California Math Triumphs VOL 1b Place Value (CALIFORNIA MATH TRIUMPHS VOL 1B)

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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: 978-0-07-878204 MHID: 0-07-878204-X 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 1B

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 1A

Place Value and Basic Number Skills

Chapter

Counting

1

Chapters 1, 2, and 3 are contained in Volume 1A. Chapters 4, 5, and 6 are contained in Volume 1B.

1-1 Counting Numbers Less Than 100 .................................4. 1NS1.1

1-2 Whole Numbers Less Than 100 . ..................................11

Standards Addressed in This Chapter 1NS1.1 Count, read, and write whole numbers to 100.

1NS1.1

Progress Check 1..............................................................18 1-3 Equal Expressions ...........................................................19 1NS1.3

1-4 Number Patterns .............................................................25 1NS1.2

Progress Check 2..............................................................32 1-5 Numbers That Make Ten ...............................................33 1NS1.4

1-6 Expanded Form for Two-Digit Numbers ...................39

1NS1.2 Compare and order whole numbers to 100 by using the symbols for less than, equal to, or greater than (<, =, >). 1NS1.3 Represent equivalent forms of the same number through the use of physical models, diagrams, and number expressions (to 20) (e.g., 8 may be represented as 4 + 4, 5 + 3, 2 + 2 + 2 + 2, 10 - 2, 11 - 3). 1NS1.4 Count and group objects in ones and tens (e.g., three groups of 10 and 4 equals 34, or 30 + 4).

1NS1.4

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Progress Check 3.............................................................46 1-7 Use Symbols to Compare Numbers ............................47 1NS1.2

1-8 Order Whole Numbers Less Than 100 ........................53 1NS1.2

Progress Check 4.............................................................59 Assessment Study Guide .....................................................................60 Chapter Test .....................................................................64

Bridal Veil Falls, El Capitan, and Half Dome, Yosemite National Park

Standards Practice ...................................................66

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Contents Chapter

Place Value

2

Standards Addressed in This Chapter 2-1 Whole Numbers to 1,000 ...............................................70 2NS1.1, 2NS1.2

2-2 Round and Compare Whole Numbers Less Than 1,000 ..............................................77 2NS1.3, 4NS1.3

Progress Check 1 .............................................................84 2-3 Whole Numbers Less Than 10,000 ...............................85 3NS1.3, 3NS1.5

2-4 Round and Compare Whole Numbers Less Than 10,000 ........................................... 91 4NS1.2, 4NS1.3

Progress Check 2 .............................................................98 2-5 Read and Write Whole Numbers in the Millions .......99 4NS1.1

4NS1.2, 4NS1.3

2-7 Order and Compare Numbers to Two Decimal Places .................................................. 111 4NS1.2, 4NS1.6

Progress Check 3 ...........................................................119 Assessment

2NS1.2 Use words, models, and expanded forms (e.g., 45 = 4 tens + 5) to represent numbers (to 1,000). 2NS1.3 Order and compare whole numbers to 1,000 by using the symbols <,=, >. 3NS1.3 Identify the place value for each digit in numbers to 10,000. 3NS1.5 Use expanded notation to represent numbers (e.g., 3,206 = 3,000 + 200 + 6). 4NS1.1 Read and write whole numbers in the millions. 4NS1.2 Order and compare whole numbers and decimals to two decimal places. 4NS1.3 Round whole numbers through the millions to the nearest ten, hundred, thousand, ten thousand, or hundred thousand. 4NS1.6 Write tenths and hundredths in decimal and fraction notations and know the fraction and decimal equivalents 1 for halves and fourths (e.g., __ = 0.5 or 2 3 7 0.50; __ = 1__ = 1.75) 4 4

Study Guide ...................................................................120 Chapter Test ...................................................................124 Standards Practice .................................................126

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2-6 Round and Compare Whole Numbers in the Millions ..............................................105

2NS1.1 Count, read, and write whole numbers to 1,000 and identify the place value for each digit.

Contents Chapter

Addition and Subtraction

3

Standards Addressed in This Chapter 3-1 Addition Facts for 0 to 5 ..............................................130 1NS2.1, 1NS2.6, 2NS2.2

3-2 Addition Facts for 6 and 7............................................137 1NS2.1, 1NS2.6, 2NS2.2

Progress Check 1 ...........................................................144 3-3 Addition Facts for 8 and 9............................................145 1NS2.1, 1NS2.5, 1NS2.7

3-4 Estimate and Add Greater Numbers......................... 151 2NS2.3, 3NS1.3, 4NS1.3, 4NS3.1

Progress Check 2 ...........................................................158 3-5 Subtraction Facts for 0 to 5...........................................159 1NS2.1, 1NS2.5, 1NS2.6

3-6 Subtraction Facts for 6 to 9...........................................165 1NS2.1, 1NS2.5, 1NS2.6

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

Progress Check 3 ...........................................................172 3-7 Subtract with Zeros .......................................................173 2NS1.2, 2NS2.3, 3NS2.1

3-8 Estimate and Subtract Greater Numbers ...................181 3NS2.1, 4NS3.1

Progress Check 4 ...........................................................189 Assessment Study Guide ...................................................................190 Chapter Test ...................................................................194

1NS2.1 Know the addition facts (sums to 20) and the corresponding subtraction facts and commit them to memory. 1NS2.5 Show the meaning of addition (putting together, increasing) and subtraction (taking away, comparing, finding the difference). 1NS2.6 Solve addition and subtraction problems with one- or two-digit numbers (e.g., 5 + 58 = ____). 1NS2.7 Find the sum of three one-digit numbers. 2NS2.2 Find the sum or difference of two whole numbers up to three digits long. 2NS2.3 Use mental arithmetic to find the sum or difference of two two-digit numbers. 3NS1.3 Identify the place value for each digit in numbers to 10,000. 3NS2.1 Find the sum or difference of two whole numbers between 0 and 10,000. 4NS1.3 Round whole numbers through the millions to the nearest ten, hundred, thousand, ten thousand, or hundred thousand. 4NS3.1 Demonstrate an understanding of, and the ability to use, standard algorithms for the addition and subtraction of multidigit numbers.

Standards Practice .................................................196 Cacti growing in Baja California Peninsula

ix CORBIS

Contents Chapter

Multiplication

4 4-1

Introduction to Multiplication 3NS2.2, 4NS4.1.............4

4-2

Multiply with 0, 1, and 10 3NS2.2, 3NS2.4, 3NS2.6 ......11 Progress Check 1...........................................................18

4-3

Multiply by 2 3NS2.2, 3NS2.4 ........................................19

4-4

Multiply by 5 3NS2.2, 3NS2.4 ....................................... 25 Progress Check 2...........................................................32

4-5

Multiply by 3 3NS2.2, 3NS2.4, 4NS3.2 ...........................33

4-6

Multiply by 4 3NS2.2, 3NS2.4, 4NS3.2, 4NS4.1 .............. 39 Progress Check 3...........................................................46

4-7

Multiply by 6 3NS2.2, 3NS2.4, 4NS3.2, 4NS4.1 ...............47

4-8

Multiply by 7 3NS2.2, 3NS2.4, 4NS3.2, 4NS4.1 .............. 53 Progress Check 4...........................................................60

4-9

Multiply by 8 3NS2.2, 3NS2.4, 4NS3.2, 4NS4.1 .............. 61

Progress Check 5...........................................................74 4-11 Multiply by 11 and 12 3NS2.2, 3NS2.4, 4NS3.2, 4NS4.1 .. 75 4-12 Perfect Squares 3NS2.2, 4NS4.1.................................... 81 Progress Check 6...........................................................88

Standards Addressed in This Chapter 2NS3.1 Use repeated addition, arrays, and counting by multiples to do multiplication. 2NS3.3 Know the multiplication tables of 2s, 5s, and 10s (to “times 10”) and commit them to memory. 3NS2.2 Memorize to automaticity the multiplication table for numbers between 1 and 10. 3NS2.4 Solve simple problems involving multiplication of multidigit numbers by one-digit numbers (3,671 × 3 = ____). 3NS2.6 Understand the special properties of 0 and 1 in multiplication and division. 4NS3.2 Demonstrate an understanding of, and the ability to use, standard algorithms for multiplying a multidigit number by a two-digit number and for dividing a multidigit number by a one-digit number; use relationships between them to simplify computations and to check results. 4NS4.1 Understand that many whole numbers break down in different ways (e.g., 12 = 4 × 3 = 2 × 6 = 2 × 2 × 3).

4-13 Multiply Large Numbers 3NS2.4, 3NS2.6, 4NS3.2 ...... 89 Assessment Study Guide ..................................................................95 Chapter Test ................................................................102 Standards Practice...............................................104

x CORBIS

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4-10 Multiply by 9 3NS2.2, 3NS2.4, 4NS3.2, 4NS4.1 .............. 67

Chapters 1, 2, and 3 are contained in Volume 1A. Chapters 4, 5, and 6 are contained in Volume 1B.

Contents Chapter

Division

5

Standards Addressed in This Chapter 5-1 Model Division .............................................................108 3NS2.2, 4NS3.2

5-2 Divide by 0, 1, and 10 ...................................................115 3NS2.2, 3NS2.6, 4NS3.2

Progress Check 1 ...........................................................122 5-3 Divide by 2 and 5 ..........................................................123 3NS2.2, 4NS3.2

5-4 Divide by 3 and 4 ......................................................... 129 3NS2.2, 4NS3.2

Progress Check 2 ...........................................................136

3NS2.2 Memorize to automaticity the multiplication table for numbers between 1 and 10. 3NS2.6 Understand the special properties of 0 and 1 in multiplication and division. 4NS3.2 Demonstrate an understanding of, and the ability to use, standard algorithms for multiplying a multidigit number by a two-digit number and for dividing a multidigit number by a one-digit number; use relationships between them to simplify computations and to check results.

5-5 Divide by 6 and 7 ..........................................................137 3NS2.2, 4NS3.2

5-6 Divide by 8 and 9 ..........................................................143 3NS2.2, 4NS3.2

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

Progress Check 3 ...........................................................150 5-7 Divide by 11 and 12.......................................................151 4NS3.2

5-8 Long Division ................................................................157 3NS2.2, 4NS3.2

Progress Check 4 ...........................................................163 Assessment Study Guide ...................................................................164 Chapter Test ...................................................................168

Badlands near Zabriskie Point, Death Valley National Park

Standards Practice .................................................170

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Contents Chapter

Integers

6

Standards Addressed in This Chapter 6-1

Model Integers ............................................................174 4NS1.8, 5NS1.5, 7NS1.2

6-2

Add Integers................................................................181 5NS2.1, 6NS2.3

Progress Check 1.........................................................188 6-3

Subtract Integers .........................................................189 5NS2.1, 6NS2.3, 7NS1.2

6-4

Add and Subtract Larger Integers .......................... 197 5NS2.1, 6NS2.3, 7NS1.2

Progress Check 2.........................................................204 6-5

Multiply Integers ........................................................205 6NS2.3, 7NS1.2, 3NS2.2, 3NS2.6

6-6

Multiply Several Integers ......................................... 211 6NS2.3, 7NS1.2, 3NS2.2, 3NS2.6

Progress Check 3.........................................................218 Divide Integers............................................................219 6NS2.3, 7NS1.2, 3NS2.2, 3NS2.6

6-8

Order of Operations with Integers.......................... 225 6NS2.3, 7NS1.2, 3NS2.2, 3NS2.6

Progress Check 4.........................................................231 Assessment

3NS2.6 Understand the special properties of 0 and 1 in multiplication and division. 4NS1.8 Use concepts of negative numbers (e.g., on a number line, in counting, in temperature, in “owing”). 5NS1.5 Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers. 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. 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. 7NS1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to wholenumber powers.

Study Guide ................................................................232 Chapter Test ................................................................236 Standards Practice...............................................238

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6-7

3NS2.2 Memorize to automaticity the multiplication table for numbers between 1 and 10.

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. 1 What is the title of Lesson 5-2? 2

Divide by 0, 1, and 10

What is the Key Concept of Lesson 4-4?

Multiply by 5 3

On what page can you find the vocabulary term fact family in Lesson 4-7?

page 47 4

What are the vocabulary words for Lesson 5-4?

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

remainder, multiple 5

How many Examples are presented in the Chapter 5 Study Guide? 8

6

What are the California Standards covered in Lesson 6-2?

4NS1.8, 5NS1.5 7

List the integers that are mentioned in exercise #34 on page 202. +129, -100, +40

8

What numbers are used in Step-by-Step Practice on page 91? 14, 89

9

On what pages will you find the Study Guide for Chapter 6?

pages 232–235 10

In Chapter 5, find the logo and Internet address that tells you where you can take the Online Readiness Quiz. It is

found on page 107. The URL is ca.mathtriumphs.com. 1

Chapter

4

Multiplication You use multiplication to plan a party. Suppose you have enough money to buy 5 pounds of lunchmeat for a party. Each pound has 16 slices. How many slices of lunchmeat do you have for your party?

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2

Chapter 4 Multiplication

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STEP

STEP

1 Quiz

Are you ready for Chapter 4? Take the Online Readiness Quiz at ca.mathtriumphs.com to find out.

2 Preview

Get ready for Chapter 4. Review these skills and compare them with what you’ll learn in this chapter.

What You Know

What You Will Learn

You know how to add.

Lesson 4-1

Examples: 2 + 2 + 2 = 6 10 + 10 + 10 + 10 = 40 300 + 300 + 300 = 900

Multiplication is repeated addition.

4 × 10 = 10 + 10 + 10 + 10 = 40

66

33 + 33 =

10 + 10 + 10 =

3

4+4+4+4=

30 16

four 10s 3 × 300 = 300 + 300 + 300 = 900 ⎫ ⎬ ⎭

2

⎫ ⎬ ⎭

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

⎫ ⎬ ⎭

three 2s

TRY IT! 1

2×3=2+2+2=6

three 300s

You know how to skip-count.

Lessons 4-3 through 4-11

Example: Skip-count by 4.

Multiples of 4 are the numbers you say when you skip-count by 4.

0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, …

0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, …

TRY IT!

The multiples of 4 are the multiplication facts below.

4

Skip-count by 2.

0, 2, 4, 6, 8, 10, 12, 14, 16, … 5

Skip-count by 5.

0, 5, 10, 15, 20, 25, 30, 35, 40, …

You know that changing the order in which you add numbers does not change the sum. 2+3=5

3+2=5

These sentences show the Commutative Property of Addition .

0×4=0 1×4=4 2×4=8 3 × 4 = 12 4 × 4 = 16 5 × 4 = 20

6 × 4 = 24 7 × 4 = 28 8 × 4 = 32 9 × 4 = 36 10 × 4 = 40

Lessons 4-4 through 4-13 5 × 4 = 20

4 × 5 = 20

These sentences show the Commutative Property of Multiplication . Changing the order in which you multiply numbers does not change the product.

3

Lesson

4-1 Introduction to Multiplication 3NS2.2 Memorize to automaticity the multiplication table for numbers between 1 and 10. 4NS4.1 Understand that many whole numbers break down in different ways.

KEY Concept factors

product

VOCABULARY

2 + 2 + 2 + 2 + 2 = 5 × 2 = 10 repeated addition

The symbols × and · are used for multiplication. Five times two can be written as 5 × 2, 5 · 2, or 5(2). You can model multiplication with an array . 2 × 5 is 2 groups of 5, or 5 × 2 is 5 groups of 2. 0OFGBDUPSJTUIF OVNCFSPGSPXT 5IFPUIFSGBDUPSJTUIF OVNCFSPGDPMVNOT

The product is the total number of rectangles in the array. 2 × 5 = 10 Another way to model the expression 2 × 5 is with a number line.

HSPVQTPG

HSPVQTPG

The product using either method is 10.

The Commutative Property of Multiplication states that the order in which you multiply the numbers does not matter. So, 2 × 5 = 5 × 2.

4

Chapter 4 Multiplication

factor a number that divides into a whole number evenly; also a number that is multiplied by another number factors product 2×3=6 array an arrangement of objects or symbols in rows of the same length and columns of the same length; the length of a row might be different from the length of a column multiplication an operation on two numbers to find their product; it can be thought of as repeated addition Example: 4 × 3 is the same as the sum of four 3s, which is 3 + 3 + 3 + 3 or 12.

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product the answer or result of a multiplication problem; it also refers to expressing a number as the product of its factors

Example 1

YOUR TURN! Draw an array to model the expression 7 × 2. Then write and model the commutative fact.

Draw an array to model the expression 6 × 3. Then write and model the commutative fact. 1. Identify the first number in the expression. 6 This represents the number of rows in the array. 2. Identify the second number in the expression. 3 This represents the number of columns in the array. Count the number of rectangles. The product of 6 × 3 = 18.

1. Identify the first number in the expression. 7 2. Identify the second number in the expression. 2 Count the number of rectangles. The product of 7 × 2 is 14 .

JTPOF GBDUPS

JTPOFGBDUPS

JTUIFPUIFSGBDUPS

3. The commutative fact for 7 × 2 = 14 is 2 × 7 = 14 .

3. The commutative fact for 6 × 3 = 18 is 3 × 6 = 18.

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Count the number of rectangles. The product of 7 and 2 is equal to the product of 2 and 7 , which is 14 . Count the number of rectangles. The product of 3 and 6 is equal to the product of 6 and 3, which is 18.

Example 2 Use a number line to model the expression 2 × 3. 1. Identify the first number in the expression. 2 This is the number of times the group is repeated. 2. Identify the second number in the expression. 3 This is the group size. 3. Draw a number line. Mark off 2 groups of 3. The product is 6.

GO ON Lesson 4-1 Introduction to Multiplication

5

YOUR TURN! Use a number line to model the expression 3 × 5.

3

1. Identify the first number in the expression.

5

2. Identify the second number in the expression. 3. Draw a number line. Mark off 3 groups of The product is

5

.

15 .

Who is Correct? Write 5 · 8 as repeated addition.

Candace 5+5+5+5+5

Juan

Rose

5+5+5+5 +5+5+5+5

8+8+8+8+8

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

Guided Practice

1

5 × 3 5 × 3 = 15; 3 × 5 = 15

3·4

2

3 · 4 = 12; 4 · 3 = 12

Step by Step Practice 3

Write 2 + 2 + 2 + 2 as a multiplication expression. Step 1 Identify the number being repeated.

2

Step 2 Count how many times the number is repeated.

4

Step 3 Write the multiplication fact.

2 the number being repeated 6

Chapter 4 Multiplication

×

4 how many times it is repeated.

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Draw an array to model each expression. Then write and model each commutative fact.

Write each repeated addition fact as a multiplication expression. Then write the commutative fact. 4

5 + 5 + 5 3 × 5 = 15; 5 × 3 = 15

5

9 + 9 2 · 9 = 18; 9 · 2 = 18

6

4 + 4 + 4 + 4 + 4 5 × 4 = 20; 4 × 5 = 20

7

3 + 3 + 3 + 3 4 · 3 = 12; 3 · 4 = 12

Step by Step Problem-Solving Practice

Problem-Solving Strategies ✓ Draw a diagram.

Solve. 8

Use logical reasoning. Solve a simpler problem. Work backward. Use an equation.

INTERIOR DESIGN Natalie and her mom are tiling a rectangular kitchen floor. Each tile is 1 foot by 1 foot. The length of the kitchen is 8 feet and the width is 14 feet. How many tiles will they need to cover the floor? Understand

Read the problem. Write what you know. The rectangular floor is 1 foot Each tile is a

8 ft

by

14 ft

.

square.

Plan

Pick a strategy. One strategy is to draw a diagram. You need to find how many tiles are needed to cover the whole floor.

Solve

Draw a rectangle. Divide it so it has 8 rows and 14 columns.

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0OFGBDUPSJTUIFOVNCFSPGDPMVNOT

0OFGBDUPS JTUIF OVNCFS PGSPXT

8

Write a multiplication fact for the array.

×

14

Write the expression as repeated addition.

14 + 14 + 14 + 14 + 14 + 14 + 14 + 14 How many tiles will Natalie and her mom need?

112

Check

Count the squares in the diagram to verify your answer. GO ON Lesson 4-1 Introduction to Multiplication

7

9

10

HEALTH Lakeesha takes a multivitamin each morning and a vitamin C tablet each night. Write a multiplication expression to show how many vitamins Lakeesha needs for a 30-day supply of vitamins. How many vitamins is this? 2 · 30; 60 vitamins Check off each step.

✔

Understand

✔

Plan

✔

Solve

✔

Check

MUSIC At Caroline’s middle school, the music teacher teaches music to each grade two times a week. If there are three grades at Caroline’s middle school, how many times does the music teacher teach each week? Write a multiplication expression to show how you found the answer.

Sample answer:

6 times; 2 × 3 Use graph paper and draw as many different rectangular arrays for the number 12 as possible.

11

Skills, Concepts, and Problem Solving

12

2×4=

2 × 4 = 8;

4×2=8 13

Copyright © by The McGraw-Hill Companies, Inc.

Use a number line to model each expression. Then write and model the commutative fact.

3·4

3 · 4 = 12;

4 · 3 = 12 8

Chapter 4 Multiplication

Draw an array to model each expression. Then write and model the commutative fact. 14

4 · 5 4 · 5 = 20; 5 · 4 = 20

15

5 × 3 5 × 3 = 15; 3 × 5 = 15

Write the multiplication expression as repeated addition. Then write the commutative fact. 16

6+6+6

3·6

17

7·4

3 · 6 = 18; 6 · 3 = 18 18

8×3

3+3+3+3+3+3+3+3

4+4+4+4+4+4+4 7 · 4 = 28; 4 · 7 = 28

19

6×5

8 × 3 = 24; 3 × 8 = 24

5+5+5+5+5+5 5 × 6 = 30; 6 × 5 = 30

Write the repeated addition as a multiplication expression. Then write the commutative fact. 20

11 + 11 + 11

3 × 11 = 33

21

2+2+2+2+2

Copyright © by The McGraw-Hill Companies, Inc.

11 × 3 = 33 22

5+5+5+5

4 × 5 = 20

5 × 2 = 10 2 × 5 = 10

23

4+4+4+4

5 × 4 = 20

4 × 4 = 16 4 × 4 = 16

Solve. 24

PACKAGING There are two different-sized packages of cinnamon rolls. One package has 8 rolls across and 2 rolls down. The other package has 3 rolls across and 4 rolls down. Which package holds more rolls? How much more?

The first package holds 4 more. 25

PUZZLES Gloria and her sister Cherise worked together on a puzzle. Gloria measured the length of the puzzle to be 10 inches and the width to be 9 inches. If each piece of the puzzle is about 1 inch square, about how many pieces are in their puzzle? Write an addition sentence to find the answer.

9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 = 90

GO ON Lesson 4-1 Introduction to Multiplication

9

Vocabulary Check Write the vocabulary word that completes each sentence.

multiplication

26

The product of two numbers indicates what operation?

27

Writing 4 × 6 = 6 × 4 is an example of the Property of Multiplication.

28

The numbers being multiplied in an expression are called

29

Writing in Math How can you verify the Commutative Property of Multiplication?

Commutative factors

.

Sample answer: Make an array for a given multiplication expression. Then interchange the order of factors. Create the array for the second expression. The number of rectangles in both arrays is the same.

Spiral Review Solve. 30

(Lesson 2-6, p. 85)

"NFSJDBT

MONEY Look at the deposit slip shown at the right. What is the amount of total deposits rounded to the nearest ten thousand?

$IFDLJOH 4BWJOHT $ISJTUNBT$MVC

4BWJOHT#BOL

".06/5 $"4) $)&$,4

0CTOBER5OTAL%EPOSITS

%BUF

/BNF T PO"DDPVOU "EESFTT

$840,000 31

%FQPTJU4MJQ

505"-

"DDPVOU/VNCFS

35,300 Write each number in word form.

(Lesson 2-5, p. 80)

32

2,503,250

two million, five hundred three thousand, two hundred fifty

33

1,770,609

one million, seven hundred seventy thousand, six hundred nine

34

3,900,000

three million, nine hundred thousand

Write the counting numbers between the following numbers.

10,11,12,13

35

9 and 14

37

13 and 15

10

Chapter 4 Multiplication

14

(Lesson 1-1, p. 4)

36

3 and 7

4,5,6

38

5 and 10

6,7,8,9

Copyright © by The McGraw-Hill Companies, Inc.

FOOD The cafeteria served thirty-five thousand, two hundred ninety-three cookies during the entire school year. What is this amount rounded to the nearest hundred?

Lesson

4-2 Multiply with 0, 1, and 10 KEY Concept The Zero Property of Multiplication states that any number times zero is zero.

VOCABULARY

5 × 0 = 0 because five groups of zero is zero. The Identity Property of Multiplication states that any number times 1 is equal to that number. 5 × 1 = 5 because five groups of one is five. The multiples of 10 are: 1 · 10 = 10 2 · 10 = 20 3 · 10 = 30 4 · 10 = 40 5 · 10 = 50

6 · 10 = 60 7 · 10 = 70 8 · 10 = 80 9 · 10 = 90 10 · 10 = 100

/PUJDFUIBUøuBOEu 8IFOZPVNVMUJQMZCZ BEEUPUIFSJHIUPGUIFPUIFS GBDUPS"TJNJMBSSVMFBMTPIPMET GPSNVMUJQMJDBUJPOCZBOE CZ u u

The multiples of 10 are the same as skip-counting by 10. 10, 20, 30, 40, 50, 60, 70, 80, 90, 100

Copyright © by The McGraw-Hill Companies, Inc.

3NS2.2 Memorize to automaticity the multiplication table for numbers between 1 and 10. 3NS2.4 Solve simple problems involving multiplication of multidigit numbers by one-digit numbers. 3NS2.6 Understand the special properties of 0 and 1 in multiplication and division.

Identity Property of Multiplication if you multiply a number by 1, the product is the same as the given number Example: 8 × 1 = 8 = 1 × 8. Zero Property of Multiplication if you multiply a number by zero, the product is zero Example: 0 × 5 = 0. expanded form the representation of a number as a sum that shows the value of each digit Example: 536 can be written as 500 + 30 + 6.

Example 1 Find 101 · 2. Estimate first. 1. Estimate. 101 · 2 → 100 · 2 = 200 2. Rewrite the problem in vertical format. 3. Multiply 2 times the digit in the ones column. 2 · 1 = 2

101 ×2 2

4. Multiply 2 times the digit in the tens column. 2 · 0 = 0 Write the product in the tens column.

101 ×2 02

5. Multiply 2 times the digit in the hundreds column. 2·1=2 Write the product in the hundreds column. 6. The product is 202. Compare to your estimate for reasonableness.

101 ×2 202

GO ON Lesson 4-2 Multiply with 0, 1, and 10

11

YOUR TURN! Find 110 · 4. Estimate first.

100 · 4 = 400

1. Estimate.

2. Rewrite the problem in vertical format. 3. Multiply 4 times the digit in the ones column. 4·0=

110 ×4

0

0

Example 2

Find the product of 111 and 3 using expanded form.

100 + 10 + 1

100 + 1 × 7 100 + 1 × 7 7 100 + 1 × 7 700 + 7

2. Write the factors in vertical form.

100 + 10 + 1 × 3

3. Multiply 3 times the ones place value. 3×1= 3

100 + 10 + 1 × 3

4. Multiply 3 times the tens place value. 10 × 3 = 30

100 + 10 + 1 × 3

5. Multiply 3 times the hundreds place value. 100 × 3 = 300

100 + 10 + 1 × 3

6. Add the products.

300 + 30 + 3 = 333

12

Chapter 4 Multiplication

440

3

30 + 3

300 + 30 + 3

Copyright © by The McGraw-Hill Companies, Inc.

5. Add the products. 700 + 7 = 707

110 ×4

1. Write the first number in expanded form.

1. Write the first number in expanded form. 100 + 1

4. Multiply 7 times the hundreds place value. 100 × 7 = 700

40

YOUR TURN!

Find the product of 101 and 7 using expanded form.

3. Multiply 7 times the ones place value. 7 × 1 = 7

5. Multiply 4 times the digit in the hundreds column. 4·1=4 Write the product in the hundreds column. 6. The product is 440 . Compare to your estimate for reasonableness.

4. Multiply 4 times the digit in the tens column. 4·1= 4

2. Write the factors in vertical form.

110 ×4

Write the product in the tens column.

Who is Correct? Find the product of 56 and 10.

Quinton

Dennis

Polly

56 × 1 = 56, so 56 × 10 = 5,600

56 1 × −−− 56

56 10 × −−−− 560

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

Guided Practice Find each product. Estimate first.

Copyright © by The McGraw-Hill Companies, Inc.

1

101 × 4

Estimate

100 × 4 = 400

1 0 1 × 4

2

111 × 2

200; 222

3

100 × 3

300; 300

4 0 4

4

110 × 5

500; 550

5

101 × 3

300; 303

Step by Step Practice 6

Find the product of 101 and 8. Step 1 Estimate. 100 × 8 = 800

101 × 8

Step 2 Rewrite the problem in vertical format. Step 3 Multiply 8 times the 8×

1

=

0

=

1

=

column.

tens

column.

0

Step 5 Multiply 8 times the 8×

ones

8

Step 4 Multiply 8 times the 8×

8

hundreds

101 × 8

08 101 × 8

columns.

8

Step 6 The product is 808 . Compare to your estimate for reasonableness.

808

GO ON

Lesson 4-2 Multiply with 0, 1, and 10

13

Find each product. 7

5×0=

0

8

10 × 1 =

10

9

8 × 10 =

80

10

1×0=

0

11

9×1=

9

12

4 × 10 =

40

13

10 × 10 = 100

14

11 × 1 =

11

15

1 × 70 =

16

0 × 20 =

0

18

110 × 5

70

Find each product using expanded form. 17

12 × 10

100 19

+

20

=

101 × 6

120

500 20

100 + 1 × 6 600 + 6 = 606 21

202 × 7

200 + 2 × 7 1,400 + 14 = 1,414 14

Chapter 4 Multiplication

+

50

32 × 10

30 + 2 × 10 300 + 20 = 320 22

41 × 10

40 + 1 × 10 400 + 10 = 410

=

550

Copyright © by The McGraw-Hill Companies, Inc.

100 + 10 × 5

10 + 2 × 10

Step by Step Problem-Solving Practice

Problem-Solving Strategies

Solve. 23

FASHION Emma plans to buy several shirts that are on sale for $10 each. Emma wants to know if she has enough money to buy 2, 3, or 4 shirts. What is the price of 2, 3, and 4 shirts? Understand

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

Read the problem. Write what you know. Emma wants to buy shirts that are $10 each.

Plan

Pick a strategy. One strategy is to look for a pattern.

Solve

List the multiples of 10: 1 × 10 = 2 × 10 =

20 , 3 × 10 =

10 ,

30 .

Look for the pattern. Notice the ones digit is a 0 ,while the tens digit is the other factor. Find the pattern. What is the price of 1 shirt? $10 2 shirts? $20 3 shirts? $30 4 shirts? $40 Does the answer make sense? Look over your solution. Did you answer the question?

Check

Copyright © by The McGraw-Hill Companies, Inc.

24

NATURE Suri planted a sapling that grows 10 inches a month. 60 inches How many inches will it grow in 6 months? Check off each step.

✔

Understand

✔

Plan

✔

Solve

✔

Check

25

COMMUNITY SERVICE The fifth graders collected 7 bags of canned goods to donate to a charity. If each bag held 10 cans, how 70 many cans did they collect in all?

26

Explain how to multiply using the expanded form of the factors in a multiplication problem.

Write one factor in expanded form. Then find the product of each of the place values and the other factor. The sum of the products is the final product.

GO ON

Lesson 4-2 Multiply with 0, 1, and 10

15

Skills, Concepts, and Problem Solving Find each product. Estimate first. 23

100 × 7

700; 700

24

101 × 3

300; 303

25

101 × 5

500; 505

26

111 × 7

700; 777

27

506 × 8

4,000; 4,048

28

409 × 6

2,400; 2,454

29

40 × 32

1,200; 1,280

30

30 × 1,001

32

701 × 4

2,804

34

50 × 71

3,550

30,000; 30,030

Find each product using expanded form.

8,440

211 × 40

33

10 ×111

35

300 × 21

6,300

36

10 × 51

510

37

401 × 30

12,030

38

111 × 90

9,990

1,110

Solve. 39

FOOD

How many hot dogs are in five packages?

5 × 10 = 50 There are 50 hot dogs in the five packages. 40

MONEY There are ten pennies in each dime. If you have six dimes, how many pennies do you have?

6 · 10 = 60 You have 60 pennies. FOOD Hot dogs are packaged in groups of 10.

16 Masterfile

Chapter 4 Multiplication

Copyright © by The McGraw-Hill Companies, Inc.

31

Vocabulary Check Write the vocabulary word that completes each sentence. 41

42

43

The Identity Property of Multiplication says that when you multiply a number by 1, the product is the same as the given number. The property that states if you multiply a number by zero, the product is zero is called the Zero Property of Multiplication

.

Writing in Math Write the fact family for 1 × 9. Explain how you know the facts in a family.

There are two multiplication facts, the given fact and its commutative fact. There are two division facts, the inverse statement for each multiplication fact.

Spiral Review Solve.

Copyright © by The McGraw-Hill Companies, Inc.

44

(Lesson 2-4, p. 91)

COMPUTERS Amado needs to create a password for his computer. He wants to use the digits 4, 9, 6, and 2. He also wants the number to be odd and round to 2,000. What four-digit number should he use as his password?

2,469 45

SAVINGS The amount of money that Tamika has invested in savings bonds rounds to $3,000 and has the digits 8, 1, 4, 2. How much has Tamika invested if the amount is an even number?

$2,814 Write the commutative fact for each expression. 46

9 + 3 = 12 3 + 9 = 12

47

(Lesson 1-5, p. 33)

2+3=5 3+2=5

48

7 + 8 = 15 8 + 7 = 15

51

53 is

Write the word before or after to make each statement true. (Lesson 1-1, p. 4)

49

9 is

after

8.

50

14 is

before

15.

after

52.

Lesson 4-2 Multiply with 0, 1, and 10

17

Chapter

4

Progress Check 1

(Lessons 4-1 and 4-2)

Use a number line to model each expression. Then write and model the commutative fact. 3NS2.2, 4NS4.1 1

2·3=6

3·2=6

Draw an array to model each expression. Then write the commutative fact. 3NS2.2 2

3 · 8 = 24

8 · 3 = 24

Write the repeated addition as a multiplication expression. Then write the commutative fact. 3NS2.2, 4NS4.1 3

9 + 9 + 9 + 9 + 9 + 9 + 9 = 7 · 9 = 63; 9 · 7 = 63

4

2+2+2+2=

5

1+1+1+1+1=

4 · 2 = 8; 2 · 4 = 8 5 · 1 = 5; 1 · 5 = 5

6

2 · 91 = 182

7

2 · 210 = 420

8

0 · 300 =

0

10

102 · 3 = 306

11

113 · 2 = 226

12

111 · 7 = 777

9

202 · 1 = 202

13

73 · 2 = 146

Solve. 3NS2.2, 3NS2.6 14

GAMES How many 1 × 1 squares are on the checkerboard shown? 64 Half of the 1 × 1 squares are black. How many is that? 32

15

COOKING To make one pizza, Tyrone uses 10 ounces of flour, 8 ounces of pepperoni, 5 ounces of pizza sauce, and 4 ounces of cheese. How much of each ingredient will Tyrone need to make 2 pizzas?

20 oz flour, 16 oz pepperoni, 10 oz of sauce, and 8 oz of cheese 18

Chapter 4 Multiplication

Photodisc/Getty Images

GAMES Checkerboards have 8 rows and 8 columns.

Copyright © by The McGraw-Hill Companies, Inc.

Find each product. 3NS2.2, 3NS2.4, 3NS2.6

Lesson

4-3 Multiply by 2 KEY Concept Multiplying by 2 is another way to write the double facts for addition. 2+2=4 6 + 6 = 12 10 + 10 = 20

3+3=6 7 + 7 = 14

4+4=8 8 + 8 = 16

5 + 5 = 10 9 + 9 = 18

Multiples of 2 are the numbers you say when you count by 2s. 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 There are several phrases that mean multiplying 9 by 2. • the product of two and nine • two times nine • double the number nine • twice the number nine

3NS2.2 Memorize to automaticity the multiplication table for numbers between 1 and 10. 3NS2.4 Solve simple problems involving multiplication of multidigit numbers by one-digit numbers.

VOCABULARY double or twice two times a number; words that indicate to multiply by two are double a number or twice a number multiple a multiple of a number is the product of that number and any whole number Example: 30 is a multiple of 10 because 3 × 10 = 30.

Copyright © by The McGraw-Hill Companies, Inc.

You should practice memorizing the multiplication facts of 2. According to the Commutative Property, the product is the same, whether the factor 2 is the first factor or the second factor.

Example 1 Find the product of 8 and 2 by using repeated addition. Then write the multiplication fact and its commutative fact. 1. What is the first factor? 8 What is the second factor? 2 2. Write an equation as repeated addition. Use the first factor as the number being added and the second factor as the number of times you add the number. 8 + 8 = 16 3. Write the multiplication fact. 8 × 2 = 16 4. Write the commutative fact. 2 × 8 = 16 GO ON Lesson 4-3 Multiply by 2

19

YOUR TURN! Find the product of 5 and 2 by using repeated addition. Then write the multiplication fact and its commutative fact. 1. What is the first factor?

5

What is the second factor?

2

2. Write an equation as repeated addition. Use the first factor as the number being added and the second factor as the number of times 5 + 5 = 10 you add the number.

5 × 2 = 10

3. Write the multiplication fact. 4. Write the commutative fact.

2

×

5

Example 2 Use an array to find the missing number that would make the equation true. 4×

?

=8

1. How many rectangles should be inside the array? 8

3. Continue making columns until there are 8 rectangles in the array. The number of columns is the missing number. How many columns do you make? 2

2 columns of 4 rows make 8 rectangles 4. The missing number is 2. The completed equation is 4 × 2 = 8.

20

Chapter 4 Multiplication

10

YOUR TURN! Use an array to find the missing number that would make the equation true. ?

6×

= 12

1. How many rectangles should be inside the array? 12 2. The factor already in the equation is 6 . The array will have six rows. 3. Continue making columns until there are 12 rectangles. How many columns do you make? 2

2

columns of 6 make 12 rectangles

rows

4. The missing number is 2 The completed equation is 6 × 2 = 12.

.

Copyright © by The McGraw-Hill Companies, Inc.

2. The factor already in the equation is 4. The array will have four rows.

=

Who is Correct? Find the product of 68 and 2.

Amber

Peter

Gigi

68 2 × −−− 204

68 2 × −−− 136

60 2 × −−− 126

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

Guided Practice Find each product by using repeated addition. 1

5×2

10

2

9×2

18

3

2×4

8

4

2 × 10

20

5

2×8

16

6

2×2

4

Draw an array to model each expression. Find the product. Then write the commutative fact.

Copyright © by The McGraw-Hill Companies, Inc.

7

8 × 2 = 16

2 × 8 = 16

8

2 × 10 = 20

10 × 2 = 20

Step by Step Practice Use an array to find the missing number that would make the equation true. 9

2×

?

= 12

Step 1 How many rectangles should be inside the array? Step 2 The array will have

2

12

rows.

Step 3 Continue making columns until there are 12 rectangles. How many columns do you use? 6

6 Step 4 2 ×

columns of

6

= 12.

2

rows make

12

rectangles GO ON

Lesson 4-3 Multiply by 2

21

Find each product by using repeated addition.

68

10

2 × 34 =

12

2 × 73 = 146

11

2 × 91 = 182

13

2 × 13 =

26

Use an array to find the missing number that would make the equation true. 14

2×

= 18

Build the array with 2 rows and a total of How many columns are there? 9

18

squares.

15

2×

3

=6

16

2×

5

= 10

17

2×

7

= 14

18

2×

4

=8

Step by Step Problem-Solving Practice

Problem-Solving Strategies ✓ Draw a diagram.

Solve. 19

Understand

Read the problem. Write what you know. There are 18 holes on the course. The score for each hole is 2 strokes.

Plan

Pick a strategy. One strategy is to draw a diagram.

Solve

Draw an array that is 18 by 2.

Count the number of squares. The par score for the 18-hole course is Check

22

You can skip-count by two 18 times.

Chapter 4 Multiplication

36

strokes.

Copyright © by The McGraw-Hill Companies, Inc.

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

The Hilltop Miniature Golf Center has a course with 18 holes. The par for each hole is two strokes. If a player gets a par score on each hole, what would be the score for the 18-hole course?

20

GAMES Marjorie and Helen have one bag of 18 marbles. They each take a marble from the bag until the bag is

9

empty. How many marbles does each girl get? Check off each step.

21

✔

Understand

✔

Plan

✔

Solve

✔

Check

BOOKS Over the summer, two students each read 7 books. How many books did they read total? 14 How do you multiply by 2?

22

Add the number to itself.

Skills, Concepts, and Problem Solving

Copyright © by The McGraw-Hill Companies, Inc.

Find each product by using repeated addition. 23

2·8

8 + 8 = 16

24

6·2

2 + 2 + 2 + 2 + 2 + 2 or 6 + 6 = 12

25

2·0

0

26

1·2

1+1=2

Use an array to find the missing number that would make the equation true. 27

2×

29

7

4

=

× 2 = 14

8

28

6

×2=

12

30

11

× 2 = 22

GO ON Lesson 4-3 Multiply by 2

23

5

31

× 2 = 10

2

32

3×

=6

34

41 × 2 =

36

3,671 × 2 =

Find each product.

66

33

33 × 2 =

35

304 × 2 = 608

82 7,342

Solve. 37

PACKAGING Paper towels come in packages of 2. How many rolls are there in 8 packages? 16

38

MONEY A dime is worth 2 nickels. If you have five dimes, how many nickels would have the same value?

5 · 2 = 10 Five dimes have the same value as 10 nickels. 39

FASHION Kraig bought 3 pairs of shoes. How many shoes does he have? 6

Vocabulary Check Write the vocabulary word that completes each sentence.

factors .

In the expression 2 · 3, 2 and 3 are

41

When you

42

Writing in Math Write three different phrases that describe the expression 2 · 4.

double a number, you multiply the number by 2.

Sample answer: the product of 2 and 4, double the number 4, twice the number 4, 2 times 4

Spiral Review 43

SLEEP On Saturday night Henry slept for 9 hours. On Sunday night, Henry slept 8 hours. How many hours did he sleep on those two nights?

17 hours

(Lesson 3-3, p. 145)

Write each number in expanded form.

8,000 + 900 + 5

44

8,905

24

Chapter 4 Multiplication

(Lesson 2-3, p. 85)

45

6,780

6,000 + 700 + 80

Copyright © by The McGraw-Hill Companies, Inc.

40

Lesson

4-4 Multiply by 5 KEY Concept Multiples of 5 are the numbers you say when you skip-count by 5. 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 5 × 4 = 20 ← commutative facts → 4 × 5 = 20 20 ÷ 5 = 4 ← related division facts → 20 ÷ 4 = 5 You can find 18 × 5 in several ways: 18 × 5 =

VOCABULARY factor a number that divides into a whole number evenly; also a number that is multiplied by another number

Consider the fact family with 4, 5, and 20.

10 + 8 × 5 −−−−− 50 + 40

3NS2.2 Memorize to automaticity the multiplication table for numbers between 1 and 10. 3NS2.4 Solve simple problems involving multiplication of multidigit numbers by one-digit numbers.

4

(Lesson 4-1, p. 4)

18 × 5 −−−− 90

fact family a group of related facts using the same numbers Example: 5 × 3 = 15, 3 × 5 = 15, 15 ÷ 5 =3, 15 ÷ 3 = 5

90

Copyright © by The McGraw-Hill Companies, Inc.

You should practice memorizing the multiplication facts of 5. According to the Commutative Property, the product is the same whether the factor 5 is the first factor or the second factor.

Example 1

YOUR TURN!

Find the product of 5 and 7 by using repeated addition. Then write the multiplication fact and its commutative fact.

Find the product of 5 and 9 by using repeated addition. Then write the multiplication fact and its commutative fact.

1. What is the first factor? 5 What is the second factor? 7

1. What is the first factor? 5 What is the second factor? 9

2. Write an equation as repeated addition. Use the first factor as the number being added and the second factor as the number of times you add the number. 7 + 7 + 7 + 7 + 7 = 35

2. Write an equation as repeated addition. Use the first factor as the number being added and the second factor as the number of times you add the number.

3. Write the multiplication fact. 5 × 7 = 35 4. Write the commutative fact. 7 × 5 = 35

9 + 9 + 9 + 9 + 9 = 45

3. Write the multiplication fact.

5 × 9 = 45

4. Write the commutative fact. 9 × 5 = 45

GO ON

Lesson 4-4 Multiply by 5

25

Example 2

YOUR TURN! Find the product of 43 and 5. Estimate first.

Find the product of 72 and 5. Estimate first. 1. Estimate.

70 × 5 = 350

2. Rewrite the problem in vertical format.

2. Rewrite the problem in vertical format.

3. Multiply 5 times the digit in the ones column. 1 5 × 2 = 10 72 Write the tens digit above the tens × 5 column and the ones digit in the 0 product under the ones column. 4. Multiply 5 times the digit in the tens column. 5 × 7 = 35 Add one 10 for 36.

1. Estimate. 40 × 5 = 200

1

72 × 5 360

5. The product is 360. Compare to your estimate for reasonableness.

3. Multiply 5 times the digit in the ones column. 5 × 3 = 15 Write the tens digit above 1 the tens column and the ones 43 digit in the product under the × 5 5 ones column. 4. Multiply 5 times the digit in the tens column. 5 × 4 = 20 Add one 10 for 21 .

5. The product is 215 . Compare to your estimate for reasonableness.

Find the product of 5 and 45.

45 5 × −−− 205

Muraco 2

45 5 × −−− 225

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

26

Chapter 4 Multiplication

Sanya

40 + 5 5 × −−−− −− 200 + 25 = 225

Copyright © by The McGraw-Hill Companies, Inc.

Who is Correct?

Charlie

1 43 × 5 215

Guided Practice Find each product by using repeated addition. 1

5×5=

25

2

5×9=

45

3

5×4=

20

4

1×5=

5

5

3×5=

15

6

8×5=

40

7

2×5=

10

8

0×5=

0

Step by Step Practice 9

Find the product of 907 and 5. Estimate first. Step 1 Estimate.

900

×5=

4,500 3

Copyright © by The McGraw-Hill Companies, Inc.

Step 2 Rewrite the problem in vertical format.

907 ×5

Step 3 Multiply 5 times the digit in the ones column. 5 × 7 = 35

5

Step 4 Multiply 5 times the digit in the tens column. 5×0= 0 Add the 3 tens above the tens column. 3+ 0 = 3 Step 5 Multiply 5 times the hundreds columns. 5 × 9 = Step 6 The product is 4,535 . Compare to your estimate for reasonableness.

3

907 ×5

35 45 .

3

907 × 5

4,535

GO ON Lesson 4-4 Multiply by 5

27

Find each product. Estimate first. 10

22 · 5 =

1

estimate:

20

· 5 = 100

2 2 5 × 1 1 0 12

202 · 5 =

1,000; 1,010

1,000; 1,010

14

504 · 5 =

2,500; 2,520

315 · 5 =

1,500; 1,575

16

108 · 5 =

500; 540

209 · 5 =

1,000; 1,045

18

550 · 4 =

2,000; 2,200

11

44 · 5 =

13

505 · 2 =

15

17

200; 220

Step by Step Problem-Solving Practice Solve. 19

PARKS There are five benches in the park. Each bench has three children sitting on it. How many children are sitting on the benches altogether? Read the problem. Write what you know. There are 5 benches. 3 children are on each bench.

Plan

Pick a strategy. One strategy is to draw a picture. Read the problem. Each rectangle represents a bench. Each circle represents a child.

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

Solve

number of benches · number of children on each bench = total children 5 3 15 . · = Check

28

Count the circles in the picture. Does the count match your answer?

Chapter 4 Multiplication

Copyright © by The McGraw-Hill Companies, Inc.

Understand

Problem-Solving Strategies ✓ Draw a picture.

20

21

BUSINESS An after-school club is going on a field trip. Parents with vans have offered to drive the students. If there are 25 students in the club, how many vans will be needed if each van 5 holds 5 students? Check off each step.

✔

Understand

✔

Plan

✔

Solve

✔

Check

PACKAGING Each box holds 5 markers. Jamira has 3 boxes. How many markers does she have?

15 Write the multiples of 5 up to 50. What pattern do you observe about the numbers in the ones place?

22

5, 10, 15, 20, 25, 30, 35, 40, 45, 50; the ones place is either a 0 or a 5

Copyright © by The McGraw-Hill Companies, Inc.

Skills, Concepts, and Problem Solving Find each product by using repeated addition. 23

5·1

1+1+1+1+1=5

24

6·5

5 + 5 + 5 + 5 + 5 + 5 = 30

25

5 · 10

10 + 10 + 10 + 10 + 10 = 50

26

0·5

0

GO ON Lesson 4-4 Multiply by 5

29

Find each product. Estimate first. 27

305 × 5

1,500; 1,525

28

41 × 5

200; 205

29

303 × 5

1,500; 1,515

30

412 × 5

2,000; 2,060

31

207 × 5

1,000; 1,035

32

406 × 5

2,000; 2,030

33

660 × 5

3,000; 3,300

34

21 × 5

100; 105

Solve. 35

MUSIC Marcus’s albums have 5 songs on each side. There are 2 sides. If Marcus has 7 albums, how many songs do they contain 70 in all?

36

LANDSCAPING Alice has 15 flowers to plant. She plans to place them in 3 rows with an equal number of flowers in each. How 5 many flowers will be in each row?

MUSIC Marcus collects vinyl albums.

Vocabulary Check Write the vocabulary word that completes each sentence. 37

Commutative Property of Multiplication 38

A number that is multiplied by another number is a(n)

39

Writing in Math Draw a picture of seating 30 students in 5 rows. How many seats in each row? Show and explain how this is related to multiplication.

9 9 9 9 9

factor 9 9 9 9 9

9 9 9 9 9

9 9 9 9 9

9 9 9 9 9

. 9 9 9 9 9

5IFSFBSFSPXT 4JYTUVEFOUTJOFBDI SPXSFTVMUTJOBUPUBM PGTUVEFOUT

There are six chairs in each row. To model multiplication, you use rectangular arrays. This array shows 5 · 6 = 30.

30 Corbis

Chapter 4 Multiplication

Copyright © by The McGraw-Hill Companies, Inc.

The property that states that the order in which two numbers are multiplied does not change the product is the

Spiral Review Solve. 40

(Lesson 2-5, p. 99)

POPULATION The population of California in 2005 was about 36,810,358. Malika wrote “thirty-six million, eight hundred ten thousand, thirty-five.” What mistake did Malika make?

Malika forgot the eight in the ones period. The number is thirty-six million, eight hundred ten thousand, three hundred fifty-eight. 41

BUSINESS A plastic manufacturer had sales of one million, twenty-four thousand, seven hundred thirty-three last year. Write 1,024,733 this number in standard form.

42

POPULATION The estimated population of the United States in 2006 was 300,435,818. Write this number in word form.

three hundred million, four hundred thirty-five thousand, eight hundred eighteen

Copyright © by The McGraw-Hill Companies, Inc.

Write the nearest hundreds each number is between.

(Lesson 2-2, p. 77)

43

320

300 and 400

44

842

800 and 900

45

133

100 and 200

46

495

400 and 500

Complete each number pattern. Explain the number pattern. (Lesson 1-4, p. 25)

47

44, 55, 66,

77 , 88;

add 11

48

15, 20, 25,

30 , 35;

add 5

49

53, 49, 45,

41 , 37;

subtract 4

50

49, 42, 35,

28 , 21;

subtract 7

Lesson 4-4 Multiply by 5

31

Chapter

4

Progress Check 3

(Lessons 4-3 and 4-4)

Draw an array to model each expression. Find the product. Then write its commutative fact. 3NS2.2 1

9 × 2 = 18

2 · 9 = 18

2

3 × 5 = 15

4

4·5

6

10 · 0

5 × 3 = 15

Multiply by using repeated addition. 3NS2.2

2+2+2=6

3

3·2

5

9·1 1+1+1+1+1+1+

5 + 5 + 5 + 5 = 20 0

1+1+1=9 Find each product. 3NS2.2 7

5×9

45

8

7×2

14

9

0×5

0

10

1·2

2

11

8·5

40

12

2·5

10

13

6·2

12

14

5·6

30

15

203 × 2 =

400; 406

16

113 × 5 =

500; 565

17

605 × 2 =

1,200; 1,210

18

111 × 5 =

500; 555

Solve. 3NS2.2, 3NS2.4 19

MONEY Five nickels have the same value as 1 quarter. How many nickels have the same value as 7 quarters?

5 ∙ 7 = 35 20

Seven quarters have the same value as 35 nickels.

HOBBIES Pooja collects salt-and-pepper shaker sets. If each set consists of 1 salt shaker and 1 pepper shaker, how many shakers does she have if she has 8 sets?

16 32

Chapter 4 Multiplication

United States Mint

Copyright © by The McGraw-Hill Companies, Inc.

Find each product. Estimate first. 3NS2.2, 3NS2.4

Lesson

4-5 Multiply by 3 KEY Concept Multiples of 3 are the numbers you say when you by 3. 3, 6, 9, 12, 15, 18, 21, 24, 27, 30

3NS2.2 Memorize to automaticity the multiplication table for numbers between 1 and 10. 3NS2.4 Solve simple problems involving multiplication of multidigit numbers by one-digit numbers. 4NS3.2 Demonstrate an understanding of, and the ability to use, standard algorithms for multiplying a multidigit number by a skip-count two-digit number and for dividing a multidigit number by a one-digit number; use relationships between them to simplify computations and to check results.

You can write the product of three and a number as 3n. Writing the letter (called a variable) next to the number indicates multiplication. Triple is a word that means to multiply by three. The expression 3n can be written in words in several ways. • • • •

the product of three and a number three times a number three multiplied by a number triple a number

You should practice memorizing the multiplication facts of 3. According to the Commutative Property, the product is the same, whether the factor 3 is the first factor or the second factor.

VOCABULARY triple a number that is multiplied by 3, or added together 3 times Example: “triple 2” is 3×2=6 factor a number that divides into a whole number evenly; also a number that is multiplied by another number Example: 3 × 4 = 12 factors

Example 1 Find the product of 3 and 9 by using repeated addition. Then write the multiplication fact and its commutative fact. 1. What is the first factor? 3 What is the second factor? 9

product the answer or result of a multiplication problem; it also refers to expressing a number as the product of its factors Example: 3 × 4 = 12

⎧ ⎨ ⎩

Copyright © by The McGraw-Hill Companies, Inc.

(Lesson 4-1, p. 4)

product (Lesson 4-1, p. 4)

2. Write an equation as repeated addition. Use the first factor as the number being added and the second factor as the number of times you add the number. 9 + 9 + 9 = 27 3. Write the multiplication fact. 3 × 9 = 27 4. Write the commutative fact. 9 × 3 = 27

GO ON Lesson 4-5 Multiply by 3

33

YOUR TURN! Find the product of 3 and 4 by using repeated addition. Then write the multiplication fact and its commutative fact. 1. What is the first factor?

3

4

What is the second factor?

2. Write an equation as repeated addition. Use the first factor as the number being added and the second factor as the number of times you add the number. 4 + 4 + 4 = 12 3. Write the multiplication fact.

3 × 4 = 12

4. Write the commutative fact. 4 × 3 = 12

Example 2 Use an array to find the missing number that would make the equation true. 3×

?

= 18

1. The factor already in the equation is 3. The array will have three rows.

6 columns of 3 rows make 18 rectangles 3. The missing number is 6. The completed equation is 3 × 6 = 18.

34

Chapter 4 Multiplication

Use an array to find the missing number that would make the equation true. 3×

?

= 15

1. The factor already in the equation is 3 . The array will have 3 rows. 2. Continue making columns until there are 15 rectangles. How many columns do you make? 5

5

columns of 3 make 15 rectangles 3. The missing number is 5 The complete equation is 3 × 5 = 15 .

rows .

Copyright © by The McGraw-Hill Companies, Inc.

2. Continue making columns until there are 18 rectangles. How many columns do you make? 6

YOUR TURN!

Who is Correct? Find the missing number. 3 ×

= 33

Ethan

Heather

Shawanna

3 × 11 = 33

3 · 10 = 33

3 · 1 = 33

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

Guided Practice Find each product by using repeated addition. 1

9×3

27

2

5×3

15

3

3×4

5

3×2

6

6

8×3

24

7

10 × 3

12 30

4

3×3

9

8

1×3

3

Draw an array to model each expression. Find the product. Then write its commutative fact.

Copyright © by The McGraw-Hill Companies, Inc.

9

3 × 8 = 24; 8 × 3 = 24

3 × 10 = 30; 10 × 3 = 30

10

Step by Step Practice 11

Find the product of 545 and 3 using expanded form. Step 1 Write the first number in expanded form.

500 + 40 + 5

Step 2 Write the numbers vertically. Find the products of each number.

500 + 40 + 5 × 3

Step 3 Multiply 3 times the ones place value. 3 5 15 × = Multiply 3 times the tens place value. 3 40 120 × = Multiply 3 times the hundreds place value. 3 500 1,500 × = Step 4 Find the sum of the products. 1,500 + 120 +

15

=

1,635

GO ON

Lesson 4-5 Multipy by 3

35

Find each product by using repeated addition. 12

3 · 122 =

13

3 · 45 =

15

3 · 307 =

17

3 · 331 =

122 + 122 + 122 = 135

366

14

3 · 211 =

633

921

16

3 · 503 =

1,509

662

18

3 · 330 =

990

Problem-Solving Strategies

Step by Step Problem-Solving Practice Solve. 19

TRAVEL Three buses were used for the field trip. Each bus will hold 35 students. How many students can go on the field trip? Understand

Read the problem. Write what you know. There are 3 buses. Each bus holds 35 students.

Plan

Pick a strategy. One strategy is to use logical reasoning.

Solve

There are 3 buses; each holds 35 students. You want to triple 35. How many students are going on the field trip? number of buses · number of students =

35

=

105 .

Use repeated addition to check your answer.

Check

36

·

total

INDUSTRY A factory that makes tennis balls puts 3 balls in each can. If a case has 24 tennis balls, how many cans are there in each case? 8 Check off each step.

✔

Understand

✔

Plan

✔

Solve

✔

Check

Chapter 4 Multiplication

Copyright © by The McGraw-Hill Companies, Inc.

3

20

Draw a diagram. ✓ Use logical reasoning. Solve a simpler problem. Work backward. Act it out.

21

SHOPPING Joe had to make three round trips to the market. The market is four miles from his house. How many miles did Joe travel to make the three round trips? 24 miles

22

Write an expression for each phrase below. Use “a number.” Evaluate the expression when

9×

Nine times a number Six times a number Triple a number

= 3.

; 27

6× 3×

for

; 18 ;9

The product of eight and a number

8×

; 24

Skills, Concepts, and Problem Solving Multiply by using repeated addition. 23

3·9=

9 + 9 + 9 = 27

24

3·3=

3+3+3=9

25

3·1=

1+1+1=3

26

0·3=

0

Copyright © by The McGraw-Hill Companies, Inc.

Use an array to find the missing number that would make the equation true. 27

3×

3

=9

28

10 ×

29

3×

4

= 12

30

3×

3

7

= 30

= 21

GO ON Lesson 4-5 Multipy by 3

37

Multiply by using repeated addition. 31

503 × 3

503 + 503 + 503 = 1,509

32

171 × 3

171 + 171 + 171 = 513

33

391 × 3

391 + 391 + 391 = 1,173

34

211 × 3

211 + 211 + 211 = 633

Solve. 35

CONSTRUCTION Zane is building a square rabbit hutch. He needs to have 3 boards for each side. How many boards will he need?

12

36

TENNIS After her tennis lesson, Alisha is putting tennis balls back into their cans. If each can holds 3 tennis balls, how many cans will she use to hold 27 balls? 9

37

NUTRITION Ti watches his diet carefully to make sure he eats at least 3 pieces of fruit each day. How many pieces of fruit does he eat in 1 week? 21

Vocabulary Check Write the vocabulary word that completes each sentence. 38

The answer in a multiplication problem is the product .

39

When you multiply a number by 3, you

40

Writing in Math Harry was asked to triple the number nine. He wrote 9 + 3 = 12. What did Harry do wrong?

triple

it.

so he should have written 3 × 9 = 27.

Spiral Review 41

INTERIOR DESIGN There is a French door between Darnell’s kitchen and living room. If there are 3 columns of window panes, how many rows are there if there are 15 panes total? (Lesson 4-1, p. 4) 5

Write true or false for each statement. If a statement is false, change the statement to make it true. (Lesson 2-6, p. 105)

true

42

349,000 < 412,000

43

810,000 < 801,000 false; 810,00 > 801,000

38

Chapter 4 Multiplication

Copyright © by The McGraw-Hill Companies, Inc.

Harry used addition instead of multiplication. Triple means to multiply by 3,

Lesson

4-6 Multiply by 4

3NS2.2 Memorize to automaticity the multiplication table for numbers between 1 and 10. 3NS2.4 Solve simple problems involving multiplication of multidigit numbers by one-digit numbers. 4NS4.1 Understand that many whole numbers break down in different ways. KEY Concept 4NS3.2 Demonstrate an understanding of, and the ability to use, standard algorithms for multiplying a multidigit Multiples of 4 are the numbers you say when you skip-count number by a two-digit number and for by 4. dividing a multidigit number by a a onedigit number; use relationships between 4, 8, 12, 16, 20, 24, 28, 32, 36, 40 them to simplify computations and to check results. Consider the fact family with 4, 7, and 28.

4 × 7 = 28 ← commutative facts → 7 × 4 = 28 28 ÷ 4 = 7 ← related division facts → 28 ÷ 7 = 7

VOCABULARY

You can find 56 × 4 in several ways. 56 × 4 = 50 + 6 × 4 −−−− 200 + 24

multiple a multiple of a number is the product of that number and any whole number (Lesson 4-3, p. 19)

2

56 × 4 −−− 224

fact family a group of related facts using the same numbers

224 Practice memorizing the multiplication facts of 4. According to the Commutative Property, the product is the same, whether the factor 4 is the first factor or the second factor.

(Lesson 3-1, p. 130)

Example 1

Copyright © by The McGraw-Hill Companies, Inc.

Draw an array to model the expression 4 × 8. Find the product. Then write the commutative fact. 1. Write the answer if you know it. Otherwise, draw an array. The first factor is a 4, so there will be 4 rows. The second factor is an 8, so there will be 8 columns. 2. Label the array 4 × 8. Count the number of rectangles. 32

4×8

3. Write the commutative fact. 4 × 8 = 32; 8 × 4 = 32 YOUR TURN! Draw an array to model the expression 4 × 7. Find the product. Then write the commutative fact. 1. Write the answer if you know it. Otherwise, draw an array. The first factor is a 4, so there will be 4 rows. The second factor is a 7, so there will be 7 columns. 2. Label the array. Count the number of rectangles. 3. Write the commutative fact. 4 × 7 = 28 ;

7

×

4

=

28 28

4

×

7 GO ON

Lesson 4-6 Multiply by 4

39

Example 2

YOUR TURN!

Find the product of 203 and 4. Estimate first.

Find the product of 103 and 4. Estimate first. 1. Estimate. 100 × 4 = 400

1. Estimate. 200 × 4 = 800 2. Rewrite the problem in vertical format. 3. Multiply 4 times the digit in the ones column. 4 × 3 = 12 1

Write the tens digit above the tens column, and the ones digit in the product under the ones column.

203 ×4 −−−− 2

4. Multiply 4 times the digit in the tens column. 4×0=0 Add in one 10 for 1.

203 ×4 −−−− 12

5. Multiply 4 times the digit in the hundreds column. 4×2=8

203 ×4 −−−− 812

1

1

6. The product is 812. Compare to your estimate for reasonableness.

2. Rewrite the problem in vertical format. 3. Multiply 4 times the digit in the ones column. 4 × 3 = 12 Write the tens digit above the tens column, and the ones digit in the product under the ones column.

1 103 ×4 −−−− 12

5. Multiply 4 times the digit in the hundreds column. 4×1= 4

103 ×4 −−−− 412

Find the product of 201 and 4.

Ciara

Jordan

Reno

201 4 × −−− 205

201 4 × −−− 842

201 4 × −−− 804

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

1

Copyright © by The McGraw-Hill Companies, Inc.

Who is Correct?

Chapter 4 Multiplication

2

4. Multiply 4 times the digit in the tens column. 4 × 0 = 0 Add in one 10 for 1 .

6. The product is 412 . Compare to your estimate for reasonableness.

40

1 103 ×4 −−−−

Guided Practice Draw an array to model each expression. Find the product. Then write the commutative fact. 1

2×4

8

2

6×4

24 ;

2 × 4 = 8; 4 × 2 = 8

;

6 × 4 = 24; 4 × 6 = 24

Find each product. 3

5×4=

20

4

4 × 10 =

6

2×4=

8

7

8×4=

9

1×4=

4

10

4×4=

40

5

3×4=

12

32

8

6×4=

24

16

11

4×9=

36

Step by Step Practice

Copyright © by The McGraw-Hill Companies, Inc.

Use an array to find the missing number that would make the equation true. 12

4×

= 36

Step 1 The factor already in the equation is will have 4 rows.

4

. The array

Step 2 Continue making columns until there are 36 rectangles. How many columns do you make? 9

9 Step 3 4 ×

columns of 4 rows make 36 rectangles

9

= 36

GO ON Lesson 4-6 Multiply by 4

41

Use an array to find the missing number that would make each equation true. 13

4=4× number of squares: 4 known factor: 4 number of rows: 4 =

14

4×

1 = 20

=

5

=

4

15

16 = 4 ×

17

315 · 4

1,200; 1,260

19

802 · 4

3,200; 3,208

Find each product. Estimate first. 16

67 · 4

18

630 · 4

280; 268 2,400; 2,520

Step by Step Problem-Solving Practice Solve. 20

Problem-Solving Strategies ✓ Draw a picture.

Understand

Read the problem. Write what you know. 4 pumpkins grow on each vine. There are 7 vines. There are 25 students.

Plan

Pick a strategy. One strategy is to draw a picture.

Solve

There are 7 vines, and each vine holds 4 pumpkins.

Use logical reasoning. Solve a simpler problem. Work backward. Use an equation.

number of pumpkins · number of vines = total number of pumpkins 4 · 7 = 28 . Are there enough pumpkins?

Yes, because 28 > 25; there will be enough Check

42

You can count the pumpkins on the vines in your picture to check your answer.

Chapter 4 Multiplication

Copyright © by The McGraw-Hill Companies, Inc.

FARMING There are an average of 4 pumpkins growing on each vine in Tina’s garden. She plans to bring one pumpkin to each of the 25 students in her class. If she has 7 vines in her garden, will she have enough pumpkins for her class?

21

22

FASHION A shoe store is having a sale. If you buy one pair of shoes, you get another pair free. If Lanoise buys 4 pairs of shoes, how many shoes will she be buying in all? 16 Check off each step.

✔

Understand

✔

Plan

✔

Solve

✔

Check

GAMES Joanne can play a video game 2 times for every dollar she inserts into the machine. If she puts $4 into the video game, how many times will she be able to play? 8 Write the fact family for 4, 6, and 24.

23

4 × 6 = 24, 6 × 4 = 24, 24 ÷ 6 = 4, 24 ÷ 4 = 6

Skills, Concepts, and Problem Solving Draw an array to model each expression. Find the product. Then write the commutative fact. 24

4×1

25

Copyright © by The McGraw-Hill Companies, Inc.

4

5×4

20 ; 5 × 4 = 20; 4 × 5 = 20

;

4 × 1 = 4; 1 × 4 = 4

Use an array to find the missing number that would make each equation true. 26

4×

28

4×

4

= 16

27

4×

2

=8

10 = 40

29

4×

0

=0

GO ON Lesson 4-6 Multiply by 4

43

30

4×

31

24 = 4 ×

= 12

=

3

=

6

Find each product. Estimate first. 32

522 × 4

2,000; 2,088

33

4 × 305

1,200; 1,220

34

111 × 4

400; 444

35

4 × 270

1,200; 1,080

Solve. PACKAGING There are 30 students in Mr. Wells’ classroom. He buys 6 boxes of markers so he will have just enough to give each student one marker. How many markers are in each box? 5

37

SOCCER Aja has soccer practice each day after school. If the soccer season is 9 weeks long, how many times will she go to practice? 45

38

SCHOOL There are 7 rows with 5 seats each in the classroom. How many empty seats will there be if there are 32 students present? 3

Vocabulary Check Write the vocabulary word that completes each sentence. 39

A group of related facts using the same numbers is a(n) fact family .

40

In the expression 4 × 4, the 4s are

44

Chapter 4 Multiplication

factors

.

Copyright © by The McGraw-Hill Companies, Inc.

36

41

Writing in Math How do you know when to regroup during multiplication?

Sample answer: When you are multiplying a multidigit number by another number, and one of the products is more than nine, you regroup to the next place value.

Spiral Review Find each product. Estimate first.

(Lesson 4-2, p. 11)

42

102 × 4

400; 408

43

401 × 2

800; 802

44

5 × 101

500; 505

45

101 × 6

600; 606

Solve. 46

PUZZLES Use the digits between 0 and 6 to make a three-digit number with digits that have a sum of 10. Write two numbers that match this description. (Lesson 3-1, p. 130)

Answers will vary. Sample numbers are 532 and 325.

Copyright © by The McGraw-Hill Companies, Inc.

47

PUZZLES Use the digits between 0 and 6 to make a three-digit number with digits that have a product of 12. Write two numbers that match this description. (Lesson 4-5, p. 33)

Answers will vary. Sample numbers are 341 and 143.

What is the missing number in each equation? 48

300,000 + 5,000 + 500 +

49

700,000 + 60,000 +

70

3,000

(Lesson 2-5, p. 99)

+ 4 = 305,574

+ 20 + 1 = 763,021

Lesson 4-6 Multiply by 4

45

Chapter

4

Progress Check 3

(Lessons 4-5 and 4-6)

Draw an array to model each expression. Find the product. Then write the commutative fact. 3NS2.2, 4NS3.2 1

3∙7

2

21 ;

3 · 7 = 21; 7 · 3 = 21

3×4

12 ;

3 × 4 = 12; 4 × 3 = 12

Find each product. 3NS2.2 3

4×9=

36

4

7×4=

28

5

0×4=

0

6

1×4=

4

7

3·5=

15

8

2·3=

6

9

6·3=

18

10

0·3=

0

Find each product. Estimate first. 3NS2.2, 3NS2.4 115 × 4 =

400; 460

12

211 × 4 =

800; 844

13

304 × 3 =

900; 912

14

3 × 322 =

900; 966

15

4 × 612 =

2,400; 2,448

16

229 × 3 =

600; 687

17

814 × 3 =

2,400; 2,442

18

918 × 4 =

3,600; 3,672

Solve. 3NS2.2, 3NS2.4 19

MONEY Four quarters are equal to one dollar. If you have 6 dollars in quarters, how many quarters equal 6 dollars?

6 · 4 = 24 Six dollars is 24 quarters. 20

MARKERS Meredith bought markers. If each box has 3 markers, how many boxes did she buy if she bought 18 markers?

6 46

Chapter 4 Multiplication

coins: United States Mint, bill: Michael Houghton/StudiOhio

Copyright © by The McGraw-Hill Companies, Inc.

11

Lesson

4-7 Multiply by 6

3NS2.2 Memorize to automaticity the multiplication table for numbers between 1 and 10. 3NS2.4 Solve simple problems involving multiplication of multidigit numbers by one-digit numbers. 4NS4.1 Understand that many whole numbers break down in different ways. KEY Concept 4NS3.2 Demonstrate an understanding of, and the ability to use, standard algorithms for multiplying a multidigit Multiples of 6 are the numbers you say when you skip-count number by a two-digit number and for by 6. dividing a multidigit number by a a one-digit number; use relationships 6, 12, 18, 24, 30, 36, 42, 48, 54, 60 between them to simplify computations and to check results. Consider the fact family with 6, 8, and 48.

6 × 8 = 48 ← commutative facts 48 ÷ 6 = 8 ← related division facts

→ 8 × 6 = 48 → 48 ÷ 8 =

VOCABULARY multiple a multiple of a number is the product of that number and any whole number Example: 30 is a multiple of 10 because 3 x 10 = 30. (Lesson 4-3, p. 19)

You can find 72 × 6 in several ways. 70 + 2 72 × 6 = × 6 −−−−− 420 + 12 1 72 × 6 432 −−− 432

fact family a group of related facts using the same numbers

You should practice memorizing the multiplication facts of 6. According to the Commutative Property, the product is the same, whether the factor 6 is the first factor or the second factor.

Copyright © by The McGraw-Hill Companies, Inc.

Example 1

(Lesson 3-1, p. 130)

YOUR TURN!

Draw an array to model the expression 6 × 7. Find the product. Then write the commutative fact. 1. Write the answer if you know it. Otherwise, draw an array. The first factor is a 6, so there will be 6 rows. The second factor is a 7, so there will be 7 columns.

6×7

Draw an array to model the expression 6 × 8. Find the product. Then write the commutative fact. 1. Write the answer if you know it. Otherwise, draw an array. The first factor is a 6, so there will be 6 rows. The second factor is an 8, so there will be 8 columns.

6

×

8

2. Label the array 6 × 7. Count the number of squares. 42

2. Label the array. Count the number of squares.

3. Write the commutative fact. 6 × 7 = 42; 7 × 6 = 42

3. Write the commutative fact. 6 × 8 = 48 ; 8 × 6 = 48

48 GO ON

Lesson 4-7 Multiply by 6

47

Example 2

YOUR TURN! Find the product of 103 and 6.

Find the product of 208 and 6. 1. Rewrite the problem in vertical format.

1. Rewrite the problem in vertical format. 2. Multiply 6 times the digit in the ones column. 6 × 3 = 18

2. Multiply 6 times the digit in the ones column. 6 × 8 = 48

4

Write the tens digit above the tens column, and the ones digit in the product under the ones column.

208 × 6 −−−− 8 4

3. Multiply 6 times the digit in the tens column. 6 × 0 = 0 Add 4 tens for 4.

208 × 6 −−−− 48

4

4. Multiply 6 times the digit in the hundreds column. 6 × 2 = 12

Write the tens digit above the tens column, and the ones digit in the product under the ones column.

208 × 6 −−−− 1,248

5. Write the answer. 1,248

3. Multiply 6 times the digit in the tens column. 6 × 0 = 0 Add one 10 for 1 . 4. Multiply 6 times the digit in the hundreds column. 6 × 1 = 6 5. Write the answer. 618

1

103 × 6 −−−−

8

1 103 × 6 −−−−

18

1 103 × 6 −−−−

618

Who is Correct? Find the product of 302 and 6.

Val

302 6 × −− −− 1,812

Loren

302 6 × −−−− 812

302 6 × −−−− 182

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

Guided Practice Draw an array to model each expression. Find the product. Then write the commutative fact. 1

48

1×6

6

; 6; 1 × 6 = 6; 6 × 1 = 6

Chapter 4 Multiplication

2

6×4

24 ; 6 × 4 = 24; 4 × 6 = 24

Copyright © by The McGraw-Hill Companies, Inc.

Shannon

Find each product. 3

6×6=

36

4

4×6=

24

5

3×6=

18

6

6×0=

0

7

8×6=

48

8

6 × 10 =

60

9

9×6=

54

10

6×7=

42

Step by Step Practice 11

Find the product of 307 and 6. Estimate first. Step 1 Estimate. 300 × 6 = 1,800 Step 2 Rewrite the problem in vertical format.

4

Step 3 Multiply 6 times the digit in the ones column. 6× 7 = 42

307 × 6 −−− 2

Step 4 Multiply 6 times the tens column. 6 × 0 = 0 Add the 4 tens above the tens column. 4 + 0 = 4

42

4

Step 5 Multiply 6 times the hundreds column. 6× 3 = 18 18

Copyright © by The McGraw-Hill Companies, Inc.

4

307 × 6 −−−

307 × 6 −−−

Step 6 The product is 1,842 . Compare to your estimate for reasonableness.

1,842

Find each product. Estimate first. 12

118 · 6 =

600; 708

13

350 · 6 =

2.400; 2,100

15

6 · 127 =

600; 762

4

1 1 8 6 × 7 0 8 14

6 · 606 =

3,600; 3,636

GO ON Lesson 4-7 Multiply by 6

49

Step by Step Problem-Solving Practice

Problem-Solving Strategies Draw a diagram.

16

CELEBRATIONS Violet invited 6 people to her birthday party. She bought 2 bags of candy that each had 24 pieces. She gave each person at the party the same number of pieces of candy. How many pieces of candy did each person get?

✓ Use logical reasoning. Solve a simpler problem. Work backward. Guess and check.

Understand

Read the problem. Write what you know. There are 2 bags of candy with 24 pieces in each. There are 6 people at the party.

Plan

Pick a strategy. One strategy is to use logical reasoning.

Solve

How can you find the total pieces of candy? Multiply the number of bags times the number of pieces in each bag. 2 × 24 = 48 How can you find the number of pieces for each person? number for each person × each person = total pieces

6

8 × = 48 There are pieces of candy. Each person will receive 8 pieces.

You can act it out using small cubes or pieces of paper.

Check

50

SURVEYS Karl was hired to survey people leaving a movie theater to see how well they liked the movie they saw. He was instructed to stop every sixth person. If Karl stopped 9 people, how many people walked by him in all? 54 Check off each step.

✔

Understand

✔

Plan

✔

Solve

✔

Check

Chapter 4 Multiplication

Copyright © by The McGraw-Hill Companies, Inc.

17

48

18

COMMUNITY SERVICE Shaq collected canned goods to donate to a food bank. Every house he visited gave him an average of 6 cans of food. If he has 36 cans of food, how many houses did he visit?

6

Bob is planting 12 seeds in a garden. He wants to plant the same number of seeds in each row. Draw arrays to show the different ways that Bob could plant the seeds.

19

Skills, Concepts, and Problem Solving Draw an array to model each expression. Find the product. Then write the commutative fact. 20

12 ; 6 × 2 = 12; 2 × 6 = 12

6×2

21

6×5

30 ; 6 × 5 = 30; 5 × 6 = 30

Copyright © by The McGraw-Hill Companies, Inc.

Find each product. 22

6·7

42

23

0·6

0

24

6·6

36

25

1·6

6

Use an array to find the missing number that would make the equation true. 26

6×

3

= 18

27

6×

4

= 24

Find each product. Estimate first. 28

114 × 6

600; 684

29

6 × 408

2,400; 2,448

30

213 × 6

1,200; 1,278

31

6 × 390

2,400; 2,340

GO ON

Lesson 4-7 Multiply by 6

51

Solve. 32

MUSIC Mr. Taylor and Ms. Parson want to arrange for their classes to sit together to sing on stage at an assembly. There are 25 students in one class and 29 in the other. If there are 6 rows on the stage, how many students should sit in each row? 9

33

HOBBIES Zion sews quilts to donate to a hospital for newborn babies. He wants to double the size of the quilts so he can give them to the children’s wing as well. The small quilts are made from 30 squares. How many squares will the larger quilts need? 60 If the larger quilts have 6 squares in each row, how many squares will be in each column? 10

Vocabulary Check Write the vocabulary word that completes each sentence.

product

34

The answer to a multiplication problem is the

35

multiple A(n) of a number is the product of that number and any whole number; for example, 30 is a multiple of 10.

36

.

Writing in Math Brad was to write an example of the commutative property. He wrote 6 × 9 = 6 × 9. What did Brad do wrong?

Brad forgot to change the order of the numbers being multiplied. He should have written 6 × 9 = 9 × 6.

Write each number in word form.

(Lesson 2-6, p. 105)

37

256,366

two hundred fifty-six thousand, three hundred sixty-six

38

1,306,101

one million, three hundred six thousand, one hundred one

Solve.

(Lesson 2-2, p. 77)

39

BUSINESS The coffee shop’s sales on Sunday contained the digits 1, 9, and 3, and rounded to $400. How much were the coffee shop’s $391 sales?

40

NUTRITION Jeremy read the doughnut package to see how many Calories were in one doughnut. Later, he did not remember the exact number, but he knew the digits were 5, 6, and 1 and that it rounded to 200. If the number was even, how many Calories were 156 there?

52

Chapter 4 Multiplication

Copyright © by The McGraw-Hill Companies, Inc.

Spiral Review

Lesson

4-8 Multiply by 7 KEY Concept Multiples of 7 are the numbers you say when you skip-count by 7. 7, 14, 21, 28, 35, 42, 49, 56, 63, 70

3NS2.2 Memorize to automaticity the multiplication table for numbers between 1 and 10. 3NS2.4 Solve simple problems involving multiplication of multidigit numbers by one-digit numbers. 4NS4.1 Understand that many whole numbers break down in different ways. 4NS3.2 Demonstrate an understanding of, and the ability to use, standard algorithms for multiplying a multidigit number by a two-digit number and for dividing a multidigit number by a one-digit number; use relationships between them to simplify computations and to check results.

You should practice memorizing the multiplication facts of 7. According to the commutative property, the factor 7 can be the first factor or the second factor.

VOCABULARY multiple a multiple of a number is the product of that number and any whole number Example: 30 is a multiple of 10 because 3 × 10 = 30. (Lesson 4-3, p. 19)

Example 1 Draw an array to model the expression 7 × 3. Find the product. Then write the commutative fact.

Copyright © by The McGraw-Hill Companies, Inc.

1. Write the answer if you know it. Otherwise, draw an array. The first factor is a 7, so there will be 7 rows. The second factor is a 3, so there will be 3 columns. 2. Label the array as a 7 × 3. Count the number of rectangles. 21 3. Write the commutative fact. 7 × 3 = 21; 3 × 7 = 21 YOUR TURN! Draw an array to model the expression 7 × 6. Find the product. Then write the commutative fact. 1. Write the answer if you know it. Otherwise, draw an array. The first factor is a 7, so there will be 7 rows. The second factor is a 6, so there will be 6 columns. 2. Label the array as a of rectangles. 42

7

×

3. Write the commutative fact.

6

. Count the number

7 × 6 = 42 ;

7

6 × 7 = 42

×

6 GO ON

Lesson 4-8 Multiply by 7

53

Example 2

YOUR TURN! Find the product of 202 and 7. Estimate first.

Find the product of 104 and 7. Estimate first.

1. Estimate. 200 × 7 =

1. Estimate. 100 × 7 = 700

1,400

2. Rewrite the problem in vertical format. 2. Rewrite the problem in vertical format. 3. Multiply 7 times the digit in the ones column. 7 × 4 = 28 Write the tens digit above the tens column, and the ones digit in the product under the ones column.

2

104 ×7 −−− 8

4. Multiply 7 times the digit in the tens column. 7 × 0 = 0 Add 2 tens for 2. 5. Multiply 7 times the digit in the hundreds column. 7×1 =7

2

104 ×7 −−− 28 2

6. The product is 728. Compare to your estimate for reasonableness.

104 ×7 −−− 728

3. Multiply 7 ones column. 7 × 2 = 14

1 Write the tens digit above the tens column, and the ones digit in the product under the ones column.

202 ×7 −−−−

4. Multiply 7 times the digit in the tens column. 7×0= 0 Add one 10 for 1 .

202 ×7 −−−−

5. Multiply 7 times the digit in the hundreds column. 7 × 2 = 14

= 56

Jeremy

Helena

Danny

7 × 56 = 392

7 × 8 = 56

7 × 7 = 49

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

54

Chapter 4 Multiplication

4

1 14

1 202 ×7 −−−−

1,414

Copyright © by The McGraw-Hill Companies, Inc.

6. The product is 1,414 Compare to your estimate for reasonableness.

Who is Correct? Find the missing number. 7 ×

times the digit in the

Guided Practice Draw an array to model each expression. Find the product. Then write the commutative fact. 1

8×7

56 ; 8 × 7 = 56; 7 × 8 = 56

7×2

2

14 ; 7 × 2 = 14; 2 × 7 = 14

Find each product. 3

7×8=

56

4

7×6=

7

2×7=

14

8

7 × 10 =

42 70

5

3×7=

21

6

7×5=

35

9

7×7=

49

10

1×7=

7

Step by Step Practice 11

Find the product of 620 and 7. Estimate first.

Copyright © by The McGraw-Hill Companies, Inc.

Step 1 Estimate. 600 × 7 =

4,200

Step 2 Rewrite the problem in vertical format. Step 3 Multiply 7 times the digit in the 7× 0 = 0 Step 4 Multiply 7 times the digit in the 7× 2 = 14

ones

column.

1

for

0

1

tens

column.

Step 5 Multiply 7 times the digit in the hundreds column. 7× 6 = 42 Add

620 × −−−7

43 .

620 × −−−7

40 1

620 × −−−7

4,340

Step 6 The product is 4,340 . Compare to your estimate for reasonableness.

GO ON Lesson 4-8 Multiply by 7

55

Find each product. Estimate first. 12

111 · 7 =

1 1 1 7 ×

700; 777

7 7 7 13

7 · 422 =

2,800; 2,954

14

202 · 7 =

1,400; 1,414

15

7 · 170 =

1,400; 1,190

16

331 · 7 =

2,100; 2,317

17

7 · 814 =

5,600; 5,698

18

521 · 7 =

3,500; 3,647

Step by Step Problem-Solving Practice

Problem-Solving Strategies ✓ Draw a picture.

Solve. 19

Use logical reasoning. Solve a simpler problem. Work backward. Act it out.

MUSIC A harp is a stringed musical instrument with 7 pedals. A music room has 9 harps. How many pedals are there? Understand

Read the problem. Write what you know. Each harp has 9 There are

7

pedals. harps.

Pick a strategy. One strategy is to draw a picture.

Solve

Draw 9 figures to show 9 harps. On each figure place a 7 to show the pedals.

Use repeated addition to find the number of pedals. 7+7+7+7+7+7+7+7+7= There are Check

56

63

63

pedals.

Does the answer make sense? Look over your solution. Did you answer the question?

Chapter 4 Multiplication

Copyright © by The McGraw-Hill Companies, Inc.

Plan

20

NATURE Rico bought saplings to plant along his back fence. The saplings have 42 branches altogether. If there are 6 saplings and each has the same number of branches, how many branches does each sapling have? Explain your strategy.

See TWE margin.

Check off each step.

21

✔

Understand

✔

Plan

✔

Solve

✔

Check

ENTERTAINMENT At a dinner party, there are 8 round tables that seat 7 people each. If there were 60 people invited to the party, will there be enough seats for everybody if they all attend? Explain.

No; there would only be seats for 56 people. Explain how multiplication is commutative. Give an example.

22

Answers will vary, but students should realize that the order in which the

Copyright © by The McGraw-Hill Companies, Inc.

factors are multiplied does not affect the product.

Skills, Concepts, and Problem Solving Draw an array to model each expression. Find the product. Then write the commutative fact. 23

7×3

21 ; 7 × 3 = 21; 3 × 7 = 21

24

7×5

35 ; 7 × 5 = 35; 5 × 7 = 35

Find each product. 25

0·7=

0

26

7·7=

49

27

7·4=

28

28

1 · 7=

7

GO ON Lesson 4-8 Multiply by 7

57

Use an array to find the missing number that would make the equation true. 29

7×

2

= 14

30

7×

9

= 63

31

7×

5

= 35

32

7×

7

= 49

Find each product. Estimate first. 33

700 × 7 =

4,900; 4,900

34

7 × 363 =

2,800; 2,541

35

109 × 7 =

700; 763

36

7 × 223 =

1,400; 1,561

Solve. SCIENCE In 14 years, a sea urchin grew a total of 7 centimeters. How many centimeters did it grow each year if it grew the same amount each year?

2 cm 38

BOOKS Sundra has 58 books. She builds a bookcase with 7 shelves. What is the number of books that each shelf should hold if each shelf holds the same number of books?

9 39

HIKING Sekio hikes an average of 4 miles per hour. If he hikes for a total of 7 hours, how many miles will he have hiked?

28 miles

58

Chapter 4 Multiplication

Dynamic Graphics Value/SuperStock

SCIENCE Sea Urchin.

Copyright © by The McGraw-Hill Companies, Inc.

37

Vocabulary Check Write the vocabulary word that completes each sentence. 40

The Commutative Property of Multiplication states that the order in which two numbers are multiplied does not change the product.

multiple

of 10 because 4 × 10 = 40.

41

Forty is a(n)

42

Writing in Math Describe an everyday situation in which you would use multiplication.

Answers will vary. Sample answers include counting things in groups or rows, the area of a room, knowing how much money it would cost to buy multiples of the same item.

Spiral Review Solve. 43

(Lesson 4-1, p. 4)

BUSINESS A new ice-cream store has a floor shaped like a rectangle. The owner finds that 3 rows of tables fit 12 tables comfortably. How many tables will be in each row?

4

Copyright © by The McGraw-Hill Companies, Inc.

44

HEALTH Ozzy has 25 marbles. He puts the marbles into 5 separate bags for his friends. If he gives an equal number of marbles to each of his 5 friends, how many marbles will each receive?

5

45

FITNESS In physical education class, Judy did 2 sets of 10 push-ups. How many push-ups did she do?

20

Order the whole numbers from least to greatest.

(Lesson 1-8, p. 53)

46

20, 10, 50, 30, 40 10, 20, 30, 40, 50

47

14, 44, 34, 42, 24 14, 24, 34, 42, 44

48

33, 39, 31, 30, 37 30, 31, 33, 37, 39

49

8, 0, 5, 15, 11, 7

Write each number in standard form and expanded form.

0, 5, 7, 8, 11, 15

(Lesson 1-6, p. 39)

50

three hundred, three

51

one hundred, one ten, seven

52

4 tens

53

9 tens and 4 ones

303; 300 + 3 40; 40

117; 100 + 10 + 7 94; 90 + 4

Lesson 4-8 Multiply by 7

59

Chapter

Progress Check 4

X 4

(Lessons 4-7 and 4-8)

Draw an array to model each expression. Find the product. Then write the commutative fact. 3NS2.2 1

42 ; 6 × 7 = 42; 7 × 6 = 42

6·7

2

24 ; 6 × 4 = 24; 4 × 6 = 24

6×4

Use an array to find the missing number that would make the equation true. 3NS2.2 3

7×

4

= 28

9

4

6×

7

6·7=

= 54

Find each product. 3NS2.2, 3NS2.4 5

0·7=

0

6

6·3=

18

42

8

6·1=

9

7 × 311 =

2,100; 2,177

10

117 × 6 =

600; 702

11

7 × 208 =

1,400; 1,456

12

401 × 6 =

2,400; 2,406

Solve. 3NS2.2, 3NS2.4 13

ENTERTAINMENT The Hamilton family is having a reunion. The organizer wants seats for all 42 people. A small table can seat 6. A large table can seat 7. If the small tables rent for $8 each, and the large tables rent for $10 each, which tables are less expensive to rent?

The smaller tables would be less expensive. 14

MACHINES An engineer designed a machine to paint car parts. If the machine paints 60 parts an hour, how many parts does it paint in 10 minutes? (Hint: There are 60 minutes in an hour.) 10

60

Chapter 4 Multiplication

Copyright © by The McGraw-Hill Companies, Inc.

Find each product. Estimate first.

6

Lesson

4-9 Multiply by 8 KEY Concept Multiples of 8 are the numbers you say when you by 8. 8, 16, 24, 32, 40, 48, 56, 64, 72, 80

3NS2.2 Memorize to automaticity the multiplication table for numbers between 1 and 10. 3NS2.4 Solve simple problems involving multiplication of multidigit numbers by one-digit numbers. 4NS4.1 Understand that many whole numbers break down in different ways. 4NS3.2 Demonstrate an understanding of, and the ability to use, standard skip-count algorithms for multiplying a multidigit number by a two-digit number and for dividing a multidigit number by a one-digit number; use relationships between them to simplify computations and to check results.

You should practice memorizing the multiplication facts of 8. According to the Commutative Property, the product is the same, whether the factor 8 is the first factor or the second factor.

Example 1

product the answer or result of a multiplication problem; it also refers to expressing a number as the product of its factors (Lesson 4-1, p. 4) factor a number that divides into a whole number evenly; also a number that is multiplied by another number

Draw an array to model the expression 8 × 5. Find the product. Then write the commutative fact. 1. Write the answer if you know it. Otherwise, draw an array.

Copyright © by The McGraw-Hill Companies, Inc.

VOCABULARY

(Lesson 4-1, p. 4)

array an arrangement of objects or symbols in rows of the same length and columns of the same length; the length of a row might be different from the length of a column (Lesson 4-1, p. 4)

The first factor is an 8, so there will be 8 rows. The second factor is a 5, so there will be 5 columns. 2. Label the array 8 × 5. Count the number of rectangles. 40 3. Write the commutative fact. 8 × 5 = 40; 5 × 8 = 40 YOUR TURN! Draw an array to model the expression 8 × 3. Find the product. Then write the commutative fact. 1. Write the answer if you know it. Otherwise, draw an array. The first factor is an 8, so there will be 8 rows. The second factor is a 3, so there will be 3 columns. 2. Label the array. Count the number of rectangles.

8

24

3. Write the commutative fact. 8 ×

3 = 24 ;

3 × 8 = 24

×

3 GO ON

Lesson 4-9 Multiply by 8

61

Example 2

YOUR TURN!

Find the missing number that would make the equation true. 8×

Find the missing number that would make the equation true. 8×

= 32

= 64

1. What number times 8 equals 32?

1. What number times 64 ?

2. The missing number is 4. 8 × 4 = 32

2. The missing number is 8×

8

8

equals

8

.

= 64

Who is Correct? Find the missing number that would make the equation true. 8× = 48

Vida 8 × ___ = 48 8 × 7 = 48

Doran

Zara

8 × ___ = 48

8 × ___ = 48

8 × 6 = 48

8 × 5 = 48

Guided Practice Draw an array to model each expression. Find the product. Then write the commutative fact. 1

62

8×5

40 ; 8 × 5 = 40; 5 × 8 = 40

Chapter 4 Multiplication

2

8×1

8

;

8 × 1 = 8; 1 × 8 = 8

Copyright © by The McGraw-Hill Companies, Inc.

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

Find each product. 3

8×8=

64

4

8×6=

48

5

8×5=

40

6

8 × 10 =

7

3×8=

24

8

8×9=

72

9

7×8=

56

10

0×8=

0

80

Step by Step Practice 11

Find the missing number that would make the equation true. 8×

= 24

Step 1 What number times 8 is 24?

3

Step 2 The missing number is is 8 ×

3

. The completed equation

= 24.

Find the missing number that would make the equation true.

Copyright © by The McGraw-Hill Companies, Inc.

12

8×

= 40

8×

5

= 40

The array has 8 rows 5 columns 40 rectangles

13

8×

2

= 16

14

8×

6

= 48

15

3×

8

= 24

16

9×

8

= 72

Find each product. Estimate first. 17

8 · 234 =

1,600; 1,872

18

541 · 8 =

4,000; 4,328

19

8 · 350 =

3,200; 2,800

20

611 · 8 =

4,800; 4,888 GO ON Lesson 4-9 Multiply by 8

63

Step by Step Problem-Solving Practice Solve. 21

BUSINESS A flower shop puts 8 flowers in each vase it sells. If the shop sold 7 vases, how many flowers did it sell? Understand

flowers. The shop

Vases

Flowers

1

8

2

16

Make a table with two columns. Title one column “vases” and the other “flowers.”

3

24

4

32

Complete the table. If 7 vases were sold, then 56 flowers were used.

5

40

6

48

7

56

Pick a strategy. One strategy is to make a table.

Plan

Solve

Draw a picture and count the flowers to check your answer.

Check

LANDSCAPING Each flat of plants has 6 rows and 8 columns of plants. If 18 plants have been planted, how many plants are left in the flat? 6 × 8 = 48; 48 - 18 = 30 Check off each step.

✔

Understand

✔

Plan

✔

Solve

✔

Check

BAKING To make one batch of oatmeal raisin cookies, Amal needs 8 ounces of raisins and 10 ounces of oatmeal. He wants to triple the batch. How many ounces of raisins and oatmeal will he need?

He needs 24 oz of raisins and 30 oz of oatmeal.

Explain how to find the missing number in a multiplication equation. Give an example.

24

Sample answer: Think: What times the factor given equals the product? 6× 64

= 54; 6 × 9 = 54

Chapter 4 Multiplication

Copyright © by The McGraw-Hill Companies, Inc.

23

Draw a diagram. Use logical reasoning. Solve a simpler problem. Work backward. ✓ Make a table.

Read the problem. Write what you know. Each vase has 8 sold 7 vases.

22

Problem-Solving Strategies

Skills, Concepts, and Problem Solving Draw an array to model each expression. Find the product. Then write the commutative fact. 25

56 ; 8 × 7 = 56; 7 × 8 = 56

8×7

26

40 ; 5 × 8 = 40; 8 × 5 = 40

5×8

Multiply. 27

8·6=

48

28

1·8=

8

29

9·8=

72

30

8·0=

0

Copyright © by The McGraw-Hill Companies, Inc.

Find the missing number that would make the equation true. 31

8×

33

8×

2 5

= 16

32

8×

= 40

34

8×

8 7

= 64 = 56

Find each product. Estimate first. 35

101 × 8 =

800; 808

36

8 × 211 =

1,600; 1,688

37

220 × 8 =

1,600; 1,760

38

8 × 103 =

800; 824

Solve. 39

EARTH SCIENCE Niyna was collecting earthworms for science class. She noticed that after 8 shovels of soil, she had found a total of 40 worms. If each shovel of soil had the same number of worms, 5 how many worms were in each shovel of soil?

40

MONEY Babette earns $8 an hour. If she works for 6 hours, how $48 much will she have earned?

41

HOBBIES Jake collects baseball caps. He keeps them in boxes that hold 8 caps each. If there are 32 boxes, how many baseball caps 256 does Jake have?

GO ON

Lesson 4-9 Multiply by 8

65

Vocabulary Check Write the vocabulary word that completes each sentence. 42

The product is the answer to a multiplication problem.

43

array In a(n) and columns.

44

Writing in Math How is multiplication related to addition?

, objects or symbols are displayed in rows

Answers may vary. Multiplication is another way to repeatedly add the same number.

Spiral Review Find each product.

(Lesson 4-3, p. 18)

144

45

2 × 72 =

47

2 × 101 =

202

46

93 × 2 =

48

120 × 2 =

186 240

Find each difference. (Lesson 3-7, p. 173) 49

2,870 - 345 =

2,525

50

562 - 89 =

51

1,598 - 798 =

800

52

4,039 - 2,026 =

Solve.

(Lesson 1-1, p. 4)

HOMES In their apartment complex, Jana lives down the hall from Tiffany. Their friend Jim lives in between them. If Jana lives in apartment number 64 and Tiffany lives in number 60, what numbers can Jim’s apartment have?

61, 62, or 63 54

FOOTBALL Ellie and Rachel go to a football game. They find seats in row G, seats 14 and 15. They put their coats on another seat to save it for their friend Aaron. Which seat numbers should they save if all three want to sit together?

seat 13 or 16 55

PUZZLES I am a counting number between 34 and 42. I have a 0 in the ones place. What number am I?

40 66

2,013

Chapter 4 Multiplication

Copyright © by The McGraw-Hill Companies, Inc.

53

473

Lesson

4-10 Multiply by 9

3NS2.2 Memorize to automaticity the multiplication table for numbers between 1 and 10. 3NS2.4 Solve simple problems involving multiplication of multidigit numbers by one-digit numbers. 4NS4.1 Understand that many whole numbers break down in different KEY Concept ways. 4NS3.2 Demonstrate an understanding of, and the ability to Multiples of 9 are the numbers you say when you skip-count use, standard algorithms for by 9. multiplying a multidigit number by a two-digit number and for dividing a 9, 18, 27, 36, 45, 54, 63, 72, 81, 90 multidigit number by a one-digit number; use relationships between them to simplify computations and to Notice the pattern in the products when multiplying a check results.

one-digit number by 9. The sum of the digits in the product is always equal to 9.

You should practice memorizing the multiplication facts of 9. According to the Commutative Property, the product is the same, whether the factor 9 is the first factor or the second factor.

VOCABULARY factor a number that divides into a whole number evenly; also a number that is multiplied by another number (Lesson 4-1, p. 4)

Example 1

Copyright © by The McGraw-Hill Companies, Inc.

Draw an array to model the expression 9 × 3. Find the product. Then write the commutative fact. 1. Write the answer if you know it. Otherwise, draw an array. The first factor is a 9, so there will be 9 rows. The second factor is a 3, so there will be 3 columns. 2. Label the array as a 9 × 3 Count the number of rectangles. 27

Building an array will work for any two factors, although it is not always practical for large numbers.

3. Write the commutative fact. 9 × 3 = 27; 3 × 9 = 27 YOUR TURN! Draw an array to model the expression 9 × 6. Find the product. Then write the commutative fact. 1. Write the answer if you know it. Otherwise, draw an array. The first factor is a 9, so there will be 9 rows. The second factor is a 6, so there will be 6 columns. 2. Label the array as a 9 × 6 . Count the number of rectangles. 54 3. Write the commutative fact. 9 × 6 = 54 ; 6 ×

9 = 54 Lesson 4-10 Multiply by 9

67

Example 2

YOUR TURN!

Find the missing number that would make the equation true. 9×

Find the missing number that would make the equation true. 9×

= 45

1. Think about the multiplication facts you have learned. What number times 9 equals 45? 5

= 63

1. What number times 9 equals 63? 2. 9 ×

7

7

= 63

2. 9 × 5 = 45

Who is Correct? Find the product of 18 and 19.

Debra 10 + 8 10 + 9 90 + 72 100 + 80 2 100 + 80 + 90 + 72 = 34

William

Adella

18 × 19 162 18 180

18 × 19 162 180 342

Guided Practice Draw an array to model each expression. Find the product. Then write the commutative fact. 1

68

9×2

18 ; 9 × 2 = 18; 2 × 9 = 18

Chapter 4 Multiplication

2

9×4

36 ; 9 × 4 = 36; 4 × 9 = 36

Copyright © by The McGraw-Hill Companies, Inc.

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

Step by Step Practice 3

Find the product of 19 and 91 using expanded form. Step 1 Write the first number in expanded form. Step 2 Write the second number in expanded form.

10 + 90

9 1

+

Step 3 Write the numbers vertically. Find the products of each number. Step 4 Multiply 1 times the ones place value. Multiply 1 times the tens place value.

1

9

×

=

×

9

×

1 × 10 = 10

Step 5 Multiply 90 times the ones place value.

90 ×

9 = 810

×

Multiply 90 times the tens place value.

Step 6 Find the sum of the products. Copyright © by The McGraw-Hill Companies, Inc.

9 1

10 + 90 + 10 +

9 1 9

10 + 90 +

9 1

10 + 9 900 + 810

90 × 10 = 900 10 +

10 + 90 +

1,729

9 + 900 + 810 =

Find each product using expanded form. 4

11 · 29 = 11 in expanded form:

10 + 1

29 in expanded form:

20 + 9

5

10 · 19 =

190

6

19 · 92 =

1,748

7

33 · 97 =

3,201

8

92 · 93 =

8,556

×

10 + 20 +

1 9

90 + 9 200 + 20 319

GO ON Lesson 4-10 Multiply by 9

69

Find each product. Estimate first. 9

31 · 9 =

270; 279

10

41 · 19 =

11

22 · 9 =

180; 198

12

76 · 9 =

Step by Step Problem-Solving Practice

800; 779 720; 684

Problem-Solving Strategies

Solve. 13

PACKAGING At an open house, a real estate agent is offering muffins and coffee. The muffins are in packages of 12. The agent purchases 19 packages. How many muffins will he have? Understand

Read the problem. Write what you know. Each package contains 12 There are 19 packages.

Plan

Draw a diagram. Use logical reasoning. ✓ Solve a simpler problem. Work backward. Make a table.

muffins.

Pick a strategy. One strategy is to solve a simpler problem. There are 19 boxes of 12 muffins each. To find the total amount of muffins, you would need to multiply the numbers.

Solve

Then mentally, multiply each number by 19. 19 ×

10

= 190

19 ×

2

=

38

It is easier to multiply mentally if you use friendly numbers and then add the products. When you multiply by 10, add a zero to the other factor. When you multiply by 2, double the other factor. Now add the product. 190 + 38 = 228 So, there are 228 muffins altogether. Check

70

Estimate the product. Does your answer seem reasonable? Look over your solution. Did you answer the question?

Chapter 4 Multiplication

Copyright © by The McGraw-Hill Companies, Inc.

Solve mentally. First think of 12 in expanded form. 10 + 2 = 12

14

HOBBIES There are 13 threads braided in a handmade bracelet. If Carla made 19 bracelets, how many threads did she use?

247

Check off each step.

15

✔

Understand

✔

Plan

✔

Solve

✔

Check

SHOPPING Florence went to a store to buy clothes for school. She bought a total of 11 pants, shorts, and skirts. She also bought a total of 19 different types of shirts and tops. Assuming that every bottom matches every top, how many outfits does Florence have?

209 Describe a method for multiplying a factor that has more than one digit (multidigit) with another multidigit factor.

16

Sample answer: Write each number in expanded form, multiply, and add the products.

Copyright © by The McGraw-Hill Companies, Inc.

Skills, Concepts, and Problem Solving Draw an array to model each expression. Find the product. Then write the commutative fact. 17

54 ; 9 × 6 = 54; 6 × 9 = 54

9×6

18

45 ; 5 × 9 = 45; 9 × 5 = 45

5×9

Find the missing number that would make each equation true. 19

9×

2

= 18

20

9×

4

= 36

21

9×

9

= 81

22

9×

3

= 27

GO ON Lesson 4-10 Multiply by 9

71

Find each product using expanded form. 23

19 · 31 =

589

24

15 · 29 =

435

25

91 · 12 =

1,092

26

92 · 44 =

4,048

27

39 · 14 =

546

28

64 · 90 =

5,760

29

900 · 55 =

49,500

30

49 · 200 =

9,800

Find each product. Estimate first. 19 × 19 =

400; 361

32

29 × 21 =

600; 609

33

11 × 91 =

900; 1,001

34

15 × 92 =

1,800; 1,380

35

39 × 40 =

1,600; 1,560

36

25 × 90 =

2,700; 2,250

37

26 × 90 =

2,700; 2,340

38

49 × 99 =

5,000; 4,851

Solve. 39

FOOD The school cafeteria prepared 14 pans of chicken. If each pan holds an average of 19 pieces of chicken, about how many 266 pieces are there in all?

40

MONEY Jai has saved $11 a week for 17 weeks. Paula has saved $19 a week for 9 weeks. Who has saved more? How much more?

Jai has saved $16 more

41

ENTERTAINMENT At a party there are 21 people who are 39 years old. How many years total have these people been alive?

819 years

72

Chapter 4 Multiplication

Copyright © by The McGraw-Hill Companies, Inc.

31

Vocabulary Check Write the vocabulary word that completes each sentence. 42

Zero Property of Multiplication The number multiplied by zero is zero.

43

The example, 8 × 1 = 8 shows the

44

Identity Property of Multiplication

states that any

.

Writing in Math Write multiplication sentences for multiplying 9 by the numbers 1 to 10. Explain any pattern that you notice in the products.

1 · 9 = 9; 2 · 9 = 18; 3 · 9 = 27; 4 · 9 = 36; 5 · 9 = 45; 6 · 9 = 54; 7 · 9 = 63; 8 · 9 = 72; 9 · 9 = 81; 10 · 9 = 90; Patterns that students notice may vary. One noticeable pattern is that the digits of each product have a sum of 9.

Spiral Review

Copyright © by The McGraw-Hill Companies, Inc.

Solve. (Lesson 3-3, p. 145) 45

PUZZLES I am an even three-digit number. The hundreds digit is 5 more than the tens digit. The tens digit is an odd counting number less than 3. The ones digit is the same as the hundreds digit. What number am I? 616

46

SCHOOL Tim took four quizzes this week in spelling. His scores are shown. What was his total for the four quizzes? 35

OUTOF OUTOF

OUTOF

OUTOF

Identify the place-value position of each underlined digit. (Lesson 2-3, p. 85)

47

5,674

49

20,004

hundreds ten-thousands

48

1,230

50

15,099

Write the whole numbers between the following numbers. 51

15 and 21

16, 17, 18, 19, 20

52

93 and 99

94, 95, 96, 97, 98

53

36 and 45 37, 38, 39, 40, 41, 42, 43, 44

54

66 and 71

ones thousands

(Lesson 1-2, p. 11)

67, 68, 69, 70 Lesson 4-10 Multiply by 9

73

Chapter

Progress Check 5

4

(Lessons 4-9 and 4-10)

Draw an array to model each expression. Find the product. Then write the commutative fact. 3NS2.2 1

9·9

81 ;

9 × 9 = 81; 9 × 9 = 81

2

32 ; 32; 8 × 4 = 32; 4 × 8 = 32

8×4

Find the missing number that would make the equation true. 3NS2.2 3

9×

5

9×

5 3

5 2

= 45

4

8×

= 27

6

8×

9

6·8=

= 40 = 16

Find each product. 3NS2.2 7

11 · 8 =

88

8

9·8=

72

48

10

0·9=

0

Find each product. Estimate first. 3NS2.2, 3NS2.4, 4NS3.2 18 × 29 =

600; 522

12

13 × 19 =

200; 247

13

28 × 27 =

900; 756

14

40 × 29 =

1,200; 1,160

Solve. 3NS2.2, 3NS2.4, 4NS4.1 15

DRAMA There are 32 people attending a school play. Arrange the chairs in equal rows. Draw pictures of how the chairs could be arranged in equal rows. Explain the arrangements you drew.

Sample answer: There could be 8 rows by 4 columns, 4 rows by 8 columns, 16 rows by 2 columns, or 2 rows by 16 columns of chairs. 16

PUZZLES Suppose the product of the digits of a two-digit number is 8. The sum of the digits is 9. Find the number.

The number is 18 or 81. 74

Chapter 4 Multiplication

Copyright © by The McGraw-Hill Companies, Inc.

11

Lesson

4-11 Multiply by 11 and 12 KEY Concept A prime number is a whole number that has exactly two factors, 1 and itself. 2, 3, 5, 7, and 11 are all prime numbers. A composite number is a whole number that has more than two factors. An example is 12, which has the factors 1, 2, 3, 4, 6, and 12. All composite numbers can be written as a product of prime numbers. This is called prime factorization . You can use a factor tree to show prime factorization. The prime factorization of 12 is 2 × 2 × 3, or 22 × 3.

Copyright © by The McGraw-Hill Companies, Inc.

The factor being multiplied is called the base . The exponent is the small number that tells how many times to use the base as a factor. 42

base

exponent

VOCABULARY prime number a whole number with exactly two factors, 1 and itself Examples: 7, 13, and 19 prime factorization a way of expressing a composite number as a product of its prime factors

3NS2.4 Solve simple problems involving multiplication of multidigit numbers by one-digit numbers. 4NS4.1 Understand that many whole numbers break down in different ways. 4NS3.2 Demonstrate an understanding of, and the ability to use, standard algorithms for multiplying a multidigit number by a two-digit number and for dividing a multidigit number by a one-digit number; use relationships between them to simplify computations and to check results.

42 = 4 × 4 = 16

composite number a whole number that has more than two factors Example: 12 has the factors 1, 2, 3, 4, 6, and 12. base the number used as the factor in an expression involving exponents Example: In 25, the base is 2. exponent the number of times a base is multiplied by itself Example: In 32, the exponent is 2.

GO ON Lesson 4-11 Multiply by 11 and 12

75

Example 1

YOUR TURN! Find the prime factorization of 20.

Find the prime factorization of 24. 1. Write the number. Under the number, write two factors of the number. In each branch, break down a number into smaller factors until all the factors are prime numbers.

1. Write the number. Under the number, write two factors of the number. In each branch, break down a number into smaller factors until all the factors are prime numbers.

/PNBUUFSIPXZPV PSEFSUIFUSFF UIFQSJNF GBDUPSJ[BUJPOPGBOVNCFS IBTUIFTBNFTFU PGGBDUPST

2. The prime factors of 24 are 2 × 2 × 2 × 3. 3. The prime factorization of 24 in exponential form is 23 × 3.

Example 2

2

×

2

×

5

.

3. The prime factorization of 20 in 2 exponential form is 2 × 5 .

YOUR TURN!

1. Rewrite the problem in vertical format. 2. Multiply 2 times the ones column. 2 × 5 = 10 35 × 12 −−−− 10 60 50 +300 420

Find the product of 42 and 11. Use the partial products method. 1. Rewrite the problem in vertical format. 2. Multiply 1 times the ones column. 1×2= 2 3. Multiply 1 times the tens column. 1 × 40 = 40 4. Multiply 10 times the ones column. 10 × 2 = 20

42 × 11 −−−−

2 40 20 +400 462

5. Multiply 10 by the tens column. 10 × 30 = 300

5. Multiply 10 by the tens column. 10 × 40 = 400

6. Find the sum of the products. 10 + 60 + 50 + 300 = 420

6. Find the sum of the products. 2 + 40 + 20 + 400 =

76

Chapter 4 Multiplication

462

Copyright © by The McGraw-Hill Companies, Inc.

Find the product of 35 and 12. Use the partial products method.

4. Multiply 10 times the ones column. 10 × 5 = 50

2. The prime factors of 20 are

3. Multiply 2 times the tens column. 2 × 30 = 60

Who is Correct? Find the product of 101 and 11. Use expanded form.

Amos

Zetta

100 + 1 0+ ×1 −−1 −−−− 100 + 1 1,000 + 10

Lexie

100 + 1 0+ ×1 −−1 −−−− 10 1,000 + 0 + 100 +1 = 1,201

100 + 1 0+ ×1 −−1 −−−− = 1,000 +2 1,002

1,000 + 100 + 10 +1 = 1,111

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

Guided Practice Find the prime factorization of each number using factor trees. 1

14

2

15

3

Find the prime factorization of 36 in exponential form.

Step 1 Write the number. Think of 2 factors whose product equals 36.

Step 2 Continue to factor until you have only prime factors. Step 3 Once all the factors are prime numbers, write the prime factorization. Use exponents when possible.

2

×

2

×

3

×

3

=

⎫ ⎬ ⎭

Copyright © by The McGraw-Hill Companies, Inc.

Step by Step Practice

2

2

×

3

2

prime factors

GO ON Lesson 4-11 Multiply by 11 and 12

77

Find the prime factorization of each number. Write in exponential form. 4

8

2

3

5

27

33

40

6

5 × 23

7

50

2 × 52

Find each product. 8

11 × 9 =

99

9

12 × 5 =

60

10

78 × 11 =

858

11

65 × 12 =

780

12

12 × 9 =

13

11 × 7 =

77

108

Step by Step Problem-Solving Practice

Problem-Solving Strategies

Solve. 14

CONSTRUCTION Derrick is building an addition onto his house. The room is 11 feet by 12 feet. How many square feet of carpeting does Derrick need to carpet his addition? Understand

Draw a picture. Use logical reasoning. ✓ Solve a simpler problem. Work backward. Make a table.

Read the problem. Write what you know.

11

by

12

feet.

Plan

Pick a strategy. One strategy is solve a simpler problem.

Solve

Write 11 in expanded form. Write the factors in vertical form. Multiply 12 times the ones place value.

10 + 1 12 −−−−

120 + 12

Multiply 12 times the tens place value. Add the products. Derrick needs Check

78

120 + 12 = 132

132 square feet

of carpet.

You can use the expanded form of 12 to check your answer.

Chapter 4 Multiplication

10 + 2 11 110 + 22 = 132

Copyright © by The McGraw-Hill Companies, Inc.

The room is

15

HOBBIES There were 12 beads in a handmade bracelet. If Carla 228 made 19 bracelets, how many beads did she use? Check off each step.

16

✔

Understand

✔

Plan

✔

Solve

✔

Check

FIELD TRIPS On Friday, 12 school groups of 40 students each visited the zoo. On Saturday, twice as many students visited the 960 zoo. How many students visited the zoo on Saturday? How do you write 123 as a product? Explain your answer.

17

Sample answer: The base is 12. The exponent is 3. So, 12 is a factor 3 times. 123 = 12 × 12 × 12 = 1,728.

Skills, Concepts, and Problem Solving Find the prime factorization of each number using factor trees. 18

52

25

19

22 × 7

28

Copyright © by The McGraw-Hill Companies, Inc.

20

23 × 3

24

21

42 2 × 3 × 7

GO ON Lesson 4-11 Multiply by 11 and 12

79

Find each product. 23

71 × 11 =

781

1,104

25

11 × 103 =

1,133

11 × 6 =

66

27

12 × 4 =

48

11 × 11 =

121

29

12 × 31 =

372

22

12 × 110 =

24

12 × 92 =

26 28

1,320

Solve. 30

ASTRONOMY Eli spent 12 minutes counting the stars in the sky. If he counted 11 stars each minute for 12 minutes, how many stars 132 did he count?

31

PHOTOS Moesha developed 9 rolls of 12-exposure film. How 108 many pictures did she develop?

Vocabulary Check Write the vocabulary word that completes each sentence.

Prime factorization

32

expresses a composite number as

a product of its prime factors. 33 34

composite number

has more than two factors.

exponent A(n) a base is multiplied by itself.

shows the number of times

Writing in Math Is 1 a composite or a prime number? Explain your reasoning.

Sample answer: neither; a prime number has exactly 2 factors, a composite number has more than 2 factors, and 1 has only 1 factor.

Spiral Review 36

ADVERTISING If 21 people waited for a bus, how many will likely read the ad? (Lesson 4-5, p. 33)

7 ADVERTISING One

Use the “make ten” strategy to find each sum. (Lesson 3-2, p. 137)

78

37

67 + 11 =

80

Chapter 4 Multiplication

Getty Images

38

53 + 9 =

62

advertiser claims that 1 out of 3 people waiting for a bus reads the advertisement on the side of the bus.

Copyright © by The McGraw-Hill Companies, Inc.

35

A(n)

Lesson

4-12 Perfect Squares

3NS2.2 Memorize to automaticity the multiplication table for numbers between 1 and 10. 4NS4.1 Understand that many whole numbers break down in different ways.

KEY Concept A multiplication problem that has two identical factors has a product that is a perfect square . 1×1=1 2×2=4 3×3=9 4 × 4 = 16

5 × 5 = 25 6 × 6 = 36 7 × 7 = 49 8 × 8 = 64

9 × 9 = 81 10 × 10 = 100 11 × 11 = 121 12 × 12 = 144

Perfect squares can be modeled using a square array. This is because the number of rows and the number of columns are the same. You can write perfect squares using exponents.

VOCABULARY perfect square a number that is the square of another number Example: 0, 1, 4, 9, 16, 25, … and so on; perfect squares can be represented by square arrays base the number used as the factor in an expression involving exponents Example: In 25, the base is 2. (Lesson 4-11, p. 75) exponent the number of times a base is multiplied by itself Example: In 32, the exponent is 2.

Copyright © by The McGraw-Hill Companies, Inc.

(Lesson 4-11, p. 75)

You should memorize which multiplication expressions make perfect squares.

Example 1 Write the multiplication expression modeled. 1. How many rows in the array?

6

2. How many columns?

6

3. Label the array.

6×6

4. Write the expression and product.

6 × 6; 36

6×6

5. Write the equation using exponents. 6 is multiplied twice, so 6 2 = 36.

GO ON Lesson 4-12 Perfect Squares

81

YOUR TURN! Write the multiplication expression modeled. 1. How many rows in the array?

8

8

2. How many columns? 3. Label the array.

4. Write the expression and product.

8

×

8

;

64

8

×

8

5. Write the equation using exponents.

82

=

64

Example 2

YOUR TURN! Evaluate the expression 52.

Evaluate the expression 42. 1. What number is the base?

1. What number is the base?

4

2. What number is the exponent?

2. What number is the exponent? 2 3. Multiply to find the value of the expression.

4 × 4 = 16

=8

Edward

Robert

6 =6+6

6 =6×6

= 12

= 36

2

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

82

Chapter 4 Multiplication

2

Copyright © by The McGraw-Hill Companies, Inc.

Evaluate the expression 6 2.

62 = 6 + 2

2

3. Multiply to find the value of the expression. 5 × 5 = 25

Who is Correct?

Carmine

5

Guided Practice Write the multiplication fact modeled.

5

1

×

5

25

=

7

2

×

7

=

49

Step by Step Practice 3

Evaluate the expression 9 2.

9

Step 1 What is the base?

2

Step 2 What is the exponent?

The exponent tells you how many times the itself .

Copyright © by The McGraw-Hill Companies, Inc.

Step 3 Multiply to find the value of the expression.

base

is multiplied by

9 × 9 = 81

Evaluate the expression. 4

10 2

base:

10 2

exponent: 10 2 =

10

10

=

100 6

82

8 × 8 = 64

6 × 6 = 36

8

32

3×3=9

52

5 × 5 = 25

10

11 2

12

1×1=1

12

42

5

12 2

7

62

9

11

12 × 12 = 144

×

11 × 11 = 121 4 × 4 = 16 GO ON Lesson 4-12 Perfect Squares

83

Step by Step Problem-Solving Practice Solve. 13

Problem-Solving Strategies ✓ Draw a diagram.

INTERIOR DESIGN Roger is buying new carpet for his room. His room is square with lengths of 12 feet. How many square feet of carpet will he need? Understand

Use logical reasoning. Solve a simpler problem. Work backward. Make a table.

Read the problem. Write what you know. The bedroom is a 12 feet.

square with lengths of

Pick a strategy. One strategy is to draw a diagram.

Plan

You can make an array to represent his room. Count the squares to find how many square feet fill the room.

Solve

If you count the squares, the sum is the same as the product of 12 × 12 .

12 × 12 = 144 So, Roger will need 144 square feet of carpet. Does your answer seem reasonable? Look over your solution. Did you answer the question?

Check

HEALTH Jocelyn has been training for a marathon. The first week she trained 1 hour, the second week she trained 4 hours, and the third week she trained 9 hours. If she continues this pattern, how many hours will she train for the marathon the fourth week?

16 hours

Check off each step.

84

✔

Understand

✔

Plan

✔

Solve

✔

Check

Chapter 4 Multiplication

Copyright © by The McGraw-Hill Companies, Inc.

14

15

SCIENCE Derrick is conducting an experiment growing mold. He measures the area it covers daily. Complete the table showing the mold’s growth.

Dimensions

16

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

Exponential Form

12

22

32

42

52

62

72

82

92

10 2

Area

1

4

9

16

25

36

49

64

81

100

Define an exponent. Explain why you would use an exponent.

An exponent tells how many times a base is multiplied by itself. It is a shortcut and saves space.

Skills, Concepts, and Problem Solving

Copyright © by The McGraw-Hill Companies, Inc.

Write the multiplication fact modeled. 17

4 × 4 = 16

18

8 × 8 = 64

19

6 × 6 = 36

20

9 × 9 = 81

GO ON Lesson 4-12 Perfect Squares

85

Write the multiplication fact modeled.

11 × 11 = 121

21

10 × 10 = 100

22

Evaluate the expression. 23

72

25

11 2

27

22

29

7 × 7 = 49

12 × 12 = 144

24

12 2

26

92

9 × 9 = 81

2×2=4

28

82

8 × 8 = 64

42

4 × 4 = 16

30

62

6 × 6 = 36

31

52

5 × 5 = 25

32

102

33

32

3×3=9

34

12

11 × 11 = 121

10 × 10 = 100 1×1=1

35

PUZZLES Write a two-digit number the sum of whose digits is 10. The number is a perfect square. 64

36

PUZZLES Write a two-digit number the sum of whose digits is 7. The number is a perfect square that is an even number. 16

37

PUZZLES Write a two-digit number the sum of whose digits is 9. The number is a perfect square that is an odd number. 81

86

Chapter 4 Multiplication

Copyright © by The McGraw-Hill Companies, Inc.

Solve.

Vocabulary Check Write the vocabulary word that completes each sentence. 38

39

40

When you multiply a factor like 4 by itself, the product is called perfect square . a(n)

exponent The multiplied by itself.

tells you how many times a base is

Writing in Math Austin was asked to evaluate the expression 62. He wrote 6 × 2 = 12. What did Austin do wrong?

The expression should be 6 2 = 6 × 6 = 36. He did not use the exponent correctly. The exponent 2 tells you to multiply the base 6 two times.

Spiral Review

Copyright © by The McGraw-Hill Companies, Inc.

Find each product.

(Lesson 4-4, p. 25)

41

6·5=

30

42

5·3=

15

43

10 · 5 =

44

5·0=

0

45

1·5=

5

46

8·5=

47

PETS Jennifer was buying supplies for her new puppy. She spent 12 dollars for a food-and-water dish, 15 dollars for a puppy bed, 9 dollars for a collar, and 10 dollars for a leash. About how much did she spend to the nearest $10? (Lesson 3-4 p. 151)

50 40

about 50 dollars Write each number in standard form. (Lesson 2-5, p. 99) 48

two hundred eight thousand, three hundred fifty

208,350 49

five hundred nine thousand, seven hundred sixty-five

509,765 Use <, =, or > to complete each statement.

(Lesson 2-4, p. 91)

50

1,950 > 1,905

51

856 < 865

52

1, 808 < 1, 818

53

98 > 89

Lesson 4-12 Perfect Squares

87

Chapter

Progress Check 6

4

(Lessons 4-11 and 4-12)

Find the prime factorization of each number. 4NS4.1, 3NS2.2 1

2×3×7

42

2

26

64

Find the prime factorization of each number. Write in exponential form. 4NS4.1, 3NS2.2 3

18

5

9

2 × 32 32

4

48

24 × 3

6

26

2 × 13

Evaluate the expression. 4NS4.1, 3NS2.2 7

72

7 × 7 = 49

8

32

3×3=9

9

62

6 × 6 = 36

10

52

5 × 5 = 25

11

2×2

4

12

11 × 11

121

13

8×8

64

14

12 × 12

144

Find each product. 4NS4.1, 3NS2.2, 3NS2.4 15

12 × 70 =

840

16

87 × 11 =

957

17

12 × 131 =

1,572

18

144 × 11 =

1,584

19

FOOD Mrs. Weber’s company is having a picnic. There are 160 expected guests. Mrs. Weber purchases 20 eight-packs of hot dogs. How many twelve-packs of hot dog buns does she need to buy to have enough for all who attend? 4NS4.1, 3NS2.2, 3NS2.4

If Mrs. Weber buys 13 packages of buns, she will only have 156 buns; she needs to buy 14 packages. 88

Chapter 4 Multiplication

Copyright © by The McGraw-Hill Companies, Inc.

Evaluate the expression. 4NS4.1, 3NS2.2

Lesson

4-13 Multiply Large Numbers

3NS2.4 Solve simple problems involving multiplication of multidigit numbers by one-digit numbers. 3NS2.6 Understand the special properties of 0 and 1 in multiplication and division. 4NS3.2 Demonstrate an understanding of, and the ability to use, standard algorithms for multiplying a multidigit number by a two-digit number and for dividing a multidigit number by a a onedigit number; use relationships between them to simplify computations and to check results.

KEY Concept Traditional Multiplication Method 1

43 × 35 215 + 1 29 1,505

Partial Products Method 43 × 35 15 200 90 1 200

5 × 3 = 15 5 × 40 = 200 30 × 3 = 90 30 × 40 = 1,200

1,505 Before you multiply, you should estimate your answer so that you can check your actual answer for reasonableness.

Example 1

Copyright © by The McGraw-Hill Companies, Inc.

1. Rewrite the problem in vertical format. 2. Multiply 2 times in the ones column. 2 × 6 = 12 3. Multiply 2 times the tens column. 2 × 30 = 60

5. Multiply 80 times the tens column. 80 × 30 = 2,400

36 × 82 −−−− 12 60 480 +2,400 2,952

Find the product of 28 and 57. Use the partial products method. 1. Rewrite the problem in vertical format. 2. Multiply 7 7 × 8 = 56

12 + 60 + 480 + 2,400 = 2,952

times the ones column.

3. Multiply 7 times the tens column. 7 × 20 = 140 28 × 57 −−−− 4. Multiply 50 times the ones 56 column. 140 50 × 8 = 400 5. Multiply 50 times the tens column. 50 × 20 = 1,000

400 +1,000 1,596

6. Add the partial products.

6. Add the partial products.

7. The product is 2,952.

estimate a number close to an exact value; an estimate indicates about how much

YOUR TURN!

Find the product of 36 and 82. Use the partial products method.

4. Multiply 80 times the ones column. 80 × 6 = 480

partial products method a way to multiply; the value of each digit in one factor is multiplied by the value of each digit in the other factor; the product is the sum of its partial products

56 =

+ 140 + 400 +

1,000

1,596

7. The product is 1,596 .

GO ON

Lesson 4-13 Multiply Large Numbers

89

Example 2

YOUR TURN!

Find the product of 76 and 14. Use the traditional multiplication method.

1. Estimate. 60 ×

1. Estimate. 80 × 10 = 800 2. Rewrite the problem in vertical form. 3. Multiply 4 times the digit in the ones column. 4 × 6 = 24 Write the tens digit above the tens column and the ones digit in the product under the ones column.

2

76 × 14 −−−− 4

2

4. Multiply 4 times the tens column. 4 × 7 = 28 Add the two tens to get 30. 5. Multiply the value of the digit in the tens place times 6. 10 × 6 = 60

7. Find the sum of the two products. 8. The product is 1,064. Compare to your estimate for reasonableness.

90

Chapter 4 Multiplication

76 × 14 −−−− 304 76 × 14 −−−− 304 60 76 × 14 −−−− 304 + 760 1,064

30

=

1,800

2. Rewrite the problem in vertical format. 3. Multiply two times the digit in the ones column. 2×6=

12

4. Multiply two times the tens column. 2×5=

10

2

1

Add the one ten(s) to get 11 . 5. Multiply each place value by the tens digit. 30 × 6 = 180 6. Multiply the value in the tens column by 30. 30 × 50 =

1 56 × 32 −−−−

56 × 32 −−−−

112 1 56 × 32 −−−−

112 80

1,500

Add one hundred(s) for 1,600 . 7. Find the sum of the two products. 8. The product is

1,792 .

Compare to your estimate for reasonableness.

56 × 32 −−−−

112 +1,680 1,792

Copyright © by The McGraw-Hill Companies, Inc.

6. Multiply the value in the tens column by 10. 10 × 70 = 700 Write the 7 in the hundreds place.

Find the product of 56 and 32. Use the traditional multiplication method.

Who is Correct? Find the product of 26 and 47.

Farris 26 47 × −−−− 42 14 24 +8 88

Vivian

Asa 26 47 × −−−− 42 140 240 + 800 1,222

26(40 + 7) = (26 · 40) + (26 · 7) = 1,040 + 182 = 1,222

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

Guided Practice Estimate each product. 1

95 × 47 Round each factor to the greatest place value. 90 × 50 Find the estimated product.

4,500

Copyright © by The McGraw-Hill Companies, Inc.

2

709 × 32 Round each factor to the greatest place value. 700 × 30 Find the estimated product.

21,000 Step by Step Practice 3

Find the product of 14 and 89. Use the traditional multiplication method. Step 1 Rewrite the problem in vertical format. Step 2 Multiply each place value by the ones digit.

3

14 × 89 −−−−

Step 3 Multiply each place value by the tens digit.

126 1 12

Step 4 Find the sum of the two products.

1,246

Step 5 The product is 1,246. Lesson 4-13 Multiply Large Numbers

91

Find each product. Use the traditional multiplication method. 4

11 · 71

11 × 71 −−−−

11 77 781

5

13 · 61 =

793

6

12 · 101 =

1,212

Find each product. Use the partial products method. 7

39 · 83 =

9

17 · 302 =

3,237 5,134

8

28 · 94 =

2,632

10

21 · 546 =

11,466

Step by Step Problem-Solving Practice

Draw a diagram.

Solve. 11

SCHOOL The teacher’s aide is helping the third-grade teacher grade tests. Each test has 45 questions. Each student took 4 tests. If there are 28 students in the class, how many test questions are there in all?

✓ Use logical reasoning. Solve a simpler problem. Work backward. Make a table.

Read the problem. Write what you know. Each student took 4 tests. Each test has 45 questions. There are 28 students in the class.

Plan

Pick a strategy. One strategy is to use logical reasoning. Find the number of questions on 4 tests, then multiply that by the number of students to find the number of test questions in all.

Solve

number of questions × number of tests = questions on 4 tests questions on 4 tests × number of students = test questions in all

Check

Does the answer make sense? Look over the solution. Did you answer the question?

Chapter 4 Multiplication

Copyright © by The McGraw-Hill Companies, Inc.

Understand

Use the values from the problem to solve. 45 × 4 = 180 180 × 28 = 5,040 There are 5,040 test questions in all.

92

Problem-Solving Strategies

12

BUSINESS Ginny is digging for clams to sell to a local seafood store. For every shovel of sand she digs up, she finds 16 small clams. If she digs 27 shovels of sand, how many clams could she 432 expect to find? If the store pays her a dime for every clam, how much will she earn? $43.20 Check off each step.

13

✔

Understand

✔

Plan

✔

Solve

✔

Check

COOKING When making meatloaf, Troy uses 12 ounces of bread crumbs. His mother uses the same recipe at her restaurant, but she multiplies the recipe to make 32 meatloaves. How many ounces of 384 ounces breadcrumbs will his mother use? Compare two different ways of multiplying 13 × 17.

14

Answers will vary. Students should compare two of the following methods: traditional, partial products, Distributive Property and expanded form.

Copyright © by The McGraw-Hill Companies, Inc.

Skills, Concepts, and Problem Solving Find each product. Use the traditional multiplication method. 15

45 · 87 =

17

112 · 15 =

3,915

1,680

5,096

16

56 · 91 =

18

13 · 144 =

1,872

20

18 × 22 =

396

22

32 × 310 =

9,920

Find each product. Use the partial products method. 19

39 × 63 =

21

232 × 16 =

2,457

3,712

Lesson 4-13 Multiply Large Numbers

93

Find each product. 23

19 × 19 =

361

24

29 × 21 =

609

25

111 × 77 =

8,547

26

25 × 312 =

7,800

Solve. 27

MUSIC If Jennifer has 63 CDs, how many total songs does Jennifer have? 693

28

FITNESS Eva runs 3 miles every day. Michelle runs 4 miles a day on weekdays. Michelle does not run on the weekends. After 45 weeks, who will have run more? By how much?

Eva will have run 45 miles more in 45 weeks than Michelle.

MUSIC Jennifer has CDs with an average of 11 songs per CD.

Vocabulary Check Write the vocabulary word that completes each sentence. 29

A number close to an exact value is called a(n) estimate . 34

Writing in Math Marissa multiplied 345 × 18. What mistake did she make?

See TWE margin.

Spiral Review 31

PUZZLES I am a number under 50 and a multiple of 8. The sum of my digits is a prime number. What numbers can I be? (Lesson 4-9, p. 61)

94

16 or 32

Chapter 4 Multiplication

Stockbyte/SuperStock

34

345 × 18 −−− 2760 +3450 6210 Or using the distributive method: 345 × 18 = 345 (10 + 8) = (345 × 10) + (345 × 8) = 3,450 + 2,760 = 6,210

Copyright © by The McGraw-Hill Companies, Inc.

30

345 × 18 −−− 2760 +345 3205

Chapter

Study Guide

4

Vocabulary and Concept Check array, p.4

Write the vocabulary word that completes each sentence.

base, p. 75

1

An arrangement of objects in rows of the same length and columns of the same length are in a(n) array .

2

A(n) an exact value.

estimate, p. 89 exponent, p. 75 factor, p. 4 multiple, p. 19 product, p. 4

3

4

estimate

is a number close to

The Zero Property of Multiplication is the property that states if you multiply a number by zero, the product is zero.

multiple of a number is the A(n) product of that number and any whole number.

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

factor

5

6

10 × 2 = 20

Copyright © by The McGraw-Hill Companies, Inc.

12 2

product

7

exponent

8

base

Lesson Review

4-1

Introduction to Multiplication (pp. 4–10)

Draw an array to model 2 × 3. Then write and model the commutative fact. 9

2 × 3 = 6; 3 × 2 = 6

Example 1 Write 4 · 9 as repeated addition. Then write the commutative fact. 1. What is the first factor? 2. What is the second factor?

10

Write 4 ∙ 5 as repeated addition. Then write the commutative fact.

5 + 5 + 5 + 5; 4 · 5 = 20; 5 · 4 = 20

4 9

3. Write an expression as repeated addition. 9+9+9+9 4. The commutative fact is

4 · 9 = 36; 9 · 4 = 36

.

Chapter 4 Study Guide

95

4-2

Multiply with 0, 1, and 10 (pp. 11–17)

Find each product. 11

8×0 0

12

16 × 1 16

13

19 × 10 190

14

10 × 0 0

Find the product. Estimate first. 15

101 × 9 900; 909

16

434 × 1 400; 434

Example 2 Find the product of 207 and 5 using expanded form. 1. Write the first number in expanded form. 200 + 7 2. Write the factors in vertical form. 3. Multiply 5 times the ones place value. 5 × 7 = 35 4. Multiply 5 times the hundreds place value. 200 × 5 = 1,000 5. Add the products. 1,000 + 35 = 1,035

4-3

200 + 7 × 5 1,000 + 35

Multiply by 2 (pp. 19–24)

Multiply by using repeated addition. 17

200 + 7 × 5 35

2·4 4+4=8 8 · 2 2+2+2+2+2+2+2+2 = 16

19

2·1 1+1=2

Find each product. 20

44 × 2 88

21

110 × 2 220

22

32 × 2 64

23

57 × 2 114

24

141 × 2 282

96

Chapter 4 Study Guide

Find the product of 43 and 2 by using repeated addition. Then write the multiplication fact and its commutative fact. 1. What is the first factor? 43 What is the second factor? 2 2. Write an equation as repeated addition. Use the first factor as the number being added and the second factor as the number of times you add the number. 43 + 43 = 86 3. Write the multiplication equation. 43 × 2 = 86 4. Write the commutative fact. 2 × 43 = 86

Copyright © by The McGraw-Hill Companies, Inc.

18

Example 3

4-4

Multiply by 5 (pp. 25–31)

Find each product. Estimate first. 25

21 × 5

100; 105 26

406 × 5

2,000; 2,030 28

Find the product of 33 and 5. Estimate first. 1. Estimate. 30 × 5 = 150

303 × 5

1,500; 1,515 27

Example 4

707 × 5

3,500; 3,535

2. Rewrite the problem in vertical format. 5 3. Multiply 5 times the ones column. 5 × 3 = 15 Write the tens digit above the tens column, and the ones digit in the answer under the ones column. 4. Multiply 5 times the digit in the tens column. 5 × 3 = 15. Add in the one ten for 16.

1

33 × 5 −−− 5 1

33 × 5 −−− 165

5. The product is 165. Compare to your estimate for reasonableness.

4-5

Multiply by 3 (pp. 33–38)

Copyright © by The McGraw-Hill Companies, Inc.

Multiply by using repeated addition. 29

3·4

4 + 4 + 4 = 12 30

Example 5

8·3

3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 = 24

Find each product. 31

15 · 3

45

32

321 · 3

963

33

402 · 3

1,206

Find the product of 3 and 6 and by using repeated addition. Then write the multiplication expression and the commutative fact. 1. What is the first factor? 3 What is the second factor? 6 2. Write an equation as repeated addition. Use the first factor as the number being added and the second factor as the number of times you add the number. 6 + 6 + 6 = 18 3. Write the multiplication expression. 3 × 6 = 18 4. Write the commutative fact. 6 × 3 = 18 Chapter 4 Study Guide

97

4-6

Multiply by 4 (pp. 39–45)

Find each product.

Example 6

16

34

4×4

35

4 × 12

48

36

1×4

4

37

9×4

36

Draw an array to find the missing number that makes 4 ×

1. The factor already in the equation is 4. The array will have four rows.

Draw an array to find the missing number that would make the equation true. 38

4×

8

= 28 true.

2. Continue making columns until there are 28 rectangles shaded. How many columns do you make? 7

= 32

7 columns of 4 rows make 28 rectangles

39

20 = 4 ×

40

Multiply by 6 (pp. 47–52)

Draw an array to find the missing number that would make the equation 6×

1

= 6 true.

Find each product. Estimate first. 41

103 × 6

600; 618

42

6 × 550

3,600; 3,300

43

362 × 6

2,400; 2,172

44

6 × 615

3,600; 3,690

98

Chapter 4 Study Guide

7

= 28.

Example 7 Find the product of 314 and 6. 1. Rewrite the problem in vertical format.

2

2. Multiply 6 times the digit in the ones column. 6 × 4 = 24

314 × 6 −−−− 4

3. Multiply 6 times the digit in the tens column. 6 × 1 = 6 Add the two tens to get 8.

314 × 6 −−−− 84

4. Multiply 6 times the digit in the hundreds column. 6 × 3 = 18

2

314 × 6 −−−− 1,884

Copyright © by The McGraw-Hill Companies, Inc.

4-7

3. 4 ×

5

4-8

Multiply by 7 (pp. 53–59)

Draw an array to find the missing number that would make the equation true. 45

7×

9

= 63

Example 8 Find the product of 213 and 7. Estimate first. 1. Estimate. 200 × 7 = 1,400 2. Rewrite the problem in vertical format. 3. Multiply 7 times the digit in the ones column. 7 × 3 = 21 Write the tens digit above 2 213 the tens column, and the × 7 ones digit in the answer −−−− 1 under the ones column.

Copyright © by The McGraw-Hill Companies, Inc.

Find each product. Estimate first. 46

800 × 7

5,600; 5,600

47

7 × 412

2,800; 2,884

48

144 × 7

700; 1,008

49

7 × 197

1,400; 1,379

4-9

213 × 7 −−−− 91

5. Multiply 7 times the digit in the hundreds column. 7 × 2 = 14

213 × 7 −−−− 1,491

6. The product is 1,491. Compare to your estimate for reasonableness.

Multiply by 8 (pp. 61–66)

Draw an array to find the missing number that would make the equation true. 50

2

4. Multiply 7 times the digit in the tens column. 7 × 1 = 7 Add the two tens to get 9.

8×

8

= 64

Example 9 Find the missing number that would make the equation true. 8×

= 56

1. What number times 8 equals 56? 2. 8 × 7 = 56

Find each product. Estimate first. 51

8 · 555

4,800; 4,440

52

823 · 8

6,400; 6,584 Chapter 4 Study Guide

99

4-10

Multiply by 9 (pp. 67–73)

Find the missing number that would make the equation true. 53

9×

2

= 18

54

9×

6

= 54

55

9×

56

9×

Example 10 Find the missing number that would make the equation true. 9×

10 = 90 3

= 81

1. Think about the multiplication facts you have learned. What number times 9 equals 81? 2. 9 × 9 = 81

= 27

Find each product. Estimate first. 57

9 · 246

1,800; 2,214

58

783 · 9

7,200; 7,047

4-11

Multiply by 11 and 12 (pp. 75–80)

Find the prime factorization of each number. Write in exponential form. 75

3×5

2

60

18

2×3

61

36

22 × 3 2

Find the prime factorization of 36. 1. Write the number. Under the number, write two factors of the number. In each branch, break down each number into smaller factors until all the factors are prime numbers.

Find each product. 62

12 × 12

144

2. The prime factors of 36 are 2 × 2 × 3 × 3.

63

9 × 11

99

64

25 × 11

275

3. Write the prime factorization of 36 in exponential form. 22 × 32

65

13 × 11

156

100

Chapter 4 Study Guide

Copyright © by The McGraw-Hill Companies, Inc.

59

2

Example 11

4-12

Perfect Squares (pp. 81–87)

Write the multiplication fact modeled.

3

66

×

3

=

9

Example 12 Evaluate the expression. 82 1. What number is the base? 8 2. What number is the exponent? 2

Copyright © by The McGraw-Hill Companies, Inc.

Evaluate the expression. 67

22

2×2=4

68

72

7 × 7 = 49

69

42

4 × 4 = 16

70

92

9 × 9 = 81

4-13

72

28 · 34 72 · 81

952 5,832

73

13 · 112

1,456

74

25 · 679

16,975

75

4. What is the value of the expression? 64

Multiply Large Numbers (pp. 89–94)

Find each product. 71

3. Write the base the number of times given by the exponent. 8 × 8

32 · 231

7,392

Example 13 Find the product of 17 and 36. Use the traditional multiplication method. 24

1. Rewrite the problem in vertical format. 2. Multiply each place value by the ones digit. 3. Multiply each place value by the tens digit. 4. Find the sum of the two products.

17 × 36 −−−− 102 +51 612

5. The product is 612. Compare to your estimate for reasonableness.

Chapter 4 Study Guide

101

Chapter

Chapter Test

4

Draw an array to model each expression. Then write and model the commutative fact. 3NS2.2, 4NS4.1 1

4×3

4 × 3 = 12 ; 3 × 4 = 12

Multiply by using repeated addition. 2

3 + 3 + 3 + 3 + 3 + 3 = 18

6×3

3

2×7

7 + 7 = 14

Find each product. 3NS2.2, 3NS2.6 4

8×0

0

5

1 × 12

7

0×1

0

8

7×2

10

3×8

24

11

4×6

13

6×6

36

14

12 × 3

12

6

3 × 10

30

14

9

11 × 3

33

24

12

8×7

56

15

9×6

54

36

Draw an array to model and find each product. 3NS2.2

27

3×9

17

5 × 11

Copyright © by The McGraw-Hill Companies, Inc.

16

55

Find each product. Estimate first. 3NS2.2, 3NS2.4

150; 135

18

27 · 5

20

406 × 2

22

219 · 4

800; 812 800; 876

19

55 · 6

360; 330

21

490 × 7

3,500; 3,430

23

723 × 8

5,600; 5,784

Find the missing number that would make the equation true. 3NS2.2 24

102

9×

9

= 81

Chapter 4 Test

25

4×

9

= 36

GO ON

Evaluate each expression. 3NS2.2, 4NS4.1 26

42

16

28

11 2

121

27

72

49

40

23 × 5

Find the prime factorization of each number. Write in exponential form. 3NS2.2, 4NS4.1 29

36

22 × 32

31

54

2 × 33

30

Solve. 3NS2.2, 4NS4.1 32

MUSIC Kai has 6 shelves in her CD stand. Each shelf holds 24 CDs. What is the maximum number of CDs that Kai’s stand holds?

144 CDs 33

MOVING When Sunde moved to a new apartment, he packed his belongings into his own pick-up truck. To make the move, he had to make 3 trips. He started at his old apartment and ended at his new apartment. The apartments were 12 miles apart. How many miles did he drive to move?

Copyright © by The McGraw-Hill Companies, Inc.

60 miles 34

FINANCE Annie wants to buy a new computer that costs about $1,500. She has been saving $35 a week for an entire year. At the end of one year, how much money will she have saved? Is that enough money for the computer? (There are 52 weeks in a year.)

$1,820; yes

Correct the mistakes. 2NS4.0, 4NS1.5, 2NS4.1 35

Carly told Janet that she knows a shortcut for multiplying by 10. She said “You add two zeros to the other factor.” Janet told her that shortcut was not right. The correct shortcut is to

add one zero to the other factor

Chapter 4 Test

103

Chapter

4

Standards Practice

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

2

Which expression has the same value as 1 + 1 + 1 + 1 + 1 + 1? 4NS1.4 A 2+3+4+5

C 10 - 5

B 1×6

D 6×2

What is the next number in this series?

3NS2.4

F 15

H 40

G 30

J 50

5

6

7

3

A 2 × 6 = 12

C 2×2=4

B 5+2=7

D 5 × 2 = 10

8

A $13

C $42

B $36

D $49

Priya is baking cookies. She makes 38 small cookies in each batch. If she bakes 9 batches, how many cookies does she make? 4NS3.2 F 47 cookies

H 224 cookies

G 114 cookies

J 342 cookies

Which numbers multiply to give a product of 132? 4NS3.2 A 4 × 18

C 12 × 11

B 18 × 11

D 8 × 12

Curtis reads 4 chapters of his book each day. How many chapters does Curtis read in a week? (7 days) 3NS2.2 F 12

H 35

G 28

J 56

Which math sentence will find the next number in this sequence? 3NS2.4 9

Find the difference of 4,081 and 3,706.

F 3 × 4 = 12

H 3 + 6 + 9 = 18

A 275

C 1,375

G 9 + 6 = 15

J 4 × 4 = 16

B 375

D 1,785

3NS2.1

GO ON 104

Chapter 4 Standards Practice

Copyright © by The McGraw-Hill Companies, Inc.

4

Emma eats a hot lunch 5 days a month. Her friend Olivia eats a hot lunch twice as often each month. Which math sentence shows how many hot lunches Olivia eats per month? 3AF1.0

An extra-large bag of potatoes is on sale for $6. If Lu buys 7 bags, how much does he spend on potatoes? 3NS2.2

10

11

12

Eric is placing 133 books on shelves. He would like to place 19 books on each shelf. How many shelves will he need? F 114

H 152

G 7

J 2,527

Which does not round to 9,000 when rounded to the nearest thousands? A 8,611

C 9,254

B 8,769

D 9,502

3NS1.3

Which three items did Kiera buy if her total was $24? 1NS2.7

Copyright © by The McGraw-Hill Companies, Inc.

4NS3.2

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

F soda, T-shirt, socks

Read each problem carefully and look at each answer choice. Eliminate answers you know are wrong. This narrows your choices even before solving the problem.

G book, CD, socks H T-shirt, CD, socks J T-shirt, CD, book

Chapter 4 Standards Practice

105

Chapter

5

Division Let’s play a game of soccer. Division helps us make up the teams. If there are 18 people who want to play, you would divide 18 by 2 to get 2 teams of 9 people.

Copyright © by The McGraw-Hill Companies, Inc.

106 Getty Images

Chapter 5 Division

STEP

STEP

1 Quiz

Are you ready for Chapter 5? Take the Online Readiness Quiz at ca.mathtriumphs.com to find out.

2 Preview

Get ready for Chapter 5. Review these skills and compare them with what you’ll learn in this chapter.

What You Know

What You Will Learn

You know how to subtract.

Lesson 5-1

Examples: 8 - 2 = 6 6-2=4 4-2=2 2-2=0

Division is repeated subtraction.

TRY IT!

24

1

32 - 8 =

2

40 - 10 =

3

16 - 4 =

12

4

20 - 1 =

19

30

8-2=6 6-2=4 4-2=2 2-2=0

Subtract 2 one time. Subtract 2 two times. Subtract 2 three times. Subtract 2 four times.

So, 8 ÷ 2 = 4

You know that addition and subtraction are inverse operations. Example: 5 + 7 = 12 12 - 7 = 5 Copyright © by The McGraw-Hill Companies, Inc.

8÷2=?

TRY IT!

Lessons 5-2 through 5-8 Multiplication and division are inverse operations . 5 × 7 = 35 35 ÷ 7 = 5

Rewrite each equation using an inverse operation.

4+5=9

5

9-5=4

6

15 + 10 = 25

25 - 10 = 15

7

64 - 31 = 33

33 + 31 = 64

8

22 + 76 = 98

98 - 76 = 22

107

Lesson

5-1 Model Division

3NS2.2 Memorize to automaticity the multiplication table for numbers between 1 and 10. 4NS3.2 Demonstrate an understanding of, and the ability to use, standard algorithms for multiplying a multidigit number by a two-digit number and for dividing a multidigit number by a one-digit number; use relationships between them to simplify computations and to check results. You use

KEY Concept

Division is the inverse operation of multiplication. multiplication facts when you divide. Division is used to make groups of equal size.

If you had eight pretzels and wanted to share them with a friend, you would divide the pretzels into two groups. The division can be written three ways. horizontal method

vertical method

fraction method

8÷2=4

4 2 8

8 __ =4 2

• The dividend in these problems is 8. • The divisor in these problems is 2. • The quotient in these problems is 4.

µ

OVNCFSPGSPXTEJWJTPS

OVNCFSPGDPMVNOTRVPUJFOU

Use multiplication to check your division. 8 ÷ 2 = 4 is correct because 2 × 4 = 8

division an operation on two numbers in which the first number is split into the same number of equal groups as the second number Example: 6 ÷ 3 means 6 is divided into 3 groups of equal size quotient the answer or result of a division problem dividend the number that is being divided divisor the number by which the dividend is being divided quotient 2 3 6

6÷3=2 dividend divisor

array objects or symbols displayed in rows of the same length and columns of the same length; the length of a row might be different from the length of a column (Lesson 4-1, p. 4)

Think of fact families when dividing. A fact family has two multiplication sentences and two division sentences. 4×2=8 2×4=8

108

Chapter 5 Division

8÷2=4 8÷4=2

inverse operations operations that undo each other Example: Multiplication and division are inverse operations.

Copyright © by The McGraw-Hill Companies, Inc.

Arrays are used to model division. In the problem 8 ÷ 2 = 4, the divisor is 2, which is the number of rows in the array. Continue making columns until there are 8 rectangles. The number of columns in the array is the quotient.

VOCABULARY

Example 1

YOUR TURN!

Draw an array to model the expression 6 ÷ 3. 1. Identify the divisor (the second number). This represents the number of rows. 3

1 row 2 rows 3 rows

2. Identify the dividend (the first number). This is the total number of rectangles in the array. 6

Draw an array to model the expression 8 ÷ 2. 1. Identify the divisor. 2 This represents the number of rows . 2. Identify the dividend. 8 total number of This is the rectangles in the array. Continue making columns until there are 8 rectangles.

Continue making columns until there are 6 rectangles.

2 columns = quotient 3. The number of columns in the array is the quotient. 2 4. Check by multiplying the quotient by the divisor. The product should be the dividend. 3×2=6 Copyright © by The McGraw-Hill Companies, Inc.

4

columns = quotient

3. The number of columns in the array is the quotient.

4

4. Check by multiplying the quotient by the divisor. 2× 4 =8

Example 2 Draw a model of 10 ÷ 2 using circles and tally marks. 1. What is the divisor?

2

Draw 2 circles. 2. What is the dividend?

10

Use tally marks to divide the 10 into 2 groups. Place a tally mark in each circle as you count until you have drawn 10 tally marks. 3. How many tally marks are in each circle? Write the problem with the quotient. 4. Check your work.

5 10 ÷ 2 = 5 2 × 5 = 10 ✓

GO ON

Lesson 5-1 Model Division

109

YOUR TURN! Draw a model of 15 ÷ 3 using circles and tally marks. 1. What is the divisor? 2. What is the dividend?

3

Draw

3

circles.

15

Use tally marks to divide the

15

into

3

groups.

Place a tally mark in each circle as you count until you have drawn 15 tally marks. 3. How many tally marks are in each circle?

5

Write the problem with the quotient. 15 ÷ 3 = 4. Check your work. 3 ×

5

5

= 15

Example 3 Write 6 ÷ 3 in two different formats. 1. What number is the divisor? 3 The divisor goes in front of the division bracket.

YOUR TURN! Write 15 ÷ 5 in two different formats.

5 15 2. Write the vertical format. 5 1. What number is the divisor? Read as,

fifteen

3. Write as a fraction. Read as,

Example 4

YOUR TURN!

Write the division facts from the fact family of 5 × 2 = 10. 1. Write the division fact using the first factor as the divisor. 10 ÷ 5 = 2 2. Write another division fact using the second factor as the divisor. 10 ÷ 2 = 5 110

Chapter 5 Division

“

fifteen

“

15 _

divided by

5

.”

5

.”

5

divided by

Write the division facts from the fact family of 7 × 2 = 14. 1. Write the division fact using the first factor as the divisor.

14

÷

7

=

2

2. Write another division fact using the second factor as the divisor.

14

÷

2

=

7

Copyright © by The McGraw-Hill Companies, Inc.

2. Write the vertical format. 3 6 Read as, “six divided by 3.” 6 3. Write as a fraction. __ 3 Read as, “six divided by 3.”

Who is Correct? Write 16 ÷ 4 in two different formats.

Ursa

Zola

_

Nicole

_

16 16 and 4 4

16 4 and 4 16

_

4 16 and 16 4

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

Guided Practice 1

Draw an array to model the expression 12 ÷ 3. Sample answer:

2

Draw a model of 12 ÷ 2 using circles and tally marks. Sample answer:

Step by Step Practice

Copyright © by The McGraw-Hill Companies, Inc.

3

Draw an array to model the expression 12 ÷ 6. Step 1 Identify the divisor.

6

This represents the number of rows. Step 2 Identify the dividend.

12

This is the total number of rectangles in the array. Continue making columns until there are rectangles.

12

Step 3 The number of columns in the array is the quotient.

2

Step 4 Check your division by multiplying the quotient by the divisor. 6 × 2 = 12 . GO ON Lesson 5-1 Model Division

111

Draw an array to model each expression. How many rows? 3 How many rectangles? 9 How many columns? 3

4

9÷3

5

8÷4

6

14 ÷ 7

7

6÷3

8

16 ÷ 2

Step by Step Problem-Solving Practice

Problem-Solving Strategies ✓ Draw a diagram.

Solve. 9

Use logical reasoning. Solve a simpler problem. Work backward. Guess and check.

GARDENS Kevin is planting a garden with 36 seeds. He wants to have 6 rows of seeds with the same number of seeds in each row. How many seeds will be in each row? Understand

Read the problem. Write what you know.

36

seeds. There will be

6

rows.

Plan

Pick a strategy. One strategy is to draw a diagram.

Solve

Make an array with the correct number of rows. Continue making columns until there are 36 rectangles. The diagram shows how the seeds will be planted. There are 6 rows and 6 columns. The number of columns represents the quotient . There will be 6 seeds in each row.

Check

112

Think about the fact family. Is the division fact part of the family?

Chapter 5 Division

Copyright © by The McGraw-Hill Companies, Inc.

There are

10

CULTURE There is a sequence of 12 animals in consecutive years of the Chinese calendar. (Each year = 1 animal.) How often does an animal occur during a 60-year period? Check off each step. 5

✔ ✔ ✔ ✔

Understand Plan Solve CULTURE

Check

Chinese calendar 11

ENTERTAINMENT José is arranging 20 chairs for a party. He wants to have 5 rows of chairs with the same number of chairs in each row. How many chairs will he put in each row? 4 chairs Explain how multiplication is the inverse operation for division.

12

Sample answer: An inverse operation undoes another operation. When you divide by a number, you are dividing a number into groups. To undo this, multiply the number of groups (divisor) by the quotient to get the dividend.

Skills, Concepts, and Problem Solving Draw an array to model each expression. Copyright © by The McGraw-Hill Companies, Inc.

13

20 ÷ 5 Sample answer:

14

8÷4

Draw a model using circles and tally marks for each expression. 15

18 ÷ 3 Sample answer:

16

9 ÷ 3 Sample answer:

Write each expression in two different formats. 17 18

25 ÷ 5

5 25

16 ÷ 8

8 16

25 _ 5

16 _ 8

GO ON Lesson 5-1 Model Division

Matthew Scherf/iStockphoto

113

Write the division facts from each fact family. 19

6 × 3 = 18

18 ÷ 6 = 3; 18 ÷ 3 = 6

20

4 × 5 = 20

20 ÷ 4 = 5; 20 ÷ 5 = 4

21

6 × 4 = 24

24 ÷ 6 = 4; 24 ÷ 4 = 6

22

2 × 11 = 22 22 ÷ 2 = 11; 22 ÷ 11 = 2

Solve. 23

PACKAGING Three packages contain 36 golf balls in all. How many golf balls are in each package if there are the same number of 12 golf balls balls in each?

24

MOVIES Eleanor received 18 movie passes for her birthday. How many times can she and 2 friends go to see a movie?

6

Vocabulary Check Write the vocabulary word that completes each sentence.

Division

25

is the same as repeated subtraction.

inverse operation

26

The

27

Writing in Math Karen wrote the division facts from the fact family 4 × 5 = 20. What mistake did Karen make? 20 ÷ 5 = 5

for division is multiplication.

20 ÷ 4 = 4

Karen forgot to use both factors. She should have written 20 ÷ 5 = 4 and 20 ÷ 4 = 5. Copyright © by The McGraw-Hill Companies, Inc.

Spiral Review Solve. 28

(Lesson 4-2, p. 11)

MONEY There are 10 dimes in each dollar. If you have 3 dollars, how many dimes do you have?

3 · 10 = 30. Three dollars have 30 dimes. Find each difference. 29 114

6,000 - 5,123

(Lesson 3-7 p. 173)

877

Chapter 5 Division

coins: United States Mint, bill: Michael Houghton/StudiOhio

30

4,000 - 2,398

1,602

Lesson

5-2 Divide by 0, 1, and 10

3NS2.2 Memorize to automaticity the multiplication table for numbers between 1 and 10. 3NS2.6 Understand the special properties of 0 and 1 in multiplication and division. 4NS3.2 Demonstrate an understanding of, and the ability to use, standard algorithms for multiplying a multidigit number by a two-digit number and for dividing a multidigit number by a one-digit number; use relationships between them to simplify computations and to check results.

KEY Concept Any number divided by itself is equal to the number itself. 3÷1 3 3 ÷ 1 = __ = 3 1 Division problems involving ten follow a pattern. 30 ÷ 10

VOCABULARY quotient the answer or result of a division problem (Lesson 5-1, p. 108)

dividend a number that is being divided (Lesson 5-1, p. 108)

30 30 ÷ 10 = ___ = 3 10

divisor the number by which the dividend is being divided

Copyright © by The McGraw-Hill Companies, Inc.

Zero divided by any number equals zero. 0÷3 0 0 ÷ 3 = __ = 0 3

quotient 2 3 6

Division by zero is not possible. 3 ÷ 0 = not possible

6÷3=2 dividend divisor

(Lesson 5-1, p. 108)

You should memorize the division rules for 0, 1, and 10.

Example 1 Draw an array to model and find 5 ÷ 1. 1. Write the answer if you know it. Otherwise, draw an array. 2. How many rectangles will be in the array? 5 3. How many rows? 1 4. Count the number of columns. 5

DPMVNOTRVPUJFOU

5. Write the quotient. 5 ÷ 1 = 5 6. Check. 5 × 1 = 5

GO ON

Lesson 5-2 Divide by 0, 1, and 10

115

YOUR TURN! Draw an array to model and find 8 ÷ 1. 1. Write the answer if you know it. Otherwise, draw an array. 2. How many rectangles will be in the array?

1

3. How many rows?

8

4. Count the number of columns. 5. Write the quotient. 8 ÷ 1 = 6. Check. 8 × 1 =

8

YOUR TURN! Find 60 ÷ 10.

Find 40 ÷ 10.

1. What number is the divisor?

1. What number is the divisor? 10 2. What number times 10 equals 40? Write the quotient. 40 ÷ 10 = 4

YOUR TURN! Find 500 ÷ 10.

Find 200 ÷ 10. 1. Rewrite the problem in vertical format. 2. Look at the first two numbers in the dividend. What number multiplied by 10 equals 20? 2

20 10 200 -20 0 -0 0

3. Subtract. Bring down the next number from the dividend. 4. What number multiplied by 10 equals 0? 0

116

Chapter 5 Division

60

1. Rewrite the problem in vertical format. 2. Look at the first two numbers in the dividend. What number multiplied by 10 equals 50? 5

50 10 500

-50 0 -0 0

3. Subtract. Bring down the next number in the dividend. 4. What number multiplied by 10 equals 0?

0

5. Subtract. 6. Check. 50 · 10 = 500

Copyright © by The McGraw-Hill Companies, Inc.

Example 3

10

2. What number times 10 equals 60? Write the quotient. 6 60 ÷ 10 = 3. Check. 6 × 10 =

3. Check. 4 × 10 = 40

6. Check. 20 · 10 = 200

DPMVNOTRVPUJFOU

8

Example 2

5. Subtract.

8

Who is Correct? Find 1,000 ÷ 10.

Emily

Liza

1,000 ÷ 10

1,000 ÷ 10

1,000 ÷ 10

= 10

= 100

= 1,000

John

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

Guided Practice Draw an array to model and find each quotient.

Copyright © by The McGraw-Hill Companies, Inc.

1

30 ÷ 10

3

2

100 ÷ 10

10

number of columns =

3

= quotient

number of columns =

10

= quotient

Step by Step Practice 3

Find 800 ÷ 10. Step 1 What number is the divisor?

10

Step 2 What number times 10 equals 800? Write the quotient. 80 800 ÷ 10 = Step 3 Check.

80

× 10 =

800

GO ON Lesson 5-2 Divide by 0, 1, and 10

117

Find each quotient. If the quotient is not possible, write not possible. 4

90

900 ÷ 10 =

Check your answer.

90

×

10

900

=

5

100 ÷ 10 =

10

6

700 ÷ 10 =

70

7

600 ÷ 10 =

60

8

400 ÷ 10 =

40

9

6÷1=

10

9÷1=

11

15 ÷ 0 = not possible

12

4 ÷ 0 = not possible

13

12 ÷ 1 =

12

14

3÷1=

15

0÷7=

0

16

0 ÷ 20 =

17

11 ÷ 11 =

18

8÷8=

6

1

9

3 0 1

Step by Step Problem-Solving Practice 19

MONEY There are 10 pennies in each dime. If you have 60 pennies, for how many dimes could you trade? Understand

Read the problem. Write what you know. There are 60 are

Plan

Solve

Check

118

10

pennies in a dime. There pennies altogether.

Pennies

Dimes

Pick a strategy. One strategy is to make a table.

10

1

20

2

Complete the table.

30

3

The table shows that 60 pennies equal 6 dimes.

40

4

50

5

Does the answer make sense? Look over your solution. Did you answer the question?

60

6

Chapter 5 Division

Copyright © by The McGraw-Hill Companies, Inc.

Solve.

20

21

CONSTRUCTION There are 10 rolls of paper towels in every jumbo package. If a shopper wants 80 rolls of paper towels, how many 8 packages should he purchase? Check off each step.

✔

Understand

✔

Plan

✔

Solve

✔

Check

FOOD The cafeteria used 120 eggs for a recipe. How many dozens of eggs did they use?

10

FOOD A dozen is 12 eggs. Explain why you cannot divide by zero.

22

Sample answer: Multiplication is used to check division. For example, in 6 ÷ 0, ask what times 0 is 6. No number times 0 is equal to 6, so division by 0 does not work. 6 ÷ 0 is undefined or impossible.

Skills, Concepts, and Problem Solving Copyright © by The McGraw-Hill Companies, Inc.

Draw an array to model and find each quotient. 23

20 ÷ 10

2

24

50 ÷ 10

5

Find each quotient. If the quotient is not possible, write not possible. 25

60 ÷ 10 =

6

26

90 ÷ 1 =

27

120 ÷ 1 =

120

28

150 ÷ 10 =

29

400 ÷ 0 = not possible

30

40 ÷ 1 =

40

31

230 ÷ 10 =

32

0 ÷ 80 =

0

23

90 15

GO ON

Lesson 5-2 Divide by 0, 1, and 10 Burke/Triolo Productions/Getty Images

119

Find each quotient. If the quotient is not possible, write not possible. 33

3 ÷ 0 = not possible

34

0 ÷ 0 = not possible

35

1÷1=

1

36

2÷1=

2

37

8÷1=

8

38

4÷1=

4

39

3÷3=

1

40

5÷5=

1

Solve. 41

MODELS Genevieve uses base-ten blocks to model a division problem. She takes the one-blocks and groups them into 10 piles of 9 blocks. What division expression models her actions?

90 ÷ 10 42

INTERIOR DESIGN Larry is laying tiles on the kitchen floor. The tiles come in boxes of 10. If each tile is one square foot, how many boxes will Larry need to tile a floor that is 80 square feet?

8 boxes FITNESS Danita started training on an exercise machine. By the third month, she trained 10 times longer each day than she did the first month. She trained for 60 minutes a day the third month. How many minutes a day did she train when she began?

6 minutes

Vocabulary Check Write the vocabulary word that completes each sentence. 44

45

120

In a division problem, the number being divided is the dividend . The division problem 12 ÷ 0 is undefined or impossible because you cannot divide by 0.

Chapter 5 Division

Copyright © by The McGraw-Hill Companies, Inc.

43

46

Writing in Math Explain why division can be thought of as repeated subtraction.

Sample answer: To find the answer to a problem such as 12 ÷ 3, keep subtracting until you get to 0. Then count the number of times you subtracted 3. 12 - 3 = 9, 9 - 3 = 6, 6 - 3 = 3, 3 - 3 = 0.

Spiral Review Solve. 47

(Lesson 3-8, p. 181)

CLUBS The members of the chess club are shown in the table to the right. About how many more females are enrolled than males?

Gender

Enrollment

female

198

male

173

There are about 200 females. There are about 170 males. There are about 30 more females than males. 48

CHARITY The amounts in dollars collected for charity by two groups are shown in the table to the right. About how much was collected in all?

Group

Amount in $

Heartmenders

5,326

All for One

8,881

Copyright © by The McGraw-Hill Companies, Inc.

Heartmenders’ amount rounds to $5,000, and All for One’s amount rounds to $9,000. The total is about $14,000.

Write true or false for each statement. 49

464 < 466

51

65 > 56

(Lesson 2-6 p. 105)

true true

For each number, write even or odd.

50

303 > 413

false

52

110 > 111

false

(Lesson 1-4, p. 25)

53

30

even

54

27

odd

55

16

even

56

63

odd

Lesson 5-2 Divide by 0, 1, and 10

121

Chapter

Progress Check 1

5

(Lessons 5-1 and 5-2)

Draw a model for each expression. 3NS2.2 1

8÷2

2

15 ÷ 5

Write the division expression represented by each model. 3NS2.2

40 ÷ 10

3

Sample answer:

14 ÷ 2

4

Sample answer:

Find each quotient. If the quotient is not possible, write not possible. 3NS2.2, 3NS2.6, 4NS3.2

6

6

70 ÷ 10 =

10

8

30 ÷ 1 =

10

70 ÷ 0 = not possible

12

35 ÷ 35 =

14

0 ÷ 10 =

60 ÷ 10 =

7

10 ÷ 1 =

9

20 ÷ 0 = not possible

11

50 ÷ 50 =

13

0÷6=

1 0

7 30 1 0

Solve. 15

HISTORY Refer to the photo caption at the right. If a decade is 10 years, how many decades ago was 23 the Declaration of Independence signed?

16

PARTIES Jeannine placed 30 items into 3 favor bags. She placed the same number of items in each bag. How 10 many items did she place in each bag?

17

MUSIC Tina plays a series of 7 notes over and over. If she plays a total of 70 notes, how many times does 10 Tina play the series?

122

Chapter 5 Division

© Photodisc/Getty Images

HISTORY The Declaration of Independence was signed about 230 years ago.

Copyright © by The McGraw-Hill Companies, Inc.

5

Lesson

5-3 Divide by 2 and 5

3NS2.2 Memorize to automaticity the multiplication table for numbers between 1 and 10. 4NS3.2 Demonstrate an understanding of, and the ability to use, standard algorithms for multiplying a multidigit number by a two-digit number and for dividing a multidigit number by a onedigit number; use relationships between them to simplify computations and to check results.

KEY Concept There are several phrases that represent dividing by 2. Consider 8 ÷ 2. • the quotient of eight and two • eight divided by two • half of eight

VOCABULARY quotient the answer or result of a division problem

Memorize the division facts of 2 and 5.

(Lesson 5-1, p. 108)

Example 1 Draw an array to model and find 8 ÷ 2.

dividend the number that is being divided (Lesson 5-1, p. 108)

1. Write the answer if you know it. Otherwise, draw an array.

divisor the number by which the dividend is being divided quotient

2 rows = divisor

2 3 6

4 columns = quotient

6÷3=2 dividend divisor

2. How many rectangles will be in the array? 8

(Lesson 5-1, p. 108)

Copyright © by The McGraw-Hill Companies, Inc.

3. How many rows? 2 4. Count the number of columns. 4 5. Write the quotient. 8 ÷ 2 = 4 6. Check. 4 × 2 = 8 YOUR TURN! Draw an array to model and find 12 ÷ 2. 1. Write the answer if you know it. Otherwise, draw an array. 2. How many rectangles will be in the array? 3. How many rows?

12

2 2

4. Count the number of columns.

6

5. Write the quotient. 12 ÷ 2 = 6 6. Check.

6

× 2 = 12

6

rows = divisor.

columns = quotient GO ON

Lesson 5-3 Divide by 2 and 5

123

Example 2 Find 64 ÷ 2. 1. Estimate. 60 ÷ 2 = 30

YOUR TURN! Find 84 ÷ 2. 5PFTUJNBUF SPVOE UIFEJWJEFOEUPUIF HSFBUFSQMBDFWBMVF

1. Estimate.

2. Rewrite the problem in vertical format.

2 64

3. Look at the first digit in the dividend. What number multiplied by 2 is 6? 3

3 2 64

Write 3 above the 6 in the dividend. 4. Multiply. Write the product under the 6 in the dividend.

3 2 64 6

5. Subtract. Bring down the 4 in the dividend. 3 2 64 -6 4 This is the new dividend. 6. What number multiplied by 2 is 4? 2 Multiply. Place the product under the 4.

8. The quotient is 32. Compare to your estimate for reasonableness.

÷2=

2. Rewrite the problem in vertical format.

2 84

3. Look at the first digit in the dividend. What number multiplied by 2 is 8? 4

2 84

4. Multiply.

6. What number multiplied by 2 is 4? 2 Multiply. 8. The quotient is 42 . Compare to your estimate for reasonableness.

Sylvia 15 85 5 -5 25 - 25 0

Chapter 5 Division

42 2 84

-8

7. Subtract.

Circle correct answer(s). Cross out incorrect answer(s). 124

4 2 84

-8

Find 85 ÷ 5.

16 85 5 -5 30 - 30 0

4

5. Subtract 8 from 8. Bring down the 4 in the dividend. 4 2 84 -8 This is the new dividend. 4

Who is Correct?

Ivan

40

Hewitt 17 85 5 -5 35 - 35 0

4 -4 0 Copyright © by The McGraw-Hill Companies, Inc.

7. Subtract.

32 2 64 -6 4 -4 0

80

Guided Practice Draw a model and find each quotient. 1

15 ÷ 5 =

3

Sample answer:

3

number of columns =

Find each quotient.

2

= quotient

12 ÷ 2 =

6

Sample answer:

6

number of columns =

= quotient

6TFUIFNVMUJQMJDBUJPO GBDUTZPVLOPXUP IFMQZPV

3

30 ÷ 5 =

6

4

45 ÷ 5 =

9

5

18 ÷ 2 =

9

6

14 ÷ 2 =

7

7

50 ÷ 5 =

10

8

20 ÷ 2 =

10

9

20 ÷ 5 =

4

10

16 ÷ 2 =

8

11

8÷2=

4

Step by Step Practice

Copyright © by The McGraw-Hill Companies, Inc.

12

Find 606 ÷ 2. Step 1 Estimate. 600 ÷

2

= 300

Step 2 Rewrite the problem in vertical format. Step 3 Look at the first digit in the dividend. What number multiplied by 2 is 6? 3 Step 4 Multiply by the divisor. Write the product under the first 6 of the dividend. Step 5 Subtract. Bring down the 0 in the dividend.

303 2 606

-6 06 -6 0

0

Step 6 What number multiplied by 2 is 0? Multiply.

Step 7 Subtract. Bring down the 6 in the dividend. Step 8 What number multiplied 2 is 6? Multiply.

3

Step 9 The quotient is 303 . Compare to the estimate for reasonableness.

GO ON

Lesson 5-3 Divide by 2 and 5

125

Find each quotient. 13

75 ÷ 5 Check your answer.

15

×

47

14

94 ÷ 2

16

115 ÷ 5

5

23

=

75

15

104 ÷ 2

52

17

365 ÷ 5

73

15 5 75

-5 25 -25 0

Step by Step Problem-Solving Practice

Problem-Solving Strategies ✓ Draw a model.

Solve. 18

FASHION If you have 28 socks, how many pairs of socks do you have? Read the problem. Write what you know. There are 2 socks in each pair of socks. There are 28 socks.

Plan

Pick a strategy. One strategy is drawing a model.

Solve

Draw 28 tallies to represent the individual socks. Circle each pair. How many groups did you have?

socks in each pair.

pairs of socks.

Use multiplication to check your answer.

Check

126

14

FASHION There are 2

MONEY There are 5 dollars in every 5-dollar bill. Felix has $45 in 5-dollar bills. How many 5-dollar bills does Felix have? 9 Check off each step.

✔

Understand

✔

Plan

✔

Solve

✔

Check

Chapter 5 Division

© Brand X Pictures/Punch Stock

Copyright © by The McGraw-Hill Companies, Inc.

Understand

There are

19

Use logical reasoning. Make a table. Solve a simpler problem. Work backward.

20

FITNESS Tyrone jogs 16 miles in 2 days. Aleesha jogs 40 miles in 5 days. Both jogged an equal distance each day. Who jogs more each day?

They both jogged the same distance each day. Describe how to divide a number with more than one digit by a single-digit number.

21

See TWE margin.

Skills, Concepts, and Problem Solving Draw a model and find each quotient. 22

7

35 ÷ 5

Sample answer:

23

44 ÷ 2

22

Sample answer:

Copyright © by The McGraw-Hill Companies, Inc.

Find each quotient. 24

18 ÷ 2 =

9

25

12 ÷ 2 =

6

26

15 ÷ 5 =

3

27

25 ÷ 5 =

5

28

45 ÷ 5 =

9

29

20 ÷ 5 =

4

30

48 ÷ 2 =

24

31

38 ÷ 2 =

19

32

65 ÷ 5 =

13

33

90 ÷ 5 =

18

34

42 ÷ 2 =

21

35

50 ÷ 2 =

25

Solve. 36

COMMUNITY SERVICE The sixth grade is collecting box tops to donate to a charity that will trade them in for cash. The students have 5 weeks to collect. If their goal is to collect 260 box tops, how many should they collect, on average, per week?

52

GO ON Lesson 5-3 Divide by 2 and 5

127

37

FOOD The cafeteria made 2 pans of lasagna and 5 pans of macaroni and cheese to serve for lunch. The lasagna is cut so that there are 60 servings altogether. The macaroni and cheese is cut so there are 125 servings total. Does one pan of lasagna have more servings than one pan of macaroni and cheese? If so, how much more?

One pan of lasagna will have 30 servings, which is 5 more than the macaroni and cheese. 38

SPORTS The after-school sports club rented the gym. It cost $125. If 5 sponsors donated an equal amount of money to cover the rental fees, then how much did each sponsor contribute?

Each sponsor donated $25.

Vocabulary Check Write the vocabulary word that completes each sentence.

half

39

If you divide something in

, it is divided into two equal parts.

40

In a division problem, the number you divide by is the

41

Writing in Math Mario divided 305 ÷ 5 this way. What mistake did Mario make?

divisor .

See TWE margin.

Spiral Review Solve. 42

(Lesson 4-7, p. 47)

GARDENS Jason sold vegetables from his garden. He had 9 customers who bought a half-dozen tomatoes each. He also sold 4 baskets each containing 6 cucumbers. How many vegetables did he sell in all? (A dozen is 12, so a half-dozen is 6.)

78 43

MODELS Jenny’s teacher gave her graph paper and asked her to represent the expression 10 × 6. How could Jenny do this?

Make an array with 10 rows and 6 columns. There will be 60 squares total. 128

Chapter 5 Division

061 305 5 30 5 5 0

Copyright © by The McGraw-Hill Companies, Inc.

601 5 305 -30 5 -5 0

Lesson

5-4 Divide by 3 and 4

3NS2.2 Memorize to automaticity the multiplication table for numbers between 1 and 10. 4NS3.2 Demonstrate an understanding of, and the ability to use, standard algorithms for multiplying a multidigit number by a two-digit number and for dividing a multidigit number by a one-digit number; use relationships between them to simplify computations and to check results.

KEY Concept When a number does not divide evenly, the part left over is called the remainder . Suppose you have 7 cookies and you want to share them with two friends.

VOCABULARY remainder the number that is left after one whole number is divided by another

There is one cookie left over. This is the remainder. When you have a remainder, sometimes it will be left as a remainder. Sometimes you can split the remainder so that each group has a whole number part and a fractional part.

multiple a multiple of a number is the product of that number and any whole number Example: 30 is a multiple of 10 because 3 × 10 = 30. (Lesson 4-2, p. 11)

The extra cookie could be divided into thirds so each person would get one-third of the cookie. Each person would get 1 of another cookie. 2 whole cookies and __ 3

Copyright © by The McGraw-Hill Companies, Inc.

Practice memorizing the division facts for 3 and 4.

Example 1

YOUR TURN!

Draw an array to model and find 8 ÷ 4. 1. Write the answer if you know it. Otherwise, draw an array. 2. How many rectangles will be in the array? 8

Draw an array to model and find 20 ÷ 4. 1. Write the answer if you know it. Otherwise, draw an array. 2. How many rectangles will be in the array? 20 3. How many rows? 4

3. How many rows? 4

DPMVNOTRVPUJFOU

4. Count the number of columns. 2

4. Count the number of columns.

5. Write the quotient. 8 ÷ 4 = 2

5. Write the quotient. 20 ÷ 4 =

6. Check. 2 × 4 = 8

6. Check.

5

× 4 = 20

5 5 GO ON

Lesson 5-4 Divide by 3 and 4

129

Example 2

YOUR TURN! Find 13 ÷ 4. Show the remainder.

Find 8 ÷ 3. Show the remainder. 1. Rewrite the problem in vertical format. 2. What number multiplied by 3 is close to 8? 3 × 3 is 9. That is too much. 3 × 2 is 6. This is close without going over. Multiply. Write the product under the dividend. 3. Subtract.

3 8 2 3 8 2 3 8 -6 2 2 R2 3 8 -6 2

4. There are no more digits. The quotient is 2. Write 2 R2. 5. Check your answer. Multiply the quotient by the divisor. Then add the remainder.

1. Rewrite the problem in vertical format.

4 13

2. What number multiplied by 4 is close to 13? 4×

3

=

12

Multiply. Write the product under the dividend. 3. Subtract. 4. The quotient is 3 with a remainder of

3 4 13

3 R1 4 13 -12

1

1

.

5. Check your answer.

3 × 4 = 12 12 + 1 = 13

2×3=6 6+2=8

Find 22 ÷ 3. Show the remainder.

Zoe 3 × 7 = 21 21 + 1 = 22 7 R1

Howie 6 R4 22 3 -18 4

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

130

Chapter 5 Division

Lacie 7 R1 22 3 -21 1

Copyright © by The McGraw-Hill Companies, Inc.

Who is Correct?

Guided Practice Draw an array to model and find each quotient. 1

18 ÷ 3 Sample answer:

number of columns =

3

2

6

= quotient

20 ÷ 4 Sample answer:

number of columns =

number of columns =

4

5

12 ÷ 4 Sample answer:

= quotient

3

= quotient

5

= quotient

15 ÷ 3 Sample answer:

number of columns =

Step by Step Practice 5

Find 13 ÷ 3. Show the remainder.

Copyright © by The McGraw-Hill Companies, Inc.

Step 1 Rewrite the problem in vertical format. Step 2 Look at the first digit in the dividend. Since the divisor is greater than the first digit, look at the first two digits. Step 3 Subtract. What number multiplied by 3 is equal to or less than 13? 4 Multiply

4

×

3

.

Write the answer under the dividend.

13 3

4 3 13

4 3 13

-12 1

Step 4 There are no more digits. Write the quotient. 13 ÷ 3 = 4 R1 Step 5 Check your answer.

4 × 12 +

3 1

= =

12 13 GO ON Lesson 5-4 Divide by 3 and 4

131

Find each quotient. Show the remainder. 6

3

14 ÷ 4 3 R2

4 14

-12 2 Check your answer.

3

×

4

=

12

12

+

2

=

14

7

11 ÷ 3 = 3 R2

8

23 ÷ 3 = 7 R2

9

18 ÷ 4 = 4 R2

10

21 ÷ 4 = 5 R1

Step by Step Problem-Solving Practice

Draw a diagram.

Solve. 11

TRIPS Twenty-one students are going on a field trip in three vans. One-third of the students will ride in each van. How many students will ride in each van? Understand

✓ Use logical reasoning. Make a table. Solve a simpler problem. Work backward.

Read the problem. Write what you know. There are

3

of

0OFUIJSENFBOTUP EJWJEFCZ

21

students.

21

will ride in each van.

Pick a strategy. One strategy is to use logical reasoning. One-third means that a whole is divided into 3 equal parts.

Solve

Divide 21 by 3. 7 students will ride in each van.

Check

Check your answer with repeated addition.

7

132

Chapter 5 Division

Michael Newman/PhotoEdit, Inc.

+

7

+

7

=

21

TRIPS Twenty-one students will go on the field trip.

Copyright © by The McGraw-Hill Companies, Inc.

_1 Plan

Problem-Solving Strategies

12

EGGS A farmer collected 7 eggs from 4 of his chickens. If each chicken produced the same number of eggs, how many eggs did each chicken produce?

1 R3 Check off each step.

13

✔

Understand

✔

Plan

✔

Solve

✔

Check

MONEY If Andre has 20 quarters, how many dollars does he have? (There are 4 quarters in a dollar.)

5 dollars 14

There are different ways to interpret a remainder. Suppose there are 25 students who are to sit in chairs that will be arranged in rows with 4 seats. 25 ÷ 4 = 6 R1 . How would you interpret the remainder?

Copyright © by The McGraw-Hill Companies, Inc.

Sample answer: The remainder is 1 student. If there are 6 rows, there would be 24 seats, which is not enough. If there are 7 rows, there will be 28 seats.

Skills, Concepts, and Problem Solving Draw an array to model and find each quotient. 15

44 ÷ 4 Sample answer:

16

24 ÷ 3 Sample answer:

17

15 ÷ 3 Sample answer:

18

16 ÷ 4 Sample answer: GO ON Lesson 5-4 Divide by 3 and 4

133

Find each quotient. Show the remainder if there is one. 19

16 ÷ 4 =

4

20

9÷3=

3

21

12 ÷ 3 =

4

22

20 ÷ 4 =

5

23

27 ÷ 3 =

9

24

36 ÷ 4 =

9

25

6÷3=

26

32 ÷ 4 =

8

27

15 ÷ 3 =

5

28

40 ÷ 4 =

10

29

12 ÷ 4 =

3

30

8÷4=

31

7 ÷ 3 = 2 R1

32

10 ÷ 3 = 3 R1

33

9 ÷ 4 = 2 R1

34

16 ÷ 3 = 5 R1

35

43 ÷ 4 = 10 R3

36

28 ÷ 3 = 9 R1

2

2

Solve. 37

Each teacher will get 3 packs plus 38

_1 of 1 pack. 3

FITNESS In 10 days, Alan walked a total of 30 miles. In the same number of days, Jennifer walked 40 miles. How much more did Jennifer walk per day than Alan?

Jennifer walked 1 mile more per day. Vocabulary Check Write the vocabulary word that completes each sentence. 39

multiple A of a number is the product of that number and any whole number.

40

remainder The divided by another.

134

Chapter 5 Division

is the number left after one whole number is

Copyright © by The McGraw-Hill Companies, Inc.

BUSINESS There are 10 packs of paper in a case. Three teachers share a case evenly. How much paper will each teacher get?

41

Writing in Math Explain how to check a division problem when it has a remainder. Use the division problem 15 ÷ 4 to show the work.

When you multiply the quotient by the divisor and add

3 4 15 12 ___ 3

the remainder, the answer should equal the dividend. The quotient is 3. The divisor is 4 and the remainder is 3. 3 × 4 + 3 = 12 + 3 = 15.

Spiral Review Solve. 42

(Lesson 4-2, p. 11)

FASHION Sharon went to a sale in which pants were $10 each and shirts were on clearance for $1 each. If Sharon spent $23 without tax, what combinations of clothing could she have purchased?

Sharon could have bought 2 pairs of pants and 3 shirts, 1 pair of pants and 13 shirts, or 23 shirts.

Copyright © by The McGraw-Hill Companies, Inc.

43

BOOKS Michael read 10 pages of his book every day for 9 days. His brother Joe read 78 pages of his book in the same 9-day period. Who read more?

Michael Use a number line to model each expression. Then write and model a number expression demonstrating the Commutative Property. (Lesson 4-1 p. 4)

44

3×2

2×3

45

3×4

4×3

Lesson 5-4 Divide by 3 and 4

135

Chapter

Progress Check 2

5

(Lessons 5-3 and 5-4)

Draw a model and find each quotient. 3NS2.2 1

7

14 ÷ 2

Sample answer:

6

18 ÷ 3

2

Sample answer:

Write the division expression represented by each model. 3NS2.2 3

24 ÷ 4

21 ÷ 3

4

Find each quotient. Show the remainder if there is one. 3NS2.2, 4NS3.2 5

35 ÷ 5 =

7

6

4÷2=

2

9

50 ÷ 5 =

10

10

18 ÷ 3 =

13

30 ÷ 4 = 7 R2

14

31 ÷ 5 = 6 R1

6

7

4÷4=

11

36 ÷ 4 =

15

1 9

11 ÷ 3 = 3 R2

8

30 ÷ 3 =

10

12

24 ÷ 3 =

8

16

33 ÷ 4 = 8 R1

17

FARMS A dairy farmer has 40 cows. He began with 16 cows and bought the rest of his herd over the past 4 years. Not including his initial herd, how many cows on average did he buy each year?

6 cows 18

BIRDS The bird club counted 36 birds on an outing to the nature center. There are 4 acres in the nature center. How many birds are there on average per acre?

9 birds 19

PHOTOS Ellis’s digital camera holds 43 pictures. Suppose he takes the same number of pictures each day. How many pictures can he take while on a 4-day vacation? How many pictures will he have left over?

He can take 10 pictures a day, and he will have 3 pictures left over. 136

Chapter 5 Division

Copyright © by The McGraw-Hill Companies, Inc.

Solve. 3NS2.2, 4NS3.2

Lesson

5-5 Divide by 6 and 7

3NS2.2 Memorize to automaticity the multiplication table for numbers between 1 and 10. 4NS3.2 Demonstrate an understanding of, and the ability to use, standard algorithms for multiplying a multidigit number by a two-digit number and for dividing a multidigit number by a onedigit number; use relationships between them to simplify computations and to check results.

KEY Concept You use the same process to divide by all single-digit numbers. Sometimes you will have remainders and sometimes you will not have remainders. 125 R4 6 754 6 1×6=6 15 12 2 × 6 = 12 34 30 5 × 6 = 30 4

107 7 749 7 49 49 0

VOCABULARY remainder the number that is left after one whole number is divided by another

1×7=7 7 × 7 = 49

(Lesson 5-4, p. 129)

multiple a multiple of a number is the product of that number and any whole number Example: 30 is a multiple of 10 because 3 x 10 = 30. (Lesson 4-2, p. 11)

A remainder must always be less than the divisor.

Memorize the division facts for 6 and 7.

Example 1

YOUR TURN!

Copyright © by The McGraw-Hill Companies, Inc.

Write the division problem represented by the model. 1. How many rectangles are in the array? 18 2. How many rows? 6

Write the division problem represented by the model. 1. How many rectangles are in the array?

42

2. How many rows?

7

UIFOVNCFSPGSPXT UIFEJWJTPS

DPMVNOTRVPUJFOU

3. Count the number of columns. This is the quotient. 3

3. Count the number of columns. This is the quotient. 6

4. Write the division problem. 18 ÷ 6 = 3

4. Write the division problem. 42 ÷ 7 = 6

5. Check. 3 × 6 = 18

5. Check.

6

×

7

=

42 GO ON

Lesson 5-5 Divide by 6 and 7

137

Example 2

YOUR TURN! Find 714 ÷ 7.

Find 606 ÷ 6. 1.

2. 3.

Estimate. Practice multiplying and 600 ÷ 6 = 100 dividing mentally before solving on paper. Rewrite the problem in vertical format. 6 606 Look at the first digit. What 1 number multiplied by 6 is 6? 6 606 1 × 6 = 6. Write 1 in the hundreds place in the quotient.

4.

Multiply. Write the product under the dividend.

5.

Subtract. Bring down the next number in the dividend.

1 6 606 -6 0

Since 0 < 6, there is not enough to divide. So, put 0 in the tens place.

10 6 606 -6 0

6.

7.

9.

What number multiplied by 6 is 6? 1 Multiply. Subtract.

10. The quotient is 101. Compare to the estimate for reasonableness.

138

Chapter 5 Division

Estimate. 700 ÷ 7 = 100

2.

Rewrite the problem in vertical format.

3.

Look at the first digit. What number multiplied by 7 is 7? 1

1 7 714

4.

Multiply. Write the product under the dividend.

5.

Subtract. Bring down the next number in the dividend.

1 7 714

-7 1

6.

Since 1 < 7, there is not enough to divide. So, put 0 in the tens place.

7.

Bring down the next number in the dividend.

8. 101 6 606 -6 06 -6 0

7 714

9.

What number multiplied by 7 is 14? 2 . Multiply. Write the product under the dividend. Subtract.

10. The quotient is 102 . Compare to the estimate for reasonableness.

10 7 714

-7 14 102 7 714

-7 14 -14 0

Copyright © by The McGraw-Hill Companies, Inc.

8.

Bring down the 6 in the dividend.

1.

Who is Correct? Find 636 ÷ 7. Show the remainder.

Gordon

Tracy 9 R6 6 63 7 -63 6

Kelly

90 R6 6 7 63 -63 6

91 6 63 7 -63 0

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

Guided Practice Write the division problem represented by the model. 1

12 ÷ 6 = 2

2

49 ÷ 7 = 7

Copyright © by The McGraw-Hill Companies, Inc.

Step by Step Practice 3

Find 20 ÷ 6. Show the remainder.

3 R2

Step 1 Rewrite the problem in vertical format.

6 20 Step 2 Look at the first digit in the dividend. Since the divisor is greater than the first digit, look at the first two digits. 3 What number multiplied by 6 is close to 20? Multiply. Write the product under the dividend.

-18 2

Step 3 Subtract. Step 4 There are no more digits. Write the quotient.

20 ÷ 6 = 3 R2

Step 5 Check your answer.

3 × 6 = 18 18 + 2 = 20 GO ON Lesson 5-5 Divide by 6 and 7

139

Find each quotient. Show the remainder. 4

2 R1

15 ÷ 7

7 15

-14 1 5

19 ÷ 6 =

3 R1

6

30 ÷ 7 =

4 R2

7

45 ÷ 7 =

6 R3

8

39 ÷ 6 =

6 R3

9

50 ÷ 7 =

7 R1

10

59 ÷ 6 =

9 R5

Step by Step Problem-Solving Practice

Problem-Solving Strategies

Solve. 11

HOBBIES Han makes jewelry. He bought beads to make 7 necklaces. He will use an equal number of beads for each necklace. If he purchased 400 beads, how many beads can he use to make each necklace? Understand

Read the problem. Write what you know. There are 400 beads to make 7 identical necklaces.

Plan

Pick a strategy. One strategy is to solve a simpler problem. When you multiply 7 by 5, the product is 35 . When you multiply 7 by 50, the product is 350 . Subtract this from 400. 50 What number multiplied by 7 is close to 50 without going over? 7 Multiply this by 7 and subtract the product from 50. remainder What is left is the . Add the factors you multiplied by 7. Along with the remainder, this is the answer.

50

+

7

=

57

R

1

So, Han can use 57 beads per necklace. Check

140

Does the answer make sense? Look over your solution. Did you answer the question?

Chapter 5 Division

Copyright © by The McGraw-Hill Companies, Inc.

Solve

Draw a diagram. Use logical reasoning. Make a table. ✓ Solve a simpler problem. Work backward.

12

13

ART The art museum has 532 pieces to place evenly into 6 exhibit halls. How many pieces will go into each exhibit hall? 88 with 4 pieces extra Check off each step.

✔

Understand

✔

Plan

✔

Solve

✔

Check

HEALTH There are 90 vitamins in a bottle. If Janelle takes 1 vitamin a day, how many weeks will a bottle of vitamins last?

12 weeks and 6 days Complete the four sections below for 28 ÷ 7 = 4.

14

Write the fact family.

Draw an array to model the division fact.

7 × 4 = 28 4 × 7 = 28 28 ÷ 4 = 7 28 ÷ 7 = 4

Copyright © by The McGraw-Hill Companies, Inc.

Write the fact in vertical and fraction forms.

4 7 28

Draw circles and tally marks to model the division fact.

28 = 4 _ 7

Skills, Concepts, and Problem Solving Write the division problem represented by each model. 15

21 ÷ 3 = 7

16

24 ÷ 4 = 6

Write each quotient. Show the remainder if there is one. 17

42 ÷ 6 =

7

18

28 ÷ 7 =

4

19

36 ÷ 6 =

6

20

21 ÷ 7 =

3

21

12 ÷ 6 =

2

22

210 ÷ 3 =

70 GO ON

Lesson 5-5 Divide by 6 and 7

141

Write each quotient. Show the remainder if there is one. 23

42 ÷ 7 =

6

24

63 ÷ 7 =

26

420 ÷ 7 =

60

27

7÷7=

29

707 ÷ 7

101

30

125 ÷ 6

9

10

25

60 ÷ 6 =

1

28

306 ÷ 6

51

20 R5

31

214 ÷ 7

30 R4

Solve. 32

FOOD There are blueberry and apple pies in the cafeteria. There are 42 pieces of each type of pie, but there is one more apple pie than blueberry pie. Each blueberry pie has one more piece than each apple pie. How many of each pie is there? How many pieces is each type of pie cut into?

There are 7 apple pies cut into 6 pieces each and 6 blueberry pies cut into 7 pieces each. Vocabulary Check Write the vocabulary word that completes each sentence. 33

remainder The is the number that is left after one whole number is divided by another.

34

multiple A(n) of a number is the product of that number and any whole number.

35

Sample answer: You can ask: ”What do I need to multiply the divisor by to get the dividend?” This is easier to do mentally than division is. Also, multiplication is useful for checking answers.

Spiral Review Solve. 36

4NJUITWJMMF (Lesson 4-8, p. 53)

GEOGRAPHY According to a map key, every inch is equivalent to 50 miles. Smithsville and Nelsonville are 7 inches apart. How far apart are the two cities?

50 × 7 = 350 miles

142

Chapter 5 Division

JO /FMTPOWJMMF

Copyright © by The McGraw-Hill Companies, Inc.

Writing In Math How does the inverse operation of division help you when solving division problems?

Lesson

5-6 Divide by 8 and 9 KEY Concept

3NS2.2 Memorize to automaticity the multiplication table for numbers between 1 and 10. 4NS3.2 Demonstrate an understanding of, and the ability to use, standard algorithms for multiplying a multidigit number by a two-digit number and for dividing a multidigit number by a one-digit number; use relationships between them to simplify computations and to check results.

As you become more comfortable with multiplication and division, try to use mental math as much as possible. Solving division problems using mostly mental math is called short division.

VOCABULARY short division division using mental math

Ask yourself: 8 times what number equals 48? 8 × 6 = 48

mental math to add, subtract, multiply, and divide in your head without using manipulatives, fingers, or pencil and paper

61 R1 8 489 Ask yourself: 8 times what number equals 9 or a number very close to 9? 8 × 1 = 8 As you practice short division, you will find ways to mark differences and remainders so that you do not have to do long division.

multiple a multiple of a number is the product of that number and any whole number Example: 30 is a multiple of 10 because 3 × 10 = 30. (Lesson 4-2, p. 11)

Memorize the division facts for 8 and 9.

Copyright © by The McGraw-Hill Companies, Inc.

Example 1

YOUR TURN!

Write the division problem represented by the model. 1. How many rectangles are in the array? 32 2. How many rows? 8

5IFEJWJTPS FRVBMTUIF OVNCFSPG SPXT

Write the division problem represented by the model. 1. How many rectangles are in the array? 27

5IFEJWJTPS FRVBMTUIF OVNCFSPG SPXT

2. How many rows? 9

3. Count the DPMVNOTRVPUJFOU number of columns. This is the quotient. 4

3. Count the number of columns. This is the quotient.

4. Write the division problem. 32 ÷ 8 = 4

4. Write the division problem.

5. Check. 4 × 8 = 32

27 5. Check.

3 3

9

÷

3

columns = quotient

3

= ×

9

=

27

GO ON

Lesson 5-6 Divide by 8 and 9

143

Example 2

YOUR TURN! Find 315 ÷ 8. Show the remainder.

Find 289 ÷ 9. Show the remainder. 1. Rewrite the problem in vertical format. 2. Look at the first digit. The divisor is greater than the first digit, so look at the first two digits. What number multiplied by 9 is close to 28? 3

289 9 3 9 289 -27 1

What number multiplied by 9 is close to 19? 2 4. 9 × 2 = 18. Write the product under the dividend. Then subtract.

3. Bring down the last digit.

32 9 289 -27 19 32 9 289 -27 19 -18 1

What number multiplied by 8 is close to 75? 9

Find 117 ÷ 9.

009 7 11 9 -81 36

Gena 012 7 11 9 -9 27

Circle correct answer(s). Cross out incorrect answer(s). 144

Chapter 5 Division

-24 7

39 315 8 -24 75

4. 9 × 8 = 72. Write the quotient 39 under the dividend. 8 315 Then subtract. -24 5. There are no more digits. 75 39 The answer is with -72 3 a remainder of . 3 39 3 Write R .

Who is Correct?

Jeff

3

315 8

Pedro 013 7 11 9 -9 27 -27 -0

Copyright © by The McGraw-Hill Companies, Inc.

5. There are no more digits. The answer is 32 with a remainder of 1. Write 32 R1.

2. Look at the first digit. The divisor is greater than the first digit, so look at the first two digits. What number multiplied by 8 is close to 31? 3

8 315

8 × 3 = 24 . Write the product under the dividend. Now use mental math. 31 - 24 = 7

9 × 3 = 27. Put the answer under the dividend. Now use mental math. 28 - 27 = 1 3. Bring down the last digit. What is it? 9

1. Rewrite the problem in vertical format.

Guided Practice Write the division problem represented by the model. 1

27 ÷ 9 = 3

16 ÷ 8 = 2

2

Step by Step Practice 3

Find 117 ÷ 9. Use short division. Step 1 Look at the first digit in the dividend. Because the 9 is greater than the 1, look at the first two digits in the dividend. Step 2 What number multiplied by 9 is close to 11 without going over? 1

Copyright © by The McGraw-Hill Companies, Inc.

9×

1

9

=

1 9 117

. Write the quotient in the tens place.

1

Step 3 11 - 9 = 2 . Write the difference next to the next digit in the dividend. Step 4 What is the third number in the dividend? Step 5 What number multiplied by 9 is close to going over? 3 9×

3

13

27 27

without

13

9 11 2 7

27 . Write the quotient in the ones place.

=

Step 6 The quotient is Check.

9 11 2 7

13 .

× 9 = 117

GO ON Lesson 5-6 Divide by 8 and 9

145

Find each quotient. Show the remainder if there is one.

12

12

9 108

×

9

4

108 ÷ 9 =

= 108

5

104 ÷ 8 =

13

6

126 ÷ 9 =

14

7

128 ÷ 8 =

16

8

117 ÷ 9 =

13

9

247 ÷ 8 =

30 R7

10

145 ÷ 9 =

16 R1

11

289 ÷ 9 =

32 R1

12

333 ÷ 8 =

41 R5

Step by Step Problem-Solving Practice

Problem-Solving Strategies Draw a diagram. Use logical reasoning. ✓ Make a table. Solve a simpler problem. Work backward.

Solve. 13

COMMUNITY SERVICE The eleventh graders were repairing houses during spring break. For every 8 hours a student worked, he or she got 1 class credit. At the end of two weeks, the students worked a total of 720 hours. Each student worked an equal number of hours. If 10 students participated, how many credits did they earn in all? Understand

Read the problem. Write what you know.

Plan

146

Pick a strategy. One strategy is to make a table.

Hours worked by each student

8

16

24

32

40

48

56

64

72

Total hours for 10 students

80

160

240

320

400

480

560

640

720

Credits earned

10

20

30

40

50

60

70

80

90

Solve

Look at the table. Find when the total hours worked is 720. The students earned 90 credits in all.

Check

Does the answer make sense? Look over your solution. Did you answer the question?

Chapter 5 Division

Copyright © by The McGraw-Hill Companies, Inc.

There are 10 students. For every 8 hours worked, they received 1 credit. They worked 720 hours total.

14

15

CONSTRUCTION A builder buys lumber by the linear foot. For each 9-foot length, she pays $5. How much does she pay for 81 $45 linear feet of lumber? Check off each step.

✔

Understand

✔

Plan

✔

Solve

✔

Check

LANDSCAPING The flower beds in front of school are placed in rows and columns of equal length. If there are 64 flowers planted in all, how many rows and how many columns are there?

8 rows and 8 columns Choose one fact family from the multiplication facts for 8 and another fact family for 9. Write all the facts for both families. Sample answer:

Copyright © by The McGraw-Hill Companies, Inc.

16

4 × 8 = 32

9 × 6 = 54

8 × 4 = 32

6 × 9 = 54

32 ÷ 4 = 8

54 ÷ 9 = 6

32 ÷ 8 = 4

54 ÷ 6 = 9

LANDSCAPING The flower beds are planted in rows and columns.

Skills, Concepts, and Problem Solving Write the division problem represented by the model. 17

32 ÷ 8

18

36 ÷ 9

GO ON Lesson 5-6 Divide by 8 and 9 Roy McMahon/Corbis

147

Find each quotient. 19

81 ÷ 9 =

9

20

24 ÷ 8 =

3

21

72 ÷ 9 =

8

22

64 ÷ 8 =

8

23

54 ÷ 9 =

6

24

16 ÷ 8 =

2

25

27 ÷ 9 =

3

26

40 ÷ 8 =

5

27

9÷9=

28

42 ÷ 8 = 5 R2

29

18 ÷ 9 =

2

30

80 ÷ 8 =

31

198 ÷ 9 =

22

32

168 ÷ 8 =

21

33

171 ÷ 9 =

19

34

136 ÷ 8 =

17

1

10

Solve. SHOPPING Bottled water costs $9 a case. 35

If Nalani has $80, how many cases of water can she buy?

36

Copyright © by The McGraw-Hill Companies, Inc.

80 ÷ 9 = 8 R8: Nalani can buy 8 cases. How many dollars will Nalani have left?

She will have $8 left. SWIMMING There are 85 students in the swim class. The game they are playing needs teams with 9 students on each team. 37

How many complete teams will there be?

85 ÷ 9 = 9 R4. They will have 9 complete teams. 38

How many more students would be needed to make another team?

They would need 5 more students to make another team.

148 Corbis

Chapter 5 Division

SWIMMING There are 85 students in the swim class.

39

BILLS During 9 months, Da Jon paid $387 for electricity at his apartment. If he paid the same amount each month, how much did he pay for 1 month?

$43 Vocabulary Check Write the vocabulary word that completes each sentence. 40

When you calculate a problem without using any tools (like paper and a pencil, or a calculator), you are performing mental math .

Short division

41

is finding out how many times one number goes into another number using mental math.

42

Writing in Math Write two different ways that you could find the answer to 90 ÷ 10.

Sample answers: Count how many times 10 can be subtracted from 90; try multiplication: 10 × ___ = 90; add 10 repeatedly until arriving at 90.

Spiral Review

Copyright © by The McGraw-Hill Companies, Inc.

43

COMMUNITY SERVICE Nancy worked 16 hours at a shelter on Saturday and Sunday. How many hours did she average per day? (Lesson 5-1, p. 108)

Find each product. 44

5 × 10

8

(Lesson 4-2, p. 11)

50

Write each number in word form. 47

45

9×1

9

46

2×0

0

(Lesson 2-5, p. 99)

2,105,303

two million, one hundred five thousand, three hundred three 48

1,307,480

one million, three hundred seven thousand, four hundred eighty 49

4,400,000

four million, four hundred thousand Lesson 5-6 Divide by 8 and 9

149

Chapter

5

Progress Check 3

(Lessons 5-5 and 5-6)

Write the division problem represented by each model. 3NS2.2, 4NS3.2 1

40 ÷ 8

2

3

27 ÷ 9

4

36 ÷ 9

48 ÷ 8

Find each quotient. Show a remainder if there is one. 3NS2.2 72 ÷ 8 =

9

6

81 ÷ 9 =

9

7

45 ÷ 9 =

5

8

56 ÷ 8 =

7

9

90 ÷ 8 =

11 R2

10

50 ÷ 9 =

5 R5

11

35 ÷ 9 =

3 R8

12

61 ÷ 8 =

7 R5

13

85 ÷ 8 =

10 R5

14

20 ÷ 8 =

2 R4

15

38 ÷ 9 =

4 R2

16

40 ÷ 9 =

4 R4

Solve. 3NS2.2, 4NS3.2 17

SHOPPING Paul is buying shirts. If Paul has $43, how many shirts can he purchase?

He can buy 5 shirts. He will have $3 left. 150

Chapter 5 Division

© Siede Preis/Getty Images

Copyright © by The McGraw-Hill Companies, Inc.

5

Lesson

5-7 Divide by 11 and 12

4NS3.2 Demonstrate an understanding of, and the ability to use, standard algorithms for multiplying a multidigit number by a two-digit number and for dividing a multidigit number by a one-digit number; use relationships between them to simplify computations and to check results.

KEY Concept When the divisor is a two-digit number, you must begin the division by looking at the first two digits in the dividend.

15 11 165 - 11 55 - 55 0

)PXNBOZ UJNFTXJMM HPJOUP

8 12 102 - 96 60 - 60 0

VOCABULARY

#FDBVTFJTHSFBUFS UIBOUIFmSTUUXPEJHJUT PGUIFEJWJEFOE MPPL BUUIFmSTUUISFFEJHJUT

Copyright © by The McGraw-Hill Companies, Inc.

(Lesson 5-1, p. 108)

dividend the number that is being divided 9 , 9 is the Example: In 3 dividend. (Lesson 5-1, p. 108) divisor the number by which the dividend is being divided Example: In 3 9 , 3 is the divisor. (Lesson 5-1, p. 108)

Memorize the division facts for 11 and 12.

Example 1

quotient the answer or result of a division problem

YOUR TURN!

Write the division modeled with this array.

Write the division modeled with this array.

1. How many rectangles are in the array? 36

1. How many rectangles are in the array? 55

5IFOVNCFS PGSPXTJT UIFEJWJTPS *OUIJT QSPCMFN UIFEJWJTPS JT

2. How many rows? 12 3. Count the number of columns. This is the quotient. 3

2. How many rows? 11 3. Count the number of columns. This is the quotient.

5

DPMVNOT RVPUJFOU

4. Write the division problem. 36 ÷ 3 = 12 5. Check. 3 × 12 = 36

5IFOVNCFS PGSPXTJT UIFEJWJTPS *OUIJT QSPCMFN UIFEJWJTPS JT

4. Write the division problem.

5

columns = quotient

55 ÷ 5 = 11

5. Check.

5 × 11 = 55

GO ON

Lesson 5-7 Divide by 11 and 12

151

Example 2

YOUR TURN! Find 220 ÷ 11.

Find 480 ÷ 12. 1. Rewrite the problem in vertical form. 2. Look at the first digit. Because the divisor is greater than the first digit, look at the first two digits. What number multiplied by 12 is 48? 4

480 12

4 12 480

2. Look at the first digit. Because the divisor is greater than the first digit, look at the first two digits. What number multiplied by 11 is 22? 2

40 12 480

3. Since 0 < 11, there is not enough to divide. So, put 0 in the ones place.

Find 74 ÷ 11.

Sinclair

Marcos

11 × 7 = 77

11 × 66 = 6

77 - 74 = 3

74 - 66 = 8

7 R3

6 R8

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

152

Chapter 5 Division

20 11 220

Copyright © by The McGraw-Hill Companies, Inc.

Who is Correct?

15 R8 74 11 -11 63 -55 08

2 11 220

4. The quotient is 20. Check. 20 × 11 = 220

4. The quotient is 40. Check. 40 × 12 = 480

Branden

220 11

Now use mental math. 22 - 22 = 0

Multiply 4 times 12. Now use mental math. 48 - 48 = 0 3. Since 0 < 12, there is not enough to divide. So, put 0 in the ones place.

1. Rewrite the problem in vertical form.

Guided Practice Write the division problem represented by the model. 1

60 ÷ 12 = 5

2

22 ÷ 11 = 2

Step by Step Practice 3

Find 12 612 .

Copyright © by The McGraw-Hill Companies, Inc.

Step 1 Look at the first digit in the dividend. Since the divisor is greater, look at the first two digits in the dividend. What number multiplied by 12 is close to 61 without going over? 5

5

12 612

5

Step 2 Multiply by the divisor. Write the product under the dividend.

12 612

-60 12

Step 3 Subtract. Bring down the 2 in the dividend. Step 4 What number multiplied by 12 is close to without going over? 1 Step 5 Subtract. Step 6 The quotient is

51 .

Check.

51

12

51

12 612

-60 12 -12 0

× 12 = 612 GO ON Lesson 5-7 Divide by 11 and 12

153

Find each quotient. 4

11 979

Check.

89

× 11 = 979

89 11 979 5

12 288

24

6

11 187

17

7

12 180

15

8

11 253

23

Step by Step Problem-Solving Practice Solve. 9

FITNESS Harold goes to the gym at 6 a.m. to work out every four days. His brother Kevin also goes to the gym at 6 a.m. to work out, but every three days. On Friday, February 1, both Harold and Kevin were at the gym. When will they both work out on the same day again? Understand

Draw a diagram. ✓ Use logical reasoning. Guess and check. Solve a simpler problem. Work backward.

Read the problem. Write what you know. Harold works out every Kevin works out every

Plan

Problem-Solving Strategies

4 3

days. days.

Pick a strategy. One strategy is to use logical reasoning.

Solve

First, list the multiples of each number. Multiples of 3: 3, 6, 9, 12, 15, . . . Multiples of 4: 4, 8, 12, 16, 20, . . . Find the first number common to both lists. That is the number of days after Friday, February 1, that the brothers will be at the gym at the same time. The brothers will both work out on Wednesday, February 13 .

Check

154

You can use a calendar to find the days each brother works out to identify the next common day.

Chapter 5 Division

Copyright © by The McGraw-Hill Companies, Inc.

Find the least common multiple of 3 and 4.

10

GARDENS The gardener charges $10 for a dozen roses. How much money will the gardener make selling the roses? Remember, a dozen equals 12.

The gardener has 11 dozen. He will make $110 selling the roses. Check off each step.

11

12

✔

Understand

✔

Plan

✔

Solve

✔

Check

BOOKS The principal needs to buy 144 books to give as rewards to the honor students. The books are packed 12 in a box. How many boxes should he buy? 12 Complete the four sections below for 60 ÷ 12 = 5. Write the fact family.

Draw an array to model the division fact.

See TWE margin. Draw circles and tally marks to model the division fact.

Copyright © by The McGraw-Hill Companies, Inc.

Write the fact in vertical and fraction forms.

Skills, Concepts, and Problem Solving Find each quotient. Show the remainder if there is one. 13

47 ÷ 11 =

4 R3

14

37 ÷ 12 =

3 R1

15

80 ÷ 11 =

7 R3

16

100 ÷ 12 =

8 R4

17

99 ÷ 11 =

9

18

96 ÷ 12 =

8

19

88 ÷ 11 =

8

20

72 ÷ 12 =

6

21

55 ÷ 11 =

5

22

48 ÷ 12 =

4

23

33 ÷ 11 =

3

24

120 ÷ 12 =

10 GO ON

Lesson 5-7 Divide by 11 and 12

155

Solve. 25

BUSINESS Elise sells the extra tomatoes from her garden each summer. If she sells a dozen tomatoes for $3, how much money does she make if she sells 108 tomatoes? $27

26

MODELS A teacher wants to buy markers for the entire fifth grade. If there are 103 students, and the boxes of markers hold 12 markers each, how many boxes should he get to have enough?

9 boxes

Vocabulary Check Write the vocabulary word that completes each sentence.

Division

27

is also called repeated subtraction.

28

A group of related facts using the same numbers is a(n) fact family .

29

Writing in Math Amanda used short division to work the problem below. What mistake did Amanda make? 0805 R3 9663 12 Check. 805 × 12 = 805 × 10 + 805 × 2 = 8,050 + 1,610 = 9,660

Amanda made a mistake checking her work. She forgot to add the remainder

Spiral Review Solve.

(Lesson 4-12, p. 81)

30

PUZZLES The sum of the digits of a two-digit number is 9. The number is a perfect square. Find the number. (There are two 36, 81 possible answers.)

31

PUZZLES The sum of the digits of a two-digit number is 13. 49 The number is a perfect square. Find the number.

Use an array to find the missing number that would make the equation true. (Lesson 4-3, p. 18) 32 156

2×

7

= 14

Chapter 5 Division

Copyright © by The McGraw-Hill Companies, Inc.

into her answer. 9,660 + 3 = 9,663, which is the correct answer.

Lesson

5-8 Long Division KEY Concept Before you start a long division problem, you should estimate so that you can check your quotient for reasonableness. To estimate, use the division facts for 1 through 10. 1,000 3 3,000

3NS2.2 Memorize to automaticity the multiplication table for numbers between 1 and 10. 4NS3.2 Demonstrate an understanding of, and the ability to use, standard algorithms for multiplying a multidigit number by a two-digit number and for dividing a multidigit number by a onedigit number; use relationships between them to simplify computations and to check results.

VOCABULARY

Round the dividend to the nearest 1,000 that is divisible by 3.

quotient the answer or result of a division problem

1,094 3 3,282 -3 028 -27 12 -12 0

(Lesson 5-1, p. 108)

dividend the number that is being divided 9 , 9 is the Example: In 3 dividend. (Lesson 5-1, p. 108) divisor the number by which the dividend is being divided Example: In 3 9 , 3 is the divisor. (Lesson 5-1, p. 108)

Copyright © by The McGraw-Hill Companies, Inc.

Estimate before you divide and compare your quotient to the estimate.

Example 1 Find 1,846 ÷ 2. Estimate first. 1. Estimate. 1,800 ÷ 2 = 900 2. Rewrite the problem in vertical form. 3. Look at the first digit. Since the divisor is greater than the first digit, look at the first two digits. What number multiplied by 2 is 18? 9 Multiply. Subtract. 4. What number multiplied by 2 is 4? 2 Multiply. Subtract.

2 1,846 9 2 1,846

92 2 1,846

5. What number multiplied by 2 is 6? 3 Multiply. Subtract.

923 2 1,846

6. What number multiplied by 2 is 2? 1 Multiply. Subtract.

9 23 2 1,846

7. The quotient is 923. Compare to the estimate for reasonableness.

GO ON

Lesson 5-8 Long Division

157

YOUR TURN! Find 2,613 ÷ 3. Estimate first. 1. Estimate. 3,000 ÷ 3 =

1,000

2. Rewrite the problem in vertical format. 3. Look at the first digit. Since the divisor is greater than the first digit, look at the first two digits. 4. Multiply. Subtract. Bring down the next number in the dividend. 5. Multiply. Subtract. Bring down the next number in the dividend.

871 3 2,613 -2 4 21 -21 03 -3 0

6. Multiply. Subtract. Bring down the next number in the dividend.

871 . 7. The quotient is Compare to the estimate for reasonableness.

Who is Correct? Find 125 ÷ 6. Show the remainder.

Rosa

Rayshaun

120 ÷ 6 = 20 020 5 6 12 12 05

020 5 6 12 12 05

20 R5

20 R1

112 5 12 6 6 65 60 5

6-5=1

112 R5

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

Guided Practice Estimate each quotient. Then divide and show the remainder. 1

182 ÷ 3

2

180 ÷ 3 = 60; 60 R2 3

5,486 ÷ 5

5,000 ÷ 5 = 1,000; 1,097 R1 158

Chapter 5 Division

316 ÷ 8

320 ÷ 8 = 40; 39 R4 4

8,583 ÷ 6

9,000 ÷ 6 = 1,500; 1,430 R3

Copyright © by The McGraw-Hill Companies, Inc.

McKenzie

Step by Step Practice Find each quotient. Estimate first. 5

9,234 ÷ 3 Step 1 Estimate. 9,000 ÷ 3 = 3,000 Step 2 Rewrite the problem in vertical form. Step 3 What number multiplied by 3 is 9? Multiply. Subtract.

3

3,078 9,234 3 -9 023 -21 24 -24 0

Step 4 Since 2 < 3, there is not enough to divide. So, put 0 in the hundreds place. Bring down the next number. Step 5 What number multiplied by 3 is close to 23? Multiply. Subtract.

7

8

Step 6 What number multiplied by 3 is 24? Multiply. Subtract.

Step 7 The quotient is 3,078 . Compare to the estimate for reasonableness.

Find each quotient. Estimate first. Show the remainder if there is one.

Copyright © by The McGraw-Hill Companies, Inc.

6

1,214 ÷ 2

7

1,000 ÷ 2 = 500; 607 8

1,080 ÷ 4

900 ÷ 3 = 300; 377 9

1,200 ÷ 4 = 300; 270 10

846 ÷ 6

994 ÷ 8

11

478 ÷ 11

500 ÷ 10 = 50; 43 R5

483 ÷ 7

700 ÷ 7 = 100; 69 13

900 ÷ 9 = 100; 124 R2 14

1,210 ÷ 5

1,000 ÷ 5 = 200; 242

600 ÷ 6 = 100; 141 12

1,131 ÷ 3

280 ÷ 9

300 ÷ 10 = 30; 31 R1 15

615 ÷ 12

600 ÷ 12 = 50; 51 R3 GO ON Lesson 5-8 Long Division

159

Step by Step Problem-Solving Practice

Problem-Solving Strategies Draw a diagram.

Solve. 16

PACKAGING The principal is buying a special pen for all of the 96 writing-club students. How many boxes does she need to buy? Understand

Read the problem. Write what you know. There are 96 writing-club students. Pens are sold by the dozen .

Plan

Pick a strategy. One strategy is to use logical reasoning.

Solve

A dozen means 12. Pens are sold in boxes of 12. For every box the principal buys, he will have pens for 12 students. This is a problem about equal grouping, which means it is a division problem.

✓ Use logical reasoning. Guess and Check. Solve a simpler problem. Work backward.

PACKAGING The pens are sold in boxes that contain one dozen.

Divide the total needed by the number of pens 96 ÷ 12 = 8 in a box. The principal needs to buy

boxes of pens.

BUSINESS Jae made 108 big cookies. He can get $10 for each dozen. How much money will he make selling the cookies?

108 ÷ 12 = 9; Jae has 9 dozen. He will make $90 selling the cookies. Check off each step.

160 Corbis

✔

Understand

✔

Plan

✔

Solve

✔

Check

Chapter 5 Division

Copyright © by The McGraw-Hill Companies, Inc.

You can use repeated subtraction to check your answer.

Check

17

8

18

CELEBRATIONS There will be 9,126 friends and relatives attending a graduation ceremony this weekend. If each graduate has 2 people attending the ceremony, how many students are graduating?

4,563 students Division is a method of repeated subtraction. Explain this sentence and illustrate it with an example.

19

See TWE margin.

Skills, Concepts, and Problem Solving Find each quotient. Show the remainder. 20

219 ÷ 4 =

22

1,655 ÷ 2 =

54 R3 827 R1

21

445 ÷ 7 =

23

3,140 ÷ 9 =

63 R4 348 R8

Find each quotient. Estimate first. Show the remainder if there is one. 24

1,768 ÷ 2

25

Copyright © by The McGraw-Hill Companies, Inc.

2,000 ÷ 2 = 1,000; 884

26

936 ÷ 4

3,000 ÷ 3 = 1,000; 1,012

27

800 ÷ 4 = 200; 234

28

1,896 ÷ 6

2,626 ÷ 8

2,700 ÷ 9 = 300; 328 R2

990 ÷ 5

1,000 ÷ 5 = 200; 198

29

1,800 ÷ 6 = 300; 316

30

3,036 ÷ 3

1,672 ÷ 8

1,600 ÷ 8 = 200; 209

31

3,578 ÷ 9

3,600 ÷ 90 = 400; 397 R5 GO ON Lesson 5-8 Long Division

161

Solve. GAMES A game that 85 students are playing needs teams with 12 students on each team. 32

How many complete teams will there be?

85 ÷ 12 = 7 R1; they will have 7 complete teams 33

How many more students would be needed to make another team?

They need 11 more students to make another team. Vocabulary Check Write the vocabulary word that completes each sentence.

divisor

34

A remainder must be less than the

.

35

When the division problem is written as a fraction, the dividend is the numerator.

36

Writing in Math Explain the first step in long division when the divisor is greater than the first digit of the dividend.

Write a zero in the quotient above the first digit in the dividend and then look at the first two digits of the dividend.

Find each quotient. Show the remainder. 37

87 ÷ 11 7 R10

Find each difference. 40

2 R5

79 ÷ 11

41

9,200 - 5,228 3,972

42

11,000 - 5,437 5,563

PUZZLES Write a four-digit even number using the digits 2, 3, 8, and 9 that rounds to 10,000. Use each digit once.

Chapter 5 Division

7 R2

39

(Lesson 2-4, p. 91)

Sample answer: 9,832

162

29 ÷ 12

(Lesson 3-7, p. 173)

6,700 - 3,698 3,002

Solve. 43

38

(Lesson 5-7, p. 151)

Copyright © by The McGraw-Hill Companies, Inc.

Spiral Review

Chapter

Progress Check 4

5

(Lessons 5-7 and 5-8)

Write the division problem represented by the model. 3NS2.2, 4NS3.2 1

33 ÷ 11 = 3

36 ÷ 12 = 3

2

Find each quotient. 3NS2.2, 4NS3.2

35 3

12 420

19 4

22

11 209

5

34

12 264

6

11 374

Find each quotient. Estimate first. Show the remainder if there is one. 4NS3.2 7

1,416 ÷ 12

8

1,200 ÷ 12 = 100; 118

Copyright © by The McGraw-Hill Companies, Inc.

9

3,949 ÷ 11

4,500 ÷ 9 = 500; 513 10

4,000 ÷ 10 = 400; 359 11

4,678 ÷ 8

4,800 ÷ 8 = 600; 584 R6

4,617 ÷ 9

6,096 ÷ 8

6,000 ÷ 10 = 600; 762 12

2,199 ÷ 9

1,800 ÷ 9 = 200; 244 R3

Solve. 3NS2.2, 4NS3.2 13

MONEY Danton has $3,210 in rolls of dimes. Each roll of dimes is worth $5. How many rolls of dimes does he have? 642

14

HOMEWORK During the school year, Ms. Little has homework checks that are worth 8 points each. In addition, a student can also earn up to 7 extra points for doing special work. Janine received the full 8 points for all of her homework checks. She has a total of 978 homework-check points. For how many homework checks has she received points? How many extra points does she have?

122 checks and 2 extra points Lesson 5-8 Long Division

163

Chapter

Study Guide

5

Vocabulary and Concept Check divisor, p. 108

Write the vocabulary word that completes each sentence.

inverse operation, p. 108 multiple , p. 129

1

quotient, p. 108 remainder, p. 129

multiple of a number is A(n) the product of that number and any whole number. Inverse operations

2

are opposite operations, which means the operations undo each other.

3

The number that follows the division sign in a division divisor . sentence is the

4

A number that is left after one whole number is divided remainder . by another is a(n)

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

dividend

5

6

24 = 3 ___

divisor 77 = 7 ___

8

11

quotient

7

7 56 8

Lesson Review

5-1

Model Division

(pp. 108–114)

8

12 ÷ 4

9

21 ÷ 7

Draw an array to model the expression 9 ÷ 3. 1. Identify the divisor in the expression. 3 This represents the number of rows. 2. Identify the dividend (the first number). 9 This is the total number of rectangles in the array.

Write each expression in two different formats. 10

55 ÷ 5

11

20 ÷ 2

164

_

5 55 55 5 20 2 20 2

_

Chapter 5 Study Guide

3. Continue making columns until there are 9 rectangles.

DPMVNOTRVPUJFOU

4. The number of columns in the array is the quotient. 3 5. Check your division by multiplying the quotient by the divisor. 3×3=9

Copyright © by The McGraw-Hill Companies, Inc.

Draw an array to model each expression.

Example 1

5-2

Divide by 0, 1, and 10

Find each quotient. 12

3÷1=

13

4 ÷ 0 = not possible

14

0÷8=

15

6÷6=

1

16

80 ÷ 10 =

8

17

120 ÷ 10 =

12

5-3

Divide by 2 and 5 (pp. 123–128)

Copyright © by The McGraw-Hill Companies, Inc.

Find each quotient.

19

18

38 ÷ 2 =

19

242 ÷ 2 = 121

20

215 ÷ 5 =

21

585 ÷ 5 = 117

5-4

43

Divide by 3 and 4 (pp. 129–135)

Find each quotient. Show the remainder. 22

19 ÷ 3

6 R1

23

26 ÷ 3

8 R2

24

35 ÷ 4

8 R3

Example 2 Find 350 ÷ 10.

3 0

(pp. 115–121)

1. What number is the divisor? 10 2. The decimal in 350 is after the 0. The divisor is 10, so move the decimal point 1 place to the left in the dividend. 3. Write the quotient. 350 ÷ 10 = 35 4. Check. 35 × 10 = 350

Example 3 Find 75 ÷ 5. 1. Estimate. 80 ÷ 5 = 16 2. Look at the first digit in the dividend. What number multiplied by 5 is close to 7? 1 3. What number multiplied by 5 is 25? 5 4. The quotient is 15. Compare to the estimate for reasonableness.

1 5 75 -5 25 15 5 75 -5 25 -25 0

Example 4 Find 14 ÷ 4. Show the remainder. 1. What number multiplied by 4 is close to 14, but not over 14? 4 × 3 = 12 This is close without going over.

3 4 14 -12 2

2. Since there are no more digits to bring down, the answer is 3 R2. 3. Check your answer. 3 × 4 + 2 = 14

25

41 ÷ 4

10 R1 Chapter 5 Study Guide

165

5-5

Divide by 6 and 7 (pp. 137–142)

Find each quotient. Show any remainders.

Example 5

26

339 ÷ 6 =

56 R3

Find 288 ÷ 6.

27

578 ÷ 7 =

82 R4

1. Estimate. 300 ÷ 6 = 50

28

29

522 ÷ 6 = 658 ÷ 7 =

87

2. Look at the first two digits. What number multiplied by 6 is close to 28? 4

94

4 × 6 is 24. Put the 4 in the tens place in the quotient. Bring down next digit. 3. What number multiplied by 6 is 48? 8 Complete the division. 4. The quotient is 48. Compare to the estimate for reasonableness.

5-6

Divide by 8 and 9

48 6 288 -24 48 -48 0

(pp. 143–149)

Write the division problem represented by the model. 30

4 6 288 -24 48

54 ÷ 9 = 6

Example 6

5IFEJWJTPS FRVBMTUIF OVNCFSPG SPXT

Find each quotient. Use short division. 31

280 ÷ 8 =

35

32

252 ÷ 9 =

28

33

376 ÷ 8 =

47

DPMVNOTRVPUJFOU

1. How many rectangles are in the array? 56 2. How many rows? 8 4. Count the number of columns. This is the quotient. 7 5. Write the division problem. 56 ÷ 8 = 7

34 166

1,053 ÷ 9 = 117 Chapter 5 Study Guide

6. Check. 8 × 7 = 56

Copyright © by The McGraw-Hill Companies, Inc.

Write the division problem represented by the model.

5-7

Divide by 11 and 12 (pp. 151–156)

Find 704 ÷ 11.

Find each quotient.

56 35

1. Look at the first two digits. What number multiplied by 11 is close to 70? Multiply. Subtract.

12 672

72 36

2. What number multiplied by 11 is 44? 4

11 792

3. The quotient is 64.

98 37

Example 7

64 11 704 -66 44 -44 0

Check. 64 × 11 = 704

12 1,176

38 38

11 418

5-8

Long Division

(pp. 157–162)

Find each quotient. Show the remainder. 39

1,668 ÷ 7

Copyright © by The McGraw-Hill Companies, Inc.

1,400 ÷ 7 = 200; 238 R2 40

1,891 ÷ 4

2,000 ÷ 4 = 500; 472 R3 41

557 ÷ 6

600 ÷ 6 = 100; 92 R5 42

778 ÷ 9

900 ÷ 9 = 100; 86 R4

6 11 704 -66 44

Example 8 Find 12,567 ÷ 8. Show the remainder. 1. Look at the first two digits. What number multiplied by 8 is close to 12? 8 × 1 = 12 1,570 Multiply. Subtract. 8 12,567 -8 2. What number multiplied by 8 is 45 close to 45? 8 × 5 = 40 -40 Multiply. Subtract. 56 -56 3. What number multiplied by 8 is 07 56? 8 × 7 = 56 Multiply. Subtract. 4. Since 7 < 8, there is not enough to divide. So put 0 in the ones place. 5. The quotient is 1,570 R 7. 6. Check your answer. 8 × 1,570 + 7 = 12,567

Chapter 5 Study Guide

167

Chapter

Chapter Test

5

Write each expression in two different formats. 4NS3.2 1

7 21

21 ÷ 7

21 _

72 ÷ 8

2

7

8 72

72 _ 8

Write the division problem represented by the model. 3NS2.2, 3NS2.4 3

4

36 ÷ 6

108 ÷ 12 = 9 5

33 ÷ 3 = 11

6

200 ÷ 10 =

20

7

11 ÷ 1 =

11

8

0÷5=

9

450 ÷ 10 =

45

10

9÷3=

3

11

8 ÷ 0 = not possible

12

14 ÷ 7 =

13

63 ÷ 9 =

14

33 ÷ 3 =

11

15

0÷2=

16

20 ÷ 10 =

17

35 ÷ 5 =

7

18

17 ÷ 5 =

19

51 ÷ 6 =

20

33 ÷ 4 =

8 R1

2 0 3 R2

7 2 8 R3

0

GO ON 168

Chapter 5 Test

Copyright © by The McGraw-Hill Companies, Inc.

Find each quotient. Show the remainder if there is one. 3NS2.2, 3NS2.6, 4NS3.2

Find each quotient. Show the remainder if there is one. 3NS2.2, 3NS2.6, 4NS3.2 21

58 ÷ 8 =

24

120 ÷ 5 =

27

2,525 ÷ 3 =

7 R2

22

57 ÷ 9 =

24

25

264 ÷ 8 =

28

2,275 ÷ 4 =

841 R2

6 R3

23

35 ÷ 2 =

33

26

324 ÷ 6 =

29

13,473 ÷ 7 = 1,924 R5

568 R3

17 R1 54

Solve. 3NS2.2, 4NS4.1 SCHOOL The principal wants to give a ribbon to every student who competes in the spelling bee. Ribbons are packaged 10 to a bundle. 30

If all 468 students compete, how many bundles of ribbons will the principal need?

47 bundles 31

How many ribbons will be left over?

2 ribbons left

Copyright © by The McGraw-Hill Companies, Inc.

32

SCHOOL Award ribbon

LIBRARY The library has 672 books that need to be placed on 8 new shelves. If each shelf is to have the same number of books, how many books will be in each group?

84 FOOD The cafeteria needs a basket of 12 rolls per table for the banquet on Friday. They have baked 320 rolls. 33

How many baskets can be filled?

26 34

How many extra rolls will there be after the baskets are filled?

8 Correct the mistakes. 3NS2.2, 3NS2.6, 4NS4.1 35

Orlando said that all fact families for multiplication and division have four equations. Give an example of a fact family with four equations. Give an example of a fact family with two equations that shows that Orlando’s statement is false.

Sample answer: 5 × 4 = 20 4 × 5 = 20 20 ÷ 4 = 5 20 ÷ 5 = 4 3×0=0 0×3=0 0÷3=0 6 × 6 = 36 36 ÷ 6 = 6 Chapter 5 Test

Getty Images

169

Chapter

5

Standards Practice

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

Which equation can be used to check 10 × 4 = 40? 3NS2.2 A 40 ÷ 8 = 5

C 4 × 4 = 16

B 10 × 10 = 100

D 40 ÷ 10 = 4

5

Ms. Wantobi is dividing counters equally into 9 jars for a math project. If she has a total of 765 counters, how many will go in each jar? 3NS2.4 A 73 counters B 84 counters

2

C 85 counters

Chandani wants to share her music CD collection with her sister. If Chandani has 150 CDs and gives her sister half of her collection, how many CDs will each sister have? 3NS2.4 F 35 CDs

H 75 CDs

G 50 CDs

J 95 CDs

D 91 counters

6

The music teacher wants to divide the band students into 12 equal groups. If there are 132 total students in band, which grouping will work? 4NS3.2 F 12 groups of 12 students

3

Jeans are on sale this week. How much would one pair of jeans cost during this sale?

G 9 groups of 15 students H 12 groups of 11 students

4"-&

+EANS FOR

J 11 groups of 10 students

7

A $16

C $21

B $18

D $162

The Lorenz family traveled 3,840 miles on their summer vacation. If they traveled the same number of miles per day and finished the trip in 8 days, how many miles per day did they travel? A 460 miles

4

Which symbol makes this math sentence true? 3NS2.4 1,113 ÷ 7

1,068 ÷ 6

F >

H <

G =

J –

3NS2.4

B 480 miles C 510 miles D 530 miles

GO ON 170

Chapter 5 Standards Practice

Copyright © by The McGraw-Hill Companies, Inc.

3NS2.4

8

Karlie reads about 40 pages per hour. If she finishes her book in 9 hours, about how many pages are in this book? H 360 pages

1

A

B

C

D

G 270 pages

J 540 pages

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

Estimate the sum by rounding to the nearest thousand. 4NS1.3 6,398 + 4,601

Copyright © by The McGraw-Hill Companies, Inc.

10

11

Directions: Fill in the circle of each correct answer.

F 180 pages

3NS2.4 9

ANSWER SHEET

A 10,000

C 11,000

8

F

G

H

J

B 10,900

D 11,900

9

A

B

C

D

10

F

G

H

J

11

A

B

C

D

Which number has a 0 in the tenthousands place? 3NS1.3 F 1,350,944

H 7,409,531

G 6,117,038

J 8,026,472

Success Strategy Read the entire question before looking at the answer choices. Watch for words like not that change the whole question.

What are the factors of the number shown in the model? 4NS4.1

A 2 × 10

C 3×5×7

B 2×3×5×7

D 2×5×5×7

Chapter 5 Standards Practice

171

Chapter

6

Integers Temperatures soar and drop in California. The record high in July is 134°F, while the record low in January is -45°F. We use integers to show temperatures above and below 0.

Copyright © by The McGraw-Hill Companies, Inc.

172

Chapter 6 Integers

Cyril Mazansky/UNEP/Peter Arnold, Inc.

STEP

STEP

1 Quiz

Are you ready for Chapter 6? Take the Online Readiness Quiz at ca.mathtriumphs.com to find out.

2 Preview

Get ready for Chapter 6. Review these skills and compare them with what you’ll learn in this chapter.

What You Know

What You Will Learn

You know whole numbers are zero and the counting numbers. Whole numbers that are less than a number are to the left of the number on the number line. Whole numbers that are greater than a number are to the right of the number on the number line.

-FTTUIBO

(SFBUFSUIBO

Copyright © by The McGraw-Hill Companies, Inc.

You know how to add whole numbers.

Lesson 6-1 Integers are whole numbers and their opposites. (...-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 …) Opposites are numbers the same distance from zero but in the opposite direction. For example, the opposite of 5 is -5.

UPUIFMFGUPG

UPUIFSJHIUPG

Lessons 6-2 and 6-4

To add positive numbers on a number line, begin at the first number. Move right the same number of spaces as the second number.

To add negative numbers on a number line, begin at the first number. Move to the left the same number of spaces as the second number.

TRY IT!

Example: -2 + (-3)

12

5+7=

5IFTVNJT #FHJOBUBOEHP MFGUQMBDFT

#FHJOBUBOEHPSJHIUTQBDFT

5IFTVNJT

173

Lesson

6-1 Model Integers KEY Concept Whole numbers are zero and the counting numbers. Opposites are numbers the same distance from zero but in the opposite direction. For example, the opposite of 4 is -4.

UPUIFMFGUPG

UPUIFSJHIUPG

JTSFBEiOFHBUJWFwOPU iNJOVTwi.JOVTwJOEJDBUFT UIFPQFSBUJPOPGTVCUSBDUJPO BOEiOFHBUJWFwJOEJDBUFTB OVNCFSMFTTUIBO

Integers are whole numbers and their opposites. …-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, … Positive numbers are numbers that are greater than zero, and negative numbers are numbers that are less than zero.

The number zero is neither positive nor negative.

4NS1.8 Use concepts of negative numbers. 5NS1.5 Identify and represent on a number line decimals, fractions, mixed numbers and positive and negative integers.

VOCABULARY integers the whole numbers and their opposites Example: …–3, –2, –1, 0, 1, 2, 3, … even number a number that can be divided by 2 Example: 2, 8, 14, 36 (Lesson 1-4, p. 26)

odd number a number that can not be evenly divided by 2; such a number has 1, 3, 5, 7, or 9 in the ones place (Lesson 1-4, p. 26)

Example 1 Graph the integers 4, -3, 0, -5, and 1 on a number line. Then write them in order from least to greatest.

negative number a number less than zero whole numbers the set of all counting numbers and zero (Lesson 1-2, p. 11)

MFTTFSOVNCFS

HSFBUFSOVNCFS

2. Write the graphed numbers in order as they appear from left to right. -5, -3, 0, 1, 4 YOUR TURN! Graph the integers 3, -2, 1, 5, and -1 on a number line. Then write them in order from least to greatest. 1. On the number line, place a dot at 3, -2, 1, 5, -1 .

2. The numbers in order from least to greatest are -2, -1, 1, 3, 5 . 174

Chapter 6 Integers

zero the number zero equals none; the number before 1 on the number line (Lesson 1-2, p. 11)

opposites two different numbers that are the same distance from 0 on a number line Example: 3 and –3

Copyright © by The McGraw-Hill Companies, Inc.

1. On the number line, place a dot at each of the numbers.

positive number a number that is greater than zero

Example 2 Use <, =, or > to compare -4 and 4.

1. Graph both numbers on the number line.

2. The number farther to the right is 4, so it is the greater number. 3. Write a comparison statement. Since –4 is less than 4, you need to use the less than symbol. -4 < 4 YOUR TURN! Use <, =, or > to compare 1 and -1.

1. Graph both numbers on the number line. 2. The number farther to the right is number. 3. Write a comparison statement. 1

1 >

, so it is the greater -1

Example 3 Write an integer to represent the sentence. “A shipwreck is 250 feet below sea level.”

Copyright © by The McGraw-Hill Companies, Inc.

1. Underline the key words. 2. Decide if the number is positive or negative. negative Imagine a number line that is vertical instead of horizontal. Sea level is “0.” Below sea level is negative. Above sea level is positive.

ĕ

3. Write the number. -250 YOUR TURN! Write an integer to represent the sentence. “A mountain climber is 375 feet above sea level.” 1. Underline key words. 2. Decide if the number is positive or negative. positive Imagine a number line that is vertical instead of horizontal. Sea level is 0. Below sea level is negative. Above sea level is positive. 3. Write the number.

375

ĕ

GO ON

Lesson 6-1 Model Integers

175

Who is Correct? Write -4, 3, 2, and -1 in order from least to greatest.

Mordecai -1, 2, 3, -4

Ashley

Maya

-1, -4, 2, 3

-4, -1, 2, 3

Circle correct answers. Cross out incorrect answers.

Guided Practice Graph the integers on a number line. Then write them in order from least to greatest. 5, -2, 1, 4, -1

1

2

4, 3, 2, -5, -2

-2, -1, 1, 4, 5

-5, -2, 2, 3, 4

3

Use <, =, or > to compare -8 and -3. Step 1 Graph both numbers on the number line. Step 2 What number is farther to the right? -3 Step 3 Write a comparison statement. -8 < -3

176

Chapter 6 Integers

Copyright © by The McGraw-Hill Companies, Inc.

Step by Step Practice

Write <, =, or > in each circle to make a true statement. 4

-1 < 0

5

-2 > -4

6

3 < 4

7

-6 > -7

8

-3 < 1

9

5 > -1

Step by Step Problem-Solving Practice Solve. 10

WEATHER The temperature in the morning was 5°F (Fahrenheit). The temperature at noon was 10°. By evening, the temperature was -5°. What was the highest temperature?

Copyright © by The McGraw-Hill Companies, Inc.

Understand

Read the problem. Write what you know. The temperature began at 5°F . Then it was 10°F . By evening, the temperature was -5°F .

Plan

Pick a strategy. One strategy is to draw a diagram. Make a line to represent a thermometer. Mark the 0. Then mark it in 5-degree increments.

Solve

Begin at 5°F. Then mark 10°F and -5° F. The highest temperature was 10°F .

Check

Problem-Solving Strategies ✓ Draw a diagram. Use logical reasoning. Make a table. Solve a simpler problem. Work backward.

"WFSUJDBMOVNCFSMJOF JTPGUFOVTFEXIFONFBTVSJOH UFNQFSBUVSF8IFOUIJTIBQQFOT UIFQPTJUJWFOVNCFSTHPVQ BOE UIFOFHBUJWFOVNCFSTHPEPXO

¡ ¡ ¡ ¡

HSFBUFSOVNCFS MFTTFSOVNCFS

Does the answer make sense? Look over your solution. Did you answer the question?

GO ON Lesson 6-1 Model Integers

177

11

SCUBA DIVING Jacob was scuba diving. He went 30 feet below the surface. What integer represents his depth?

Jacob would be 30 feet below sea level, or –30. Check off each step.

12

✔

Understand

✔

Plan

✔

Solve

✔

Check

MONEY Theresa owes her mother $68. What integer represents how much Theresa still owes her mother? -$68 How does graphing integers on a number line help in comparing them?

13

The order the integers are placed on a number line from left to right is the order of their value from least to greatest.

Skills, Concepts, and Problem Solving Graph the integers on a number line. Then write them in order from least to greatest. -9, 8, 2, -5, 1

15

17

19

-6, -3, -2, 4, 10 178

Chapter 6 Integers

-5, -3, 0, 2, 4

-3, 10, -2, 4, -6

-5, -3, -1, 0, 5

2, 0, -3, 4, -5

18

-4, -1, 5, 8, 9

-3, -1, 5, 0, -5

-9, -5, 1, 2, 8 16

9, -4, 5, -1, 8

7, -4, -5, 3, -2

-5, -4, -2, 3, 7

Copyright © by The McGraw-Hill Companies, Inc.

14

Write the integers from least to greatest. 20

-18, -10, 20, 14, -13 -18, -13, -10, 14, 20

21

-10, -15, 5, 25, -25 -25, -15, -10, 5, 25

22

68, -42, 91, -19, 35 -42, -19, 35, 68, 91

Write the integers from greatest to least. 23

-105, -106, 100, 50, -35 100, 50, -35, -105, -106

24

-805, -900, 500, 450, -350 500, 450, -350, -805, -900

25

244, -301, 187, -24, -256 -301, -256, -24, 187, 224

Use <, =, or > to compare each pair of numbers. 26

14 and -16

27

14 > -16 28

25 and -25

-9 <-7 29

Copyright © by The McGraw-Hill Companies, Inc.

25 > -25 30

0 and -6

0 > -6

-9 and -7

-98, and 99

-98 < 99 31

3 and 0

3>0

Solve. 32

WEATHER The temperature at noon was 18°F. What integer represents the temperature?

18°F 33

FINANCES You spend $25. What integer represents your money?

-$25 34

MUSIC Esteban downloaded sixteen songs to his MP3 player. What integer represents the songs?

16

GO ON Lesson 6-1 Model Integers

179

Vocabulary Check Write the vocabulary word that completes each sentence. 35

negative

A(n)

Integers

36

number is a number less than 0. are the whole numbers and their opposites.

37

Numbers that are the same distance from 0 on a number line are opposite integers.

38

Writing in Math Explain how to list integers from greatest to least.

Sample answer: Graph them on a number line and then write the integers as they appear from left to right.

Spiral Review Solve. 39

(Lesson 4-13, p. 89)

FASHION Danny is making belts to sell to local clothing stores. He sells them for $11 each. Last year he sold a total of 432 belts. How much money did he earn last year?

$4,752 40

BASKETBALL The high-school basketball team scored 75 points three games in a row. How many points total did they score?

41

GARDENS Rachel is buying seeds for her garden. She bought 25 packets of herb seeds. Each packet holds about 35 seeds. She also bought 18 packets of flower seeds. Each flower packet holds about 105 seeds. How many more flower seeds than herb seeds did Rachel buy?

1,015 flower seeds

Write expression or equation for each of the following.

(Lesson 1-3, p. 19)

42

5+4=9

equation

43

2 + 10

44

3+1-2

expression

45

3-2=5-4

180

Chapter 6 Integers

expression equation

Copyright © by The McGraw-Hill Companies, Inc.

225 points

Lesson

6-2 Add Integers

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. 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.

KEY Concept The answer to an addition problem is called the sum. To add positive numbers on a number line, begin at the first number. Move right the same number of spaces as the second number.

VOCABULARY

To add negative numbers on a number line, begin at the first number. Move left the same number of spaces as the second number.

Inverse Property of Addition for any number, the sum of that number and its opposite is zero

Example: -2 + (-3) 5IFTVNJT

(Lesson 3-1, p. 130)

#FHJOBUBOEHPMFGUTQBDFT

Commutative Property of Addition the order in which two numbers are added does not change the sum Example: 12 + 15 = 15 + 12 (Lesson 3-1, p. 130)

The sum of an integer and its opposite is always zero. This is the Inverse Property of Addition . 5 + (-5) = 0 or (-5) + 5 = 0

sum the answer or result of an addition problem (Lesson 3-1, p. 145)

Copyright © by The McGraw-Hill Companies, Inc.

The Commutative Property of Addition also applies when adding integers.

Example 1 Find the sum of -4 and 3. Use a number line. 1. Graph the first number. 2. From the first number, go right on the number line.

YOUR TURN! Find the sum of -2 and 5. Use the number line. 1. Graph the first number. 2. From the first number, go the number line.

right

on

3

3. You are at -1 on the number line.

3. You are at

4. Write the sum. -4 + 3 = -1

4. Write the sum. -2 + 5 =

on the number line.

3

Lesson 6-2 Add Integers

181

Example 2 Find the sum of -5 and 3. Use counters. 1. Use five negative counters to represent the first number.

ĕ

ĕ

ĕ

2. Use three positive counters to represent the second number.

3. A zero pair is made up of 1 negative and 1 positive counter. You can make 3 zero pairs.

ĕ

ĕ

ĕ

ĕ ĕ

ĕ ĕ

1MBDFUISFFZFMMPXDPVOUFST UPSFQSFTFOU 5IFO QMBDFmWFSFEDPVOUFST UPSFQSFTFOUĕ

1BJSUIFQPTJUJWFBOE OFHBUJWFDPVOUFST5IFO SFNPWFBMM[FSPQBJST

4. There are two negative counters left. 5. What is the sum? -2 YOUR TURN! Find the sum of -3 and 8. Use counters. 1. What type of counters will you use to represent the first number?

three negative counters 2. What type of counters will you use for the second number?

eight positive counters

ĕ

ĕ

3. How many zero pairs can you make?

3

9ĕ 9ĕ 9ĕ

9 9 9

five positive counters 5. What is the sum?

5

182

Chapter 6 Integers

Copyright © by The McGraw-Hill Companies, Inc.

4. What is left over?

ĕ

Example 3

YOUR TURN!

What is the opposite of -3? Use it to show the Inverse Property of Addition. 1. Graph the number. What number is the same distance from zero as -3? 3

What is the opposite of 2? Use it to show the Inverse Property of Addition. 1. Graph the number. What number is the same distance from zero as 2? -2

2. Write an example of the inverse property using the two numbers. -3

2. Write an example of the inverse property using the two numbers.

+3=0

"OFBTZXBZUPmOE UIFPQQPTJUFPGB OVNCFSJTUPTXJUDI UIFTJHO

-2 = 0

2+

Who is Correct? Find the sum of 7 and -3.

Roxy

Copyright © by The McGraw-Hill Companies, Inc.

7 + (-3) = 4

Sierra

Evan

7 + (-3) = -4

7 + (-3) = -10

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

Guided Practice Find each sum. Use counters. 1

3

3

4 + (–1) =

2

9

-10 + 7 =

-3

9 9 9 9 9 9 9 9 9 9 9 9 9 9

8 + (–5) =

3

9

4

-9

-6 + (–3) =

9 9 9 9 9

9 9 9 9 9

GO ON

Lesson 6-2 Add Integers

183

Step by Step Practice 5

Find the sum of -6 and -3. Use the number line. Step 1 Graph the first number. Step 2 From the first number, move left .

Step 3 Where are you on the number line?

-9

Step 4 Write the sum.

-6 + (-3) = -9

Find each sum. Use the number line. 6

–6 + 5 =

-1

7

–3 + 0 =

-6

Start at

and move

8

5

places.

-3

right

–4 + (–2) =

-6

9

3 + (–3) =

0

10

2 + (–3) =

11

–6 + (–1) =

-7

12

–3 + (–4) =

13

5 + (–7) =

14

8 + (–9) =

-2

-1 -7 -1

What is the opposite of each number? Use it to show the Inverse Property of Addition. 15

14

17

-92

184

-14; 14 + (-14) = 0

16

-48

92; -92 + 92 = 0

18

5

Chapter 6 Integers

48; -48 + 48 = 0 -5; 5 + (-5) = 0

Copyright © by The McGraw-Hill Companies, Inc.

Find each sum.

Step by Step Problem-Solving Practice Solve. 19

20

Problem-Solving Strategies ✓ Draw a picture. Use logical reasoning. Make a table. Solve a simpler problem. Work backward.

HIKING Two friends start out on a day-long hike on a hill. They begin at an elevation of 500 feet. Their hike takes them to an altitude that is 200 feet lower than where they began. What is their current altitude? Understand

Read the problem. Write what you know. The hikers begin at an altitude of 500 ft . They walk down 200 ft feet in altitude.

Plan

Pick a strategy. One strategy is to draw a picture.

Solve

Begin at 500 feet and go down 200 feet. 500 + (–200) = 300 ft

Check

Does the answer make sense? Look over your solution. Did you answer the question?

IJLFCFHJOT GFFU IJLFFOET GFFU

GAMES Tammy puts 10 quarters into a video game. She uses $1 of her credit. How much credit does she have left?

$2.50 + (-$1.00) = $1.50

.

Copyright © by The McGraw-Hill Companies, Inc.

Check off each step.

21

✔

Understand

✔

Plan

✔

Solve

✔

Check

HEALTH Julio had a fever of 102ºF. At the end of the day, his temperature had gone down 2 degrees. What was his temperature at the end of the day?

100°F

22

Give examples of the Inverse Property and Commutative Property of Addition. Explain each property.

See TWE margin. GO ON Lesson 6-2 Add Integers

185

Skills, Concepts, and Problem Solving Find each sum. Use counters. 23

-1

5 + (–6) =

ĕ

24

ĕ

ĕ

ĕ

ĕ

9 9 9 9 9

9ĕ 9ĕ ĕ ĕ 9 ĕ 9ĕ 9

ĕ

2

–2 + 4 =

ĕ

ĕ

9ĕ 9ĕ

9

9

Find each sum. Use the number line. 25

-4

–7 + 3 =

26

27

-3

2 + (–5) =

-5

–2 + (–3) =

28

-4

–3 + (–1) =

What is the opposite of each number? Use it to show the Inverse Property of Addition. 6

31

-28

28; -28 + 28 = 0

30

–5

5; -5 + 5 = 0

32

37

-37; 37 + (-37) = 0

34

–64 + (–35) =

Find each sum.

-4

33

18 + (–22) =

35

–72 + 41 =

-31

36

–105 + 80 =

37

–55 + 15 =

-40

38

–32 + (–12) =

-99 -25

Solve. 39

TRAVEL Thirty–two people got on a bus. Ten people got off at the next stop. How many people were on the bus then?

32 + (–10) = 22; there were 22 people on the bus. 186

Chapter 6 Integers

-44

Copyright © by The McGraw-Hill Companies, Inc.

-6; 6 + (-6) = 0

29

40

BUILDINGS John was on the second floor. He got on the elevator and went up 4 floors. What floor was John on then?

John was on the sixth floor. Vocabulary Check Write the vocabulary word that completes each sentence. 41

The property that states that the order in which numbers are added does not affect the sum is the Commutative Property of Addition

42

For any number, zero plus that number is the number. This is the Identity Property of Addition .

43

Writing in Math Explain what is wrong with how the sum of -7 and -3 is found on the number line shown on the right.

.

The mistake is that the second number is a negative. You should go to the left to add a –3. The answer is –10, not –4.

Spiral Review

Copyright © by The McGraw-Hill Companies, Inc.

Write the numbers from least to greatest. 44

15, –5, –2, 12, -5, 2, 12, 15

45

89, –73, –99, 52 -99, -73, 52, 89

Solve. 46

(Lesson 6-1, p. 174)

(Lesson 5-2, p. 115)

GRAPHING While making a scale for her graph, Sarah wants each side of each square to represent 10 years. If she plots a point that represents 70 years, how many squares high will it be?

70 ÷ 10 = 7

47

FOOD The cafeteria manager wants to ensure there is one piece of pizza for every student. If he cuts each pizza into 10 slices, how many whole pizzas will he need to have enough for 114 students?

12; because 11 would not be enough.

42

PACKAGING Cinnamon rolls come in packages of 7 or 8. What combination of packages is needed for exactly 30 cinnamon rolls?

2 of each package

Lesson 6-2 Add Integers

187

Chapter

6

Progress Check 1

(Lessons 6-1 and 6-2)

Graph the integers on a number line. Then write them in order from least to greatest. 4NS1.8, 5NS1.5 1

-8, -2, 3, -3 -8, -3, -2, 3

2

6, 2, -5, 0

-5, 0, 2, 6

Write <, =, or > in each circle to make a true statement. 4NS1.8 3

-6 < -4

5 > -5

4

0 > -1

5

Find each sum. Use the number line. 5NS2.1, 7NS1.2, 6NS2.3 6

-3 + (–6) =

-9

8

-5 + 3 =

6

8 + (-2) =

7

-2

Find each sum. Use counters. 5NS2.1 9

6 + (-7) =

-1

10

ĕ

ĕ

ĕ

ĕ

ĕ ĕ

ĕ

ĕ

ĕ

ĕ ĕ

What is the opposite of each number? Use it to show the Inverse Property of Addition. 5NS2.1 11

-10 10; (-10) + 10 = 0

12

-8

8; (-8) + 8 = 0

13

6 (-6); 6 + (-6) = 0

Solve. 4NS1.8, 5NS2.1, 6NS2.3 14

FOOTBALL Terry’s team was at their own 20-yard line during a football game. They lost 15 yards. What yard line were they on for the next play? 5-yard line

15

HEALTH Every 10 steps burns 5 Calories. Every apple adds 35 Calories. If Paulina ate 1 apple and took 10 steps, how many calories would she have gained or lost? gained 30 Calories or lost -30 calories

188

Chapter 6 Integers

Copyright © by The McGraw-Hill Companies, Inc.

9ĕ 9ĕ 9 9 ĕ ĕ 9 9 9 9ĕ 9 9 9ĕ 9ĕ

-5

-2 + (-3) =

Lesson

6-3 Subtract Integers KEY Concept Subtraction is defined as adding the opposite of a number. The first step in subtracting integers is to rewrite the subtraction expression as an addition expression. 2 - 5 can be written as the addition expression 2 + (-5). You can use the number line to show the sum.

4UBSUBUUIFmSTUOVNCFS 4JODFUIFOFYUOVNCFSJTOFHBUJWF NPWFUPUIFMFGUQMBDFT5IFTVNJTĕ

PQQPTJUF ĕĕ

ĕ ĕ TBNFSFTVMU

The absolute value of a number is the distance the number is from zero. All absolute values are positive numbers or zero. The symbol for absolute value of the number x is ⎪x⎥. ]]BOE]ĕ]

Copyright © by The McGraw-Hill Companies, Inc.

An easy way to think of absolute value of a negative number is to drop the negative sign. Finding the absolute value of numbers can help you when adding and subtracting integers. Subtracting Integers Subtraction

Rewritten as Addition

Signs

Answer

-3 - 5

-3 + (-5 )

same

-8

3-5

3 + (-5 )

different

-2

-3 - (-5)

-3 + 5

different

2

3 - (-5)

3+5

same

8

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. 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. 7NS1.2 Add, subtract, multiply and divide rational numbers and take positive rational numbers to wholenumber powers.

VOCABULARY absolute value the distance between a number and 0 on a number line Example: The absolute value of both -5 and 5 is 5. subtraction an operation on two numbers that tells the difference when some or all are taken away; subtraction is also used to compare two numbers (Lesson 3-5, p. 159)

opposites two different numbers that are the same distance from 0 on a number line Example: 3 and -3 (Lesson 6-1, p. 174)

GO ON Lesson 6-3 Subtract Integers

189

Rewrite subtraction problems as adding the opposite. The rules given above for determining the sign of an answer are used for both addition and subtraction problems.

Example 1 Find the difference of -4 and -3. Use the number line. 1. Write the subtraction expression. -4 - (-3) 2. To subtract integers, add the opposite. Write the addition expression. -4 + 3 This is the new expression.

3. Graph the first number. 4. The sign of the second integer is positive. Move right on the number line. 5. You are at -1 on the number line. 6. Write the difference. -4 - (-3) = -1 YOUR TURN! Find the difference of -1 and 2. Use the number line. 1. Write the subtraction expression.

-1 - 2

3. Graph the first number.

negative 4. The sign of the second integer is left Move on the number line. -3 on the number line. -1 - 2 = -3 6. Write the difference. 5. You are at

190

Chapter 6 Integers

.

Copyright © by The McGraw-Hill Companies, Inc.

2. To subtract integers, add the opposite. -1 + (- 2) Write the addition expression. This is the new expression.

Example 2 Find the difference of -5 and -2. Use counters. 1. Write the subtraction expression. -5 - (-2) 2. To subtract integers, add the opposite. Write the addition expression. -5 + 2 This is the new expression. 3. Use five negative counters to represent the first number. 4. Use two negative counters to represent the second number. 5. You can make 2 zero pairs.

1MBDFmWFSFEDPVOUFST UPSFQSFTFOUĕ 5IFOQMBDFUXPZFMMPX DPVOUFSTUPSFQSFTFOU BEEJOHUXP

ĕ ĕ

ĕ

6. There are three negative counters left.

ĕ

7. Write the difference. -5 - (-2) = -3

ĕ

YOUR TURN! Find the difference of -2 and -3. Use counters. 1. Write the subtraction expression.

-2 - (-3)

2. To subtract integers, add the opposite. -2 + 3 Write the addition expression. This is the new expression. 3. What type of counters will you use for the first number? Copyright © by The McGraw-Hill Companies, Inc.

two negative counters

ĕ

4. What type of counters will you use for the second number?

ĕ

three positive counters 5. How many zero pairs can you make? 6. What is left over? 7. Write the difference?

1 positive counter

2

9ĕ

9ĕ

9 9

-2 - (-3) = 1

GO ON Lesson 6-3 Subtract Integers

191

Example 3

YOUR TURN! Which number has the greater absolute value?

Which number has the greater absolute value?

5 or -7

-6 or 3

5 units

1. How far is 5 from 0? So, ⎥ 5⎥=5

1. -6 is 6 units from 0. So, ⎥-6⎥=6 2. 3 is 3 units from 0. So, ⎥ 3⎥=3

2. How far is –7 from 0? So, ⎥-7⎥=7

3. Which integer has the greater absolute value? -6

7 units

3. Which integer has the greater absolute -7 value?

VOJUT VOJUT

VOJUT VOJUT

Who is Correct? Find the difference of -1 and 5. Use counters.

Tammy

Traci

ĕ

ĕ

ĕ

ĕ

ĕ

ĕ

ĕ

ĕ ĕ

ĕ

ĕ

ĕ

ĕ

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

Guided Practice Find each difference. Use the number line. 1

7 - (-5) =

12

192

Chapter 6 Integers

2

-2 - (-5) =

3

Copyright © by The McGraw-Hill Companies, Inc.

ĕ

Brice

Step by Step Practice 3

Find the difference of -6 and -3. Use the number line.

-6 - (-3)

Step 1 Write the subtraction expression.

Step 2 To subtract integers, add the opposite. Write the addition expression. -6 + 3 This is the new expression. Step 3 Graph the first number.

positive

Step 4 The sign of the second integer is Which direction will you go on the number line? right

.

Step 5 Where are you on the number line? -3 Step 6 Write the difference. -6 - (-3) = -3

Find each difference. Use the number line.

Copyright © by The McGraw-Hill Companies, Inc.

4

6 - (-4)

addition sentence: sum:

6 6

+ +

4 4

=

10

5

-3 - 1 =

-4

6

Find each difference. Use counters. 6 7 9-3= 9 9 9

-6 - (-7) =

1

8

-5 - (-4) = 9 9 9 9

-1

GO ON Lesson 6-3 Subtract Integers

193

Find each difference. 9

12 - (-7) =

19

10

22 - (-11) =

11

-15 - 16 =

-31

12

-13 - 21 =

-34

Step by Step Problem-Solving Practice

Problem-Solving Strategies Draw a diagram. Use logical reasoning. ✓ Make a table. Solve a simpler problem. Act it out.

Solve. 13

FINANCES Sue bought a stock for $7 a share. The first month the stock gained $2. The next month the stock lost $5. How much is her stock worth now? Understand

Plan

Pick a strategy. One strategy is to make a table.

Solve

Follow the changes in the stock’s value using the table. $7 + $2 + (-$5) = $4

Change ($)

$7 +2

$9

-5

$4

Check off each step.

✔

Understand

✔

Plan

✔

Solve

✔

Check

HIKING At the beginning of a week-long hike, Dewane was 200 feet below sea level. At the end of the week, he had reached a mountain peak at 1,000 feet above sea level. What is the difference in the altitudes?

1,200 ft 194

Chapter 6 Integers

Copyright © by The McGraw-Hill Companies, Inc.

WEATHER The low temperature one winter was 10 degrees below 0°F. In summer, the high temperature was 90°F. What is the change in temperature from winter to summer?

90 - (-10) = 90 + 10 = 100; there is 100 degree difference.

15

Stock value

You can use counters to check your answers.

Check

14

Read the problem. Write what you know. The stock was bought for $7 . The stock then gained $2 and lost $5 in value.

33

Is the statement true or false? “The absolute value of a number depends on its direction from zero.” Explain.

16

The statement is false. The absolute value depends on the distance from 0, not the direction, because distance is always a positive number.

Skills, Concepts, and Problem Solving Find each difference. Use the number line. 17

1

-10 - (-11) =

18

ĕ ĕĕ ĕ ĕ ĕ ĕ ĕ ĕ ĕ ĕ ĕ

-11

-6 - 5 =

ĕ ĕĕ ĕ ĕ ĕ ĕ ĕ ĕ ĕ ĕ ĕ

Find each difference. Use counters. 19

2

-4 - (-6) = 9 9 9 9

20

-9

-5 - 4 =

9 9 9 9

ĕ ĕ ĕ

ĕ

ĕ ĕ

ĕ

ĕ

ĕ

Copyright © by The McGraw-Hill Companies, Inc.

Which number has the greater absolute value? 21

-9 or 9

same

22

-8 or -12

-12

5 or -4

23

5

Find each difference. 24

-8 - (-2) =

26

-2 - 51 =

-6 -53

-76

25

-44 - 32 =

27

-110 - (-70) =

-40

Solve. 28

TRAVEL Cheyenne vacationed in California at 40 feet below sea level. When she returned to Colorado, she was 1,750 feet above sea level. What is the difference in elevations?

1,790 feet

29

NUTRITION Before exercising, Jan ate a granola bar that had 275 Calories. While exercising, Jan burned 525 Calories. What is her net gain of Calories?

-250 Calories

GO ON

Lesson 6-3 Subtract Integers

195

Vocabulary Check Write the vocabulary word that completes each sentence. 30

Subtraction is defined as adding the

31

absolute value The is from zero.

32

opposite

.

of a number is the distance the number

Writing in Math How are the words positive and negative used differently in math than in everyday situations?

See TWE margin.

Spiral Review What is the opposite of each number? Use it to show the Inverse Property of Addition. (Lesson 6-2, p. 181) 33

-18

18; 18 + (-18) = 0

34

4

35

-31

31; 31 + (-31) = 0

36

17

Solve.

-4; -4 + 4 = 0 -17; -17 + 17 = 0

(Lesson 5-3, p. 123)

BOOKS Together, Rasheeka and her friend read 18 books over the summer. If they each read the same number of books, how many books did they read? 9

38

FOOD Sheila’s manager asked her to take a survey of the customers in the pizza parlor. She was to ask customers at every other table to name their favorite pizza. If there were 16 tables with customers in the restaurant, how many tables did she approach?

8

Write an equation demonstrating the Commutative Property and draw rectangular arrays to model each product. (Lesson 4-1, p. 4) 39

196

3·7

7·3=3·7

Chapter 6 Integers

Copyright © by The McGraw-Hill Companies, Inc.

37

Lesson

6-4 Add and Subtract Groups of Integers KEY Concept To add and subtract integers, begin by rewriting each subtraction as addition. Then add from left to right. 5-6

+2

⎧ ⎨ ⎩

= 5 + (-6) + 2

You can also use mental math to work the problem.

= -1 + 2

5NS2.1 Add, sub tract, multiply and divide with decimals; add with negative integers, subtract positive integers from negative integers; and verify the reasonableness of the results. 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. 7NS1.2 Add, subtract, multiply and divide rational numbers and take positive rational numbers to wholenumber powers.

VOCABULARY

=1 When there are parentheses in a problem, perform the operations in the parentheses first. Then add or subtract from left to right. Follow the rules for subtraction and rewrite each subtraction expression as an addition expression.

Copyright © by The McGraw-Hill Companies, Inc.

Use the Associative Property of Addition to make solving problems easier. The order in which you group numbers does not change the answer. Group the Associative (3 + 6) + (-6) = 3 + (6 + (-6)) 6 and -6 Property of together. Addition =3+0

Inverse Property of Addition

By the inverse property, their sum is zero.

=3

Identity Property of Addition

The sum of 0 and a number is the number.

Associative Property of Addition the property that states that the way addends are grouped does not change the sum Example: (4 + 5) + 2 = 4 + (5 + 2) opposites two different numbers that are the same distance from 0 on a number line (Lesson 6-1, p. 174)

absolute value the distance between a number and 0 on a number line (Lesson 6-3, p. 189)

Example 1 Use the Associative Property of Addition to find the missing number. (4 + 13) + 7 = 4 + (

+ 7)

1. The numbers are in the same order. 2. Did the parentheses move to different numbers? yes 3. What is the missing number? 13 4. In order to simplify, what must be done first? 13 + 7

GO ON

Lesson 6-4 Add and Subtract Groups of Integers

197

YOUR TURN! Use the Associative Property of Addition to find the missing number. (2 + 15) + 5 = 2 + (

+ 5)

1. Are the numbers in the same order? yes 2. Did the parentheses move to different numbers? yes 3. What is the missing number? 4. What must be added first?

15

15 + 5

Example 2 Simplify 5 + (-6) - 7 + 3 - (-2). 1. Rewrite each subtraction as addition. 5 + (-6) -7 + 3 - (-2) = 5 + (-6) + (-7) + 3 + 2 2. Find the sum from left to right. = 5 + (-6) + (-7) + 3 + 2 = (-1) + (-7) + 3 + 2 =

(-8) + 3 + 2 (-5) + 2

=

(-3)

YOUR TURN! Simplify 4 + (-3) - 2 + 5 - (-1). 1. Rewrite each subtraction as addition. 4 + (-3) -2 + 5 - (-1) = 4 + (-3) + (-2) + 5 + 1 2. Find the sum from left to right. = 4 + (-3) + (-2) + 5 + 1

198

=

1 + (-2) + 5 + 1

=

(-1) + 5 + 1

=

4+1

=

5

Chapter 6 Integers

Copyright © by The McGraw-Hill Companies, Inc.

=

Example 3 Simplify 6 × 5 - (3 - (-4 )) + (-7). 1. Rewrite each subtraction as addition. 6 × 5 - (3 - (-4 )) + (-7) = 6 + 5 - (3 + 4) + (-7) 2. Simplify the expression in the parentheses. 6 + 5 - (3 + 4) + (-7) = 6 + 5 + 7 + (-7) 3. Simplify from left to right.

= 6 + 5 + 7 + (-7) =

11 + 7 + (-7)

=

18 + (-7)

=

11

YOUR TURN! Simplify 10 - 7 - (5 - (-6 )) + (-4). 1. Rewrite each subtraction as addition.

6

10 - 7 - (5 - (-6 )) + (-4) = 10 + ( -7 ) - (5 +

) + (-4)

2. Simplify the expression in the parentheses. 10 + ( -7 ) - (5 +

6

) + (-4) = 10 + ( -7 ) -

Copyright © by The McGraw-Hill Companies, Inc.

3. Simplify from left to right.

11

+ (-4)

= 10 + (-7) - 11 + (-4) =

3

- 11 + (-4)

-8 + (-4) -12

= =

Who is Correct? Simplify 9 + (-8) - 5 + 7 - (-3).

Allan = =

3) 9 + (-8) - 5 + 7 - (- 12 - 3 1 8

Petra

Kiele

3) 9 + (-8) - 5 + 7 - (3 + 7 = 9 + (-8) + (-5) + 3 + 7 + + (-5) 1 = 3 + 7 + -4 = 3 + 3 = 6 =

3) 9 + (-8) - 5 + 7 - ((-3) + 7 = 9 + (-8) + (-5) + (-3) + 7 + (-5) + 1 = (-3) + 7 + -4 = (-3) + 3 = 0 =

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

GO ON

Lesson 6-4 Add and Subtract Groups of Integers

199

Guided Practice Use the Associative Property of Addition to find the missing number. 1

(9 + 12) + 8 = 9 + ( ___ + 8)

12

2

(8 + 35) + 5 = 8 + ( ___ + 5)

35

Step by Step Practice 3

Simplify 6 + (-7) + 8 -4 -(-3). Step 1 Rewrite each subtraction as addition. 6 + (-7) + 8 - 4 - (- 3) =

6 + (-7) + 8 + (-4) + 3

Step 2 Find the sum from left to right. 6 + (-7) + 8 + (-4) + 3 =

6

Simplify. -24 + (-5) - (-71)

rewritten sentence: -24 + (-5) + 71 sum:

14

5

7 - (-4) - (-3)

7

11 - 5 - (13) + 5

9

18 - 2 + (9 + (-4 )) + (-9)

11

8 - 7 + (5 - 14) + 25

200

Chapter 6 Integers

-2

17

12

42 15

6

-1 + 5 - (-11)

8

-3 - (-52) - (-9) - 6

10

22 + 9 - (5 - (-2 )) + 12

12

56 - 9 + (9 - (-3 )) + (-8)

52

36

51

Copyright © by The McGraw-Hill Companies, Inc.

4

Step by Step Problem-Solving Practice

Problem-Solving Strategies ✓ Draw a diagram.

Solve. 13

WEATHER The temperature in the summer in Death Valley, California, is about 120°F during the day. In the evening, the temperature drops 35°. What is the temperature in the evening? Understand

Use logical reasoning. Guess and check. Solve a simpler problem. Use an equation.

Read the problem. Write what you know. The temperature is 120°F It then goes down 35° .

.

Pick a strategy. One strategy is to draw a diagram.

Plan

Make a number line to represent a thermometer. Use the number line to find the change in temperature.

Solve

'SPN¡' NPWFEPXO¡

Begin at 120°F and move down 35°. 120°F - 35° = 120°F + (-35°) = 85°F Does the answer make sense? Look over your solution. Did you answer the question?

Check

Copyright © by The McGraw-Hill Companies, Inc.

14

TRANSPORTATION A bus picked up 15 people at the first stop and 10 people at the second stop. Then, 5 people got off the bus at the third stop. Finally, 8 more people got off at the fourth stop. How many people are on the bus now?

15 + 10 - 5 - 8 = 12; there are 12 people still on the bus. Check off each step.

15

✔

Understand

✔

Plan

✔

Solve

✔

Check

FINANCES Elizabeth bought stock for $200. The first week, the stock fell $55. The second week, the stock rose $30. What is the stock worth now? Write an integer that represents the loss or gain.

$175; -$25

GO ON Lesson 6-4 Add and Subtract Groups of Integers

201

Explain how the Associative Property of Addition helps you to add integers.

16

Sample answer: It makes the math easier because you can group numbers together that are easier to add.

Skills, Concepts, and Problem Solving Use the Associative Property of Addition to find the missing number. 17

(15 + 18) + 12 = 15 + ( 18

+ 12)

18

11 + (14 + 13) = (11 + 14) +

13

19

(101 + 13) + 9= 101 + (13 + 9)

Simplify. 20

88 + (-12) - 44 + 36

68

21

-15 - (-11) + 100 + (-99)

22

111 + (-23) - 16 + 9

81

23

-27 + (-45) + (-77) - (-8) -141

24

9 + 3 - (-5) - 29 - (-1) -11

25

-14 + (-9) - (-81) - 17

26

(-23) + 17 - (-6)

27

11 - 35 + (-150) -174

28

-126 - (-13) + 4 -109

29

93 - (-5) + (-22)

30

106 - 115 + 10 - (-14) + (-2)

31

29 - (-6) - 141 + (-9) - (-2) -113

32

88 + 11 + (-58) + 2

33

152 - (-89) + -56 - 12 + (-65)

43

13

Solve. 34

ROLLER COASTER A roller coaster climbs a 129-foot hill and then descends 100 feet. If the roller coaster climbs back up 40 feet, what is its elevation? 69 feet

35

TRAVEL A crop-dusting plane flies at an altitude of 630 feet. It then descends 227 feet when it turns on the duster. When it is done, the plane ascends 180 feet. At what altitude is the plane flying then? 583 ft

202

Chapter 6 Integers

41

76

108

Copyright © by The McGraw-Hill Companies, Inc.

0

-3

36

HIKING Refer to the photo caption shown at the right. If the surface of a gorge is at 307 feet below sea level, how far below sea level is the base of the gorge? 1,360 ft

Vocabulary Check Write the vocabulary word that completes each sentence. 37

Associative Property of Addition The states that the grouping of the addends does not change the sum. Opposites

HIKING A backpacker hikes into a gorge that is 1,053 feet below the surface.

38

are two different integers that are the same distance from 0 on a number line.

39

Writing in Math Explain how absolute value helps you add integers. Give at least three examples.

See TWE margin.

Spiral Review Which number has the greater absolute value? 40

Copyright © by The McGraw-Hill Companies, Inc.

42

9 or 8

9

-6 or 6 Both are the same.

Find each quotient. Show the remainder.

(Lesson 6-3, p. 189)

41

-3 or -4

43

1 or -2

-4 -2

(Lesson 5-4, p. 129)

44

47 ÷ 5 =

9 R2

45

15 ÷ 2 =

46

17 ÷ 5 =

3 R2

47

63 ÷ 10 =

Solve.

7 R1 6 R3

(Lesson 4-4, p. 25)

48

FASHION Nina likes to coordinate her purse with her outfits. She buys most of her purses from a second-hand store that sells them for $5 each. If Nina has 9 purses she has purchased there, how much has she spent? $45

49

HORTICULTURE Claire has 5 flower bouquets with 6 stems each. Tara has 6 flower bouquets with 5 stems each. Who has more flower stems? They have the same number of flower stems.

50

INTERIOR DESIGN Thomas is tiling the floor in his kitchen. Each box holds 5 square feet of tile. If he has 8 boxes of tile, how many square feet can he cover? 40 sq ft Lesson 6-4 Add and Subtract Groups of Integers

age fotostock/SuperStock

203

Chapter

Progress Check 2

6

(Lessons 6-3 and 6-4)

Find each difference. Use the number line. 5NS2.1 1

2

-3 - (-5)

2

17

14 - (-3)

Find each difference. Use counters. 5NS2.1 3

-6 - (-4) ĕ

ĕ ĕ

ĕ ĕ

ĕ

-2

4

9ĕ 9ĕ 9ĕ

ĕ

ĕ

9 9 9 9

9ĕ

4

-1 - (-5) ĕ

9ĕ

9

Which number has the greater absolute value? 5NS2.1, 6NS2.3, 7NS1.2 5

16 or -3

7

-6 or 4

16

6

1 or 1 Both

are the same.

8

-5 or -9

-9

10

63 - 103 + (-117 - (-12)) + 3 -142

12

(89 - (-77)) + 17 - (-5) + 19 207

-6

Simplify. 5NS2.1, 6NS2.3

12

-5 + (-7) - (-111) - 87

11

(55 - (-13)) + 22 - (-8) + 12 110

Use the Associative Property of Addition to find the missing number. 5NS2.1, 6NS2.3 13

(170 + (-14)) + (-52) = 170 + (-14 + (-52))

14

-77 + (-6 + (-17)) = (-77 + (-6) ) + (-17)

15

(14 + (-93)) + 670 = 14 + (-93 + 670 )

16

(7 + (-13)) + 56 = 7 + ( (-13) ) + 56)

Solve. 5NS2.1, 6NS2.3, 7NS1.2 17

TRAVEL Matthew walked down the North Rim of the Grand Canyon until he was at an elevation of 1,325 meters. Then he hiked back up 477 meters. At what elevation is he?

1,590 meters 204 Corbis

Chapter 6 Integers

TRAVEL The North Rim of the Grand Canyon is about 2,438 meters above sea level.

Copyright © by The McGraw-Hill Companies, Inc.

9

Lesson

6-5 Multiply Integers KEY Concept When you multiply integers, the numbers that you are multiplying are called factors . The answer is the product . To find the product, multiply the absolute values of the numbers and then determine the correct sign of the answer. If the signs are the same, then the sign of the product is positive. 4 × 4 = 16 and -4 × (-4) = 16 If the signs are different, then the sign of the product is negative. 3 × (-2) = -6 and -3 × 2 = -6 Multiplication is repeated addition. It can be shown on a number line. -2 × 3 is (-2) + (-2) + (-2) or -6

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. 7NS1.2 Add, subtract, multiply and divide rational numbers and take positive rational numbers to whole-number powers. 3NS2.2 Memorize to automaticity the multiplication table for numbers between 1 and 10. 3NS2.6 Understand the special properties of 0 and 1 in multiplication and division.

VOCABULARY product the answer or result to a multiplication problem; it also refers to expressing a number as the product of its factors (Lesson 4-1, p. 4)

ĕ ĕĕ ĕ ĕ ĕ ĕ ĕ ĕ ĕ ĕ ĕ HSPVQTPGĕ

Copyright © by The McGraw-Hill Companies, Inc.

All of the multiplication properties for whole numbers apply to integers. The Zero Property of Multiplication states that any number times zero is zero. -5 × 0 = 0 The Identity Property of Multiplication states that any number times 1 equals that number. -5 × 1 = -5 The Commutative Property of Multiplication states that the order in which the numbers are multiplied does not matter.

factor a number that divides into a whole number evenly; also a number that is multiplied by another number (Lesson 4-1, p. 4)

multiplication an operation on two numbers to find their product; it can be thought of as repeated addition Example: 4 × 3 is the same as the sum of four 3s, which is 3 + 3 + 3 + 3, or 12. (Lesson 4-1, p. 4)

-2 × 5 = 5 × (-2) The Associative Property of Multiplication states that the product stays the same when you change the grouping of the numbers. (2 × (-3)) × 3 = 2 × (-3 × 3) Use the Distributive Property to simplify addition and multiplication problems. 8(10 + (-2)) = (8 · 10) + (8 · (-2))

GO ON Lesson 6-5 Multiply Integers

205

Example 1

YOUR TURN! Find 4 × (-3). Use a number line.

Find 3 × (-6). Use a number line. 1. Identify the first number in the expression. 3 This is the number of times the group is repeated.

2. Identify the second number in the -3 expression. This is the group size.

2. Identify the second number in the expression. -6 This is the group size. 3. Draw a number line. Mark off 3 groups of -6.

ĕ ĕ ĕ ĕ ĕ ĕ ĕ ĕ ĕ ĕ ĕ ĕ

1. What is the first number in the 4 expression? This is the number of times the group is repeated.

3. Draw a number line. Mark off 4 groups of -3.

ĕ ĕĕ ĕ ĕ ĕ ĕ ĕ ĕ ĕ ĕ ĕ HSPVQTPGĕ

HSPVQTPGĕ

4. The signs are different, so the product is negative. 5. Write the product. -18

Find 5 × (-4) by multiplying absolute values. 1. Find the absolute value of each. ⎪5⎥ = 5 and ⎪-4⎥ = 4 2. Multiply the absolute values of the numbers. 5 × 4 = 20 3. The signs are different, so the product is negative. 4. Write the product with the sign. -20

5. Write the product.

-12

YOUR TURN! Find 9 × (-3) by multiplying absolute values. 1. Find the absolute value of each.

⎪9⎥ = 9 and ⎪-3⎥ = 3 2. Multiply the absolute values of the numbers. 9 3 27 × = 3. Are the signs the same or different? different Will the product be positive or negative? negative 4. Write the product with the sign.

-27

206

Chapter 6 Integers

Copyright © by The McGraw-Hill Companies, Inc.

Example 2

4. Are the signs the same or different? different Will the product be positive or negative? negative

Example 3 Simplify. Name the property. (2 × (-5)) × 1 = 2 × (-5 × 1) Associative Property of Multiplication = 2 × (-5)

Identity Property of Multiplication

= -10 YOUR TURN! Simplify. Name the property. (-8 × 0) = (0 × (-8))

Commutative Property of Multiplication Zero Property of Multiplication

=0

Who is Correct? Find 3 × (-5).

Jessica 3 × (-5) = -2

Martha

Sherman

3 × (-5) = -15

3 × (-5) = 15

Copyright © by The McGraw-Hill Companies, Inc.

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

Guided Practice Find each product. Use a number line. 1

2 × (-4)

-8 ĕ ĕ ĕ ĕ ĕ ĕ ĕ ĕ ĕ ĕ HSPVQTPGĕ

2

3 × (-5) -15

ĕ ĕ ĕ ĕ ĕ ĕ ĕ ĕ ĕ ĕ ĕ ĕ ĕ ĕ ĕ ĕ

HSPVQTPGĕ

GO ON Lesson 6-5 Multiply Integers

207

Step by Step Practice 3

Find (-6) × (-4). Step 1 Find the absolute value of each. ⎥-6⎥ = 6 and ⎥-4⎥ = 4 Step 2 Multiply the absolute values of the numbers. 6 4 24 × =

same

Step 3 Are the signs the same or different?

Step 4 Will the product be positive or negative? positive Step 5 Write the product with the sign.

24

Step 6 Check the sign to make sure it is correct. (-) × (-) = (+)

Same signs mean the product will be positive. Find each product. 4

-9 × (-3) absolute value: ( 9 ) × ( 3 ) = ( 27 ) product: 27 sign: positive

5

4 × (-5) -20

6

6×2

12

7

-8 × 4 -32

Step by Step Problem-Solving Practice

Understand

Read the problem. Write what you know. It lost a value of $2 a month for 3 months.

Plan

Pick a strategy. One strategy is to draw a number line to represent the value of the stock.

Solve

Use the number line to find the change in value.

Check

208

Use logical reasoning. Guess and check. Solve a simpler problem. Work backward.

ĕ ĕ ĕ ĕ ĕ ĕ ĕ

Add the numbers in a different order or grouped differently. Your sum should match your answer.

Chapter 6 Integers

21

HSPVQTPGĕ

Copyright © by The McGraw-Hill Companies, Inc.

FINANCES Renee’s stock lost $2 each month for 3 months. How much has the stock lost in value?

Draw a number line. Mark off 3 groups of -2 The stock has lost $6 in value.

-3 × (-7)

Problem-Solving Strategies ✓ Draw a diagram.

Solve. 9

8

10

EXERCISE After finishing his workout, Tony’s heart rate decreased by 2 beats per minute for each of the next 4 minutes. How much did Tony’s heart rate drop in 4 minutes?

-2 × 4 = -8; Tony’s heart rate dropped 8 beats.

11

✔

Understand

✔

Plan

✔

Solve

✔

Check

FOOTBALL The football team lost 7 yards on 3 plays in a row. How many yards did they lose altogether?

-7 × 3 = -21; the team lost 21 yards running the 3 plays. Explain how you multiply integers. Give examples.

12

See TWE margin

Skills, Concepts, and Problem Solving

Copyright © by The McGraw-Hill Companies, Inc.

Find each product. Use a number line. 13

-7

7 × (-1)

14

-2 × 4

-8

HSPVQPG

HSPVQTPG

Find each product by multiplying absolute values.

-6

15

3 × (-2)

18

2 × (-7) -14

16

-9 × 1

19

3×8

-9 24

17

-8 × (-5)

40

20

-6 × (-5)

30

Find the missing number. Name the property. 21

(4 × 5) × (-5) = 4 × (

22

-9 ×

0

5

× (-5)) Associative Property of Multiplication

= 0 Zero Property of Multiplication

GO ON Lesson 6-5 Multiply Integers

209

Solve. 23

FINANCES Kaolin bought a stock for $10 a share. The stock lost $3 for each of the next 2 months. How much did each share of stock lose in value?

-3 × 2 = -6; the stock lost $6.

24

WEATHER The temperature was 68ºF in the evening. It dropped 3º every hour over an 8-hour period. What was the temperature in the morning?

8 × (-3) = -24; the temperature dropped 24º and 68ºF -24º = 44ºF. Vocabulary Check. Write the vocabulary word that completes each sentence. 25

The Identity Property of Multiplication states that when you multiply a number by 1, the product is the same as the given number.

Multiplication

26 27

can also be thought of as repeated addition.

Writing in Math Katie worked the following problem. What mistake did she make?

-8 × (-4) = -32

The mistake was made using the signs. The signs of the two numbers are the same, so the product should have a positive value.

Use the Associative Property of Addition to find the missing number. (Lesson 6-4, p. 197) 28

(34 + (-23)) + 45 = 34 + ( -23 + 45)

Find each product. 30

32

31

13 × 78 = 1,014

(Lesson 3-6, p. 165)

PHOTOS Karissa downloaded 46 pictures from her camera to her computer. She put 15 pictures in a folder named “Birthday Party.” How many pictures did not go in the “Birthday Party” folder?

31 210

-27 + (8 + 4) = (-27 + 8) +

(Lesson 4-10, p. 67)

93 × 12 = 1,116

Solve.

29

Chapter 6 Integers

4

Copyright © by The McGraw-Hill Companies, Inc.

Spiral Review

Lesson

6-6 Multiply Several Integers KEY Concept When you multiply several integers, you multiply from left to right. If there are an even number of negative integers, the product is a positive number. -4 × 3 × (-2) = 24

Two numbers have negative signs. 2 is an even number. The product is positive.

If there are an odd number of negative signs, the product is a negative number. -3 × 2 × (-1) × (-5) = -30

Three numbers have negative signs. 3 is an odd number. The product is negative.

Copyright © by The McGraw-Hill Companies, Inc.

This is the exponent.

54 = 5 × 5 × 5 × 5 = 625 (-5)4 = (-5) × (-5) × (-5) × (-5) = 625 (-2)3 = (-2) × (-2) × (-2) = -8

exponent the number of times a base is multiplied by itself Example: in 25, 5 is the exponent. order of operations rules that tell what order to use when evaluating an expression: (1) Simplify operations within parentheses (2) Simplify terms with exponents (3) Multiply and divide from left to right (4) Add or subtract from left to right even number a number that can be divided by 2

In an expression with an exponent, the sign depends on the number of negative integers. The order of operations is a set of rules that tell you which operation to perform first within expressions. 1. Simplify operations in parentheses. 2. Simplify terms with exponents. 3. Multiply and divide from left to right. 4. Add and subtract from left to right.

VOCABULARY

(Lesson 4-11, p. 75)

When an integer has an exponent , you multiply the integer by itself the number of times shown by the exponent. This is the base.

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. 7NS1.2 Add, subtract, multiply and divide rational numbers and take positive rational numbers to wholenumber powers.

(Lesson 1-4, p. 25)

odd number a number that can not be divided evenly by 2; such a number has 1, 3, 5, 7, or 9 in the ones place (Lesson 1-4, p. 25)

Always work from left to right when simplifying.

GO ON Lesson 6-6 Multiply Several Integers

211

Example 1 Simplify (-4) × (-1) × (-2) × 3. 1. There are 3 negative signs. Is that number even or odd? odd The sign of the product will be negative. 2. Multiply the absolute values of the numbers. 4 × 1 × 2 × 3 = 24 3. Write the product with the sign. -24

YOUR TURN! Simplify (-5) × 2 × (-1) × 6. 1. How many negative signs in the problem? 2 Is that number even or even What will be the sign of odd? the product? positive 2. Multiply the absolute values of the numbers. 5 × 2 × 1 × 6

60

3. Write the product with the sign.

Example 2 Find (-4)³. 1. The exponent is 3. 2. The base is -4. 3. Write the expression as a multiplication problem. (-4) × (-4) × (-4) 4. What is the sign of the product? negative

=

60

YOUR TURN! Find (–3)5. 1. What is the exponent? 2. What is the base?

5

-3

3. Write the expression as a multiplication problem. -3 × ( -3 ) × ( -3 ) × ( -3 ) × ( -3 ) 4. What will be the sign of the product?

negative

5. What is the product? -243

Example 3 Simplify 5 × (1 + (-5)) × 3. Simplify according to the order of operations. There are no exponents in the problem.

YOUR TURN! Simplify 22 × (4 - (-2)) × -6. Simplify according to the order of operations. Are there exponents in the problem? yes 22 × (4 - (-2)) × (-6)

5 × (1 + (-5)) × 3 =

5 × (-4) × 3

=

=

-20 × 3

=

=

-60

=

Do the operation within the parentheses. Do the multiplication from left to right. 212

Chapter 6 Integers

4

×

6

24

× (-6)

-144

× (-6)

Copyright © by The McGraw-Hill Companies, Inc.

5. The product is -64.

Who is Correct? Simplify -1 × (6 + (–3)) × -4.

Santiago

-1 × (6 + (-3)) × -4

Arthur

Betty -1 × (6 + (-3)) × -4

-1 × (6 + (-3)) × -4

= -6 + 12

= -3 × -4

= -3 × -4

=6

= 12

= -12

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

Guided Practice Find each product. 1

32

3

×

3

=

9

3

25

2

×

2

×

2

2

×

2

×

2

4

43 =

×

4

×

4

=

64

32

Step by Step Practice

Copyright © by The McGraw-Hill Companies, Inc.

4

Simplify -3 × 4 × (-2) × (-3). Step 1 How many negative signs are in the expression? Step 2 Is that number even or odd?

3

odd

Step 3 What will be the sign of the product? negative Step 4 Multiply the absolute values of the numbers. 3 × 4 × 2 × 3 = 72 Step 5 Write the product with the sign. -72

Simplify. 5

-8 × 2 × (-5) × 6 = 480

6

2 × 3 × (-5) = -30

7

-4 × (-3) × (-7) × (-2) = 168

8

3 × 2 × 6 × (-10) = -360

9

4 × (-1) × (-2) × (-6) = -48

Number of negative integers: Sign of product: positive

2

GO ON Lesson 6-6 Multiply Several Integers

213

Step by Step Problem-Solving Practice

Problem-Solving Strategies

Solve. 10

TRAVEL A bus picked up 40 people from the airport. It stopped at 5 hotels. Seven people got off the bus each time. How many people were left on the bus? Understand

Draw a diagram. Use logical reasoning. Solve a simpler problem. Work backward. ✓ Make a table.

Read the problem. Write what you know. There are 40 people on the bus. The bus stops 5 times. It lets off 7 people each time. Hotel

People

Make a table with two columns. Title one column Hotel and the other column People.

0

40

1

33

Complete the table. If 7 people get off the bus at each of 5 hotels, how many people were left on the bus?

3

Pick a strategy. One strategy is to make a table.

Plan

Solve

number of people who got off the bus

-7 × 5 = -35

number of people left on the bus

40 + (-35) =

2

4 5

26 19 12 5

5

So, 5 people were left on the bus.

11

WEATHER The temperature drops 5 degrees each hour. How much will the temperature drop at the end of 4 hours?

-5 × 4 = -20; the temperature will drop 20 degrees. Check off each step.

214

✔

Understand

✔

Plan

✔

Solve

✔

Check

Chapter 6 Integers

Copyright © by The McGraw-Hill Companies, Inc.

Does the answer make sense? Look over your solution. Did you answer the question?

Check

12

NATURE John threw a pebble in the pond. From the surface, the pebble fell toward the bottom of the pond at a rate of 2 feet per second. Where was the pebble in relation to the surface of the pond after 6 seconds?

The pebble was 12 feet under the surface. What is the missing number in the equation? Explain.

13

5×

× 4 = -20

The correct answer is negative 1. The product is negative and is equal to the product of the other two numbers, so the third factor is -1.

Skills, Concepts, and Problem Solving

Copyright © by The McGraw-Hill Companies, Inc.

Simplify.

48

14

-9 × (-3) × (-1) = -27

15

4 × (-6) × (-2) =

16

–2 × 5 × (-2) × 5 × (-1) = -100

17

4 · (-9) · (-2) =

72

18

-3 · (-8) · (-2) = -48

19

(4)(-5)(-2)(2) =

80

20

(4)(-8)( -2)(-1) = -64

21

-2(3) × 2(-5) =

60

23

(-3)2 =

25

43 =

Find each product.

6 × 6 = 36

22

62 =

24

(-2)3 = (-2) × (-2) × (-2) = -8

(-3) × (-3) = 9

4 × 4 × 4 = 64

Simplify. 26

5 × (1 + (-6)) × 2 = -50

27

(-3)2 × 1 × (-4 + 2) = -18

28

3 × (1 + (-3)) × 22 = -24

29

(-2)3 × (-2 + 3) × -2 =

16

30

3 × (1 + (-4)) × 32 = -81

31

-23 × (-3 + 3) × (-1) =

0 GO ON

Lesson 6-6 Multiply Several Integers

215

Solve. 32

SCHOOL Thirty students got on the school bus. The bus made 7 stops. Four students got off the bus at each stop. How many students are left on the school bus?

7 × (-4) = -28; 30 + (-28) = 2; there are two students left on the bus. 33

WEATHER The outside temperature is 50°F and rising at a rate of 5 degrees per hour. How long will it be before the temperature reaches 70°F?

4 hours 34

TRAVEL Gloria and Isabelle took a tram from the top of a mountain. The tram went downhill 8 feet each minute. How much had the tram gone downhill after 30 minutes?

SCHOOL A school bus picked up 30 students from school.

-8 × 30 = -240; the tram went downhill 240 feet. Vocabulary Check Write the vocabulary word that completes each sentence. Give an example.

36

37

The number of times a base is multiplied by itself is the exponent .

order of operations The evaluating expressions.

tells what order to follow when

Writing in Math Explain how you evaluate integers with exponents. Give an example.

Determine the base and the exponent. In the problem 5², the 5 is the base and the 2 is the exponent. The exponent tells how many times to multiply the base by itself. Determine the sign by the following rule: If the base is negative and the exponent is odd, the sign is negative. Otherwise, the sign is positive.

216 Corbis

Chapter 6 Integers

Copyright © by The McGraw-Hill Companies, Inc.

35

Spiral Review Simplify. Name each property. 38

(Lesson 6-5, p. 205)

3 × (2 × (-9)) = (3 × 2) × (-9) Associative Property of Multiplication

= 6 × (-9) = -54 -1 ) + (4 × 5) = -4 + 20 = 16

Distributive Property

39

4(-1 + 5) = (4 ×

40

-2 ×

41

-7 × 1 =

42

NATURE A snail fell into a well that had a depth of 10 feet. The snail could climb 2 feet each day, but at night would slip back 1 foot. How many days until the snail gets out of the well?

0

=0

Zero Property of Multiplication Identity Property of Multiplication

-7

GFFU

The snail gained 1 foot each day. After 8 days, he will be 2 feet from the top of the well. By the end of the next day he will be out of the well, so he will not slip back.

Copyright © by The McGraw-Hill Companies, Inc.

He will be out in 9 days.

Solve.

(Lesson 5-5, p. 137)

43

BUSINESS A store had sales of $1,750 in a 7-hour period. What $250 were the sales per hour?

44

FOOD Two cheese pizzas and two sausage pizzas were cut into 28 pieces. The cheese pizzas were cut into an equal number of pieces, and the sausage pizzas were cut into a different but equal number of pieces. How many pieces were in each pizza? Assume each pizza had over 5 pieces.

Six pieces for one type of pizza and 8 for the other type of pizza.

Find each product.

(Lesson 4-5, p. 33)

45

3 × 101 = 303

46

113 × 3 = 339

47

402 × 3 = 1,206

48

3 × 206 = 618 Lesson 6-6 Multiply Several Integers

217

Chapter

6

Progress Check 3

(Lessons 6-5 and 6-6)

Find each product. Use a number line. 6NS2.3, 7NS1.2, 3NS2.2 1

3 × (-3) = -9

2

ĕ ĕĕ ĕ ĕ ĕ ĕ ĕ ĕ ĕ ĕ ĕ

4 × (-2) = -8

HSPVQTPG

HSPVQTPG

Find each product by multiplying absolute values. 6NS2.3, 7NS1.2, 3NS2.2

12

3

-3 × (–4) =

5

-2 × 5 = -10

4

1 × (-8) =

-8

6

-7 × -8 =

56

8

8 (3 + 4) × 2 = 112

10

-4 (7 + 3) × 4 = -160

Simplify. 6NS2.3, 7NS1.2, 3NS2.2, 3NS2.6 7

2 × (1 + (-3)) × 5 = -20

9

2 × (5 × 2) + 3 =

23

Find each product. 6NS2.3, 7NS1.2, 3NS2.2, 3NS2.6 (-5)2 =

13

(-4)3 = -64

15

3 × (-2) × (-5) =

30

9

12

32 =

14

-10 × 2 × 3 × (-2) = 120

16

2 × 5 × 4 × (-2) = -80

Solve. 6NS2.3, 7NS1.2, 3NS2.2, 3NS2.6 17

TRAVEL Refer to the photo caption at the right. Keesha walked down the mountain to an elevation of 1,150 meters. Then she hiked back up 250 meters to her campsite. At what elevation is she?

1,100 meters

18

FINANCES Mr. Washington bought stock for $20 dollars a share. Each share lost $2 for each of the next 4 months. How much has each share lost in value? TRAVEL The height of the mountain is

-2 × 4 = -8; each share lost $8.

218 Corbis

Chapter 6 Integers

2,000 meters above sea level.

Copyright © by The McGraw-Hill Companies, Inc.

25

11

Lesson

6-7 Divide Integers

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. 7NS1.2 Add, subtract, multiply and divide rational numbers and take positive rational numbers to wholenumber powers.

KEY Concept Division is the inverse operation for multiplication. You use the multiplication facts whenever you divide. The sign rules are the same as the rules for multiplication.

VOCABULARY

If the signs are the same, then the quotient is positive. 6÷3=2 (-6) ÷ (-3) = 2

division an operation on two numbers in which the first number is split into groups that are the size of the second number Example: 12 ÷ 3 means 12 is divided into 3 groups of equal size.

If the signs are different, then the quotient is negative. (-6) ÷ 3 = -2 6 ÷ (-3) = -2

(Lesson 5-1, p. 108)

Division by zero is impossible. 3 ÷ 0 is undefined.

quotient the answer or result of a division problem (Lesson 5-1, p. 108)

dividend the number that is being divided 2 ← quotient divisor → 4 8 ← dividend

Example 1 Copyright © by The McGraw-Hill Companies, Inc.

Find 18 ÷ (-6).

(Lesson 5-1, p. 108)

1. The signs are different. The quotient will be negative.

divisor the number by which the dividend is being divided Example: In 3 9 , 3 is the divisor. (Lesson 5-1, p. 108)

2. Find the absolute value of each. ⎪18⎥ = 18 and ⎪-6⎥ = 6 3. Divide the absolute values of the numbers. 18 ÷ 6 = 3 4. Write the quotient with a negative sign. -3 YOUR TURN! Find –36 ÷ 9.

different

1. Are the signs the same or different? 2. Find the absolute value of each.

⎪-36⎥ = 36 and ⎪9⎥ = 9

3. Divide the absolute values of the numbers. 4. Write the quotient with the sign.

36

÷

9

=

4

-4 GO ON Lesson 6-7 Divide Integers

219

Example 2

YOUR TURN!

-10 Find _____. -5

-45 Find _____. -3

1. The signs are the same. The quotient will be positive.

1. Are the signs the same or different?

2. Divide the absolute values of the numbers. 10 ÷ 5 = 2

2. Will the quotient be positive or negative?

3. Write the quotient. 2

3. Divide the absolute values of the numbers.

same

positive

45

÷

3

15

=

15

4. Write the quotient.

Who is Correct? Simplify -30 ÷ (-5).

Horace

Lisa

-30 ÷ (–5)

-30 ÷ (-5)

-30 ÷ (-5)

= -35

= -6

=6

Simon

Guided Practice Find each quotient.

5

1

5÷1=

3

–5 ÷ (-1) =

5

12 ÷ (-2) =

7

-12 ÷ 2 =

220

-5

2

–5 ÷ 1 =

5

4

5 ÷ (-1) =

-6

6

–12 ÷ (-2) =

8

12 ÷ 2 =

-6

Chapter 6 Integers

-5

6

6

Copyright © by The McGraw-Hill Companies, Inc.

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

Step by Step Practice 9

Find -16 ÷ 4.

different

Step 1 Are the signs the same or different?

negative

Step 2 Will the quotient be positive or negative? Step 3 Divide the absolute values of the numbers.

16 ÷ 4 = 4 Step 4 Write the quotient.

-4

Check the sign to make sure it is correct.

(-) ÷ (+) = (-)

Find each quotient. 10

30 ÷ (-3) signs: (+) ÷ (-) =

-10

Copyright © by The McGraw-Hill Companies, Inc.

quotient:

11

24 ÷ (-8) = -3

13

-16 ÷ (-4) =

15

-18 ÷ 3 = -6

17

-20 ÷ (–2) =

19

-4 = ___

21

28 ___ =

23

-100 _____ =

2

-4

-25

(-)

4

12

-12 ÷ 3 = -4

5

14

15 ÷ 3 =

16

33 ÷ (–11) = -3

18

-25 ÷ (-5) =

-2

20

10 ___ =

-7

22

-36 ____ =

-4

24

-50 ____ =

5

4

10

-2

9

-10

5

-5

GO ON Lesson 6-7 Divide Integers

221

Step by Step Problem-Solving Practice

Problem-Solving Strategies ✓ Use a model.

Solve. 25

Use logical reasoning. Make a table. Guess and check. Solve a simpler problem.

FINANCES Mr. Brown lost $240 in the last 4 months on his stocks. If he lost the same amount each month, what was his loss per month? Understand

Read the problem. Write what you know. Mr. Brown lost period of time.

$240 in a

4

-month

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

Plan

Draw a fraction bar worth a total of $240. Divide it into 4 equal parts. How much is each part worth? Solve

Divide $240 by 4. $240 ÷ 4 = $60 So, Mr. Brown lost $60 per month.

Check

Multiply to check your answer.

$60 26

× 4 $240

FOOTBALL A football team lost 33 yards in 3 plays. If the team lost the same number of yards each play, what was the loss per play?

-33 ÷ 3 = -11; the average loss per play was 11 yards.

27

✔

Understand

✔

Plan

✔

Solve

✔

Check

WEATHER The temperature dropped a total of 40 degrees in 5 hours. How much did the temperature drop per hour if the temperature dropped by the same amount each hour?

-40 ÷ 5 = -8; the temperature dropped an average of 8 degrees per hour. 28

Explain how to divide integers.

Determine the sign. Divide the absolute value of the numbers and write the correct sign. 222

Chapter 6 Integers

Copyright © by The McGraw-Hill Companies, Inc.

Check off each step.

Skills, Concepts, and Problem Solving Find each quotient. 29

-8 ÷ 4 =

-2

31

–10 ÷ (-2) =

33

-22 = ____

-11

35

–32 ____ =

4

37

-72 ÷ 8 =

39

48 ÷ 6 =

2

-8

5

-9 8

-3

30

9 ÷ (-3) =

32

12 ÷ 6 =

34

45 ___ =

-5

36

14 = ___

2

38

-60 ÷ (–6) =

10

40

-56 ÷ (-7) =

8

-9

7

2

Solve.

Copyright © by The McGraw-Hill Companies, Inc.

41

MONEY Tyler wants to save $100. If he wants to save the money in 10 weeks, how much should he save each week?

100 ÷ 10 = 10; Tyler needs to save $10 a week. 42

WEATHER The temperature at the base of the mountain is 78ºF. At the top of the mountain, which is 5,000 feet high, the temperature is 53ºF. Rodney is driving up to the peak of the mountain. How many degrees will the temperature decrease for every 200 feet he travels?

¡'

GFFU

¡'

GFFU

Every 200 feet that Rodney drives up the mountain, the temperature will decrease 1ºF.

Vocabulary Check Write the vocabulary word that completes each sentence. 43

The dividend is the number being divided into.

44

A(n) negative number is less than zero.

GO ON Lesson 6-7 Divide Integers

223

45

Writing in Math Jean worked the following problem. What mistake did she make?

She used the wrong sign. The signs of the dividend and

-32 ÷ (-4) = -8

divisor are the same, so the quotient is positive.

Spiral Review Find each product.

(Lesson 6-6, p. 211)

46

32 =

3×3=9

47

(–5)2 =

(-5) × (-5) = 25

48

(–4)3 =

(-4) × (-4) × (-4) = -64

49

23 =

2×2×2=8

Solve. 50

(Lesson 5-1, p. 106)

HOBBIES Ti is planting 35 flowers. He wants to have 5 rows of flowers. How many flowers will he put in each row?

35 ÷ 5 = 7 51

Casey read 1 book every 2 weeks. Terrance read 1 book every 2 weeks. So, they read at the same rate. Write an equation expressing the Commutative Property and draw rectangular arrays to model each product. (Lesson 4-1, p. 4) 52

2·8 2·8=8·2

Solve. 54

224

53

3·4

3·4=4·3

(Lesson 3-6, p. 165)

CHARITY The food pantry had 189 cans of food on the shelf before Thanksgiving. After Thanksgiving there were only 22 cans left. How many cans were given away for Thanksgiving? 167 Chapter 6 Integers

Copyright © by The McGraw-Hill Companies, Inc.

BOOKS Casey read a total of 6 books over the 12 weeks of summer. Terrance read a total of 20 books during the 40 weeks of the school year. Who read faster?

Lesson

6-8 Order of Operations with Integers KEY Concept The order of operations is the set of rules that tell operation to perform first within expressions.

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. 7NS1.2 Add, subtract, multiply and divide rational numbers and take positive rational numbers to wholenumber powers. 3NS2.2 Memorize to automaticity the you which multiplication table for numbers between 1 and 10. 3NS2.6 Understand the special properties of 0 and 1 in multiplication and division.

1. Simplify operations in parentheses. 2. Simplify terms with exponents. 3. Multiply and divide. Do the operations from left to right. 4. Add and subtract. Do the operations from left to right. 5 × (-3)2 ÷ (-2 + (-3)) × (-2) = 5 × (-3)2 ÷ (-5) × (-2) Simplify parentheses. = 5 × 9 ÷ (-5) × (-2) = 45 ÷ (-5) × (-2)

Simplify exponents. Multiply or divide.

= (-9) × (-2) = 18

VOCABULARY even number a number that can be divided by 2 (Lesson 1-4, p. 25)

odd number a number that can not be divided evenly by 2; such a number has 1, 3, 5, 7, or 9 in the ones place (Lesson 1-4, p. 25)

Use the order of operations to simplify problems in this lesson.

positive number a number greater than zero (Lesson 6-1, p. 174) negative number a number less than zero

Copyright © by The McGraw-Hill Companies, Inc.

(Lesson 6-1, p. 174)

Example 1 Simplify (-50) ÷ (-5) ÷ (-2). -50 ÷ (-5) ÷ (-2) =

10 ÷ (-2)

=

-5

.

Divide left to right. The quotient is positive because both numbers are negative. Divide. The quotient is negative because the signs are different.

YOUR TURN! Simplify 200 ÷ (-4) ÷ (-5). 200 ÷ (-4) ÷ (-5) =

-50 ÷ (-5)

=

10

Divide from left to right. The quotient is negative because one was positive and the other negative.

positive Divide. The quotient is the signs are the same because

. GO ON

Lesson 6-8 Order of Operations with Integers

225

Example 2 6 ÷ 2 × (-5) ÷ (-3)

Simplify 6 ÷ 2 × (-5) ÷ (-3). The problem is multiplication and division only. Work from left to right.

=

3 × (-5) ÷ (-3)

=

-15 ÷ (-3)

=

5

YOUR TURN! 20 ÷ (-10) × (-4) ÷ (-2)

Simplify 20 ÷ (–10) × (–4) ÷ (–2). The problem is multiplication and division only.

=

Work from left to right.

= =

-2 × (-4) ÷ (-2) 8

÷ (-2)

-4

Example 3 Simplify 6 × (-2) 2 ÷ (1 + (-3)) × (-2). 6 × (-2)2 ÷ (1 + (-3)) × (-2) =

6 × (-2)2 ÷ (-2) × (-2) 6 × 4 ÷ (-2) × (-2)

=

24 ÷ (-2) × (-2)

=

-12 × (-2)

=

24

Simplify expressions involving exponents. Do the multiplication and division from left to right.

YOUR TURN! Simplify 4 × (-3) 2 ÷ (1 - (-5)) × (-1). 4 × (-3) 2 ÷ (1 - (-5)) × (-1) = = = = =

226

2 4 × (-3) ÷ 6 × (-1)

6

4×9÷

36

× (-1)

÷ 6 × (-1)

6

× (-1)

-6

Chapter 6 Integers

Do the operations inside the parentheses. Simplify expressions involving exponents. Multiply and divide left to right.

Copyright © by The McGraw-Hill Companies, Inc.

=

Do operations inside parentheses first.

Who is Correct? Simplify (-2)² × (-4 - (-2)) ÷ 2.

Rebecca

Tavaris

÷2 (-2)2 × (-4 - (-2)) 2 ÷ ) +2 4 = (-2)2 × (= (-2)2 × (-2) ÷ 2 = 4 × (-2) ÷ 2 = -8 ÷ 2 = -4

÷2 (-2) 2 × (-4 - (-2)) 2 2 = (-2) × -6 ÷ = 4 × -6 ÷ 2 = -24 ÷ 2 = -12

Nick

÷2 (-2) × (-4 - (-2)) 2 = 4 × (-4 +2) ÷ = 4 × (-2) ÷ 2 = -4 ÷ -1 =4 2

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

Guided Practice Simplify.

3

1

18 ÷ (-2) ÷ (-3) =

3

-12 ÷ (-2) ÷ (-3) =

5

6×4÷2+5=

-2

17

2

-9 ÷ (-1) ÷ (-9) =

4

2 × 4 × 5 + 10 =

6

10 ÷ 2 × 5 =

-1

50

25

Copyright © by The McGraw-Hill Companies, Inc.

Step by Step Practice 7

Simplify 6 ÷ 3 + (-2) 2 - (1 + (-3)) × (-2). Step 1 Are there parentheses? yes Simplify the expression in the parentheses. 6 ÷ 3 + (-2)2 - (-2) × (-2) Step 2 Are there exponents in the problem? yes Simplify the expression involving the exponents. 6÷3+

4

- (-2) × (-2)

Step 3 Is there division or multiplication? yes Perform the division and multiplication from left to right.

2

+4-

4

Step 4 Is there addition or subtraction? yes Perform the addition and subtraction from left to right. 6 + (-4) =

2

GO ON

Lesson 6-8 Order of Operations with Integers

227

Simplify. 8

5 × (4 - (-6)) ÷ 2 + 5 = 5 × 10 ÷ 2 + 5 =

50 ÷ 2 + 5

=

25 + 5

=

30

9

(-3) 2 + 1 × (-4 - 2) =

11

3 × (1 + (-6)) + 4 2 =

13

(-3) 2 + 3 × (-6 + 2) =

3 1 -3

10

(-2) 3 ÷ 2 × (-2 + 3) × (-2) =

12

5 × (4 + (-6)) ÷ 2 + 5 =

14

(-4)2 ÷ 8 ÷ 2 =

Step by Step Problem-Solving Practice

Read the problem. Write what you know. Ticket prices are $12.95, $7.95, and $10.95 . There are 4 adults, 3 children, and 2 seniors.

Plan

Pick a strategy. One strategy is to make a table. Number of Tickets 4 3 2

Price $12.95 $ 7.95 $10.95

Multiply the number of tickets times the price. Add the products. 4 × $12.95 + 3 × $7.95 + 2 × $10.95

$51.80 +

$21.90 = $97.55 Kareem’s family will spend $97.55 to go to the zoo. Check 228

$23.85

+

Add the prices for 4 adults, 3 children, and 2 seniors.

Chapter 6 Integers

Copyright © by The McGraw-Hill Companies, Inc.

Understand

Type of Ticket Adult Children Seniors

1

Use logical reasoning. Guess and check. Solve a simpler problem. Act it out.

ENTERTAINMENT Kareem’s family is going to the zoo. Admission costs $12.95 for adults, $7.95 for children, and $10.95 for seniors. How much will it cost for Kareem’s family of 4 adults, 3 children, and 2 seniors to go to the zoo?

Solve

0

Problem-Solving Strategies ✓ Make a table.

Solve. 15

8

16

TRAVEL Theresa’s luggage cannot exceed 65 pounds. She has 2 suitcases. Each one weighs about 22 pounds. Her golf clubs weigh about 15 pounds. How much does Theresa’s luggage weigh? Is it within the weight limit?

59 pounds; yes Check off each step.

17

✔

Understand

✔

Plan

✔

Solve

✔

Check

5(F-32) TEMPERATURE You can use the expression________ to convert 9 degrees Fahrenheit to degrees Celsius. Find the temperature to the nearest degrees Celsius when a thermometer shows 80°F.

27°C Write the order of operations. Use your own words.

18

See TWE margin.

Copyright © by The McGraw-Hill Companies, Inc.

Skills, Concepts, and Problem Solving Simplify.

4

16 ÷ (-2) ÷ (-2) =

21

-28 ÷ (-7) ÷ (-1) =

23

4 × (-6) ÷ (-2) =

12

24

-2 × 5 ÷ (-2) ÷ 5 × (-1) =

25

45 ÷ (-9) · (-2) =

10

26

-3 · (–8) ÷ (-2) = -12

27

-1 × 9 ÷ 3 =

28

9 × (9 – 7) ÷ 2 + 5 =

29

(-3) 2 + 1 × (-4 - 2) =

30

3 × (1 + (-8)) + 3 2 = -12

20

-4

-3 3

22

-8 ÷ (-1) ÷ (-8) =

-1

19

- 9 ÷ (-3) × (-2) =

-6 -1

14

GO ON Lesson 6-8 Order of Operations with Integers

229

Solve. 31

5(F - 32) TEMPERATURE Use the expression _________ to convert 25°F to 9 degrees Celsius. Find the temperature to the nearest degrees Celsius when a thermometer shows 80°F.

-4°C 32

THEATER At the Steubenville Community Theater, tickets to an evening movie are priced $10.65 for adults and $6.95 for children and students. How much does it cost for a family of 3 adults and 6 children to see an evening movie?

$74.55 Vocabulary Check Write the vocabulary word that completes each sentence.

even

33

A number that is divisible by 2 is a(n) number.

34

A(n)

35

Writing in Math Explain why multiplying three negative integers yields a negative product.

positive

number is greater than zero.

Three are an odd number of negatives, so that means the answer is negative. Example: -1 × -1 = 1; 1 × -1 = -1 Copyright © by The McGraw-Hill Companies, Inc.

Spiral Review Solve. 36

(Lesson 4-11, p. 75)

ADVERTISING Maria spent 12 minutes counting the commercials she saw on television. If she counted 36 commercials in those 12 minutes, how many commercials did she count per minute?

3

37

BUSINESS Rhyaina’s sister set up a lemonade stand. She sold a dozen glasses of lemonade every hour the stand was open. If she sold lemonade for 8 hours, how many glasses did she sell?

96

Find each sum. 38 230

7 + 13 + 8

(Lesson 3-3, p. 145)

28

Chapter 6 Integers

39

11 + 25 + 9

45

40

12 + 16 + 8

36

Chapter

6

Progress Check 4

(Lessons 6-7 and 6-8)

Find each quotient. 3NS2.2, 6NS2.3 1

12 ÷ 12 =

3

-2 ÷ (-1) =

5

1

-1

2

-14 ÷ 14 =

2

4

4 ÷ (-1) =

32 ÷ (-4) =

-8

6

-12 ÷ (-3) =

7

56 ÷ (-8) =

-7

8

-20 ÷ 2 = -10

9

-8 = ___

10

16 = ___

-4

2

-4

4

-8

-2

Copyright © by The McGraw-Hill Companies, Inc.

Simplify. 3NS2.2, 6NS2.3

5

11

-120 ÷ (-40) ÷ (-3) =

-1

12

75 ÷ (-5) ÷ (-3) =

13

42 ÷ 7 × (-2) ÷ (-3) =

4

14

8 × (-2) 2 ÷ (3 - (-1)) × -9 = -72

Solve. 6NS2.3, 7NS1.2, 3NS2.2, 3NS2.6 15

LUNCH Suppose a student spends $3 each day on lunch for 5 days. What integer represents how much the student spent on lunch?

-$15 16

PUZZLES What is the missing number in the equation? -35 ÷

5

× 6 = -42 Lesson 6-8 Order of Operations with Integers

231

Chapter

Study Guide

6

Vocabulary and Concept Check absolute value, p. 189

Write the vocabulary word that completes each sentence. Not all vocabulary terms will be used.

Associative Property of Addition, p. 197

1

A number greater than zero is a(n) positive number .

2

A number not evenly divisible by 2 is a(n) odd number .

integers, p. 174 Inverse Property of Addition, p. 181 negative number, p. 174 opposite, p. 174

3

order of operations The is a set of rules that tells what order to follow in evaluating an expression.

4

…-3, -2, -1, 0, 1, 2, 3 … are

order of operations, p. 211 positive number, p. 174

integers

.

whole numbers

5

0, 1, 2, 3, 4 … are

.

6

Two different numbers that are the same distance from 0 on a opposite number line are numbers.

7

absolute value The a number and 0 on a number line.

8

Write the name of the property shown below.

Inverse Property of Addition

Lesson Review

6-1

Model Integers

(pp. 174–180)

Example 1

Write <, =, or > in each circle to make a true statement. 9

3 > -3

Graph both numbers on the number line.

10

Use <, =, or > to compare -5 and 5.

Since -5 is less than 5, you need to use the “less than” symbol. -5 < 5

232

The number farthest to the right is 5, so it is the greater number.

2 > -4

Chapter 6 Study Guide

Copyright © by The McGraw-Hill Companies, Inc.

3 + (-3) = 0

is the distance between

6-2

Add Integers

(pp. 181–187)

Example 2

Find each sum. Find the sum of 5 and -2. Use the number line.

11

-1 + 6 =

5

12

3 + (-7) =

-4

13

-7 + (-2) =

14

8 + (-4) =

15

9+3=

16

7 + (-10) =

Graph the first number.

-9

From the first number, go left on the number line.

4

12

-3

You are at 3 on the number line. Write the sum. 5 + (-2) = 3

6-3

Subtract Integers

(pp. 189–196)

Copyright © by The McGraw-Hill Companies, Inc.

Find each difference.

-7

17

-5 - 2 =

18

1 - (-3) =

19

6-8=

20

-3 - (-8) =

Which number has the greater absolute value?

4

⎪-4⎥ or ⎪2⎥

-2 5

Which number has the greater absolute value? 21

6 or -9

22

-5 or -3

Example 3

-4 is 4 units from 0. 2 is 2 units from 0. Which integer has the greater absolute value? -4 VOJUT VOJUT

-9 -5

Chapter 6 Study Guide

233

6-4

Add and Subtract Groups of Integers (pp. 197–203)

Simplify. 23

8 + (-2) - 2 + 7 - (-9)

20 24

Example 4 Simplify - 4 + (-9) - 3 + 6 - (-1). Rewrite the subtraction as addition. -4 + (-9) - 3 + 6 - (-1) = -4 + (-9) + (-3) + 6 + 1 Find the sum from left to right.

14 + (-8) + 5 - (-4) - 3

-4 + (-9) + (-3) + 6 + 1

12 25

21 + 5 -(-3) - 6 + 6

24

6-5

Multiply Integers

=

-13 + (-3) + 6 + 1

=

-16 + 6 + 1

=

-10 + 1

=

-9

Example 5

(pp. 205–210)

Find each product. 26

8 × (-3) =

-24 27

-9 × (-2) =

28

Find the absolute value of each. ⎥-7⎥ = 7 and ⎥-4⎥ = 4 Multiply the absolute values of the numbers. 7 × 4 = 28 The signs are the same. The product is positive. Write the product with the sign. 28

-2 × (-15) =

30

6-6

Multiply Several Integers (pp. 211–217)

Simplify. 29

22 × (5 - (-3)) × 6 = 192

30

-3 × (4 - (-1)) × 6 = -90

31

2 × (5 - (-2)) × (-5) = -70

32

32 × (-5 + 14) × (-1) = -81

234

Chapter 6 Study Guide

Example 6 Simplify 8 × (2 + (-5)) × (-3). Simplify according to the order of operations. There is one set of operations inside parentheses. 8 × (2 + (-5)) × (-3) =

8 × (-3) × (-3)

=

-24 × -3

=

72

Do the operation in parentheses. Do the multiplication from left to right.

Copyright © by The McGraw-Hill Companies, Inc.

18

Find -7 × (-4) by multiplying absolute values.

6-7

Divide Integers

Example 7

(pp. 219–224)

Find each quotient. 33

Find 27 ÷ (-3).

-44 ÷ (-11) = Find the absolute value of each. ⎥ 27⎥ = 27 and ⎥-3⎥ = 3 The signs are different. The quotient will be negative.

4 34

-12 = ____ -6

Divide the absolute values of the numbers. 27 ÷ 3 = 9

2 35

Write the quotient with a negative sign. -9

-35 ÷ 7 =

-5 36

36 ____ = -12

-3

6-8

Order of Operations Example 8 with Integers (pp. 225–230)

Copyright © by The McGraw-Hill Companies, Inc.

Simplify. 37

8 + (2 × 4) × 33 ÷ 9 =

32 39

4 × (-3)2 ÷ (1 - (-3)) × (-1)

9 ÷ 3 × (-15 ÷ (-3)) =

15 38

Simplify 4 × (-3)² ÷ (1 - (-3)) × (-1).

6 + (3 × 5) ÷ 7 + 22 =

= 4 × (-3)2 ÷ 4 × (-1)

Do the operation in parentheses.

= 4 × 9 ÷ 4 × (-1)

Multiply the exponents.

= 36 ÷ 4 × (-1)

Multiply from left to right.

= =

9 × (-1) –9

7 40

16 ÷ 4 × (-28 ÷ 7) =

-16

Chapter 6 Study Guide

235

Chapter

6 1

Chapter Test

Graph -7, -1, 4, 0, 6, -3. Then write the numbers in order from least to greatest. 4NS1.8, 5NS1.5

-7, -3, -1, 0, 4, 6

Write <, =, or > in each circle to make a true statement. 4NS1.8 2

-2 > –5

3

-8 < 2

5

(-3) + 6 =

Find each sum. 5NS2.1, 5NS1.5

-7

3

4

-5 + (-2) =

6

What is the opposite of 8? Use it to show the Inverse Property of Addition. 5NS2.1

-8; 8 + (-8) = 0

Find each difference. 5NS2.1, 6NS2.3 -1 - (-4) =

3

8

-3 - (-1) =

-2

Which number has the greater absolute value? 5NS2.1, 7NS1.2 9

-12 or -9 = -12

-3

10

-3 or 2 =

12

2 2 + (42 ÷ 6) × (–12) ÷ 7 =

14

-6 × (-3) = -18

Simplify. 5NS2.1, 6NS2.3, 7NS1.2 11

75 - 52 + (-200 - (-15)) + 30 = -132

-8

Find each product. 5NS2.1, 6NS2.3 13

236

2 × (-5) = -10

Chapter 6 Test

GO ON

Copyright © by The McGraw-Hill Companies, Inc.

7

Divide. 6NS2.3 15

-72 ÷ 9 =

-8

16

-36 = ____ -4

9

FOOTBALL On the first play of their drive, the Tomahawk team started from their opponent’s 30-yard line and gained 5 yards. On the second play of their drive, they lost 2 yards. On the third play, the Tomahawks gained 8 yards. On what yard line were they?

(0"-

17

(0"-

Solve. 6NS2.3, 7NS1.2

41-yard line 18

HEALTH Missy went to the dentist. The cost was $128. Her dental insurance was expected to pay $80 of the balance. Missy wrote the dentist a check for $48. Later, Missy found out that her insurance actually paid $88. Assuming Missy had no prior balance on her dental account other than this visit, what was her balance after both payments were received?

She had an account balance of -$8. In other words, the

Copyright © by The McGraw-Hill Companies, Inc.

dentist owed her $8.

Correct the mistakes. 5NS2.1, 6NS2.3, 7NS1.2 19

WEATHER The weather forecaster on channel 15 said, “It is currently 27°F. Overnight there is a cold front expected, and the temperature is expected to drop by approximately 40°F. That will bring us to a low temperature of 13°F.” What is wrong with this prediction?

27°F - 40°F = -13°F; the temperature will be 13° below zero. 20

MONEY John purchased a credit card at his cafeteria for $25. He bought food that totaled $6.25. John told his mom that he had a balance of $6.25. What is wrong with John’s answer?

He spent $6.25, but the credit he had remaining was actually $25 - $6.25, or $18.75.

Chapter 6 Test

237

Chapter

Standards Practice

6

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

Which letter on the number line represents -3? 5NS1.5 "

#

2

%

A A

C C

B B

D D

LaShaun went scuba diving in the ocean. He dove to 55 feet below sea level to explore a cave. Then he climbed 19 feet to take pictures of some jellyfish. Where is LaShaun in relation to sea level? 5NS2.1 F -74 feet

H 36 feet

G -36 feet

J 74 feet

6

7

Joaquin owed his brother $15. He also owed his mother $22 and his best friend $9. If he earns $65 next week, how much money will he have left after he pays what he owes? 5NS1.2 A $19

C $49

B $46

D $111

8

Due to a construction project, a small restaurant loses $4,000 per day in business. If this pattern continues for the next 3 days, how much money will this restaurant have compared to its normal sales? 6NS2.3 A -$12,000

C -$8,000

B -$10,000

D -$4,000

Find the product of -11, -8, and -4. F –352

H 88

G –322

J 352

A football team lost 45 yards on 3 plays. If the team lost the same number of yards on each play, which integer shows the yards lost per play? 6NS2.3 A -15 yards

C -10 yards

B -12 yards

D -8 yards

Use the order of operations to solve this problem. 7NS1.2

__

-12 (-3)(-9 + 6) 52 - (-11) 4

Which symbol makes this math sentence true? 6NS2.3

-46,051 - 27,389

238

6NS2.3

=

F -15

H 3

G -3

J 15

-55,028 - (-18,412)

F <

H >

G =

J -

Chapter 6 Standards Practice

GO ON

Copyright © by The McGraw-Hill Companies, Inc.

3

$

5

9

ANSWER SHEET

Find the estimated difference by rounding to the nearest hundred thousands place. 4NS1.3

Directions: Fill in the circle of each correct answer.

3,781,027 -1,694,783

10

Copyright © by The McGraw-Hill Companies, Inc.

11

12

1

A

B

C

D

2

F

G

H

J

A 2,000,000

C 2,090,000

3

A

B

C

D

B 2,086,000

D 2,100,000

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

What is the product of 108 and 4? F 72

H 112

G 104

J 432

3NS2.4

Enrique is baking cookies. He wants to make 153 large cookies to sell for a school fund-raiser. If he can make 9 cookies per batch, how many batches will he need to bake? 3NS2.4 A 14 batches

C 16 batches

B 15 batches

D 17 batches

Success Strategy When checking your answers, do not change your mind on your answer choice unless you misread the question. Your first choice is often the right one.

Kaylie’s math quiz scores are shown in the table. What is her average quiz score? 7NS1.2 Quiz

Score

1

95

2

84

3

91

4

82

F 82

H 86

G 85

J 88 Chapter 6 Standards Practice

239

Index A

D

absolute value, 189, 197

difference, 189, 197

addition integers, 181–187, 197–203

Distributive Property of Multiplication, 89–94

Answer sheet 105, 171, 239

dividend, 108, 115, 123, 151, 157, 219

array, 4, 61, 108 Assessment, 102–103, 168– 169, 236–237 Associative Property of Addition, 197 Associative Property of Multiplication, 89–94

B base (of a power), 75, 81, 211

C

Chapter Preview, 3, 107, 173 Chapter Test, 102–103, 168– 169, 236–237 Commutative Property of Addition, 181 Commutative Property of Multiplication, 4–10, 11–17, 25–31, 33–38, 39–45, 47–52, 53–59, 61–66, 67–73, 75–80, 81–87, 89–94

divisor, 108, 115, 123, 151, 157, 219 double (twice), 19

E even number, 174, 211, 225 expanded form, 11, 89–94 exponent, 75, 81, 211

F fact family, 25–31, 33–38, 39– 45, 47–52, 53–59, 61–66, 67– 73, 75–80, 81–87, 89–94, 108– 114, 115–121, 123–128, 129– 135, 137–142,143–149, 151– 156, 157–162 factor, 4, 11, 19, 25, 33, 39, 61, 67, 81, 205

I

composite number, 75–80

integers, 174–180, 181, 189

Correct the Mistakes, 103, 169, 237

inverse operations, 108

240

Index

Inverse Property of Addition, 181

Key Concept, 4, 11, 19, 25, 33, 39, 47, 53, 61, 67, 75, 81, 89, 108, 115, 123, 129, 137, 143, 151, 157, 174, 181, 189, 197, 205, 211, 219, 225

L long division, 157–162

M mental math, 143 multiple, 11, 19, 39, 47, 53, 129, 137, 143 multiplication integers, 205–217 whole numbers, 4–10, 11– 17, 25–31, 33–38, 39–45, 47–52, 53–59, 61–66, 67– 73, 75–80, 81–87, 89–94

N negative number, 174, 219, 225 Number Sense, 4, 11, 19, 25, 33, 39, 47, 53, 61, 67, 75, 81, 89, 108, 115, 123, 129, 137, 143, 151, 157, 174, 181, 189, 197, 205, 211, 219, 225

O odd number, 174, 211, 225 opposites, 174, 181, 189, 197 order of operations, 211

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

California Mathematics Content Standards, 4, 11, 19, 25, 33, 39, 47, 53, 61, 67, 75, 81, 89, 108, 115, 123, 129, 137, 143, 151, 157, 174, 181, 189, 197, 205, 211, 219, 225

division integers, 219–224 whole numbers, 108–114, 115–121, 123–128, 129– 135, 137–142,143–149, 151–156, 157–162, 219

K

P perfect square, 81 positive number, 174, 219, 225 prime factor, 75–80 prime factorization, 75–80 prime number 75–80 Problem-Solving, see Step-byStep Problem Solving product, 4–10, 11–17, 25–31, 33–38, 39–45, 47–52, 53–59, 61–66, 67–73, 75–80, 81–87, 89–94, 205 Progress Check, 18, 32, 46, 60, 74, 88, 122, 136, 150, 163, 188, 204, 218, 231

Q quotient, 108, 115, 123, 151, 157, 219

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

R Real-World Applications advertising, 80, 196, 230 art, 141 astronomy, 80 baking, 64 basketball, 180 birds, 136 books, 23, 58, 135, 155, 196, 224 buildings, 187 business, 29, 31, 52, 59, 64, 87, 93, 134, 156, 160, 217, 230 celebrations, 50, 161 charity, 121, 224 chemistry, 237 clubs, 121 community service, 14, 51, 127, 146, 149, 237 computers, 16 construction, 38, 78, 119, 147 cooking, 18, 94 culture, 113 earth science, 65 eggs, 133

entertainment, 57, 60, 72, 113 farming, 43 farms, 136 fashion, 14, 24, 43, 135, 180, 203 field trips, 79 finance, 103 finances, 179, 194, 201, 208, 210, 218, 222 fitness, 59, 94, 120, 127, 134, 154 food, 10, 15, 72, 88, 119, 128, 142, 169, 187, 217 football, 66, 188, 209, 222 games, 18, 23, 43, 162, 185 gardens, 112, 128, 155, 180 geography, 142 graphing, 93, 187 health, 8, 59, 84, 141, 185, 188, 209, 237 hiking, 58, 185, 194, 203 history, 122 hobbies, 32, 52, 65, 71, 79, 140, 224 homes, 66, 103 homework, 163 horticulture, 203 industry, 36 interior design, 7, 38, 84, 120, 203 landscaping, 30, 64, 147 library, 169 machines, 60 models, 120, 128, 156 money, 10, 15, 24, 32, 46, 65, 72, 114, 118, 163, 178, 237 music, 8, 52, 56, 94, 103, 122 nature, 14, 57, 215, 217 nutrition, 38, 52, 195, 223 packaging, 9, 24, 29, 44, 70, 187, 196 parks, 28 parties, 122 pets, 87 photos, 38, 80, 136, 210 population, 31 puzzles, 9, 45, 66, 73, 74, 86, 94, 156, 162, 231 savings, 16 school, 45, 73, 169, 216, 231 science, 58, 85

scuba diving, 178 shopping, 37, 71, 148 sleep, 24 soccer, 45, 80 surveys, 50 swimming, 148 tennis, 38, 46 transportation, 201 travel, 36, 186, 195, 202, 204, 214, 216, 218 trips, 132 weather, 177, 179, 194, 201, 202, 210, 214, 216, 222, 223, 237 Reflect, 8, 14, 23, 29, 37, 43, 51, 57, 64, 71, 79, 85, 94, 113, 119, 127, 133, 141, 147, 155, 161, 178, 185, 195, 202, 209, 215, 222, 230 remainder, 129, 137

S short division, 143 signed numbers Spiral Review, 10, 16, 24, 31, 38, 45, 52, 59, 66, 73, 80, 86, 94, 114, 121, 128, 135, 142, 149, 156, 162, 180, 187, 196, 203, 210, 217, 224, 230 Standards Practice, 104–105, 170–171, 238–239 Statistics, Data Analysis, and Probability Step-by-Step Practice, 6, 13, 21, 27, 35, 41, 49, 55, 63, 69, 77, 83, 92, 111, 117, 125, 131, 139, 145, 153, 159, 176, 183, 193, 200, 208, 213, 221, 228 Step-by-Step Problem Solving Practice, 7–8, 14, 22–23, 28–29, 36, 43, 50, 56–57, 64, 70–71, 78–79, 84, 93, 112–113, 118– 119, 126, 132–133, 140–141, 146–147, 154–155, 160, 177– 178, 185, 194, 201, 208–209, 214, 222, 229 Draw a diagram, 7, 22, 36, 84, 112, 177, 185, 201, 208 Draw a model, 127 Draw a picture, 28, 43, 56 Look for a pattern, 14

Index

241

Make a table, 64, 118, 146, 194, 214, 229 Solve a simpler problem, 70, 78, 140 Use a model, 222 Use logical reasoning, 50, 93, 132, 154, 160 Study Guide, 95–101, 164–167, 232–235 subtraction integers, 189–196, 197–203 Success Strategy, 105, 171, 239 sum, 181

T triple, 33

V

W

Vocabulary, 4, 11, 19, 25, 33, 39, 47, 53, 61, 67, 75, 81, 89, 108, 115, 123, 129, 137, 143, 151, 157, 174, 181, 189, 197, 205, 211, 219, 225

Who is Correct?, 6, 12, 21, 26, 35, 40, 48, 54, 62, 68, 77, 82, 91, 111, 117, 130, 139, 144, 152, 158, 176, 183, 192, 199, 207, 213, 220, 227

Vocabulary and Concept Check, 95, 164, 232

Writing in Math, 10, 16, 24, 31, 38, 45, 52, 59, 66, 73, 80, 86, 94, 114, 121, 128, 135, 142, 149, 156, 162, 180, 187, 196, 203, 210, 216, 224, 230

Vocabulary Check, 10, 16, 24, 30, 38, 45, 52, 58, 66, 73, 80, 86, 94, 114, 120, 128, 134, 142, 149, 156, 162, 179, 187, 196, 203, 210, 216, 223, 230

Z zero, 11–17, 174

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

242

Index