole circle? What is this number called?

2. How many parts are shaded? What is this number called? . 3. Write the fraction. ______

Numerator

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Denominator

Example 2

YOUR TURN!

Write a fraction that represents the number of circles in the set.

Write a fraction that represents the number of turtles in the set.

1. There are 7 objects in the set. 7 is the denominator. 2. There are 2 circles in the set. 2 is the numerator. 3. Write the fraction. 2 __ 7

1. How many animals are in the set altogether? What is this number called?

2. How many turtles are in the set? What is this number called?

3. Write the fraction. ______

Numerator Denominator

GO ON Lesson 1-1 Parts of a Whole and Parts of a Set (l)Dorling Kindersley/Getty Images, (r)Stockdisc/PunchStock

5

Who is Correct? Write a fraction to represent the number of Xs in the set.

Reina 9 __

Tate

Toshi

9

9

4 __

5

5 __

Circle correct answer(s). Cross out incorrect answer(s).

Guided Practice Write a fraction to represent each situation. 1

the shaded region of the square number of shaded parts ______ ______________________ = number of equal parts

2

the number of suns in the set number of objects in the set

3

2 . Use equal parts of Draw a picture to model the fraction __ 3 a whole. number of shaded parts ______ = ______________________ number of equal parts

6

Chapter 1 Parts of a Whole

Draw a figure. Divide it into 3 equal parts. Shade 2 parts.

Copyright © by The McGraw-Hill Companies, Inc.

number of suns in set _________________________ = ______

Step by Step Practice 4

Write a fraction to represent the number of people in the group with their arms raised. Step 1 Count to find the denominator. people make up the whole group. Step 2 Count to find the numerator. people have their arms raised. numerator Step 3 Write the fraction: ____________ = ______ denominator

Draw a picture to model the fraction. Use equal parts of a whole. 4 5 __ 7 Draw a whole with

Copyright © by The McGraw-Hill Companies, Inc.

Shade

equal parts.

parts.

Draw a picture to model the fraction. Use a set of objects. 3 6 __ 4 Write a fraction to represent each situation. 7

What fraction of the set of balls are the footballs?

8

What fraction of the set of balls are neither baseballs nor footballs?

9

What fraction represents the shaded part of the rectangle?

GO ON Lesson 1-1 Parts of a Whole and Parts of a Set (t)David Woolley/Getty Images, (bl bcl)Getty Images, (bcr br)CORBIS

7

Step by Step Problem-Solving Practice Solve. 10

Problem-Solving Strategies ✓ Draw a picture.

SNACKS Jennifer brought cookies to her after-school meeting. She gave 5 friends 1 cookie each. She had 2 cookies left. What fraction of the cookies was left? Understand

Look for a pattern. Guess and check. Act it out. Work backward.

Read the problem. Write what you know. Five friends received There are

cookie each.

cookies left.

Plan

Pick a strategy. One strategy is to draw a picture. Draw a circle to represent the cookie that each friend received. Draw 2 shaded circles to represent the cookies that were left.

Solve

Count the total number of circles. This is the number for the whole. It is the of the fraction. There are 2 cookies left. This is the number for the part. It is the of the fraction. part numerator ______ = ____________ = ______ whole

ONLINE SHOPPING Juan’s family ordered jackets online. Two jackets were blue, one was red, and four were green. What fraction of the jackets ordered was not blue? Check off each step. Understand Plan Solve Check

12

8

SCHOOL For a school party, Mr. Gomez brought 24 fruit bars. His 19 students ate 1 bar each. What fraction of the fruit bars was eaten? Chapter 1 Parts of a Whole

Copyright © by The McGraw-Hill Companies, Inc.

Look at the numbers in the problem. Does the solution answer the question? Is it reasonable?

Check

11

denominator

Explain how the word not affected your answer for Exercise 12.

13

Skills, Concepts, and Problem Solving Write a fraction to represent each shaded region. 14

15

16

17

0 in.

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18

1

19

Draw a picture to model each fraction. Use equal parts of a whole or set. 20

2 __

22

5 __

24

5

6 4. Draw two pictures to model the fraction __ 5 Use equal parts of a whole for one picture and parts of a set for the other picture.

21

3 __

23

2 __

3

4

GO ON Lesson 1-1 Parts of a Whole and Parts of a Set

Bonhommet/PhotoCuisine/Corbis

9

Solve. 25

SHOPPING José had a $100 gift certificate. He spent $41 on shoes and $32 on pants. What fraction of the gift certificate did he use?

26

WORDS

27

SCHOOL Lakesha’s score on her math exam is shown at the right. What fraction of the questions did she answer incorrectly?

28

PETS My friend told me that of her dog’s 5 puppies, 2 are females. How many female puppies are there? __ 5 How many puppies are there altogether?

What fraction of the letters in California are consonants?

Vocabulary Check each sentence. 29

Write the vocabulary word that completes

In a , the top number is the and the bottom number is the

, .

The

is an entire amount or object.

31

Writing in Math In your own words, describe what the numerator and denominator of a fraction represent. Be sure to use the words numerator and denominator.

Draw a picture to model each fraction. Use equal parts of a whole or set. 3 2 4 32 __ 33 __ 34 __ 5 8 4

Chapter 1 Parts of a Whole

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30

10

23 out of 25 correct!

Lesson

1-2 Recognize, Name, and Compare Unit Fractions KEY Concept The pizzas shown are the same size, but are cut into pieces that are different sizes.

1 1 one piece = __ one piece = __ 6 8 Each piece is a unit fraction . A fraction with 1 in the numerator is a unit fraction. Compare the sizes. The pieces of pizza on the left are larger. 1 1 > __ __ 6 8

Copyright © by The McGraw-Hill Companies, Inc.

To compare unit fractions, compare the denominators. The greater the denominator, the more parts. The more parts, the smaller each part is. The fraction with the lesser number in the denominator is the greater unit fraction.

VOCABULARY unit fraction A fraction that has 1 in its numerator 1 1 Example: __ or __ 7 3 numerator the number above the bar in a fraction that tells how many equal parts are being used 3 ← numerator __ 4 (Lesson 1-1, p. 4)

denominator the number below the bar in a fraction that tells how many equal parts there are in the whole set 3 __ 4 ← denominator (Lesson 1-1, p. 4)

1 _

1 >_ 6 8

6 < 8 so the unit fraction

2NS4.1 Recognize, name, and 1 compare unit fractions from ___ 12 1 __ to . 2 4NS1.7 Write the fraction represented by a drawing of parts of a figure; represent a given fraction by using drawings; and relate a fraction to a simple decimal on a number line.

1 is greater. 6

1 unit

Denominator

When you divide something into equal pieces, the more pieces you 1 is greater divide it into, the smaller each piece will be. That is why __ 6 1. than __ 8

GO ON

Lesson 1-2 Recognize, Name, and Compare Unit Fractions Envision/Corbis

11

Example 1 Show where you make cuts to create the 1 1 1 unit fractions , , and . 2 4 8

__

_

1 unit

1. Make one cut to create two equal parts. Shade one part. 1. The shaded part is the unit fraction __ 2 2. Cut each half into two equal parts so that there are four equal parts in all. Shade (in a different color) one 1. part to show the unit fraction __ 4 3. Cut each fourth into two equal parts so that there are eight equal parts in all. Shade (in a different color) one 1. part to show the unit fraction __ 8 YOUR TURN! Show where you make cuts to create the 1 1 1 unit fractions , , and . 2 4 8

__

_

1 unit

2. Cut each half into two equal parts so that there are four equal parts in all. Shade one part of the fourths. What unit fraction does the shaded part represent?

3. Cut each fourth into two equal parts so that there are eight equal parts in all. Shade one part. What unit fraction does the shaded part represent?

12

Chapter 1 Parts of a Whole

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1. Make one cut to create two equal parts. Shade one part. What unit fraction does the shaded part represent?

Example 2 Name the unit fraction that represents the shaded region in each figure. 1 unit

1. There are 3 equal parts. The unit fraction is __ 3 1. There are 6 equal parts. The unit fraction is __ 6 1. There are 12 equal parts. The unit fraction is ___ 12 YOUR TURN! Name the unit fraction that represents the shaded region in each figure.

1 unit

There are

equal parts.

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The unit fraction is

There are

equal parts.

The unit fraction is

There are

.

.

equal parts.

The unit fraction is

.

GO ON Lesson 1-2 Recognize, Name, and Compare Unit Fractions

13

Example 3

YOUR TURN!

Compare. Write <, =, or > to make a true statement.

Compare. Write <, =, or > to make a true statement.

1 __

1 __

1 __

1 __

3

4

7

9

1. The denominators of the fractions are 3 and 4. 2. Write a comparison statement for the denominators.

1. What are the denominators of the fractions?

3<4 2. Write a comparison statement for the denominators.

Recall that the smaller denominator makes the greater unit fraction.

7

9

3. Write a comparison statement for the fractions. 1 1 __ __ 7 9 1 3

1 4

3. Write a comparison statement for the fractions.

1 > __ 1 __ 3

4

Name the unit fraction for one piece of pizza that is cut into 10 pieces.

Damian

Lloyd

Emma

10

10

2

5 ___

1 ___

1 __

Circle correct answer(s). Cross out incorrect answer(s).

Guided Practice 1

14

Write the unit fraction that represents the shaded region.

Chapter 1 Parts of a Whole

Copyright © by The McGraw-Hill Companies, Inc.

Who is Correct?

2

Show a unit fraction in each circle. Name each unit fraction. Which unit fraction is greater?

Step by Step Practice 3

1 and ___ 1. Show where you make cuts to create the unit fractions __ 5 10 Shade each unit fraction. 1 . How many parts do Step 1 The unit fraction is __ 5 you need to create?

1 unit

Step 2 Show the cuts using lines. Shade one part. What unit fraction is shaded?

Copyright © by The McGraw-Hill Companies, Inc.

Step 3 Draw lines to divide each fifth in half. Shade one part. What unit fraction is shaded?

Show where you make cuts to create each unit fraction. 1 4 __ How many equal parts do you need to create? 2

5

1 ___

6

10

1 __ 6

Compare. Write <, =, or > to make each a true statement. 1 1 1 1 __ ___ 7 ___ 8 __ 8 12 4 10 9

1 __

1 __

3

8

10

1 ___

1 __

12

5

GO ON

Lesson 1-2 Recognize, Name, and Compare Unit Fractions

15

Step by Step Problem-Solving Practice

Problem-Solving Strategies ✓ Draw a diagram.

Solve. 11

1 of a book. Manu has read __ 1 Tina has read __ 5 8 of the same book. Who has more left to read? READING

Understand

Read the problem. Write what you know.

Guess and check. Act it out. Solve a simpler problem. Work backward.

Tina has read of the book. Manu has read of the book. Plan

Pick a strategy. One strategy is to draw a diagram. Draw a rectangle to 1 and __ 1 on separate rectangles. represent the whole book. Show __ 5 8

Solve

The shaded parts represent the part of the book already read. 1. 1 is larger than __ __ 5

has more of the book left to read. Does your answer make sense?

Check

12

8 has read more of the book.

FARMING One-third of the wheat field has been harvested. One-fourth of the cornfield has been picked. The fields are the same size. Which field has had more harvested? Check off each step.

Plan Solve Check In your own words, explain how to compare unit fractions.

13

Skills, Concepts, and Problem Solving Show where you make cuts to create each unit fraction. 14

1 __

16

Chapter 1 Parts of a Whole

4

15

1 __ 3

Copyright © by The McGraw-Hill Companies, Inc.

Understand

Compare. Write <, =, or > to make each a true statement. 1 1 1 1 1 __ __ 16 __ 17 __ 18 ___ 8 3 5 4 12 20

1 __

1 __

2

4

21

1 ___

1 ___

12

10

22

1 __ 8

1 __

1 __

7

8

19

1 __

1 __

2

5

23

1 ___

1 __

11

9

Solve. 1 of his daily fruit servings at breakfast. William eats __ 3 1 of his daily fruit Yancy eats __ servings at breakfast. Who eats more 4 of his fruit servings at meals other than breakfast?

24

NUTRITION

25

BUSINESS Mr. Barber orders office paper each month. The order 1 blue paper, and __ 1 pink paper. Of which 1 white paper, __ includes __ 2 3 9 type of paper does Mr. Barber order the least amount?

Vocabulary Check sentence.

Write the vocabulary word that completes each

In a unit fraction, the

27

When you compare unit fractions, look at the

28

Writing in Math Examine the unit fractions at the right. In your own words, what can you say is true when the denominator of a unit fraction increases? Use the words denominator and increases.

Copyright © by The McGraw-Hill Companies, Inc.

26

is always 1. . 1 5

1 4

1 3

1 6

1 7

1 8

Spiral Review SCHOOL

_

Mark scored 18 on a quiz. 20

29

How many points was the quiz worth?

30

How many points did he get?

(Lesson 1-1, p. 4)

Lesson 1-2 Recognize, Name, and Compare Unit Fractions

17

Chapter

Progress Check 1

1

(Lessons 1-1 and 1-2)

Write a fraction to represent each unshaded region. 1

2

3

4

Draw a picture to model each fraction. Use equal parts of a whole or set. 5 6 5 __ 6 __ 5 7

Write the unit fraction that represents each shaded region or part. 7

8

Write <, =, or > to make 1 1 __ 10 __ 8 5

each a true statement. 1 1 ___ 11 __ 3 10

1 __ 2

Solve. 13

WORDS

What fraction of the letters in California are vowels?

14

WORDS

What fraction of the letters in Mississippi are i’s?

15

FITNESS Nina swam the length of the pool 5 times. What fraction of her swim does one length represent?

16

FOOD Bo has one dozen apples to give his friends. Each person got one apple. What fraction of Bo’s apples did each person get?

18

Chapter 1 Parts of a Whole

12

1 ___

1 ___

12

11

Copyright © by The McGraw-Hill Companies, Inc.

Compare. 1 9 ___ 11

Lesson

1-3 Representing Fractions KEY Concept When the numerator and denominator of a fraction are equal, the fraction equals 1.

2NS4.3 Know that when all fractional parts are included, such as four-fourths, the result is equal to the whole and to one. 4NS1.7 Write the fraction represented by a drawing of parts of a figure; represent a given fraction by using drawings; and relate a fraction to a simple decimal on a number line.

VOCABULARY fraction a number that represents part of a whole or part of a set (Lesson 1-1, p. 4) numerator the number above the bar in a fraction that tells how many equal parts are being used (Lesson 1-1, p. 4) denominator the number below the bar in a fraction that tells how many equal parts are in the whole or set

You can model fractions using fraction tiles, fraction circles, and diagrams.

Example 1

YOUR TURN!

Copyright © by The McGraw-Hill Companies, Inc.

_

1 Use fraction tiles to form a rectangle 5 equal to 1. Write the fraction that equals 1.

1 1 5

1 5

1 5

5

5 __ =1 5

_

1 Draw fraction pieces to form a circle 8 equal to 1. Write the fraction that equals 1. 1 8

1 5

1 5

1 tiles next to each other. It takes 1. Line the __ 5 1 to equal the length of five green tiles __  5 one blue tile (1). 1 tiles = 1 2. Five __

(Lesson 1-1, p. 4)

1 pieces does it take 1. How many __ 8 to equal one whole circle? 1 pieces = 1 2. Eight __ 8 ______ = 1

GO ON Lesson 1-3 Representing Fractions

19

Example 2

YOUR TURN!

Write the unit fraction that represents the shaded region. Then write a fraction with the same denominator that equals 1.

Write the unit fraction that represents the shaded region. Then write a fraction with the same denominator that equals 1.

1. Six wedges form a whole circle. 1. 2. The unit fraction is __ 6 6 __ 3. = 1 6

1. How many pieces form the entire strip? 2. What is the unit fraction? 3. ______ = 1

Who is Correct? Write a fraction that is equal to one.

Caroline

Eduardo

6 =1 ___

1 __ =1

10

6

Sal 6 __ =1 6

Guided Practice Write the unit fraction that represents the shaded region or part of the set. Then write a fraction with the same denominator that equals 1. number of shaded parts ______ ______________________ = number of equal parts

1

2

20

number of shaded parts ______ ______________________ = number of equal parts

Chapter 1 Parts of a Whole

______ = 1

______ = 1

Copyright © by The McGraw-Hill Companies, Inc.

Circle correct answer(s). Cross out incorrect answer(s).

Step by Step Practice 3

1 fraction tiles to form a rectangle equal to 1. Write the Use __ 8 fraction that equals 1.

1 1 8

Step 1 Line up the left edge of the rectangle that represents 1 with a fraction tile. 1 fraction tiles to form a rectangle equal to 1. Step 2 Use __ 8 1 tiles are needed? How many __ 8 Step 3

1 tiles = 1 __ 8

______ = 1

Use each given size of fraction tile to form a rectangle equal to 1. Write the fraction that equals 1. 1 4 __ fraction tiles 3 1 tiles are needed to make 1? How many __ 3 Copyright © by The McGraw-Hill Companies, Inc.

The fraction is 5

1 fraction tiles __ 2

. 6

1 fraction tiles ___ 12

Write the unit fraction that represents the shaded region or part of the set. Then write a fraction with the same denominator that equals 1. 7

8

9

10

GO ON Lesson 1-3 Representing Fractions

21

Step by Step Problem-Solving Practice

Problem-Solving Strategies ✓ Use a model.

Solve. Use fraction tiles or fraction circles. 11

BAKING Natalie baked biscuits for a family dinner. Her brother ate 2 biscuits. The 3 other people ate one biscuit each. There were no biscuits left over. Write a fraction to represent one biscuit. Understand

Read the problem. Write what you know. Natalie’s brother ate 3 other people ate There were

Plan

Look for a pattern. Guess and check. Solve a simpler problem. Work backward.

biscuits. biscuit each.

biscuits left over.

Pick a strategy. One strategy is to use a model. +

Solve biscuits her brother ate

= biscuits others ate

biscuits in all

Use 1 fraction tile to model the whole batch of 1 tiles to model the biscuits her biscuits. Use __ 5 brother ate.

1 5

1 5

1 tiles to model the biscuits the 3 other Use __ 5 people ate.

1 1 5

1 5

1 5

1 5

1 5

One biscuit = ______. Check

22

Does your answer make sense? Does your answer represent 1 biscuit?

Chapter 1 Parts of a Whole

Copyright © by The McGraw-Hill Companies, Inc.

1

12

3 SCHOOL Rico wrote __ of a book report last night. He wrote 7 6 pages. How many pages will Rico’s completed report be? Check off each step.

Understand Plan Solve Check 13

RECREATION Mr. Thad has a group of 9 students who play foursquare at recess. Each day, only 8 students can play. Mr. Thad made a schedule so one player sits out each day. What fraction of the students sits out each day?

14

COMMUNITY SERVICE

Postcards need to be handed out for an 1 of the upcoming blood drive. Tamika and Jacob each hand out __ 8 cards. More volunteers are needed so that each person does the same amount of work as Jacob and Tamika. How many more volunteers are needed?

Copyright © by The McGraw-Hill Companies, Inc.

15

What can you say is true about every unit fraction? Be sure to use the words numerator and denominator.

Skills, Concepts, and Problem Solving Write the unit fraction that represents each shaded region or part of the set. Then write a fraction with the same denominator that equals 1. 16

17

18

19

GO ON Lesson 1-3 Representing Fractions Getty Images

23

Use each given size of fraction tile to form a rectangle equal to 1. Write the fraction that equals 1. 20

1 fraction tiles __

21

1 fraction tiles __

22

1 fraction tiles __

23

1 fraction tiles __

4 6

8 3

Use fraction circles to identify each unit fraction shown. How many of these fractions equal 1? 24

25

26

27

Circle the fraction in each group that is equal to one whole. 1 __

2 __

3 __

3

3

3

30

8 __

1 ___

4 __

8

10

8

29

4 __

5 __

1 __

5

5

5

31

7 __

11 ___

17 ___

9

12

17

Solve. 32

NATURE Rafael planted 16 lettuce plants in his garden. One morning he found a rabbit eating one of the plants. What fraction of lettuce plants did the rabbit eat?

33

JOBS Keshia drives a package delivery truck. Each day she loads the truck so that packages in the back half of the truck get delivered before lunch. Packages in the front half of the truck are delivered after lunch. What fraction of packages does Keshia deliver by the end of the day?

24

Chapter 1 Parts of a Whole

Copyright © by The McGraw-Hill Companies, Inc.

28

Vocabulary Check Write the vocabulary word that completes each sentence. 1 . 34 ___ is an example of a 11 35

A whole.

is a number representing some part of a

36

Writing in Math Christine was asked to divide a circle into thirds 1 . Christine drew the picture below. and show the unit fraction __ 3 1. She said since the denominator is 3, the shaded region represents __ 3 What is the problem with Christine’s picture of the unit fraction 1 ? Use the circle next to Christine’s to show how to correctly shade __ 3 1. the unit fraction __ 3

Spiral Review Copyright © by The McGraw-Hill Companies, Inc.

Write a fraction to represent each shaded region or part of a set. 37

38

39

40

41

(Lesson 1-1, p. 4)

1 nuts. Kareem FOOD Carly made a pound of trail mix that was __ 1 nuts. Whose5trail mix had made a pound of trail mix that was __ 8 more nuts? (Lesson 1-2, p. 11)

Lesson 1-3 Representing Fractions

25

Chapter

1

Study Guide

Vocabulary and Concept Check denominator, p. 4

Write the vocabulary word that completes each sentence.

fraction, p. 4

1

is a number that represents part of a A(n) whole or part of a set.

2

The number that represents how many equal parts into which an object is divided is called the .

3

The term that refers to the entire set or object is . the

4

The number that represents how many equal parts of the object are being used is called the .

5

A(n)

numerator, p. 4 unit fraction, p. 11 whole, p. 4

has a numerator equal to 1.

Label each diagram below. Write the correct vocabulary term in each blank. 6

7

1-1

Parts of a Whole and Parts of a Set

Write a fraction to represent each situation. 8

9

26

the shaded region of the hexagon

the number of hearts in the set

Chapter 1 Study Guide

(pp. 4–10)

Example 1 Write a fraction to represent the shaded region of the rectangle. There are 12 equal parts of the whole. This number is the denominator. There are 3 parts shaded. This number is the numerator. Write the fraction. 3 12

_

Copyright © by The McGraw-Hill Companies, Inc.

Lesson Review

1-2

Recognize, Name, and Compare Unit Fractions

Compare. Write < , =, or > to make each a true statement. 10

1 __

1 __

6

5

12

1 __

1 __

8

5

14

1 __

1 ___

9

10

16

1 ___

1 ___

20

25

1 11 __ 7

1 13 ___ 1 15 __ 5

1 17 ___ 18

Example 2

1 ___

Compare. Write <, =, or > to make a true statement.

11 1 ___

_1

_1

15

2

3

10

(pp. 11–18)

The denominators of the fractions are 2 and 3.

1 __ 2

Write a comparison statement for the denominators. 2 < 3

1 __

Recall that the smaller denominator makes the greater unit fraction. Write a comparison statement for the fractions. 1 > 1 2 3

3

_ _

1-3

Representing Fractions

Copyright © by The McGraw-Hill Companies, Inc.

Use each given size of fraction tile to form a rectangle equal to 1. Write the fraction that equals 1. 18

1 fraction tiles __

19

1 fraction tiles __

20

1 fraction tiles ___

21

1 fraction tiles __

22

1 fraction tiles __

23

1 fraction tiles __

(pp. 19–25)

Example 3 1 Use __ fraction tiles to form a rectangle 2 equal to 1. Write the fraction that equals 1.

1

3 4

10 2 5 6

1 2

1 tiles next to each other. It takes two Line the __ 2 1 to equal the length of one red pink tiles __ 2 tile (1).

()

1 tiles = 1 Two __ 2 2 __ = 1 2

Chapter 1 Study Guide

27

Chapter

1

Chapter Test

Write a fraction to represent each shaded region or part of a set. 1

2

Draw a picture to show each fraction. Use equal parts of a whole or parts of a set. 5 1 3 __ 4 __ 5 6 5

3 __ 3

Fill in the blanks to make each fraction equal 1. ______

7

______

9

6 ______

1

20 14 8 ______

Copyright © by The McGraw-Hill Companies, Inc.

6

Write the unit fraction to represent each shaded region. 10

11

GO ON 28

Chapter 1 Test

Compare. Write < , =, or > to make each a true statement. 1 1 1 1 __ __ 12 __ 13 __ 9 8 3 6 14

1 ____

1 ____

125

100

15

1 ___

1 ___

13

15

Solve. 16

17

18

MUSIC The Deerfield band has 60 members. Nineteen are trumpet players. What fraction does this represent?

1 of the students buy pizza. On CAFETERIA On Wednesdays, __ 1 of the same number 4of students buy sandwiches. Which Fridays, __ 2 item is bought by more students?

FINANCE Ava made a $50 payment on her phone bill of $78. What fraction does this represent?

Copyright © by The McGraw-Hill Companies, Inc.

Correct the mistakes. 19

Eliza had a pizza that was cut into 8 equal slices. When Eliza asked her friend Cora how much pizza she would like, Cora said one-fourth. Eliza gave her four slices. How many slices should Eliza have given Cora?

20

When asked what fraction of letters in Sacramento are a’s, Carol answered 2. What was Carol’s mistake?

Chapter 1 Test

29

Chapter

1

Standards Practice

Select the best answer and fill in the corresponding circle on the sheet at right. 1

Selma shaded one section for each day she exercised. Which fraction of these days did she exercise?

5 C _ 12 3 D _ 4

1 A _ 2 5 B _ 7

2

4

Which does NOT equal one whole? 6 12 H _ F _ 6 12 8 4 G _ J _ 9 4

5

The Tigers beat the Royals in two out of three baseball games. Which shows the fraction of the games the Tigers won? 3 1 C _ A _ 3 5 2 2 B _ D _ 5 3

6

Which fraction is equal to one whole?

_

Which model represents 6 ? 8 F

H

G

J

3

Which symbol makes the sentence true? 1□1 5 3

_ _

7

1 H _ 2 6 J _ 9

Which fraction does the shaded part of the model represent?

A > B = C < D +

1 A _ 4 3 B _ 5

3 C _ 10 2 D _ 5

GO ON 30

Chapter 1 Standards Practice

Copyright © by The McGraw-Hill Companies, Inc.

5 F _ 5 3 G _ 8

8

Which symbol makes the sentence true? 1□ 1 9 12

12

_ _

F >

H <

G =

J +

Joey played 3 levels of his video game. There are 7 levels in all. Which shows the fraction of levels Joey played? 3 F _ 7 G 7 7

4 H _ 7 3 J _ 10

_

ANSWER SHEET 9

Which fraction does the model represent?

Copyright © by The McGraw-Hill Companies, Inc.

7 A _ 10 7 B _ 12

10

7 C _ 14 5 D _ 14

How many tiles like the one shaded below are there in 1?

F 1

H 12

G 10

J 24

Directions: Fill in the circle of each correct answer. 1

A

B

C

D

2

F

G

H

J

3

A

B

C

D

4

F

G

H

J

5

A

B

C

D

6

F

G

H

J

7

A

B

C

D

8

F

G

H

J

9

A

B

C

D

10

F

G

H

J

11

A

B

C

D

12

F

G

H

J

Success Strategy 11

Which symbol makes the sentence true? 1□1 6 3

_ _

A >

C -

B =

D <

Read each question completely. Then think of the answer in your head before looking at the answer choices.

Chapter 1 Standards Practice

31

Chapter

2

Equivalence of Fractions

1 of a pie, You see fractions every day, such as __ 3 1 of the cost, __ 2 of an 2inch __ of your team, __ 3 3 4 of rain, and so on. To compare fractions, you find equivalent fractions.

Copyright © by The McGraw-Hill Companies, Inc.

32

Chapter 2 Equivalence of Fractions

Boden/Ledingham/Masterfile

STEP

STEP

1 Quiz

2 Preview

Get ready for Chapter 2. Review these skills and compare them with what you’ll learn in this chapter.

What You Know

What You Will Learn

You know that two items can look different and yet be equal.

Lessons 2-1 and 2-5

five $10 bills

Copyright © by The McGraw-Hill Companies, Inc.

Are you ready for Chapter 2? Take the online readiness quiz at ca.mathtriumphs.com to find out.

Equivalent fractions have the same value. 2 are equivalent 1 and __ For example, __ 2 4 fractions.

$50

You know that there are 1 hamburgers below. 3 __ 2

Lesson 2-2

You know how to order the coins below from least to greatest.

Lessons 2-3 and 2-4

A mixed number has a whole number part and a fraction part. 1 whole number fraction 3__ 2

Order fractions by finding equivalent fractions that have common denominators . Then order the fractions from least to greatest. 3 1 , __ 2 , __ 1 , __ __ 6 2 3 4 6 ___ 9 8 ___ 2 , ___ ___ , , 12 12 12 12

33 (bkgd)Boden/Ledingham/Masterfile, (t)Michael Houghton/StudiOhio, (b)Mark Ransom/RansomStudios

Lesson

2-1 Equivalent Fractions and Equivalent Forms of One KEY Concept Equivalent fractions are fractions that have the same value . 1 unit

1 2 1 is equivalent to 2 . 2 4

2NS4.3 Know that when all fractional parts are included, such as four-fourths, the result is equal to the whole and to one. 3NS3.1 Compare fractions represented by drawings or concrete materials to show equivalency and to add and subtract simple fractions in context. 4NS1.5 Explain different interpretations of fractions, for example, parts of a whole, parts of a set, and division of whole numbers by whole numbers; explain equivalence of fractions.

VOCABULARY equivalent fractions fractions that represent the same number 6 3 Example: __ = __ 4 8 value the amount of a number

1 4 2 is equivalent to 3 . 4 6 1 6 3 is equivalent to 4 . 6 8

equivalent forms of one different expressions that represent the same number 2 Example: __ 2

1 8 4 is equivalent to 6 . 8 12 1 12 6 . So, 12 = 24 = 36 = 48 = 12

An equivalent form of one is any nonzero number divided by itself.

3 =1 3

7 =1 7

12 =1 12

When a fraction has the same numerator and denominator (except zero), the fraction always equals 1. Remember that any number multiplied by 1 or divided by 1 is equal to itself. 2 3 3 3 6 __ × 1 = __ → __ × __ = __ 4

4

4

2

8

3 6 So, __ = __ 8 4

You can name an endless number of fractions that are equivalent to any fraction. 34

Chapter 2 Equivalence of Fractions

Copyright © by The McGraw-Hill Companies, Inc.

2 =1 2

Example 1

YOUR TURN!

Complete the models to name an equivalent fraction.

Complete the models to name an equivalent fraction.

1 = ______ __ 3

1 = ______ __

6

2

1. What can you multiply the denominator 3 by to get a denominator of 6?

1. What can you multiply the denominator 2 by to get a denominator of 8?

2. Multiply 3 by 2 to get 6. So, multiply the 2 , an equivalent form of one. fraction by __ 2

2. What fraction should you multiply by to get an equivalent fraction?

1 _____ 1 · __ 2 2 3 __ = = __ 3

3 · 42

6

2. 1 = __ So, __ 3 6

1= 1 = ___ 1 · ______ = ______ So, __ __ 2

Example 2 Copyright © by The McGraw-Hill Companies, Inc.

8

_6. 8

10

2. Multiply the original fraction by another 3 equivalent form of one, such as __. 3 8 · 43

24

10

8 8 ·______ = ______ ___ = ____

16

· __ 3 18 6=6 ______ __ = ___

8 _ .

1. Multiply the original fraction by an 2. equivalent form of one, such as __ 2

· __ 2 3 6=6 12 ______ __ = ___

8

.

2

Name two fractions equivalent to

1. Multiply the original fraction by an 2. equivalent form of one, such as __ 2 8 · 42

8

YOUR TURN!

Name two fractions equivalent to

8

2·

10 ·

2. Multiply the original fraction by another 3 equivalent form of one, such as __. 3 8 8 ·______ = ______ ___ = ____ 10

10 ·

6 18 6 12 and __ = ___. So, __ = ___ 8 16 8 24 8 8 So, ___ = ______ and ___ = ______. 10 10

GO ON

Lesson 2-1 Equivalent Fractions and Equivalent Forms of One

35

Who is Correct? Write an equivalent fraction for

Palba

_ _ _

10 = 10 · 2 = 20 30 15 · 2 15

10 _ . 15

Felix

Troy

_ _ _

_ _ _

10 = 10 · 10 = 100 225 15 · 15 15

10 = 10 ÷ 5 = 2 3 15 ÷ 5 15

Circle correct answer(s). Cross out incorrect answer(s).

Guided Practice 1

3

Complete the models to name an 2 1 = ______ equivalent fraction. __ 4

2

Complete the models to name two 4. fractions equivalent to __ 6

1 shaded. This circle shows __ 4

3

Circle One = 8

1

Circle Two = 6

2

Circle Three = 6

Step by Step Practice 4

8. Name a fraction equivalent to ___ 10 Step 1 Choose an equivalent form of one by which to multiply the fraction.

36

Chapter 2 Equivalence of Fractions

8 8 ·______ = ______ ___ = ____ 10

10 ·

2

Circle Four = 8

Copyright © by The McGraw-Hill Companies, Inc.

Which fractional part of a circle 1? at the right is equal to __ 4

Complete to name an equivalent fraction. 5

6 2 = ______ __

7

8 1 = ______ __

3

2 6 _____ · ______ = ______ 3

2

6

4 = ______ __

8

2 = ______ 12 __

6

18

3

Step by Step Problem-Solving Practice Solve. 9

Taneesha and Zoe each ate the same amount 4 of her pizza. If Taneesha’s of their own pizzas. Zoe ate __ 6 pizza is divided into 12 equal pieces, how much did Taneesha eat?

NUTRITION

Understand

Problem-Solving Strategies ✓ Draw a diagram. Look for a pattern. Guess and check. Solve a simpler problem. Work backward.

Read the problem. Write what you know. Zoe ate of her pizza. Taneesha ate an amount.

Copyright © by The McGraw-Hill Companies, Inc.

Plan

Pick a strategy. One strategy is to draw a picture. 4. Draw 6 equal parts. Shade to show __ 6

Draw another figure with twice as many parts. 4. Shade to equal __ 6 Solve

Count the total parts and the shaded parts of the second figure. 4 ______ = __ 6

8 Taneesha ate ___ of her pizza. 12 Check

Does the answer make sense?

GO ON Lesson 2-1 Equivalent Fractions and Equivalent Forms of One

37

10

FOOD Sharon’s dad baked two same-sized pans of brownies for Sharon’s birthday party. One pan is cut into 12 equal pieces. The other pan is cut into 36 equal pieces. One serving from the pan of 12 pieces consists of 1 piece. How many pieces from the second pan equal one serving? Check off each step. Understand Plan Solve Check

11

12

5 Thomas receives __ of a 100-foot kite string. Jordan 9 2 of a different 100-foot receives __ string. Did they receive the same 3 amounts of string? Explain.

KITES

9 4 is equivalent to ___ because Lakimbre said that __ 5 10 4 + 5 = 9 and 5 + 5 = 10. Explain what was incorrect in Lakimbre’s thinking. Be sure to use the terms numerator and denominator in your explanation.

Copyright © by The McGraw-Hill Companies, Inc.

Skills, Concepts, and Problem Solving Complete each model to name an equivalent fraction. 13

14

2 = ______ __ 5

38

Chapter 2 Equivalence of Fractions

2 = ______ __ 3

Complete to name an equivalent fraction. 15

1 = ______ ___ 11

55

16

6 ______ __ =

18

2 __

20

5 __

22

1 ___

7

14

Name two equivalent fractions for each fraction. 17

1 __

19

3 __

21

3 __

2 5 4

3 7 10

Copyright © by The McGraw-Hill Companies, Inc.

Solve. 23

CONSTRUCTION Pam had two pieces of wood the same length. She cut the first piece into 6 equal parts. She cut the second piece into 10 equal parts. How many parts from the second piece of wood equal the same length as 3 parts from the first piece of wood?

24

HOBBIES Tonya and Lois sewed same-sized quilts. Lois’s quilt has 36 pieces; 9 of those are yellow. Tonya’s quilt has 12 yellow pieces that cover the same area as Lois’s 9 yellow pieces. How many pieces are in Tonya’s entire quilt?

25

SPORTS Two bicycle wheels are shown at the right. One has 4 spokes, and the other has 16. Shade in the area on the second tire to show the equivalent fraction for the shaded area of the first. Complete to show the equivalent fraction. 1 = ______ ______

GO ON Lesson 2-1 Equivalent Fractions and Equivalent Forms of One

39

Vocabulary Check sentence. 26

Write the vocabulary word that completes each

A(n) divided by itself.

is any nonzero number are fractions that have the same value.

27 28

Writing in Math Write a letter to your friend Olivia describing equivalent forms of one.

Spiral Review Compare. Write <, =, or > to make each a true statement. 29

1 __

1 __

2

4

30

1 __

1 ___

5

11

31

(Lesson 1-3, p. 19)

1 __

1 __

3

4

NUTRITION Katrina cut a cantaloupe into eight equal-sized sections. Her family ate three sections for breakfast. What part of the cantaloupe was eaten for breakfast? Has more of the cantaloupe been eaten or more saved for a later time?

34

CLEANING Mr. Ling has a house with 14 windows. He washed 9 of the windows Saturday morning before lunch. What part of the windows does he need to wash after lunch? Is that more or less than he has already washed?

Write a fraction to represent each situation.

(Lesson 1-1, p. 4)

35

the number of hands showing five fingers up

36

the number of open mailboxes

40

Chapter 2 Equivalence of Fractions

(bl)Guy Grenier/Masterfile, (br)David Young-Wolff/Photo Edit

37

the number of blue jerseys

1 __

1 __

7

2

Copyright © by The McGraw-Hill Companies, Inc.

33

32

Lesson

2-2 Mixed Numbers and Improper Fractions KEY Concept When the numerator of a fraction is greater than the denominator, it means you have more parts than the number of parts into which the whole was divided. In other words, you have more than one whole.

2NS4.3 Know that when all fractional parts are included, such as four-fourths, the result is equal to the whole or to one. 4NS1.5 Explain different interpretations of fractions, for example, parts of a whole, parts of a set, and division of whole numbers by whole numbers; explain equivalence of fractions. 5NS1.5 Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers.

VOCABULARY =

7 4

1

=

+

3 4

A fraction in which the numerator is greater than the 7 is an denominator is called an improper fraction . __ 4 improper fraction.

Copyright © by The McGraw-Hill Companies, Inc.

3 7 of a pizza is the same as 1__ pizzas. The You can see that __ 4 4 3 number 1__ is called a mixed number . 4

improper fraction a fraction with a numerator that is greater than or equal to the denominator 17 5 Examples: ___, __ 3 5 mixed number a number that has a whole number part and a fraction part Example: 1 6__ fraction 2 whole number

The numbers below the number line show two groups of fractions. Notice how the numerators differ from the denominators. 1 2

0

1 4

2 4

1

3 4

4 4

2

12

1 5 4

6 4

7 4

8 4

The numerators are less The numerators are greater than than the denominators. or equal to the denominators.

The values of the fractions to the right of 1 are greater than 1. These can be written as improper fractions or mixed numbers.

When solving a problem, you usually write your answer as a mixed number, not an improper fraction. GO ON Lesson 2-2 Mixed Numbers and Improper Fractions Envision/Corbis

41

Example 1 Change 3

_2 to an improper fraction using drawings. 5

1. Model the mixed number using circles.

5 __

5 __

5 __

2 __

5

5

5

5

2. 2. Count the number of fifths that make 3__ 5 There are 17 shaded fifths in the model. 17 . Write 17 fifths as ___ 5 YOUR TURN! Change 2

_3 to an improper fraction using drawings. 4

1. Model the mixed number using circles.

42

Chapter 2 Equivalence of Fractions

Copyright © by The McGraw-Hill Companies, Inc.

3 2. Count the number of fourths that make 2__. 4 How many shaded fourths are in the model? 3 Write 2__ as an improper fraction. 4

Example 2 Change

11 _ to a mixed number using fraction strips. 3

1. Model the improper fraction using strips. Mark strips to have three equal sections. wholes

3=1 __ 3

fraction

3=1 __

3=1 __

3

2 __

3

3

2. Write the whole number part of the mixed number.

3

3. Write the fraction part of the mixed number.

2 __

4. Write the whole number and fraction parts together. 2. The mixed number is 3__ 3

3 2 3__ 3

YOUR TURN! Change

17 _ to a mixed number using fraction strips. 7

1. Model the improper fraction using strips. Mark strips to have equal sections. Shade in each fraction strip. Copyright © by The McGraw-Hill Companies, Inc.

wholes

7= __ 7

fraction

7= __

______

7

2. Write the whole number part of the mixed number. 3. Write the fraction part of the mixed number. 4. Write the whole number and fraction parts together. What is the mixed number?

GO ON Lesson 2-2 Mixed Numbers and Improper Fractions

43

Example 3 Write 4

_2 as an improper fraction. 3

1. Find how many thirds are in 4. Multiply 4 by 3.

4 × 3 = 12

2. Add the number of thirds in the fraction. This is how many thirds are in the improper fraction.

12 + 2 = 14

3. Write the total number of thirds as an improper fraction.

2 = ___ 14 4__ 3 3 4×3+2 12 + 2 2 14 4__ = __________ = _______ = ___ 3 3 3 3

4. Write Steps 1–3 as one process. YOUR TURN!

the whole number of the mixed number

3 Write 4_ as an improper fraction. 5

4×

1. Find how many fifths are in 4. Multiply 4 by 5.

= the denominator of the mixed number

2. Add the number of fifths in the fraction. This is how many fifths are in the improper fraction. +3= 3. Write the total number of fifths as an improper fraction. +

+

4× ________________ = _____________ = ______ 5

5

5

Example 4 Write

64 _ as a mixed number. 3

1. Divide the numerator by the denominator. 2. The quotient is the whole number part of the mixed remainder number. The fraction part is written as __________. divisor 1 64 ___ = 21__ 3 3

44

Chapter 2 Equivalence of Fractions

denominator and divisor

quotient numerator

remainder

Copyright © by The McGraw-Hill Companies, Inc.

4. Write Steps 1–3 as one process.

YOUR TURN! Write

32 _ as a mixed number. 5

1. Divide the numerator by the denominator. 2. What is the quotient? This is the whole number part of the mixed remainder number. Write the fraction part as __________. divisor

Example 5 Graph 2

32 ___ = 5

YOUR TURN!

_3 on a number line.

Graph 3

4

1. Write the mixed number as an improper fraction. 3 2 × 4 + 3 8 + 3 11 2__ = _________ = _____ = ___ 4 4 4 4 2. Draw a number line with marks in fourths. 3 3. Draw a dot for 2__. 4

_2 on a number line.

3 1. Write the mixed number as an improper fraction. × + 2 = _____________________ = 3__ 3 +

______________ = ______

3

24

Copyright © by The McGraw-Hill Companies, Inc.

3 4

4 4

5 4

6 4

7 4

8 4

9 4

10 4

11 4

12 4

13 4

2. Draw a number line with marks in thirds. 2. 3. Draw a dot for 3__ 3 2

33 3 3

4 3

5 3

6 3

7 3

8 3

9 3

10 3

11 3

12 3

13 3

Who is Correct? Write 3

_3 as an improper fraction. 4

Belinda

_ _ _ _

3 3·4+3 = 3 = 4 4 12 + 3 = 15 4 4

Aretha

_ _ _ _

3·4+3 = 3 3 = 4 4 21 = 7 4 4

Amato

_ _ _ _

3 3·4+3 = 3 = 3 4 21 = 7 3 3

GO ON

Lesson 2-2 Mixed Numbers and Improper Fractions

45

Guided Practice 1

1 to an improper fraction using drawings. Change 3__ 2

1 = ______ 3__ 2

2

14 to a mixed number using drawings. Change ___ 4

14 = ___

______

4

Step by Step Practice 3

2 as an improper fraction. Write 9__ 3 Step 1 Find out how many thirds are in 9. Add the number of thirds in the fraction. × + Step 2 Write the total number of thirds as an improper fraction.

Write each mixed number as an improper fraction. 4

· + 3 1__ = __________________ = ______ 7

5

3 2__ = ______ 8

6

8 2__ = ______ 9

7

1 = ______ 7__ 2

10

31 ___ =

Write each improper fraction as a mixed number. 8

46

7= __

______

4

Chapter 2 Equivalence of Fractions

9

14 = ___ 6

______

3

______

Copyright © by The McGraw-Hill Companies, Inc.

· + 2 = __________________ = ______ 9__ 3

11

7 on the number line. Graph 1__ 8 8 8

9 8

10 8

11 8

12 8

13 8

14 8

15 8

Step by Step Problem-Solving Practice NUMBER SENSE Akiko had friends over for quesadillas. The quesadillas were divided into thirds. The friends ate a total of 22 pieces. Write an improper fraction and a mixed number for the total amount of quesadillas the friends ate. Understand

18 8

Draw a diagram. Look for a pattern. Guess and check. Act it out. ✓ Work backward.

Read the problem. Write what you know. The quesadillas are cut into The friends ate

.

pieces.

Plan

Pick a strategy. One strategy is to work backward. Start by writing the improper fraction. Use the improper fraction to write the mixed number.

Solve

Write the improper fraction representing the amount the friends ate.

Copyright © by The McGraw-Hill Companies, Inc.

17 8

Problem-Solving Strategies

Solve. 12

16 8

______

number of pieces eaten number of pieces in each quesadilla

Divide the numerator by the denominator to write the mixed number. 3 22 -

The friends ate ______ or Check

______ quesadillas.

You can draw the quesadillas to verify your answer.

GO ON

Lesson 2-2 Mixed Numbers and Improper Fractions

47

13

CRAFTS Janice had ribbon for a craft project. She cut each ribbon into 8 equal parts. She used 23 of the parts. Write the number of ribbon parts she used as an improper fraction and a mixed number. Check off each step. Understand Plan Solve Check

14

15

MANUFACTURING FunForAll Games packages 3 number cubes in each game box. At the end the day, one production line had 14 number cubes left. Write the number of game boxes that can be completed with the leftover number cubes as an improper fraction and a mixed number. 3 11 . Explain your diagram Draw a diagram showing 2__ = ___ 4 4 using the words improper fraction, mixed number, numerator, and denominator.

Change each mixed number to an improper fraction using drawings. 16

5 2__ = ______ 6

48

Chapter 2 Equivalence of Fractions

17

2 = ______ 3__ 3

Copyright © by The McGraw-Hill Companies, Inc.

Skills, Concepts, and Problem Solving

Write each improper fraction as a mixed number. 18

5 __ =

19

2

11 = ___ 2

20

10 ___ =

23

3 2__ = 5

3

Write each mixed number as an improper fraction. 21

1= 1__ 2

22

2= 4__ 3

Solve. 24

1 on a number line. Graph 2__ 5

Copyright © by The McGraw-Hill Companies, Inc.

4 5

5 5

6 5

7 5

8 5

9 5

10 5

11 5

12 5

25

CARPENTRY Jason had pieces of wood that were the same length. He cut each piece into 6 equal parts. He used 15 of the parts to make a birdhouse. Write the number of pieces of wood he used as an improper fraction and a mixed number.

26

FOOD Each pie at a party was cut as shown. At the party, 33 pieces of pie were eaten. Write the number of pies that were eaten as both an improper fraction and a mixed number.

Vocabulary Check each sentence. 27

13 5

14 5

Write the vocabulary word that completes

A(n) is a fraction with a numerator that is greater than or equal to the denominator.

28

3 The number 3__ is called a(n) 4

29

Writing in Math

.

1 pound of coffee. A coffee shop charges $2 for __ 4 3 __ How much do they charge for 1 pounds of coffee? Explain. 4

Spiral Review Name two equivalent fractions. 30

1 __ 5

(Lesson 2-1, p. 34)

31

1 __ 3

Lesson 2-2 Mixed Numbers and Improper Fractions Bonhommet/PhotoCuisine/Corbis

49

Chapter

Progress Check 1

2

(Lessons 2-1 and 2-2)

Name two equivalent fractions. 1

1 __

3

2 __

5

3 5

2

3 __

4

4 ___

4 11

Complete the model to name two 2. fractions equivalent to __ 3

Write each improper fraction as a mixed number. 6

9 __ =

7

13 ___ =

8

15 ___ =

9

12 = ___

10

33 ___ =

11

40 ___ =

2

7

5 8

4 7

Write each mixed number as an improper fraction. 1= 6__ 2

13

2= 8__ 3

14

1= 2__ 7

15

3 5__ = 4

16

5 1__ = 8

17

3 10__ = 7

Graph each mixed number on a number line. 18

2 1__ 4

1 4

0

19

1 1__ 8

2

1 2 4

3 4

4 4

5 4

6 4

7 4

8 4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

6 2 miles. Lydia swam __ miles. Did they Ruri swam 1__ 3 3 swim the same length?

20

HOBBIES

50

Chapter 2 Equivalence of Fractions

Copyright © by The McGraw-Hill Companies, Inc.

12

Lesson

2-3 Least Common Denominator (LCD) and Greatest Common Factors KEY Concept A multiple of a number is the product of that number and any whole number. To find the least common multiple (LCM) of two or more numbers, list the multiples of each number. Then find the least multiple the numbers have in common. Multiples of 8

Multiples of 6

Multiples of 12

8×1=8

6×1=6

12 × 1 = 12

8 × 2 = 16

6 × 2 = 12

12 × 2 = 24

8 × 3 = 24

6 × 3 = 18

12 × 3 = 36

8 × 4 = 32

6 × 4 = 24

12 × 4 = 48

The least common number in all three lists is 24. The LCM is 24. The least common denominator (LCD) of a set of fractions is the LCM of their denominators. 3 1 5 , and ___ is 24. The LCD for the fractions __, __ 8 6 12

Copyright © by The McGraw-Hill Companies, Inc.

The greatest common factor (GCF) of two whole numbers is the greatest number that is a factor of both numbers. Factors of 42

Factors of 54

1 × 42 2 × 21 3 × 14 6×7

1 × 54 2 × 27 3 × 18 6×9

The greatest common number in both lists is 6. The GCF of 42 and 54 is 6. Every composite number can be written as a product of prime numbers . A prime number is a whole number greater than one whose only factors are one and itself.

4NS1.5 Explain different interpretations of fractions, for example, parts of a whole, parts of a set, and division of whole numbers by whole numbers; explain equivalence of fractions.

VOCABULARY least common multiple (LCM) the least whole number greater than 0 that is a common multiple of two or more numbers Example: The LCM of 2 and 3 is 6. least common denominator (LCD) the least common multiple of the denominators of two or more fractions, used as a denominator 1 Example: The LCD of ___ 12 1 and __ is 24. 8 greatest common factor (GCF) the greatest number that is a factor of two or more numbers Example: The GCF of 30 and 75 is 15. composite number a number greater than 1 with more than two factors Example: 4 and 6 prime number any whole number greater than 1 with exactly two factors, 1 and itself Examples: 7, 13, and 19 prime factorization expressing a composite number as a product of its prime factors Example: 30 = 3 × 2 × 5

GO ON Lesson 2-3 Least Common Denominator (LCD) and Greatest Common Factors

51

The prime factorization of a number is the product of its prime factors. One way to find the prime factorization is to use a factor tree. Write the number that is being factored at the top.

Continue to factor any number that is not a prime number.

20

12 3

4 2

5

4 2

12 = 3 × 2 × 2

2

Choose any pair of whole number factors.

2

20 = 5 × 2 × 2

Example 1

YOUR TURN!

Find the least common denominator (LCD) 1 1 1 of , , and . 4 5 8

Find the least common denominator (LCD) 1 1 1 of , , and . 2 5 10

Find the LCM of the denominators.

Find the LCM of the denominators.

1. List the multiples of each denominator.

1. List the multiples of each number.

__

_

2. Circle the numbers that are common to all three lists. The least of these numbers is the LCM. 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40 , 44, … 5: 5, 10, 15, 20, 25, 30, 35, 40 , 45, … 8: 8, 16, 24, 32, 40 , 48, … The LCM of 4, 5, and 8 is 40. 1 , and __ 1 is 40. 1 , __ The LCD of __ 8 4 5

52

Chapter 2 Equivalence of Fractions

_

2: 5: 10: 2. Circle the numbers that are common to all three lists. The least of these numbers is the LCM. 2: 5: 10: What is the LCM of 2, 5, and 10? 1 , __ 1 , and ___ 1? What is the LCD of __ 2 5 10

Copyright © by The McGraw-Hill Companies, Inc.

4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, … 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, … 8: 8, 16, 24, 32, 40, 48, …

__

Example 2

YOUR TURN!

Find the GCF of 20 and 36 by using prime factors.

Find the GCF of 25 and 60 by using prime factors.

1. Write the prime factorization.

1. Write the prime factorization.

20 5

36 4

6

25 6

3

5

60 6

10

3

2. The common prime factors are 2 and 2. So, the GCF of 20 and 36 is 2 × 2 or 4.

2. The common prime factor is

.

So, the GCF of 25 and 60 is

.

Example 3

YOUR TURN!

Find the GCF of 42 and 56 by making a list.

Find the GCF of 63 and 77 by making a list.

Copyright © by The McGraw-Hill Companies, Inc.

1. List the factors by pairs. Factors of 42

Factors of 56

1 , 42

1 , 56

2 , 21

2 , 28

3, 14

4, 14

6, 7

7 ,8

2. The common factors are 1, 2, 7, and 14. The GCF of 42 and 56 is 14.

1. List the factors by pairs. Factors of 63

Factors of 77

2. The common factors are The GCF of 63 and 77 is

. .

GO ON Lesson 2-3 Least Common Denominator (LCD) and Greatest Common Factors

53

Who is Correct? Find the LCD of

11 _5, _4, and _ . 6 9

12

Vera Factors of 6: 1, 2, 3, 6 Factors of 9: 1, 3, 9 6, 12 Factors of 12: 1, 2, 3, 4, 3. is D LC e Th 3. is The GCF

Pamela

King

6 x 9 x 12 = 648. The LCD is 648.

, 24, 30, 36 Multiples of 6: 6, 12, 18 , 27, 36 18 Multiples of 9: 1, 9, 24, 36 , 12 Multiples of 12: 1, is 36. D LC The LCM is 36. The

Circle correct answer(s). Cross out incorrect answer(s).

Guided Practice 1

Find the LCM of 3, 5, and 6.

Find the LCM of 10, 4, and 8.

2

Multiples of 10:

Multiples of 5:

Multiples of 4:

Multiples of 6:

Multiples of 8:

Step by Step Practice 3

Find the GCF of 3, 12, and 36 by using prime factors. Step 1 Write the prime factorization. Step 2 The common prime factor is . So, the GCF of 3, 12, and . 36 is

54

Chapter 2 Equivalence of Fractions

3

12

36

Copyright © by The McGraw-Hill Companies, Inc.

Multiples of 3:

Find the GCF of each set of numbers. 4

3 and 9 Factors of 3: Factors of 9:

5

10 and 15

6

48 and 60

8

3, 6, and 7

Find the LCM of each set of numbers. 7

2, 5, and 7

Step by Step Problem-Solving Practice

Problem-Solving Strategies Look for a pattern.

Solve. 9

MONEY The list below shows the amounts of money that the club leader collected. Each member paid the same amount to attend a field trip. What is the most the trip could cost per member? Explain. Understand

Copyright © by The McGraw-Hill Companies, Inc.

✓ Use logical reasoning.

Read the problem. Write what you know. The club leader collected: Each member paid the amount.

Plan

Pick a strategy. One strategy is to use logical reasoning. You need to know the amount each member paid. Logical reasoning tells you that you must find the greatest common factor of the amount of money collected each day.

Solve

Write the factors of each amount. Factors of 42

Factors of 12

The GCF of 42, 12, and 60 is Check

Solve a simpler problem. Work backward. Act it out.

.

Money Collected from Members Monday

$42

Tuesday

$12

Wednesday

$60

Factors of 60

.

Did you answer the question? If each member paid , could the club leader collect the amounts listed? GO ON Lesson 2-3 Least Common Denominator (LCD) and Greatest Common Factors

55

10

FOOD Su wants to ship baked goods to her cousin. She wants to ship 14 oatmeal squares, 12 granola bars, and 8 lemon bars. She can pack only one type of baked good in each box. She must pack the same number of baked goods in each box. What is the greatest number of baked goods that Su can pack in each box? Check off each step. Understand Plan Solve Check

11

5 1 , __ 1 , and __ CARPENTRY Three boards have thicknesses of __ inch. 2 4 8 What is the common denominator of the measures of the thickness of the boards?

Explain the relationship between the LCD and the LCM. Use an example.

12

Find the LCM of each set of numbers. 13

2, 3, 8

14

4, 5, 6

15

5, 6, 9, 10

16

3, 6, 18

56

Chapter 2 Equivalence of Fractions

Copyright © by The McGraw-Hill Companies, Inc.

Skills, Concepts, and Problem Solving

Find the LCD of each set of fractions. 3 5 17 __ and __ 6 4

18

5 1 and ___ __ 8

12

19

3 3 1 , __ __ , and __

20

3 __ 2 __ , 1 , and __

21

3 __ 3 7 , and ___ __ , 2 , __

22

3 __ 5 2 __ , 1 , __, and __

23

3 __ 3 5 2 , __ __ , , and ___

24

3 7 5 3 __ __ , , ___ , and ___

3 4

5

4 3 9

3 4 8

12

12

4 3

9

4 3 6

9

4 5 10

12

Solve.

Copyright © by The McGraw-Hill Companies, Inc.

25

FLOWERS Rayna is planting flowers. She has enough flowers to plant 6, 7, or 14 in each row. What is the least number of flowers she could have?

BASKETBALL Use the table at the right for Exercises 26 and 27. The numerator is the number of free throws made and the denominator is the number of free throws attempted. 26

If you were to rewrite the stats to be fractions with the least common denominator (LCD), what is the LCD of these fractions?

Last Night’s Basketball Stats Player

27

Douglas

9 ___

Simpson

1 __

Torres

4 __

Rollins

8 ___

Rewrite each fraction with the LCD. 9 ___ = ______ 12

4 = ______ __ 6

1 = ______ __ 5

8 ___ = ______ 10

Free Throws

12 5 6 10 GO ON

Lesson 2-3 Least Common Denominator (LCD) and Greatest Common Factors

57

Vocabulary Check each sentence.

Write the vocabulary word that completes

28

is expressing a composite number as a product of its prime factors.

29

The for two or more fractions is the least common multiple of denominators of the fractions.

30

Writing in Math Explain the difference between finding the LCM and the GCF of two numbers.

Spiral Review Write each improper fraction as a mixed number. 5 11 31 __ = 32 ___ = 2 2

(Lesson 2-2, p. 41)

Write each mixed number as an improper fraction. 1 2 34 1__ = 35 4__ = 2 3

33

10 ___ =

36

3 2__ = 5

3

(Lesson 2-2, page 41)

Solve. 40

1 of the people are under the age POPULATION In Wallyville, __ 1 of the 7people are under the age of 10. In of 10. In Hoppsburg, __ 3 1 of the people Salston, __ are under the age of 10. Which has the 8 least fraction of people under the age of 10? (Lesson 1-3, p. 19)

41

INVITATIONS On Monday, Hannah told Austin that their wedding invitations had to be mailed Friday. They decided to work an equal amount of time each night to get the invitations ready. What fraction of the invitations do they need to finish each night?

58

Chapter 2 Equivalence of Fractions

Copyright © by The McGraw-Hill Companies, Inc.

Use each given size of fraction tiles to form a rectangle equal to 1. Write the fraction that equals 1. (Lesson 1-2, p. 11) 1 1 1 37 __ fraction tiles 38 __ fraction tiles 39 __ fraction tiles 5 2 6

Lesson

2-4 Compare and Order Fractions KEY Concept One way to compare fractions is to look at models. < means “is less than” > means “is greater than” 1 2

>

1 4

one half

is greater than

one fourth

= means “is equal to”

Another way to compare fractions is to find equivalent fractions that have the same denominators. Then compare the numerators. Fractions with common denominators are fractions that have the same denominator. Look again at the pizza slices.

_1 4

3NS3.1 Compare fractions represented by drawings or concrete materials to show equivalency and to add and subtract simple fractions in context. 6NS1.1 Compare and order positive and negative fractions, decimals, and mixed numbers and place them on a number line.

VOCABULARY common denominators the same denominator (bottom number) used in two or more fractions equivalent forms of one different expressions that represent the same number 3 Example: __ 3 (Lesson 2-1, p. 34)

least common multiple (LCM) the least whole number greater than one that is a common multiple of each of two or more numbers Example: The LCM of 3 and 5 is 15.

Copyright © by The McGraw-Hill Companies, Inc.

(Lesson 2-3, p. 51)

32 × __ 2 1 = 1______ __ = __ 2

2 × 42

4

Because the denominators are equal, compare the numerators. 2 > __ 1 __ 4 4 To order fractions, rename the fractions using their LCD. Then compare the numerators of the fractions.

Recall that you can write equivalent fractions by multiplying by an equivalent form of one . GO ON Lesson 2-4 Compare and Order Fractions Envision/Corbis

59

Example 1

_

_

3 2 Use <, =, or > to compare and . 8 3 Shade the models given. 3 1. The circle on the left has 8 sections. Use it to model __. 8 Shade 3 sections. 2. The circle on the right has 3 sections. Use it to 2 . Shade 2 sections. model __ 3 3. Compare the shaded areas. The symbol < means “is less than.” 3 __ __ <2 8 3

3 __ 8

<

2 __ 3

YOUR TURN!

_

_

2 4 Use <, =, or > to compare and . 5 7 Shade the models given. 1. How many sections does the top fraction strip have? 2 . How many sections should you shade? Use it to model __ 5 2. How many sections does the bottom fraction strip have?

3. Compare the shaded areas. Use <, =, or > to write a statement. 2 4 __ __ 5 7

60

Chapter 2 Equivalence of Fractions

Copyright © by The McGraw-Hill Companies, Inc.

4 . How many sections should you shade? Use it to model __ 7

Example 2

YOUR TURN!

Use <, =, or > to compare

_2 and _4.

_5 and _2.

5 3 Rename the fractions using a common denominator.

6 3 Rename the fractions using a common denominator.

1. Find the LCM of 3 and 5.

1. Find the LCM of 6 and 3.

Multiples of 3

Multiples of 5

3×1=3

5×1=5

3×2=6

5 × 2 = 10

3×3=9

5 × 3 = 15

Multiples of 6

2. Rename each fraction as an equivalent fraction with the same denominators.

3 × 5 = 15 The LCD is 15.

5 = ____ 5 ·______ = ______ __

2. Rename each fraction as an equivalent fraction with a denominator of 15. · __ 5 10 2 2 3 __ = ______ = ___ 3

3 · 45

5 · 43

6

6·

2 = ____ 2 ·______ = ______ __

15

3

· __ 12 4 4 3 __ = ______ = ___ 5

Multiples of 3

What common denominator should you use?

3 × 4 = 12

Copyright © by The McGraw-Hill Companies, Inc.

Use <, =, or > to compare

15

3. Compare the numerators of the equivalent fractions.

10 < 12

Write a statement using the equivalent fractions.

10 ___ ___ < 12

4. Replace the equivalent fractions with the original fractions.

2 < __ 4 __

15 3

15

5

3·

3. Compare the numerators of the equivalent fractions. Write a statement using the equivalent fractions. 4. Replace the equivalent fractions with the original fractions.

GO ON Lesson 2-4 Compare and Order Fractions

61

Example 3 Order the fractions

7 _5, _3, and _ from least to greatest. 6 8

12

1. Find the LCM of 6, 8, and 12.

Multiples of 6: 6, 12, 18, 24, 30, … Multiples of 8: 8, 16, 24, 32, … Multiples of 12: 12, 24, 36, …

The least number in all three lists of multiples is 24. The LCM is 24. 2. Write equivalent fractions that have 24 as their denominators. 5 5 · __ 4 3 20 __ = ______ = ___ 6 · 44

6

· __ 3 3 3 9 __ = ______ = ___ 8 · 43

8

24

24

7 · __ 2 7 14 3 ___ = _______ = ___ 12

3. Compare the numerators of the equivalent fractions.

12 · 42

24

9 < 14 < 20

4. Order the fractions from least to greatest. 20 3 5 9 14 < ___ 7 < __ ___ < ___ means that __ < ___ .

8 12 6 24 24 24 5 3 7 __ , . So, from least to greatest, the fractions are __, ___ 8 12 6 YOUR TURN! Order the fractions

4 _2, _7, and _ from least to greatest. 3 9

18

1. Find the LCM for the denominators.

What is the least number in all three lists of multiples? The LCM is . 2. Write equivalent fractions that have 2· 2 __ = __________ = ______ 3

3·

in the denominators.

7· 7 __ = _________ = ______ 9

9·

4· 4 ___ = __________ = ______ 18

18 ·

3. Compare the numerators of the equivalent fractions. <

<

4. Order the fractions from least to greatest. means that So, from least to greatest, the fractions are 62

Chapter 2 Equivalence of Fractions

. .

Copyright © by The McGraw-Hill Companies, Inc.

Multiples of 3: Multiples of 9: Multiples of 18:

Who is Correct? Use <, =, or > to compare

_4 and _4. 5

8

Abigail

Erina

32 ·8 =_ _4 = 4_ 40 5·8 5 20 5 · =_ _4 = 4_ 40 5 · 8 8 20 32 > _ _

_4 > _4

40

8

5

Anton

_ _

4 4 4 = 4, so 5 = 8

40

Circle correct answer(s). Cross out incorrect answer(s).

Guided Practice Use <, =, or > to compare the fractions. Shade the models given.

Copyright © by The McGraw-Hill Companies, Inc.

1

5 __

3 __

6

4

2

3 __

4 ___

5

10

Step by Step Practice 3

2 and __ 4 . Rename the fractions Use <, =, or > to compare __ 6 9 using a common denominator. Step 1 Find the LCM of 6 and 9. Step 2 Rename each fraction to an equivalent fraction. ·

2 ____________ ______ __ = = 6

·

·

4 ____________ ______ __ = = 9

·

Step 3 Compare the numerators. Write a statement using the equivalent fractions. Step 4 Replace the equivalent fractions with the original fractions.

2 __

4 __

6

9 GO ON

Lesson 2-4 Compare and Order Fractions

63

Order the fractions from least to greatest. 5 5 3 4 __, __, and ___ 6 8 12

5

5 2 , and __ 1 , __ __ 2 3

9

·

______ = ____________ = ______

· ·

______ = ____________ = ______

· ·

______ = ____________ = ______

·

Step by Step Problem-Solving Practice Solve. 6

Peter and Paul are twins. They are very close 1 inches tall, and Paul is 5 feet in height. Peter is 5 feet 2__ 4 5 2__ inches tall. Who is taller? 8 FITNESS

Understand

inches tall.

Draw a bar divided into 4 equal parts. Draw another bar of equal length. Divide it into 8 equal parts. 5 1 and __ . Compare the shaded parts. Shade __ 8 4 5 1 5 feet 2__ 5 feet 2__ 8 4 So, Paul is taller than Peter. Check

64

Does the answer make sense? Did you answer the question?

Chapter 2 Equivalence of Fractions

Copyright © by The McGraw-Hill Companies, Inc.

Pick a strategy. One strategy is to draw a diagram. Compare the two heights. Both heights are 5 feet and 2 inches. Only compare the fractional parts of each boy’s height.

Solve

Look for a pattern. Guess and check. Act it out. Solve a simpler problem.

Read the problem. Write what you know. Peter is 5 feet inches tall. Paul is 5 feet

Plan

Problem-Solving Strategies ✓ Draw a diagram.

7

FOOD The school cafeteria has two same-sized pans. One has a taco casserole that is cut into 36 equal pieces. Another has macaroni and cheese that is cut into 24 equal pieces. At lunch, 24 pieces of the taco casserole are eaten, and 21 pieces of the macaroni are eaten. Which pan has more left in it? Check off each step. Understand Plan Solve Check

8

5 6 On Wednesday, it rained __ inch in Columbus and ___ inch 8 12 in Cleveland. Which city got more rain?

WEATHER

5 3 Sarah said that __ is greater than __ because 5 > 3 and 8 4 8 > 4. Draw diagrams. Explain if Sarah is correct.

9

Copyright © by The McGraw-Hill Companies, Inc.

Skills, Concepts, and Problem Solving Use <, =, or > to compare the fractions. Shade the models given. 5 4 __ 10 ___ 8 10

11

6 ___

3 __

10

7

GO ON Lesson 2-4 Compare and Order Fractions

65

Use <, =, or > to compare the fractions. Shade the models given. 3 1 1 4 __ __ 12 __ 13 __ 7 4 4 8

Order the fractions from least to greatest. 14

5 1 and __ __

15

5 1 and ___ __

16

3 1 , and __ 1 , __ __

17

1 , and __ 2 1 , __ __

18

3 2 , __ 7 , and ___ 1 , __ __

19

5 1 , __ 2 1 , __ __ , and __

20

3 __ 3 5 2 , __ __ , , and ___

21

3 ___ 5 1 , __ __ , 7 , and ___

6

4

3 4

5

2 3 9 3 4 8

12 12

8

12

2 3

9

2 3 6

2 5 10

9

12

Solve. 22

5 3 Kelly needs 2__ cups of flour to make a cake. She has 2__ 8 4 cups. Does she have enough? Explain. BAKING

ENTERTAINMENT Clara and Gwen went to the county fair. There was a pie-eating contest. The top three contestants ate 5 7 pies, respectively. Which contestant 11 pies, and 5__ 5__ pies, 5___ 6 8 12 ate the most pie? Who ate the least?

24

SPORTS

Felipe and Jamal each collected 6 soccer balls from the 1. 1 > __ field. Divide the set of soccer balls to show that __ 2 3 Circle 1 of the set of 6 soccer balls. 2

66

Circle 1 of the set of 6 soccer balls.

Chapter 2 Equivalence of Fractions

Getty Images

3

Copyright © by The McGraw-Hill Companies, Inc.

23

Vocabulary Check sentence.

Write the vocabulary word that completes each

25

A(n) two or more fractions.

is the same denominator in

26

The is the least number greater than 1 that is a common multiple of each of two or more numbers.

27

Writing in Math

3 9 1 , __ Graph __ , and ___ on the number line below. 2 5 10 Explain your reasoning. 0

1

Spiral Review

Copyright © by The McGraw-Hill Companies, Inc.

Solve.

(Lesson 2-1, p. 34)

28

BAKING Ollie baked a pumpkin pie and a blueberry pie. Both pies were the same size. He cut the pumpkin pie into 12 equalsized pieces. He cut the blueberry pie into 8 equal-sized pieces. How many pieces of the pumpkin pie equal one-fourth of the pie? How many pieces of the blueberry pie equal one-fourth of the pie?

29

GROCERIES A 5-pound bag of medium-sized potatoes has an average of 20 potatoes. A 2-pound bag of small potatoes has an average of 16 potatoes. How big are the potatoes in each bag?

Draw a picture to model each fraction. Use equal parts of a whole or sets. (Lesson 1-1, p. 4) 2 7 30 __ 31 ___ 3 10

32

6 __ 6

33

1 __ 4 Lesson 2-4 Compare and Order Fractions

67

Chapter

Progress Check 2

2 1

(Lessons 2-3 and 2-4)

Write the prime factorization for 56.

Find the GCF of each set of numbers. 2

24 and 36

3

12, 20, and 32

5

3 _5_ and ___

Find the LCD of each set of fractions. 4

_1_ and _5_

2

6

8

20

Use <, =, or > to compare the fractions. Rename the fractions using a common denominator. 3 2 7 ___ _1_ 6 __ 7 __ 9 5 10 4 Order the fractions from least to greatest. _1_ and _2_

10

_2_, _2_, and _3_

12

3 3 _3_, _1_, ___ , and __

3

5

3 5

8 4 10

4

5

9

7 _5_ and ___

11

_4_, _2_, and _3_

13

9 7 , _1_, and ___ _4_, ___

9

18

9 3

5 20 2

4

10

Solve. 14

$"5&(03: Housing

One art museum features four types of paintings. 3 Modern art is _1_ of the collection. African art is ___ of the 2 16 2 is early collection. Asian art is _1_ of the collection and ___ 32 4 European. Which type of art has the least number of ART

pieces in the museum? 68

Chapter 2 Equivalence of Fractions

Food and Health Care Savings

2 5

/

15

1 6

/

Entertainment

#6%(&5 1 3

/

MONEY Mrs. Hernandez’s monthly budget is shown at the right. The most money is spent on food and health care. Which category has the next most money spent?

Copyright © by The McGraw-Hill Companies, Inc.

8

Lesson

2-5 Simplify Fractions

3NS3.1 Compare fractions represented by drawings or concrete materials to show equivalency and to add and subtract simple fractions in context. 4NS1.5 Explain different interpretations of fractions for example, parts of a whole, parts of a set, and division of whole numbers by whole numbers; explain equivalence of fractions.

KEY Concept A fraction is in simplest form or lowest terms when the numerator and the denominator of the fraction have no common factor other than 1. How many fourths of this pizza are left to eat?

2 4

VOCABULARY

How much of this pizza is left to eat?

1 2

simplest form a fraction in which the numerator and the denominator have no common factor greater than 1 3 Example: __ is the 5 6 simplest form of ___. 10

simplest form

greatest common factor (GCF) the greatest number that is a factor of two or more numbers Example: The greatest common factor of 12, 18, and 30 is 6.

Copyright © by The McGraw-Hill Companies, Inc.

1 is written in simplest form The pizzas are equal in size. __ 2 because there is no number that will evenly divide into both the numerator and the denominator.

Any fraction can be written in simplest form by dividing the numerator and denominator by the GCF. You can also simplify using models and prime factorization.

Example 1

_

8 Write in simplest form. Use models. 10 Shade an equivalent area and name the simplified fraction.

fraction

=

simplest form

8 ___

=

4 __

1. Shade the circle segments so the shading in both circles covers the same amount of each circle. 2. Count the total number of parts and the shaded parts of the circle on the right. Write the equivalent fraction.

10

5 GO ON

Lesson 2-5 Simplify Fractions Envision/Corbis

69

YOUR TURN! 3 Write in simplest form. Use models. 15 Shade an equivalent area and name the simplified fraction.

_

fraction

=

simplest form

3 ___

=

______

1. Shade the circle segments so the shading in both circles covers the same amount of each circle. 2. Count the total number of parts and the shaded parts of the circle on the right. Write the equivalent fraction.

15

Example 2 Write

12 _ in simplest form. Divide by the GCF.

Remember a factor is a number that is multiplied by another number to get a product. Example: 2 and 3 are factors of 6.

20

1. List all the factors of the numerator and the denominator. Factors of 12: 1, 2, 3, 4, 6,12

Factors of 20: 1, 2, 4, 5, 10, 20

2. The greatest common factor (GCF) is the greatest factor the numbers have in common. The greatest number in both lists is 4. Because 4 is the greatest number that will evenly divide both the numerator and the denominator, 4 is the GCF.

20

Both can be divided by 2 and 4.

÷ __ 4 3 12 12 3 ___ = __ = ________ 20

20 ÷ 44

5

YOUR TURN! Write

15 _ in simplest form. Divide by the GCF. 50

1. List all the factors of the numerator and the denominator. Factors of 15:

Factors of 50:

2. What is the greatest number in both lists? The GCF of 15 and 50 is . 3. Divide the numerator and denominator by the GCF. 15 4. What is the simplest form of ___? 50 70

Chapter 2 Equivalence of Fractions

15 ÷______ ______ 15 = ______ ___ = 50

50 ÷

Copyright © by The McGraw-Hill Companies, Inc.

3. Divide the numerator and denominator by the GCF. 3 12 in simplest form is __ 4. So, ___ . 20 5

12 ___

Example 3 Write

Write the number that is to be factored at the top.

18 _ in simplest form. Use prime factorization. 30

1. Use a factor tree to write the numerator as a product of prime numbers. 2. Use a factor tree to write the denominator as a product of prime numbers.

30

18 3

3. Replace the numerator with its prime factors. Replace the denominator with its prime factors. Find all equivalent forms of 1. 18 = _________ 3· 2·3 ___

6 2

3 3

10 2

5

Choose any pair of whole number factors of 18 and 30.

Continue to factor each number that is not a prime number.

3· 2·5

30

4. Eliminate all equivalent forms of 1. ·2·3 18 = 3 ________ ___

18 __ 3 ___ =

3·2·5

30

5

30

YOUR TURN! Write

12 _ in simplest form. Use prime factorization. 42

Copyright © by The McGraw-Hill Companies, Inc.

1. Use a factor tree to write the numerator as a product of prime numbers.

12

42

2. Use a factor tree to write the denominator as a product of prime numbers. 3. Replace the numerator with its prime factors. Replace the denominator with its prime factors. Find all equivalent forms of 1. ·

·

·

·

12 = _________________ ___ 42

4. Eliminate all equivalent forms of 1. 12 = ___ 42

GO ON Lesson 2-5 Simplify Fractions

71

Who is Correct? Write

18 _ in simplest form. Use any method. 45

Julio

Doug

_ _ _ _ _

·3·2 18 = 3_ _ 3·3·5 45 2 18 = _ _

18 = 18 ÷ 9 = 2 45 ÷ 9 5 45 18 = 2 5 45

5

45

Liza

162 ·9 =_ _ 18 = 18 _ 405 45 · 9 45 162 _ 405

Circle correct answer(s). Cross out incorrect answer(s).

Guided Practice 1

6 Write __ in simplest form. Use models. Shade an equivalent area 8 and name the simplified fraction. =

_6_

=

8

2

simplest form

______

Write _4_ in simplest form. Divide by the GCF. 8 ÷

______ = ______________ = ______

÷

3

8 Write ___ in simplest form. Divide by the GCF. 18 ÷

______ = ______________ = ______

÷ 72

Chapter 2 Equivalence of Fractions

Copyright © by The McGraw-Hill Companies, Inc.

fraction

Step by Step Practice 4

12 in simplest form. Use prime factorization. Write ___ 16 Step 1 Write the numerator as a product of prime numbers. Step 2 Write the denominator as a product of prime numbers. 16

12 4

4

4

Step 3 Replace the numerator with its prime factors. Replace the denominator with its prime factors. Find and eliminate all equivalent forms of 1. ·

·

·

·

______ = _________________________________ = ______

·

3 12 is __ . Step 4 The simplest form of ___ 16 4

Copyright © by The McGraw-Hill Companies, Inc.

Write each fraction in simplest form. Use prime factorization. 5

6

·

·

·

·

12 = _________________ = ______ ___ 30

10 ___ 15

7

15 ___ 18

Write each fraction in simplest form. Divide by the GCF. 10 24 9 ___ 10 ___ 25 32

8

21 ___

11

21 ___

28

49

GO ON Lesson 2-5 Simplify Fractions

73

Step by Step Problem-Solving Practice

Problem-Solving Strategies

Solve. 12

LANDSCAPING Doris wants a border around her flower bed. She can choose brown or red bricks of the same size. If she uses brown bricks, she will use 54 of the 72 brown bricks she has. Write the fraction of the brown bricks she will use in simplest form. Understand

Look for a pattern. ✓ Use logical reasoning. Solve a simpler problem. Work backward. Draw a diagram.

Read the problem. Write what you know. She would use ______ of the brown bricks.

Plan

Pick a strategy. One strategy is to use logical reasoning. Simplify the fraction that shows how many brown bricks she would use.

Solve

Find the GCF of 54 and 72. Divide the numerator and denominator by the GCF. Factors of 54: 1, 2, 3, 6, 9,  18, 27, 54

÷

______ = _____________ = ______

÷ Check

Use another method. Use prime factorization. Simplify the fraction. Does your answer make sense? Did you answer the question?

74

Chapter 2 Equivalence of Fractions

Copyright © by The McGraw-Hill Companies, Inc.

Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12,  18, 24, 36, 72

13

James and Nat run on the track after school each day. 18 16 James runs ___ of a mile in 4 minutes. Nat runs ___ of a mile in 27 24 the same time. James says he ran faster. Is he correct? Explain. FITNESS

Check off each step. Understand Plan Solve Check 14

SCHOOL Omar correctly answered 54 out of 60 questions on his last test. What fraction of the questions did he answer correctly? Simplify your answer to its simplest form. What fraction of the questions did he answer incorrectly? Simplify your answer to its simplest form. Explain which method used for simplifying fractions you prefer and why.

Copyright © by The McGraw-Hill Companies, Inc.

15

Skills, Concepts, and Problem Solving Write each fraction in simplest form. Use models. Shade an equivalent area and name the simplified fraction. 16

3 ______ __ =

17

6

8 ___ = ______ 12

Write each fraction in simplest form. Use the GCF. 18

36 ___ = 60

19

32 ___ = 40

20

60 ____ = 144

21

45 ___ = 90

GO ON Lesson 2-5 Simplify Fractions

75

Write each fraction in simplest form. Use prime factorization. 22

18 ___ = 27

23

25 ___ =

24

30

16 ___ = 48

25

27 = ___ 39

Solve. 26

FOOD Ti took 48 cookies to a picnic. He brought 8 cookies home. Write the fraction of cookies that were eaten in simplest form.

27

SPORTS Mario and Byron are playing a game where they each get 10 chances to throw a basketball into a hoop. After 4 of the balls they throw. playing 6 games, they both make __ 5 How many throws do they each make?

Vocabulary Check sentence.

Write the vocabulary word that completes each

28

A fraction in is a fraction in which the numerator and the denominator have no common factor greater than 1.

29

The is the greatest number that divides evenly into two or more numbers.

30

Writing in Math

Spiral Review Find the LCM of each set of numbers. 31

2, 10, and 25

Solve. 33

76 Corbis

(Lesson 2-4, p. 59)

32

2, 3, and 7

(Lesson 2-3, p. 51)

3 A potato soup recipe needs 2__ cups of milk. A 5 2 cups of milk. broccoli soup recipe needs 2__ Which recipe requires 3 more milk? COOKING

Chapter 2 Equivalence of Fractions

Copyright © by The McGraw-Hill Companies, Inc.

Suppose that you had a fraction that had a 12x . What do you think this fraction is in symbol in it, such as ____ 15x simplest form? Explain your reasoning.

4 The boys each made __ 5 of the baskets attempted.

Chapter

2

Study Guide

Vocabulary and Concept Check common denominators, p. 59 composite numbers, p. 51 equivalent forms of one, p. 34

Write the vocabulary word that completes each sentence. 1

The least common multiple of the denominators (bottom numbers) of two or more fractions is the .

2

The is the least whole number greater than 0 that is a common multiple of two or more numbers.

3

A(n) is any whole number with exactly two factors, 1 and itself.

equivalent fractions, p. 34 greatest common factor (GCF), p. 51 improper fraction, p. 41 least common denominator (LCD), p. 51 least common multiple (LCM), p. 51

.

4

prime factorization, p. 51

5

The greatest number that is a factor of two or more numbers is known as the .

6

A fraction is in when the numerator and the denominator have no common factor greater than 1.

prime number, p. 51 simplest form, p. 69 value, p. 34

Copyright © by The McGraw-Hill Companies, Inc.

__34 = __68 is an example of

mixed number, p. 41

Label each diagram below. Write the correct vocabulary term in each blank. 7

8

9 Chapter 2 Study Guide

77

Lesson Review

2-1

Equivalent Fractions and Equivalent Forms of One (pp. 34–40)

Complete to name an equivalent fraction.

Example 1

10

Complete to name an equivalent fraction.

2 = ______ __ 6

3

11

_1 = _2 4

8

Ask yourself, “What can I multiply the denominator 4 by to get 8”?

3 = ______ __ 4

8

Multiply 4 by 2 to get 8. Multiply the fraction 2. by __ 2 · __ 2 1 1 2 3 __ = ______ = __ 4 · 42 8 4 1 = __ 2. So, __ 4 8

Mixed Numbers and Improper Fractions

Name two equivalent fractions. 12

13

1 __

(pp. 41–49)

Example 2

_2 as an improper fraction.

2

Write 3

1 __

Multiply. 3 × 4 Add. 12 + 2 Write the total number of fourths as an improper fraction.

4

14

1 as an improper fraction. Write 4__ 5

15

73 Write ___ as a mixed number. 4

78

Chapter 2 Study Guide

4

3 · 4 + 2 12 + 2 14 ________ = _______ = ___ 4 4 4

Copyright © by The McGraw-Hill Companies, Inc.

2-2

2-3

Least Common Denominator (LCD) and Greatest Common Factors (pp. 51–58)

Find the least common multiple (LCM) of each set of numbers. 16

2, 4, and 7

17

2, 5, and 6

18

9, 8, and 36

19

4, 26, and 52

Example 3 Find the least common multiple (LCM) of 3, 4, and 6. List the multiples of each number. Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, … Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, … Multiples of 6: 6, 12, 18, 24, … Find the numbers that are common in all three lists. The least of these numbers is the LCM. Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, … Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, … Multiples of 6: 6, 12, 18, 24, … The LCM of 3, 4, and 6 is 12.

Copyright © by The McGraw-Hill Companies, Inc.

Find the greatest common factor (GCF) of each set of numbers. 20

14 and 70

21

20 and 75

22

48 and 80

23

45 and 120

Example 4 Find the greatest common factor (GCF) of 32 and 56 by using prime factors. 32

56

4 2

8 2

2

7 4

2

8 2

2

4 2

2

The common factors are 2, 2, and 2. So, the GCF of 32 and 56 is 2 × 2 × 2 or 8.

Chapter 2 Study Guide

79

2-4 24

2. 1 and __ Use <, =, or > to compare __ 6 8 Shade the models given.

_1 6

25

_2 8

1 and __ 2. Use <, =, or > to compare __ 3 9 Rename the fractions using a common denominator.

_1 3

26

Compare and Order Fractions (pp. 59–67)

_2 9

3 5 11 Order the fractions __, __, and ___ 4 6 12 from least to greatest.

Example 5

_

3 4 Use <, =, or > to compare and __. Shade the 5 4 models given.

The circle on the left has four sections. 3 Use it to model __. Shade 3 sections. 4 The circle on the right has five sections. 4 . Shade 4 sections. Use it to model __ 5 Compare the shaded areas. 3<4 Use <, =, or > to write a statement. __ 4 5

_

Example 6

__

_

5 4 7 Order the fractions , , and from least to 8 9 12 greatest. Find the LCM of 8, 9, and 12.

3 ___ 7 , __ __ , 11 8 4 14

Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, … Multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, … Multiples of 12: 12, 24, 36, 48, 60, 72, 84, … The LCD of the fractions is 72.

28

8 ___ 3 4 ___ , , ___

Write equivalent fractions.

30 15 45

5 5 45 · __ 9 3 __ = ___ = ______ 8

8 · 49

72

32 · __ 8 3 4 =4 ______ __ = ___ 9

9 · 48

7 · __ 6 42 7 3 ___ = ___ = _______ 12

12 · 46

72

Compare the numerators. 32 < 42 < 45 Order the fractions from least to greatest. 5 45 means that __ 32 < ___ 42 < ___ 4 < ___ 7 < __ ___ 72

80

Chapter 2 Study Guide

72

72

9

12

8

72

Copyright © by The McGraw-Hill Companies, Inc.

27

2-5 29

Simplify Fractions

(pp. 69–76)

6 Write __ in simplest form. Shade 8 an equivalent area and name the simplified fraction.

Example 7 Write

6 _ in simplest form. Use models. Shade

10 an equivalent area and name the simplified fraction.

Shade the circle segments so the shading in both circles covers the same amount of each circle.

fraction = 6 __ 8

=

simplest form ______

Copyright © by The McGraw-Hill Companies, Inc.

Write each fraction in simplest form. 20 30 ___ = 28

Count the total number of parts and the shaded parts of the circle on the right. Write the equivalent fraction. fraction = simplest form 6 3 ___ __ = 5 10

Example 8 Write

15 _ in simplest form. Divide by the GCF. 18

31

14 = ___ 35

List all the factors of the numerator and of the denominator.

32

18 ___ = 50

Factors of 15: 1,  3, 5, 15 Factors of 18: 1, 2,  3, 6, 9, 18

33

9 ___ =

The greatest number in both lists is 3. The GCF of 15 and 18 is 3.

34

35

36

7 = ___ 56

Divide the numerator and denominator by the GCF.

÷ __ 5 3 15 = 15 ________ ___ = __ 18

18 ÷ 4 3

6

15 5 So, ___ in simplest form is __. 6 18

21 = ___ 33

Chapter 2 Study Guide

81

Chapter

Chapter Test

2

Name two equivalent fractions. 1

1 __

2

6

Write each mixed number as an improper fraction. 2 1 3 3__ = 4 2__ = 5 5

9 ___ 12

5

Write each improper fraction as a mixed number. 23 7 ___ = 8 4

4= 7__ 9

6

1= 4__ 4

27 = ___ 12

Use <, =, or > to compare the pairs of fractions. Rename the fractions using a common denominator. 9

1 __

2 __

5

9

10

3 __

5 ___

8

12

Find the least common multiple (LCM) of each set of numbers. 3, 7, and 9

Order the fraction sets from least to greatest. 1 4 3 13 __, __, and __ 9 5 2

15

12

2, 3, and 8

14

3 6 2 , __ __ , and __ 3 5

Copyright © by The McGraw-Hill Companies, Inc.

11

8

12 in simplest form. Use prime factorization. Write ___ 18 18

12 4

9

GO ON 82

Chapter 2 Test

16

18 Write ___ in simplest form. Divide by the GCF. 20

Copyright © by The McGraw-Hill Companies, Inc.

Solve. 17

ENTERTAINMENT At a reception, two cakes were the same size. One cake was cut into 36 equal pieces. The other cake was cut into 48 equal pieces. At the end, there was exactly the same amount of each cake remaining. If 12 pieces of the 36-piece cake were left, then how many pieces of the 48-piece cake were left?

18

CONSTRUCTION A carpenter had a stack of boards. He cut each board into 4 equal pieces. He used 23 of the pieces. Write an improper fraction and a mixed number for the number of board pieces he used.

19

BUSINESS Matthew has a paper route. At the end of each week he must collect the money due from his customers. He was able to collect $75 of the $120 due. What fraction of the total money did he collect? What fraction of the amount due does he still need to collect? Simplify both answers to lowest terms.

Correct the mistakes. 20

Tyler took 36 cupcakes to school for his classroom party. He brought one dozen cupcakes home. He said that only 1 of the cupcakes were eaten. What mistake did Tyler make? __ 3

Chapter 2 Test Eri Morita/Getty Images

83

Chapter

2

Standards Practice

Choose the best answer and fill in the corresponding circle on the sheet at right. 1

Marnie and Yago shared a pizza. The pizza was cut into 8 equal pieces. If Yago ate 4 of the pieces, what fraction of the pizza did he eat? 7 1 C __ A __ 2 8 3 B __ 4

2

3

D 2

_

Which fraction is equal to 2 ? 3 3 F __ 9

4 H __ 5

4 G __ 6

7 J ___ 10

6

A >

C <

B =

D +

84

3 G 3__ 4

1 J 3__ 2

Chapter 2 Standards Practice

3 1 , __ 7 , __ B __ 8 2 4

3 , __ 1 7 , __ D __ 4 8 2

_

Rachel finished 4 of her homework 6 before dinner. Her sister Sonia finished 6 of her homework before 8 dinner. Which math sentence is correct? 6 6 < __ 4 4 > __ H __ F __ 6 8 8 6 6 4 = __ J __ 6 8

Which fraction is equal to the fraction at point C on the number line?

1 10

0

8

5 H 3__ 8

3 __ 1 , __ C __ ,7 2 4 8

"

Which mixed number does the model represent?

6 F 4__ 6

3 __ 7 , __ A __ ,1 8 4 2

6 4 < __ G __ 6 8

7

6

___

2 10

#

$

3 10

4 10

% 5 10

1 A __ 4

2 C __ 5

3 B ___ 10

3 D __ 5

6 10

_

Write 8 in simplest form. 10 4 F __ 5

1 H __ 2

2 G __ 3

2 J __ 5

GO ON

Copyright © by The McGraw-Hill Companies, Inc.

12

Order these fractions from least to greatest: 7 , 1 , 3 . 8 2 4

_

Which symbol makes the sentence true?

_7 □ _5

4

5

9

10

Copyright © by The McGraw-Hill Companies, Inc.

11

Which fraction does the model represent?

7 A __ 9

5 C __ 8

8 B ___ 12

7 D ___ 12

Which shows one-fifth written in fraction form? 5 F __ 1

1 H ___ 15

1 G __ 5

1 J ___ 50

Which fraction is equal to the number at point C on the number line? " 0

12

#

$

1 2 3 4 5 6 7 8 9 10 10 10 10 10 10 10 10 10

1

5 A __ 5

8 C ___ 12

4 B __ 5

1 D __ 2

ANSWER SHEET Directions: Fill in the circle of each correct answer. 1

A

B

C

D

2

F

G

H

J

3

A

B

C

D

4

F

G

H

J

5

A

B

C

D

6

F

G

H

J

7

A

B

C

D

8

F

G

H

J

9

A

B

C

D

10

F

G

H

J

11

A

B

C

D

12

F

G

H

J

Success Strategy If you do not know the answer to a question, go on to the next question. Come back to the problem, if you have time. You might find another question later in the test that will help you figure out the skipped problem.

Which symbol makes the sentence true?

_1 □ _1 2

12

F >

H <

G =

J +

Chapter 2 Standards Practice

85

Index A Answer sheet, 31, 85 ascending order, 59–67

models, 19–25, 34–40, 41–49 ordering, 59–67 simplest form, 69–76 simplifying, 69–76 unit, 11–17

Assessment, 28–29, 82–83

C California Mathematics Content Standards, 4, 11, 19, 34, 41, 51, 59, 69

G

N number line, 41–49, 50, 67 Number Sense, 4, 11, 19, 34, 41, 51, 59, 69 numerator, 4–10, 11–17, 19–25, 51–58, 59–67, 69–76

greater than, 59–67

O

greatest common factor (GCF), 51–58, 69–76 order, 59–67

I

Chapter Preview, 3, 33

order fractions, 59–67

Chapter Test, 28–29, 82–83 common denominators, 51–58, 59–67

improper fraction, 41–49

K

compare, 59–67 composite number, 51 Correct the Mistakes, 29, 83

descending order, 59–67

equivalent forms of one, 34–40, 59–67

Progress Check, 18, 50, 68

L least common denominator (LCD), 51–58 least common multiple (LCM), 51–58, 59–67 less than, 59–67 like fractions, 67–69

M

equivalent fractions, 34–40, 51–58, 59–67, 69–76

F fraction, 4–10, 11–17, 19–25 common denominators, 51–58, 59–67 comparing, 59–67 equivalent, 34–40, 51–58, 59–67, 69–76 improper fractions, 41–49 least common denominator (LCD), 51–58 like, 59–67 mixed numbers, 41–49 86

Index

Problem-Solving. See Step-byStep Problem Solving

Manipulatives fraction circle, 5, 9, 10, 11, 12, 13, 14, 15, 17, 18, 20, 24, 25, 34, 35, 36, 60, 63, 65, 69, 75, 78, 80, 81 fraction strip (tiles), 3, 5, 6, 7, 9, 11, 13, 14, 15, 16, 19, 20, 22, 34, 35, 43, 60, 63, 65, 75 Math Reasoning. See Step-byStep Problem Solving mixed number, 41–49 money, 3

R Real-World Applications baking, 22, 66, 67 baseball, 49 basketball, 57 business, 17, 83 cafeteria, 29 carpentry, 49 cleaning, 40 community service, 23, 49 construction, 39, 83 cooking, 48, 76 crafts, 48 entertainment, 66, 83 farming, 16 finance, 29 fitness, 18, 64, 75 flowers, 56 food, 18, 25, 38, 49, 56, 65, 76 food service, 83 groceries, 67 hobbies, 39, 50, 83 invitations, 49 jobs, 24

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E

prime factorization, 51–58 prime number, 51–58

Key Concept, 4, 11, 19, 34, 41, 51, 59, 69

D denominator, 4–10, 11–17, 19–25, 51–58, 59–67, 69–76

P

kites, 38 landscaping, 74 manufacturing, 48 money, 55, 56 music, 29 nature, 24 number sense, 47 nutrition, 17, 37, 40 online shopping, 8 pets, 10 population, 58 reading, 16 school, 8, 10, 17, 23, 75 shopping, 10 snacks, 8 sports, 39, 66, 76 weather, 65 words, 10, 18 Reflect, 9, 16, 23, 38, 48, 56, 65, 75

S

Standards Practice, 30–31, 84–85 Step-by-Step Practice, 7, 15, 21, 36, 46, 54, 63, 73 Step-by-Step Problem Solving Practice, 8, 16, 22–23, 37–38, 47–48, 55–56, 64–65, 74–75 Draw a diagram, 16, 37, 64 Draw a picture, 8 Use a model, 22 Use logical reasoning, 55, 74 Work backward, 47

V value, 34 Vocabulary, 4, 11, 19, 34, 41, 51, 59, 69 Vocabulary and Concept Check, 26, 77 Vocabulary Check, 10, 17, 25, 40, 49, 57, 66, 76

W

Study Guide, 26–27, 77–81 Success Strategy, 31, 85 whole, 4–10

U unit fraction, 11–17

Who is Correct?, 6, 14, 20, 36, 45, 54, 63, 72 whole numbers, 41–48 Writing in Math, 10, 17, 25, 40, 49, 57, 67, 76

simplest form, 69–76

Copyright © by The McGraw-Hill Companies, Inc.

Spiral Review, 17, 25, 40, 49, 57, 67, 76

Index

87