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302 Unit 5 Fractions, Decimals, and Percents

Advance PreparationFor Part 1, make a transparency of the chart on Math Masters, page 137. For the optional Readiness

activity in Part 3, each student will need 6 strips of colored paper, each 2" by 8 1

_ 2 ". These can be prepared

ahead of time or during the activity.

Teacher’s Reference Manual, Grades 4–6 pp. 38, 44, 45, 60–74, 88, 89

Key Concepts and Skills• Find equivalent fractions using a length

model. [Number and Numeration Goal 5]

• Compare fractions to the benchmarks 0, 1

_ 2 ,

and 1. [Number and Numeration Goal 6]

• Order fractions from least to greatest.

[Number and Numeration Goal 6]

• Solve fraction number stories using a

number-line model.

[Operations and Computation Goal 4]

• Use fraction sticks to add fractions.

[Operations and Computation Goal 4]

Key ActivitiesStudents use the Fraction-Stick Chart to find

equivalent fractions and compare fractions to

the benchmarks 0, 1

_ 2 , 1, and 1

1

_ 2 . They use

the fraction-stick model to compare pairs of

fractions, find equivalent fractions, and

continue their exploration of fraction addition.

Ongoing Assessment: Recognizing Student Achievement Use journal page 129. [Number and Numeration Goal 6]

Ongoing Assessment: Informing Instruction See page 305.

Key Vocabularybenchmark � fraction stick � equivalent fractions

MaterialsMath Journal 1, pp. 129–132

Study Link 5�2

transparency of Math Masters, p. 137 � slate

� Geometry Template � straightedge (optional)

Playing Fraction Top-ItStudent Reference Book, p. 316

Fraction Cards (Math Journal 2,

Activity Sheets 5–7)

Students practice comparing fractions

by playing Fraction Top-It.

Math Boxes 5�3Math Journal 1, p. 133

Students practice and maintain skills

through Math Box problems.

Study Link 5�3Math Masters, p. 131

Students practice and maintain skills

through Study Link activities.

READINESS

Making Fraction Strips6 strips of colored paper, each 2" by 8

1

_ 2 "

Students use a length model to explore

comparing and ordering fractions by making

fraction strips.

ENRICHMENTExploring a Fraction-Stick ChartMath Masters, p. 130

Students explore relationships between

the numerators and denominators of

equivalent fractions.

ELL SUPPORT

Building a Math Word BankDifferentiation Handbook, p. 142

Students add the term equivalent fractions

to their Math Word Banks.

Teaching the Lesson Ongoing Learning & Practice

132

4

Differentiation Options

Comparing and Ordering Fractions

Objectives To review equivalent fractions; and to provide

experience with comparing and ordering fractions.e

��

eToolkitePresentations Interactive Teacher’s

Lesson Guide

Algorithms Practice

EM FactsWorkshop Game™

AssessmentManagement

Family Letters

CurriculumFocal Points

Common Core State Standards

302_EMCS_T_TLG1_G5_U05_L03_576825.indd 302302_EMCS_T_TLG1_G5_U05_L03_576825.indd 302 2/14/11 1:20 PM2/14/11 1:20 PM

Date Time

Math Message

Decide whether each of these measurements is closer to the benchmark 0, 1

_ 2 , or

1 inch. Circle the closest benchmark.

1. 1

_ 8 inch is closest to . . . 0 inches. 1

_ 2 inch. 1 inch.

2. 15 _ 16

inch is closest to . . . 0 inches. 1 _ 2 inch. 1 inch.

3. 5 _ 8 inch is closest to . . . 0 inches. 1

_ 2 inch. 1 inch.

4. 3 _ 8 inch is closest to . . . 0 inches. 1

_ 2 inch. 1 inch.

5. Explain your solution for Problem 4.

Sample answer: I know that 3 _

8 is only 1 _

8 away

from 4 _ 8 , which is 1

_ 2 . So 3 _ 8 must be closest to 1

_ 2 .

Ordering Fractions

For each problem below, write the fractions in order from least to greatest.

6. 6

_ 8 , 3

_ 8 , 5

_ 8 , 8

_ 8 3

_ 8 ,

5

_ 8 ,

6

_ 8 ,

8

_ 8

7. 2 _ 7 , 2

_ 9 , 2 _ 5 , 2

_ 12

2

_ 12 ,

2 _ 9

, 2 _ 7

, 2 _ 5

8. 2 _ 3 , 1

_ 4 , 1 _ 3 ,

3

_ 4 1 _ 4

, 1 _ 3

, 2 _ 3

, 3

_ 4

9. 3

_ 5 , 4

_ 10 , 9

_ 20 , 1

_ 25 1

_ 25

, 4

_ 10

, 9

_ 20 ,

3

_ 5

10. 3

_ 7 , 1

_ 10 , 7 _ 8 ,

5

_ 7 1

_ 10

, 3

_ 7 ,

5

_ 7 ,

7 _ 8

11. 5

_ 9 , 2 _ 5 , 1

_ 6 , 9

_ 10 1 _ 6

, 2 _ 5

, 5

_ 9 ,

9

_ 10

12. 4 _ 8 , 4

_ 7 , 3

_ 5 , 4 _ 9

4 _ 9

, 4 _ 8

, 4 _ 7

, 3

_ 5

�

Comparing and Ordering FractionsLESSON

5�3

EM3MJ1_G5_U05_121-163.indd 129 1/20/11 8:34 AM

Math Journal 1, p. 129

Student Page

Lesson 5�3 303

Getting Started

1 Teaching the Lesson

▶ Math Message Follow-Up WHOLE-CLASSDISCUSSION

(Math Journal 1, p. 129)

Remind students that a benchmark is a well-known count or measure that can serve as a reference point when estimating. When working with fractions, the benchmarks 0, 1 _

2 , and 1 are often

used. Discuss how students applied their knowledge of numerators, denominators, and benchmarks to tell if each measurement is closest to 0, 1 _

2 , or 1.

Ongoing Assessment: Journal

Page 129

Problem 5 �

Recognizing Student Achievement

Use journal page 129, Problem 5 to assess students’ understanding of the

structure of fractions. Students are making adequate progress if their explanations

correctly represent the relationship between the numerator and denominator and

the use of benchmarks such as 0, 1 _ 2 , and 1.

[Number and Numeration Goal 6]

▶ Ordering Fractions PARTNER ACTIVITY


On the board or a transparency, draw 4 horizontal lines in a row to order the fractions from Problems 1–4 of the journal page. Volunteers explain which of the Math Message fractions is least and which is greatest. 1 _ 8 is the least because it is the closest to 0; 15 _ 16 is the greatest because it is closest to 1. Write these fractions on the first and last lines, respectively. Ask: Since 5 _ 8 and 3 _ 8 are equally close to 1 _ 2 , how do we decide where to write them? Both fractions are eighths, and 5 is more than 3, which means that 3 _ 8 is the smaller fraction so it is closer to 0.

Mental Math and Reflexes Math MessageUse the benchmarks 0,

1

_ 2 ,

and 1 to answer Problems 1 –5 on journal page 129.

Study Link 5�2 Follow-Up Allow students five minutes to compare answers and correct any errors. Have volunteers share their mixed-number problems.

Have students use the rulers on their Geometry Templates.

Find 2 1

_ 2 inches on a ruler. How many half-inches is that? 5

Find 6 cm on a ruler. How many 1

_ 2 cm is that? 12

Find 2 4

_ 8 inches on a ruler. How many quarter-inches is that? 10

Find 3 1

_ 2 cm on a ruler. How many

1

_ 2 cm is that? 7

1

_ 2 inch is what fraction of 2

1

_ 2 inches?

1

_ 5

3

_ 4 inch is what fraction of 2

1

_ 2 inches?

3

_ 10

303-307_EMCS_T_TLG1_G5_U05_L03_576825.indd 303303-307_EMCS_T_TLG1_G5_U05_L03_576825.indd 303 2/14/11 1:26 PM2/14/11 1:26 PM


Student Page


Student Page

Fraction-Stick ChartLESSON

5�3

Date Time

Which is larger:

� 47 �or

� 45 �?

Which is larger:

� 47 �or

� 38 �?

Which is larger:

�1 72 �or

� 46 �?

Which is larger:

1� 23 �

or� 43 �?

Which is closer to 1

� 12 �:1

� 13 �or

1� 25 �

?

Which is

� 25 �closest to:

0or

� 12 �or

1?

Which is

�1 36 �closest to:

0or

� 12 �or

1?

Which is

� 58 �closest to:

0or

� 12 �or

1?

� 23 ��

—6

� 23 ��

—9

� 23 ��

—12

� 34 ��

—8

� 34 ��

—12� 34 �

�—

16

� 21 06 ��

—4

� 18 4��

—4

� 16 1��

—12

1� 35 �

�—

51

� 12 ��

—8

1� 34 �

�—

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128

227

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129

6

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1.

4.

7.

10

.

2.

5.

8.

11

.

3.

6.

9.

12

.

Fill in the blanks. Circle the correct answ

er.

Adding with Fraction SticksLESSON

5�3

Date Time

A whole stick is worth 1.

1. Use the fraction sticks to find equivalent fractions.

a. �18� � —

16b. —

8 � �1126� � —

4c. —

8 � �34� � —

16

d. �12� � —

4 � —8 � —

16e. —

2 � �44� � —

8 � —16

2. Use the fraction sticks to add fractions with the same denominator.

Example: �18� � �

28� � � �

38�

a. �24� � �

14� � �

b. �136� � �1

96� � �

c. �116� � �1

56� � �1

86� � �

3. Use the fraction sticks to add fractions with different denominators.

a. �12� � �

14� � �

b. �12� � �

38� � �

c. �58� � �

14� � �

d. �14� � �

78� � �1

26� � �

�1146�, or �

78�

�1126�, or �

34�

�34�

1682842

126362

� 1

� 8 eighths

� 16 sixteenths

� 2 halves

� 4 quarters

�78�

�2106�, 1�1

46�, or 1�

14�

�34�

�78�


Explain that examining numerators and denominators is the first step when comparing and ordering fractions. Refer students to journal page 129.

Examples:

� Ask students to describe the fractions in Problem 6. The denominators are the same. So we know that all the unit fractions are the same size. Only the number of pieces (the numerators) need to be compared.

� Ask students to describe the fractions in Problem 7. The numerators are the same. Now what do we know? There are the same number of pieces for each fraction. So only the size of the pieces (the denominators) needs to be compared. Remind students that the smaller the denominator is, the larger the piece is.

� Ask students to describe the fractions in Problem 8. There are two different denominators; there are two unit fractions. First compare the unit fractions 1 _ 3 and 1 _ 4 . The smaller denominator is the larger fraction. The remaining fractions are within one piece of 1—that is, 3 _ 4 is 1 _ 4 away from 1, and 2 _ 3 is 1 _ 3 away from 1.

Since 1 _ 3 is greater than 1 _ 4 , that makes 2 _ 3 farther away from 1.

Ask partners to complete the problems on the journal page. When most students are finished, discuss their strategies for Problems 9–12, including the following:

� Problem 9: The least fraction is 1 _ 25 . That leaves the other three to compare. Change each to an equivalent fraction with a denominator of 20.

� Problem 10: One of these fractions is close to 0, one is close to 1, and the other two are close to 1 _ 2 .

� Problem 11: One way to compare a fraction to 1 _ 2 is to see if the numerator is more or less than 1 _ 2 of the denominator. Two of the fractions are greater than 1 _ 2 . Their order is determined because 5 _ 9 is close to 1 _ 2 and 9 _ 10 is close to 1. Of the two fractions that are less than 1 _ 2 , the denominators are close to each other, and one numerator is 2 times the numerator of the other.

� Problem 12: One fraction equals 1 _ 2 (the numerator is 1 _ 2 of the denominator). The others are close to 1 _ 2 , and 4 _ 9 is less than 1 _ 2 .

The other two are greater than 1 _ 2 but only by 1 _ 2 of a piece each—that is, 4 _ 7 is 1 _ 2 of a seventh greater than 1 _ 2 , and 3 _ 5 is 1 _ 2 of a fifth greater than 1 _ 2 . Because fifths are greater than sevenths,

3 _ 5 is greater than 4 _ 7 .

▶ Introducing the

WHOLE-CLASS ACTIVITY

Fraction-Stick Chart(Math Journal 1, p. 130; Math Masters, p. 137)

Use a transparency of the chart from Math Masters, page 137 to demonstrate how to use the Fraction-Stick Chart.

EM3cuG5TLG1_303-307_U05L03.indd 304EM3cuG5TLG1_303-307_U05L03.indd 304 12/4/10 7:59 AM12/4/10 7:59 AM

Lesson 5�3 305

1. Skip-Counting with FractionsPlease refer students to journal page 130. Explain that a fraction stick is a model for the whole, or the ONE, that shows unit fractions for the interval between 0 and 1. Each row of the Fraction-Stick Chart combines 2 fraction sticks to show the interval from 0 to 2, divided into unit fractions for a particular denominator.

Example: The third row shows two sticks, each divided into thirds. There are 6 pieces in this row, each labeled 1 _ 3 . The pieces can be used to count by thirds.

2. Finding Equivalent FractionsThe Fraction-Stick Chart on Math Masters, page 137 can be used to find equivalent fractions. Survey the class for examples of pairs of fractions that name the same part of a whole. Emphasize that these are equivalent fractions and list the examples on the board. For example, 3 _ 6 is equivalent to 1 _ 2 .

Example: Find equivalent fractions for 2 _ 3 .

Step 1: The denominator is 3, so use the thirds stick to locate the fraction 2 _ 3 . Count the pieces from left to right. The right edge of the second piece is 2 _ 3 .

Step 2: Place one edge of a straightedge at 2 _ 3 , that is, along the right edge of the second 1 _ 3 piece. The straightedge should be parallel to the sides of the Fraction-Stick Chart. Now look for other fraction sticks on the chart along the straightedge.

On the sixths stick, the straightedge touches the right edge of a piece. Count the sixths-stick pieces from left to right. The straightedge is at the end of the fourth piece, which is 4 _ 6 . So 4 _ 6 is equivalent to 2 _ 3 .

On the ninths stick, the straightedge touches the end of the sixth piece, which is 6 _ 9 . So 6 _ 9 = 2 _ 3 .

On the twelfths stick, the straightedge touches the end of the eighth piece, which is 8 _ 12 . So 8 _ 12 = 2 _ 3 . The fractions 2 _ 3 , 4 _ 6 , 6 _ 9 , and 8 _ 12 are equivalent. They name the same distance on the Fraction-Stick Chart.

On the other sticks, the straightedge cuts through some of the pieces, so 2 _ 3 cannot be written as an equivalent fraction using the denominator on those sticks.

3. Comparing FractionsThe Fraction-Stick Chart on Math Masters, page 137 can be used to compare fractions, for example, compare 4 _ 9 and 3 _ 8 . (See margin.)

Step 1: The denominator of the first fraction is 9, so use the ninths stick to locate 4 _ 9 . Count the pieces from left to right. The right edge of the fourth piece is 4 _ 9 . Place the straightedge along this edge.

NOTE A fraction stick is a narrow rectangle

divided into pieces that represents fractions.

Sometimes it is helpful to make physical

fraction sticks, as in Part 3 of this lesson.

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17

eighthsninths

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03

23

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43

53

63

The third row of the Fraction-Stick Chart

Ongoing Assessment: Informing Instruction

Watch for students who confuse the fraction

labels with the actual end of the fraction stick.

Remind students that the labels are in the

center of a portion, not at its end. Have

students draw a light pencil line along the

straightedge and use the line instead of the

straightedge to identify equivalent portions.



Student Page


Student Page

Fraction Number StoriesLESSON

5�3

Date Time

Shade the fraction sticks to help you solve these fraction number stories. Write a number model for each story.

1. Chris made pizza dough with �58� cup of white flour and �

14� cup of whole wheat flour.

a. How much flour did he use in all? cup

b. Number model:

2. Sheryl’s puppy weighed 1�12� pounds

when it was born. After two weeks,the puppy had gained �

38� pounds.

a. How much did the puppy weigh after two weeks? pounds

b. Number model:

3. Shade the fraction sticks to solve the number model. Then write a fraction numberstory that fits the number model.

a. �34� � �

58� =

b. Number story:

4. Make up your own fraction number story. Draw and shade fraction sticks to solve it.Write a number model for your story.

a. Number story:

b. Number model:

c. Solution:

1�38�

1�78�

�78�

Answers vary.

1�12� � �38� � 1�78�

�58� � �14� � �78�

Answers vary.

Math Boxes LESSON

5�3

Date Time

3. Where 32 students vacationed, ...

a. what fraction of the students traveled within their state?

b. what fraction traveled to Europe?

c. what fraction traveled to Canada or Mexico?

�1312�

Canadaor Mexico

8

Europe4

anotherstate

6

stayedhome 3

within state11

Vacation Travel

1. Write a 10-digit numeral that has

9 in the tens place,3 in the millions place,5 in the billions place,7 in the hundred-millions place,1 in the thousands place, and6 in all other places.

, , ,

Write the numeral in words.

six hundred ninety-sixhundred sixty-one thousand, sixty-three million, six Five billion, seven hundred

6961663675

2. Write each fraction as a whole number or a mixed number.

a. �147� �

b. �234� �

c. �52� �

d. �98� �

e. �352� � 6�25�

1�18�

2�12�

84�14�

4. Divide.

a. 21�4�9�3� ∑

b. 35�6�2�3� ∑ 17 R28 23 R10

5. Find the following landmarks for this set ofnumbers: 929, 842, 986, 978, 869, 732,898, 986, 900, 899, 986, 920, 842

a. Minimum:

b. Maximum:

c. Mode:

d. Range:

e. Median:

732

900254986986

�342�, or �18�

�382�, or �14�

4

22–24

62 63

59 125

119


Step 2: Locate 3 _ 8 (the right edge of the third piece on the eighths stick). Since 3 _ 8 is to the left of 4 _ 9 , it is less than 4 _ 9 . Conversely, 4 _ 9 is to the right of 3 _ 8 , so it is greater than 3 _ 8 . Ask partners to complete journal page 130. Ask volunteers to explain their solutions.

▶ Adding with Fraction Sticks PARTNER ACTIVITY


The five fraction sticks at the top of journal page 131 provide a length model for fraction work. Note that the denominators are restricted to 1, 2, 4, 8, and 16. This model will be used to formally introduce the addition of fractions. All these problems can be solved visually. The correct amount of shading for any fraction can be decided by using the appropriate fraction stick at the top of the page.

▶ Solving Fraction Number Stories

INDEPENDENT ACTIVITY


Students solve the fraction number stories on journal page 132. Circulate and assist. Encourage students to use the benchmarks 0, 1 _ 2 , and 1 when explaining their solutions and assessing the reasonableness of answers. Discuss the answers and write the strategies on the board.

2 Ongoing Learning & Practice

▶ Playing Fraction Top-It PARTNER ACTIVITY

(Student Reference Book, p. 316; Math Journal 2,

Activity Sheets 5–7)

Students practice comparing fractions by playing Fraction Top-It. Students use the Fraction Cards that were stored in Lesson 5-1.

▶ Math Boxes 5�3



Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 5-1. The skill in Problem 5 previews Unit 6 content.

Writing/Reasoning Have students write a response to the following: Explain how you converted the fractions to mixed numbers in Problem 2. Sample answer: For the whole number part, I found how many groups of the denominator were in the numerator. The fraction part was what was left.

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STUDY LINK

5�3 Fraction-Stick Problems

Name Date Time

Shade the fraction sticks to help you find equivalent fractions.

1. �12� � —8

2. �34� � —16

3. —4 � �28� � —16

Shade the fraction sticks to help you solve the addition problems.

4. �14� � �

34� �

5. �12� � �

28� �

6. �12� � �

34� �

Shade the fraction sticks to help you solve the fraction number stories.

7. Joe was baking a cake. He added �34� cup of white sugar and �

38� cup of brown

sugar. How much sugar did he use in all?

(unit)

8. On the back of this page, write a number story using fractions. Then write anumber model to show how you solved it.

41

4

12

�44� � 1

Practice

59

�68� � �

34�

�54� � 1�

14�

�98�, or 1�

18� cups

9. 3��8�9�1� 10. 6��8�9�1� →

11. 12��8�9�1� → 12. 24��8�9�1� → 37 R374 R3

148 R3297

Math Masters, p. 131

Study Link Master

LESSON

5�3

Name Date Time

Fraction-Stick Chart

1. Using the Fraction-Stick Chart, list all the fractions that are equivalent to �

12�.

a. What pattern do you notice in the numerators for these fractions?Numerators increase by 1 until the last one, which increases by 2.

b. What pattern do you notice in the denominators for these fractions?Denominators increase by 2 until the lastone, which increases by 4.

c. Are the patterns complete?

d. What fraction is missing that would make the pattern complete?

2. Using the Fraction-Stick Chart, list all the fractions that are equivalent to �13�.

a. What pattern do you notice in these fractions?

b. Use this pattern to find the next 3 fractions that are equivalent to �13�.

Numerators increase by 1 and denominators

No

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50�, �1

62�, �1

86�

increase by 3.

Math Masters, p. 130

Teaching Master

Lesson 5�3 307

▶ Study Link 5�3


(Math Masters, p. 131)

Home Connection Students find equivalent fractions, add fractions, and solve fraction number stories using fraction sticks.

3 Differentiation Options

READINESS

SMALL-GROUP ACTIVITY

▶ Making Fraction Strips 15–30 Min

To explore comparing and ordering fractions using a length model, have students make fraction strips. Students cut 6 strips of paper, each 2" by 8 1 _ 2 " and then fold and label them to represent halves, thirds, fourths, fifths, sixths, and eighths. (See margin.)

� To fold into thirds, fold one end in so the doubled parts and the single part are the same size.

� To fold into sixths, fold a strip into thirds and then in half.

� For fifths, fold the two outside edges of the strip in toward the middle but not together. Fold them so the doubled parts and the space between them look like three equal parts. Now fold the doubled parts back the other way at the point where the folded ends first came down.

Have students label the fractions on the folds. As they finish, ask students to fold two strips so only the unit fraction shows and then make a comparison statement. For example, 1 _ 5 is less than 1 _ 3 .

ENRICHMENT PARTNER ACTIVITY

▶ Exploring a Fraction-Stick Chart 15–30 Min

(Math Masters, p. 130)

To apply students’ understanding of fractions, have them explore the relationships between the numerators and denominators of equivalent fractions. When they finish, have them share one of their discoveries.

ELL SUPPORT

SMALL-GROUP ACTIVITY

▶ Building a Math Word Bank 15–30 Min

(Differentiation Handbook, p. 142)

To provide language support for fractions, have students use the Word Bank Template found on Differentiation Handbook, page 142. They write the term equivalent fraction, draw pictures related to the term, and write other related words. See the Differentiation Handbook for more information.

When folded into thirds, the edge of the folded

end divides the rest into halves.

When folded into fifths, the two folded ends are

equal in size to the space in the middle.


Comparing and Ordering Fractions - Everyday Math - Login · 2". These can be prepared ... comparing...

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