California math triumphs place value and basic number skills volume 1b

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


Photo Credits
coins: United States Mint, bills: Michael Houghton/StudiOhio; Cover Jupiter
Images; iv (tl bl br) File Photo, (tc tr) The McGraw-Hill Companies, (cl c) Doug
Martin, (cr) Aaron Haupt; v (1 2 3 4 6 7 8 9 11 12) The McGraw-Hill Companies,
(5 10 13 14) File Photo; vii Dynamics Graphics Group/Creatas/Alamy; viii Digital
Vision/PunchStock; ix x CORBIS; xi Digital Vision/PunchStock; xii Digital Vision/
Getty Images; 2–3 Getty Images; 15 Masterfile; 18 Photodisc/Getty Images;
58 Dynamic Graphics Value/SuperStock; 80 Getty Images; 94 Stockbyte/
SuperStock; 106–107 Getty Images; 113 Matthew Scherf/iStockphoto;
119 Burke/Triolo Productions/Getty Images; 122 Photodisc/Getty Images;
127 Brand X Pictures/PunchStock; 132 Michael Newman/PhotoEdit, Inc.;
147 Roy McMahon/CORBIS; 148 CORBIS; 150 Siede Preis/Getty Images;
161 CORBIS; 169 Getty Images; 172 Cyril Mazansky/UNEP/Peter Arnold, Inc.;
203 age fotostock/SuperStock; 204 216 218 CORBIS

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

iv
(tl)File Photo, (tc tr)The McGraw-Hill Companies, (cl c)Doug Martin, (cr)Aaron Haupt, (bl bc)File Photo

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

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

v
(L to R 1 2 3 4 6 7 8 9 11 12)The McGraw-Hill Companies, (5 10 13 14)File Photo


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

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

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

vii
Dynamics Graphics Group/Creatas/Alamy


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

viii
Digital
Digital Vision/PunchStock
Vision/PunchStock

Giant Redwoods, Sequoia National Park

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

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

Poppy meadow in the Santa Ynez Mountains

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

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

xi
Digital Vision/PunchStock


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

xii
Digital Vision/Getty Images

Big Sur Coast

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

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?

Copyright © by The McGraw-Hill Companies, Inc.

2

Chapter 4 Multiplication

Getty Images


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.

16

⎫

4+4+4+4=

⎫

3

3 × 300 = 300 + 300 + 300 = 900
⎬

10 + 10 + 10 =

four 10s
⎭

2

30

⎬

4 × 10 = 10 + 10 + 10 + 10 = 40

66

33 + 33 =

⎭

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.
0OF
GBDUPS
JT
UIF
OVNCFS
PG
SPXT
5IF
PUIFS
GBDUPS
JT
UIF
OVNCFS
PG
DPMVNOT

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.

           


HSPVQT
PG



HSPVQT
PG


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.

Copyright © by The McGraw-Hill Companies, Inc.

           

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 .


JT
POF
GBDUPS


JT
POF
GBDUPS



JT
UIF
PUIFS
GBDUPS

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

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

Copyright © by The McGraw-Hill Companies, Inc.

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.

Copyright © by The McGraw-Hill Companies, Inc.

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.

Copyright © by The McGraw-Hill Companies, Inc.

0OF
GBDUPS
JT
UIF
OVNCFS
PG
DPMVNOT

0OF
GBDUPS

JT
UIF

OVNCFS
PG
SPXT

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

0CTOBER
5OTAL
%EPOSITS 



%BUF

/BNFT

PO
"DDPVOU
"EESFTT

$840,000
31

%FQPTJU
4MJQ

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

/PUJDF
UIBU
øu



BOE

u



8IFO
ZPV
NVMUJQMZ
CZ

BEE

UP
UIF
SJHIU
PG
UIF
PUIFS

GBDUPS
"
TJNJMBS
SVMF
BMTP
IPMET

GPS
NVMUJQMJDBUJPO
CZ

BOE
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