California math triumphs fractions and decimals volume 2a

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

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Photo Credits
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x Daryl Benson/Masterfile; 2–3 Larry Dale Gordon/Getty Images; 3 (t)Michael
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1 2 3 4 5 6 7 8 9 10 055/027 16 15 14 13 12 11 10 09 08 07

California Math Triumphs
Volume 2A


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

Chapter 3 Operations with Fractions
2B
Chapter 4 Positive and Negative Fractions and Decimals

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

Volume 3 Ratios, Rates, and Percents
3A
Chapter 1 Ratios and Rates
3A
Chapter 2 Percents, Fractions, and Decimals
3B
Chapter 3 Using Percents
3B
Chapter 4 Rates and Proportional Reasoning
Volume 4 The Core Processes of Mathematics
4A
Chapter 1 Operations and Equality
4A
Chapter 2 Math Fundamentals
4B
Chapter 3 Math Expressions
4B
Chapter 4 Linear Equations
4B
Chapter 5 Inequalities
Volume 5 Functions and Equations
5A
Chapter 1 Patterns and Relationships
5A

Chapter 2 Graphing
5B
Chapter 3 Proportional Relationships
5B
Chapter 4 The Relationship Between
Graphs and Functions
Volume 6 Measurement
6A
Chapter 1 How Measurements Are Made
6A
Chapter 2 Length and Area in the Real World
6B
Chapter 3 Exact Measures in Geometry
6B
Chapter 4 Angles and Circles
iii


Authors and Consultants
AUTHORS

Frances Basich Whitney

Kathleen M. Brown

Dixie Dawson

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

Math Curriculum Staff Developer
Washington Middle School
Long Beach, California

Math Curriculum Leader
Long Beach Unified
Long Beach, California

Philip Gonsalves

Robyn Silbey

Kathy Vielhaber

Mathematics Coordinator
Alameda County Office of Education
Hayward, California

Math Specialist
Montgomery County Public Schools
Gaithersburg, Maryland

Mathematics Consultant
St. Louis, Missouri

Viken Hovsepian
Professor of Mathematics
Rio Hondo College
Whittier, California

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

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

Instructional Planning
and Support

ELL Support and
Vocabulary

Beatrice Luchin

ReLeah Cossett Lent

Mathematics Consultant
League City, Texas

Author/Educational Consultant
Alford, Florida

iv

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

Math Teacher
California College Preparatory Academy
Oakland, California

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

Eppie Leamy Chung

Monica S. Patterson

Teacher
Modesto City Schools
Modesto, California

Educator
Aspire Public Schools
Modesto, California

Judy Descoteaux

Rechelle Pearlman

Mathematics Teacher
Thornton Junior High School
Fremont, California

4th Grade Teacher
Wanda Hirsch Elementary School
Tracy, California

Paul J. Fogarty

Armida Picon

Mathematics Lead
Aspire Public Schools
Modesto, California

5th Grade Teacher
Mineral King School
Visalia, California

Lisa Majarian

Anthony J. Solina

Classroom Teacher

Cottonwood Creek Elementary
Visalia, California

Lead Educator
Aspire Public Schools
Stockton, California

vi

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

Bobbi Anne Barnowsky


Volume 2A Fractions and Decimals
Chapter

Parts of a Whole

1

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

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

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

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

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

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

Bixby Creek Bridge on Highway 1, south of Carmel

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

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

Recognize, name, and
1
1
compare unit fractions from ___ to __.
12 2
2NS4.3 Know that when all
fractional parts are included, such as fourfourths, the result is equal to the whole and
to one.
4NS1.5 Explain different interpretations

of fractions, for example, parts of a whole,
parts of a set, and division of whole
numbers by whole numbers; explain
equivalence of fractions (see Standard
4.0).
4NS1.7 Write the fraction represented
by a drawing of parts of a figure; represent
a given fraction by using drawings; and
relate a fraction to a simple decimal on a
number line.

vii
Roy Ooms/Masterfile


Contents
Chapter

Equivalence of Fractions

2

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

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

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

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

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

Assessment

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

viii
Daryl Benson/Masterfile

3NS3.1 Compare fractions represented
by drawings or concrete materials to show
equivalency and to add and subtract
1
simple fractions in context (e.g., __ of a
2
2
pizza is the same amount as __ of another

4
3
pizza that is the same size; show that __ is
8
1
larger than __).
4
4NS1.5 Explain different interpretations
of fractions, for example, parts of a whole,
parts of a set, and division of whole
numbers by whole numbers; explain the
equivalence of fractions (see Standard
4.0).
5NS1.5 Identify and represent
on a number line decimals, fractions,
mixed numbers, and positive and negative
integers.
6NS1.1 Compare and order
positive and negative fractions, decimals,
and mixed numbers and place them on a
number line.

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

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

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

Contents
Chapter

Operations with Fractions

3

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

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

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

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

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

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

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

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

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

Progress Check 3 .............................................................45
Assessment
Study Guide .....................................................................46

Chapter Test .....................................................................50

San Diego Harbor

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

ix
Jeremy
Jeremy Woodhouse/Masterfile
Woodhouse/Masterfile


Contents
Chapter

Positive and Negative Fractions
and Decimals

4

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

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

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

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

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

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

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

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

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

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

x
Daryl Benson/Masterfile

3NS3.4 Know and understand that fractions and
decimals are two different representations of the
1
same concept (e.g., 50 cents is __ of a dollar,
2
3
__
75 cents is of a dollar).
4
4NS1.6 Write tenths and hundredths in decimal
and fraction notations and know the fraction and
decimal equivalents for halves and fourths
3
1
2
(e.g., _ = 0.5 or 0.50; __ = 1_ = 1.75).
4
2
4
4NS1.7 Write the fraction represented by a drawing
of parts of a figure; represent a given fraction by using
drawings; and relate a fraction to a simple decimal on
a number line.
2NS5.1 Solve problems using
combinations of coins and bills.
2NS5.2 Know and use the decimal
notation and the dollar and cent symbols for money.
5NS1.5 Identify and represent on a
number line decimals, fractions, mixed numbers, and
positive and negative integers.

6NS1.1 Compare and order positive and
negative fractions, decimals, and mixed numbers and
place them on a number line.
4NS2.0 Students extend their use and
understanding of whole numbers to the addition and
subtraction of simple decimals.
5NS2.0 Students perform calculations and solve
problems involving addition, subtraction, and simple
multiplication and division of fractions and decimals.
7NS1.2 Add, subtract, multiply, and divide
rational numbers (integers, fractions, and terminating
decimals) and take positive rational numbers to
whole-number powers.
5NS2.1 Add, subtract, multiply, and divide
with decimals; add with negative integers; subtract
positive integers from negative integers; and verify the
reasonableness of the results.
4NS1.8 Use concepts of negative numbers
(e.g., on a number line, in counting, in temperature, in
“owing”).
6NS2.3 Solve addition, subtraction,
multiplication, and division problems, including
those arising in concrete situations, that use positive
and negative integers and combinations of these
operations.

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

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

Standards Addressed
in This Chapter


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

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

1 What is the title of Chapter 1?
2

What is the Key Concept of Lesson 2-1?

3

What are the four steps of problem solving?

4

What are the vocabulary words for Lesson 2-5?

5

How many Examples are presented in Lesson 1-3?

6

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

7

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

8

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

9

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

10

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

1

Chapter

1

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

Copyright © by The McGraw-Hill Companies, Inc.

2

Chapter 1 Parts of a Whole

Larry Dale Gordon/Getty Images


STEP

STEP

1 Quiz

2 Preview

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

What You Know

What You Will Learn

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

Lessons 3-1 and 3-2

1
1
2

1
4

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

Copyright © by The McGraw-Hill Companies, Inc.

Each person gets an equal-sized piece.

1
2

1
4

1
4

1
4

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

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

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


Lesson

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

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

3
number
of
red
parts
1
______________________ = __
number of colors in flag 3

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

5
__
6 ← denominator
whole
the entire amount or
object

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

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

Chapter 1 Parts of a Whole

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

Copyright © by The McGraw-Hill Companies, Inc.

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

color, or __
3

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


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

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

YOUR TURN!

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

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

Numerator

Copyright © by The McGraw-Hill Companies, Inc.

Denominator

Example 2

YOUR TURN!

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

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

1. There are 7 objects in the set.

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

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

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

3. Write the fraction. ______

Numerator
Denominator

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

5


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

Reina
9
__

Tate

Toshi

9

9

4
__

5

5
__

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

Guided Practice
Write a fraction to represent each situation.
1

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

=
number of equal parts

2

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

3

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

6

Chapter 1 Parts of a Whole

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

Copyright © by The McGraw-Hill Companies, Inc.

number of suns in set
_________________________

= ______


Step by Step Practice
4

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

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

Copyright © by The McGraw-Hill Companies, Inc.

Shade

equal parts.

parts.

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

What fraction of the set of balls are the footballs?

8

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

9

What fraction represents the shaded part of the rectangle?

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

7


Step by Step Problem-Solving Practice
Solve.
10

Problem-Solving Strategies
✓ Draw a picture.

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

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

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

cookie each.

cookies left.

Plan

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

Solve

Count the total number of circles. This is the
number for the whole. It is the

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

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

12

8

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

Copyright © by The McGraw-Hill Companies, Inc.

Look at the numbers in the problem. Does the

solution answer the question? Is it reasonable?

Check

11

denominator


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

13

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

15

16

17

0
in.

Copyright © by The McGraw-Hill Companies, Inc.

18

1

19

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

2
__

22

5
__

24

5

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

21

3
__

23

2
__

3

4

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

Bonhommet/PhotoCuisine/Corbis

9


Solve.
25

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

26

WORDS

27

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

28

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

What fraction of the letters in California are consonants?

Vocabulary Check
each sentence.
29

Write the vocabulary word that completes

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

,
.

The

is an entire amount or object.

31

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

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

Chapter 1 Parts of a Whole

Copyright © by The McGraw-Hill Companies, Inc.

30

10

23 out of 25 correct!

Lesson

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

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

Copyright © by The McGraw-Hill Companies, Inc.

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

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

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

1
_

1
>_
6 8

6 < 8 so the unit
fraction

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

1
is greater.
6

1 unit

Denominator

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

GO ON

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

11


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

__

_

1 unit

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

__

_

1 unit

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

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

12

Chapter 1 Parts of a Whole

Copyright © by The McGraw-Hill Companies, Inc.

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


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

1.
There are 3 equal parts. The unit fraction is __
3
1.
There are 6 equal parts. The unit fraction is __
6
1.
There are 12 equal parts. The unit fraction is ___

12
YOUR TURN!
Name the unit fraction that represents the shaded region in each figure.

1 unit

There are

equal parts.

Copyright © by The McGraw-Hill Companies, Inc.

The unit fraction is

There are

equal parts.

The unit fraction is

There are

.

.

equal parts.

The unit fraction is

.

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

13


Example 3

YOUR TURN!

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

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

1
__

1
__

1
__

1
__

3

4

7

9

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

1. What are the denominators of the
fractions?

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

Recall that the smaller denominator
makes the greater unit fraction.

7

9

3. Write a comparison statement for the
fractions.
1

1
__
__
7
9
1
3

1
4

3. Write a comparison statement
for the fractions.

1 > __
1
__
3

4

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

Damian

Lloyd

Emma

10

10

2

5
___

1
___

1
__

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

Guided Practice
1

14

Write the unit fraction that represents the shaded region.

Chapter 1 Parts of a Whole

Copyright © by The McGraw-Hill Companies, Inc.

Who is Correct?

2

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

Step by Step Practice
3

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

1 unit

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

Copyright © by The McGraw-Hill Companies, Inc.

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

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

5

1
___

6

10

1
__
6

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

4
10
9

1
__

1
__

3

8

10

1
___

1
__

12

5

GO ON

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

15