California math triumphs fractions and decimals volume 2a

4NS1.7 Write the fraction represented by a drawing of parts of a fi gure; represent a given fraction by using drawings; and relate a fraction to a simple decimal on a number line... 12

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

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California Math Triumphs

iii

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

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

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

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

5A Chapter 1 Patterns and Relationships

5A Chapter 2 Graphing

5B Chapter 3 Proportional Relationships

5B Chapter 4 The Relationship Between

Graphs and Functions

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

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Authors and Consultants

iv

AUTHORS

Frances Basich Whitney

Project Director, Mathematics K–12

Santa Cruz County Offi ce of Education

Capitola, California

Kathleen M Brown

Math Curriculum Staff Developer Washington Middle School Long Beach, California

Dixie Dawson

Math Curriculum Leader Long Beach Unifi ed Long Beach, California

CONSULTANTS

Assessment

Donna M Kopenski, Ed.D.

Math Coordinator K–5

City Heights Educational Collaborative

San Diego, California

Instructional Planning and Support

Beatrice Luchin

Mathematics Consultant League City, Texas

ELL Support and Vocabulary

ReLeah Cossett Lent

Author/Educational Consultant Alford, Florida

Dinah-Might Activities, Inc.

San Antonio, Texas

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California Advisory Board

v

Carol Cronk

Mathematics Program Specialist

San Bernardino City Unifi ed

School District

San Bernardino, California

Audrey M Day

Classroom Teacher Rosa Parks Elementary School San Diego, California

Jill Fetters

Math Teacher Tevis Jr High School Bakersfi eld, California

Grant A Fraser, Ph.D.

Professor of Mathematics California State University, Los Angeles

Los Angeles, California

Eric Kimmel

Mathematics Department Chair

Frontier High School

Bakersfi eld, California

Donna M Kopenski, Ed.D.

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

Michael A Pease

Instructional Math Coach Aspire Public Schools Oakland, California

Chuck Podhorsky, Ph.D.

Math Director City Heights Educational Collaborative San Diego, California

Arthur K Wayman, Ph.D.

Professor Emeritus

California State University, Long

Beach

Long Beach, California

Frances Basich Whitney

Project Director, Mathematics K–12 Santa Cruz County Offi ce of Education

Capitola, CA

Mario Borrayo

Teacher Rosa Parks Elementary San Diego, California

Melissa Bray

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

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

Bonnie Awes

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

Kathleen M Brown

Math Curriculum Staff Developer Washington Middle School Long Beach, California

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California Reviewers

vi

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

Bobbi Anne Barnowsky

Monica S Patterson

Educator Aspire Public Schools Modesto, California

Rechelle Pearlman

4th Grade Teacher Wanda Hirsch Elementary School Tracy, California

Armida Picon

5th Grade Teacher Mineral King School Visalia, California

Anthony J Solina

Lead Educator Aspire Public Schools Stockton, California

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Volume 2A Fractions and Decimals

of a set and parts of a whole.

2NS4.1 Recognize, name, and compare unit fractions from _1

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 fi gure; represent

a given fraction by using drawings; and relate a fraction to a simple decimal on a number line.

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

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viii

Chapter

2 Equivalence of Fractions

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

to one.

3NS3.1 Compare fractions represented

by drawings or concrete materials to show equivalency and to add and subtract simple fractions in context (e.g., 1

2 of a pizza is the same amount as 2

4 of another pizza that is the same size; show that 3

8 is larger than 1

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

Alabama Hills, Owens Valley

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ix

San Diego Harbor

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

Chapter

3-1 Add Fractions with Like Denominators .4

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.

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4-9 Operations with Positive

and Negative Numbers 121

4NS1.7 Write the fraction represented by a drawing

of parts of a fi gure; represent a given fraction by using drawings; and relate a fraction to a simple decimal on

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.

Antelope Valley

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1

Let’s Get Started

Use the Scavenger Hunt below to learn where things are

located in each chapter

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

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2 Chapter 1 Parts of a Whole

Chapter

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

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3

readiness quiz at ca.mathtriumphs.com to find out

them with what you’ll learn in this chapter

You know that half of one dollar is

50 cents Half of 50 cents is a quarter,

Each person gets an equal-sized piece

Lessons 3-2 and 3-3

The pizza is divided into 8 equal parts The entire pizza can berepresented by the fraction 8

8 Each person gets 1

2 of the pizza is greater than 1

8 of the pizza

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4 Chapter 1 Parts of a Whole

the number above the bar

in a fraction that tells

how many equal parts are being used

the number below the bar

how many equal parts are

in the whole or the set

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.

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,

and blue Each color of the flag represents 1

3 of the whole flag.

number of red partsnumber of colors in flag = 1

3

Notice that the “whole” is the area of all of the flag Each

color, or 1

3 , represents an equal area of the flag.

A fraction can also name part of a set

In a chess set that contains 32 pieces, 16 of the pieces are

pawns Among all the chess pieces, 16 _

32 are pawns.

16 _

32 =

number of pawns _

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

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Lesson 1-1 Parts of a Whole and Parts of a Set 5

Example 2

Write a fraction that represents the number

of circles in the set.

1 There are 7 objects in the set

What is this number called?

2 How many turtles are in the set?

3 Write the fraction

Numerator

Denominator

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

2 How many parts are shaded?

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9

Toshi

5

9

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 suns in set

number of objects in the set =

3 Draw a picture to model the fraction 2

3 Use equal parts of

Draw a figure Divide

it into 3 equal parts

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GO ON

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

Step 3 Write the fraction: numerator

denominator =

Step by Step Practice

Draw a picture to model the fraction Use equal parts of a whole.

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?

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Solve

meeting She gave 5 friends 1 cookie each She had

2 cookies left What fraction of the cookies was left?

Understand Read the problem Write what you know

Five friends received cookie each

There are 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

whole = numerator

denominator =

Check Look at the numbers in the problem Does the

solution answer the question? Is it reasonable?

Step by Step Problem-Solving Practice

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

His 19 students ate 1 bar each What fraction of the fruit bars

was eaten?

Problem-Solving Strategies

✓ Draw a picture

Look for a pattern.

Guess and check.

Act it out.

Work backward.

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13 Explain how the word not affected your answer for

Exercise 12

Skills, Concepts, and Problem Solving

Write a fraction to represent each shaded region

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Solve.

and $32 on pants What fraction of the gift certificate did he use?

the right What fraction of the questions did she answer

incorrectly?

23 out of 25 correct!

2

5 are females How many female puppies are there?

How many puppies are there altogether?

Vocabulary Check Write the vocabulary word that completes

each sentence

30 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

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Lesson 1-2 Recognize, Name, and Compare Unit Fractions 11

7

numerator

how many equal parts are being used

3

how many equal parts there are in the whole set 3

The pizzas shown are the same size, but are cut into pieces

that are different sizes

one piece = 1

6 one piece = 1

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

6 > 1

8

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.

6 < 8 so the unit fraction 1 is greater

1 unit

When you divide something into equal pieces, the more pieces you

divide it into, the smaller each piece will be That is why 1

6 is greaterthan 1

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.

GO ON

Denominator

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_ 4 , and _ 1

1 Make one cut to create two equal parts Shade one part

The shaded part is the unit fraction 1

2

2 Cut each half into two equal parts so that there are

four equal parts in all Shade (in a different color) one

part to show the unit fraction 1

4

3 Cut each fourth into two equal parts so that there are

eight equal parts in all Shade (in a different color) one

part to show the unit fraction 1

8

YOUR TURN!

Show where you make cuts to create the

unit fractions _ 1 2 , _ 1 4 , and _ 1 8

1 unit

1 Make one cut to create two equal parts Shade one part

What unit fraction does the shaded part represent?

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?

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There are equal parts

The unit fraction is

GO ON

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2 Write a comparison statement 3 < 4

for the denominators

Recall that the smaller denominator

makes the greater unit fraction

1

3

1 4

3 Write a comparison statement 1

3 > 1

4 for the fractions

YOUR TURN!

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

1

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2 Show a unit fraction in each circle Name each unit fraction Which

unit fraction is greater?

3 Show where you make cuts to create the unit fractions 1

5 and 1 _

10 Shade each unit fraction

Step 1 The unit fraction is 1

5 How many parts do 1 unityou need to create?

Step 2 Show the cuts using lines Shade one

part What unit fraction is shaded?

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.

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Solve.

5 of a book Manu has read 1

8

of the same book Who has more left to read?

Tina has read of the book Manu has read of the book

Plan Pick a strategy One strategy is to draw a diagram Draw a rectangle to

represent the whole book Show 1

5 and 1

8 on separate rectangles

Solve The shaded parts represent the part of the

book already read

1

5 is larger than 1

8 has read more of the book

has more of the book left to read

Step by Step Problem-Solving Practice

of the cornfield has been picked The fields are the same size Which field

has had more harvested? Check off each step

Understand

Plan

Solve

Check

13 In your own words, explain how to compare unit fractions

Show where you make cuts to create each unit fraction.

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

3 of his daily fruit servings at breakfast

Yancy eats 1

4 of his daily fruit servings at breakfast Who eats more

of his fruit servings at meals other than breakfast?

includes 1

2 white paper, 1

3 blue paper, and 1

9 pink paper Of which type of paper does Mr Barber order the least amount?

Vocabulary Check Write the vocabulary word that completes each

sentence.

26 In a unit fraction, the is always 1

27 When you compare unit fractions, look at the

28 Writing in Math Examine the

unit fractions at the right In your

own words, what can you say is true

when the denominator of a unit

fraction increases? Use the words

denominator and increases

Spiral Review

SCHOOL Mark scored _ 18 20 on a quiz

29 How many points was the quiz worth?

30 How many points did he get? (Lesson 1-1, p 4)

1 4

1 8

1 3

1 6

1 5

1 7

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of her swim does one length represent?

got one apple What fraction of Bo’s apples did each person get?

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Lesson 1-3 Representing Fractions 19

in a fraction that tells how many equal parts are being used (Lesson 1-1, p 4)

denominator

in a fraction that tells how many equal parts are

in the whole or set

(Lesson 1-1, p 4)

KEY Concept

2NS4.3 Know that when all fractional parts are included, such

as four-fourths, the result is equal

to the whole and to one.

4NS1.7 Write the fraction represented

by a drawing of parts of a figure; represent a given fraction by using drawings; and relate a fraction to a

simple decimal on a number line.

Representing Fractions

When the numerator and denominator of a fraction are equal,

the fraction equals 1

Example 1

Use _ 1 5 fraction tiles to form a rectangle

equal to 1 Write the fraction that equals 1.

1 8

1 How many 1

8 pieces does it take

to equal one whole circle?

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Example 2

Write the unit fraction that represents the

shaded region Then write a fraction with

the same denominator that equals 1.

1 Six wedges form a whole

1 How many pieces form the entire strip?

2 What is the unit fraction?

6 = 1

Sal

6

6 = 1

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

Guided Practice

Write the unit fraction that represents the shaded region or part of the

set Then write a fraction with the same denominator that equals 1.

1 number of shaded parts

2

number of shaded parts

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3 Use 1

8 fraction tiles to form a rectangle equal to 1 Write the

fraction that equals 1

1

1 8 Step 1 Line up the left edge of the rectangle that represents

1 with a fraction tile

Use each given size of fraction tile to form a rectangle equal to 1

Write the fraction that equals 1.

Write the unit fraction that represents the shaded region or part of the

set Then write a fraction with the same denominator that equals 1.

GO ON

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Solve Use fraction tiles or fraction circles.

brother ate 2 biscuits The 3 other people ate one biscuit

each There were no biscuits left over Write a fraction to

represent one biscuit

Natalie’s brother ate biscuits

3 other people ate biscuit each

There were biscuits left over

Plan Pick a strategy One strategy is to use a model

biscuits her biscuits biscuitsbrother ate others ate in allUse 1 fraction tile to model the whole batch of biscuits Use 1

5 tiles to model the biscuits her brother ate

Look for a pattern.

Guess and check.

Solve a simpler problem.

Work backward.

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7 of a book report last night He wrote

6 pages How many pages will Rico’s completed report be?

Check off each step

Understand Plan

Solve Check

four-square at recess Each day, only 8 students can play Mr Thad made

a schedule so one player sits out each day What fraction of

the students sits out each day?

upcoming blood drive Tamika and Jacob each hand out 1

8 of the cards More volunteers are needed so that each person does the

same amount of work as Jacob and Tamika How many more

volunteers are needed?

15 What can you say is true about every unit fraction?

Be sure to use the words numerator and denominator

Write the unit fraction that represents each shaded region or part of

the set Then write a fraction with the same denominator that equals 1.

GO ON

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Use each given size of fraction tile to form a rectangle equal to 1

Write the fraction that equals 1.

Use fraction circles to identify each unit fraction shown How many of

these fractions equal 1?

morning he found a rabbit eating one of the plants What fraction

of lettuce plants did the rabbit eat?

the truck so that packages in the back half of the truck get

delivered before lunch Packages in the front half of the truck are

delivered after lunch What fraction of packages does Keshia

deliver by the end of the day?

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Vocabulary Check Write the vocabulary word that completes each

36 Writing in Math Christine was asked to divide a circle into thirds

and show the unit fraction 1

3 Christine drew the picture below

She said since the denominator is 3, the shaded region represents 1

3 What is the problem with Christine’s picture of the unit fraction

1

3 ? Use the circle next to Christine’s to show how to correctly shade

the unit fraction 1

8 nuts Whose trail mix had more nuts? (Lesson 1-2, p 11)

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Write the vocabulary word that completes each sentence.

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

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

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

4 The number that represents how many equal parts

of the object are being used is called the

5 A(n) has a numerator equal to 1.

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

Lesson Review

1-1 Parts of a Whole and Parts of a Set (pp 4–10)

Write a fraction to represent

each situation.

8 the shaded region of the hexagon

9 the number of hearts in the set

Example 1

Write a fraction to represent the shaded region of the rectangle.

There are 12 equal parts

of the whole This number

is the denominator

There are 3 parts shaded

This number is the numerator

Write the fraction _ 123

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Chapter 1 Study Guide 27

1-3 Representing Fractions (pp 19–25)

Use each given size of fraction

tile to form a rectangle equal

to 1 Write the fraction that

denominators 2 < 3Recall that the smaller denominator makes the greater unit fraction Write a comparison statement for the fractions 1_

2 > _ 1

3

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trumpet players What fraction does this represent?

fraction does this represent?

Correct the mistakes

19 Eliza had a pizza that was cut into 8 equal slices When Eliza

asked her friend Cora how much pizza she would like, Cora said

one-fourth Eliza gave her four slices How many slices should

Eliza have given Cora?

20 When asked what fraction of letters in Sacramento are a’s, Carol

answered 2 What was Carol’s mistake?

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Chapter

GO ON

1 Selma shaded one section for each day

she exercised Which fraction of these

days did she exercise?

5 The Tigers beat the Royals in two out

of three baseball games Which shows the fraction of the games the Tigers won?

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