533A Chapter 9

About the MathProfessional Development

LESSON AT A GLANCE

Interactive Student Edition

Personal Math Trainer

Math on the Spot Video

iTools: Fractions

Professional Development Videos

Teaching for DepthIn this lesson, students apply what they have learned about comparing fractions to ordering fractions. Students use fraction strips and are taught to look at elements of the fractions that can be used to compare and order. For example, students apply the reasoning skills they have already learned to compare fractions to ordering unit fractions. Since all unit fractions have the numerator 1, students can use reasoning to compare the denominators and order the fractions.

Students order three fractions by comparing two fractions at a time to determine the greater and lesser fraction. They can then use the results of that comparison to compare one of the first two fractions to the third fraction. By comparing in this way, students identify the greatest and least fractions and can then order the third fraction between the greatest and least fractions.

Compare and Order Fractions

LESSON 9.5

Learning ObjectiveCompare and order fractions by using models and reasoning strategies.

Language ObjectiveStudents do a quick write in their Math Journal explaining how you can compare and order fractions.

MaterialsMathBoard, color pencils

F C R Focus:Common Core State Standards

3.NF.A.3d Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Also 3.NF.A.1

MATHEMATICAL PRACTICES (See Mathematical Practices in GO Math! in the Planning Guide for full text.)MP3 Construct viable arguments and critique the reasoning of others. MP4 Model with mathematics. MP5 Use appropriate tools strategically. MP6 Attend to precision.

F C R Coherence:Standards Across the GradesBefore2.NBT.A.4

Grade 33.NF.A.3d

After4.NF.A.2

F C R Rigor:Level 1: Understand Concepts....................Share and Show ( Checked Items)Level 2: Procedural Skills and Fluency.......On Your Own, Practice and HomeworkLevel 3: Applications..................................Think Smarter and Go Deeper

F C R For more about how GO Math! fosters Coherence within the Content Standards and Mathematical Progressions for this chapter, see page 505J.

FOCUS COHERENCE RIGOR

ENGAGE1

Lesson 9.5 533B

Daily RoutinesCommon Core

1 23 4 Fluency BuilderMaterials Daily Facts Practice Worksheets (see eTeacher Resources), scissors

• Provide groups of students with the Daily Facts Practice Worksheets for division facts through 10. Have students cut out each card.

• Students should shuffle the cards and place them face down in front of the group. Students take turns selecting a card and giving the quotient. If the student is correct, he or she receives 1 point. If the student is incorrect, he or she receives no points.

• A student should be appointed as the scorekeeper. The first student to reach 10 points wins.

How can you compare and order fractions?

with the Interactive Student Edition

Essential QuestionHow can you compare and order fractions?

Making ConnectionsDistribute unit fraction strips to small groups of students. Ask the students to compare the fraction strips. Then, invite students to discuss ordering.

• What does it mean to order? arrange in a certain way

• What are some ways you can order objects? from biggest to smallest or smallest to biggest numbers? from least to greatest or greatest to least

Learning ActivityWhat is the problem the students are trying to solve? Connect the story to the problem.

• What information do you know? the family brought ingredients for tacos; 3 _ 6 cup lettuce, 3 _ 4 cup cheese, 3 _ 8 cup tomatoes

• What do you need to do to solve the problem? compare and order the fractions to see which ingredient the family brought the most of

Literacy and Mathematics• Have students work in small groups to discuss how a recipe

is similar to other organized lists. Have students record an explanation of why using an organized list is important when cooking.

Literature ConnectionFrom the Grab-and-Go™ Differentiated Centers Kit

Students read about how to find equal parts to write fractions.

Pizza Parts!

Problem of the Day 9.5The campers went on a river trip. There were 5 rafts with 4 people in each. There were 2 canoes with 2 people in each. How many people in all went on the trip?

Interactive Student EditionMultimedia Glossary e

24 people

Common Core Fluency Standard 3.OA.C.7

1 23 4 Pages 86–87 in Strategies and Practice for Skills and Facts Fluency provide additional fluency support for this lesson.

EXPLORE2

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Lesson 9.5

Chapter 9 533

So, the distances in order from least to greatest are

_ mile, _ mile, _ mile.

Sierra, Tad, and Dale ride their bikes to school.

Sierra rides 3 _ 4 mile, Tad rides 3 _ 8 mile, and Dale

rides 3 _ 6 mile. Compare and order the distances

from least to greatest.

Compare and Order Fractions Essential Question How can you compare and order fractions?

Activity 1 Order fractions with the same numerator.

Materials ■ color pencil

You can order fractions by reasoning about the size of unit fractions.

STEP 1 Shade one unit fraction for each fraction strip.

_ is the longest unit fraction.

_ is the shortest unit fraction.

STEP 3 Continue shading the fraction strips so that three unit fractions are shaded for each strip.

Are the shaded fourths still the longest? _

Are the shaded eighths still the shortest? _3 _ 4 mile is the __ distance. 3 _ 8 mile is the __ distance. 3 _ 6 mile is between the other two distances.

STEP 2 Shade one more unit fraction for each fraction strip.

Are the shaded fourths still the longest? _

Are the shaded eighths still the shortest? _

• Circle the fractions you need to use.

• Underline the sentence that tells you what you need to do.

• The more pieces a whole is divided into, the smaller the pieces are.

• The fewer pieces a whole is divided into, the larger the pieces are.

Number and Operations—Fractions—3.NF.A.3d Also 3.NF.A.1

MATHEMATICAL PRACTICESMP3, MP4, MP5, MP6

shortestlongest

yes

yes

yes

yes

1 _ 4

1 _ 8

3 _ 8

3 _ 6

3 _ 4

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0 1

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Name

Write the fractions in order from least to greatest.

1. 4 __ 6

, 6 __ 6

, 3 __ 6

2. 2 __ 3

, 2 __ 6

, 2 __ 4

Think: When the numerators are the same, look at the denominators to compare the size of the pieces.

3. 1 __ 4

, 1 __ 8

, 1 __ 2

4. 3 __ 4

, 0 __ 4

, 2 __ 4

You can use a number line to compare and order fractions.

Order 5 _ 8 , 2 _

8 , and 7 _

8 from least to greatest.

Since you are comparing eighths, use a number line divided into eighths.

Step 1 Draw a point on the number line to show 5 _ 8 .

Step 2 Repeat for 2 _ 8

and 7 _ 8

.

Step 3 Fractions increase in size as you move right

on the number line. Write the fractions in order

from left to right.

So, the order from least to greatest is 2 _ 8 , 5 _

8 , 7 _

8 .

Compare and Order Fractions

Lesson 9.5Reteach

0 _ 4 , 2 _

4 , 3 _

4

2 _ 6 , 2 _

4 , 2 _

3

1 _ 8 , 1 _

4 , 1 _

2

3 _ 6 , 4 _

6 , 6 _

6

9-13 ReteachChapter Resources© Houghton Mifflin Harcourt Publishing Company

Name

7. Ashley is another runner, and she has completed 7 __ 8

of

the race. Is she closest to the finish line? Explain your answer.

Race to the Fraction Line

Use the table for 1–7.

Race ResultsRunners Jean Shannon Sally Julie Rachel

Fraction of Race Completed After 30 Minutes

3 _ 8

3 _ 4

1 _ 4

2 _ 4

5 _ 8

1. Who is closest to the finish line? What fraction of the race has she run?

2. Who is farthest from the finish line?

3. List Jean, Shannon, and Sally in order from the closest to the finish line to the farthest.

4. List Shannon, Julie, and Rachel in order from the farthest from the finish line to the closest.

5. List all the fractions of the race completed in order from closest to the finish line to the farthest.

6. List all the runners in order from farthest from the finish line to the closest.

Lesson 9.5Enrich

3 _ 4

, 5 _ 8 , 2 _ 4

, 3 _ 8 , 1 _ 4

Shannon

Julie, Rachel,

Sally, Jean,

Shannon

Julie, Rachel,

Sally

Shannon, Jean,

She is closest to the fi nish line because

7 _ 8 is greater than all the other fractions.

She has only 1 _ 8 of the race left to run.

SallyShannon; 3 _ 4

9-14 EnrichChapter Resources© Houghton Mifflin Harcourt Publishing Company

DifferentiatedInstruction1

2

3

533 Chapter 9

LESSON 9.5

Enrich 9.5Reteach 9.5

Unlock the Problem MATHEMATICAL PRACTICES

In today’s lesson, you will use what you learned about comparing fractions to order fractions.

Activity 1MP4 Model with mathematics. Guide students through the steps as they shade three unit fractions on each fraction strip. Discuss with students that each fraction strip represents the same-size whole.• Can you compare fractions that represent

different-size wholes? Why or why not? No, because when comparing fractions, I need to compare to the same-size whole.

• Which strips represent the greatest fraction? the strips that are the longest

• Which strips represent the least fraction? the strips that are the shortest

Discuss with students that to order fractions, they can compare fractions two at a time. So, they can compare 3 _ 4 and 3 _ 8 to determine 3 _ 4 is greater than 3 _ 8 . Then, they compare 3 _ 4 to 3 _ 6 to determine that 3 _ 4 is the greater fraction. From these two comparisons, students can then use reasoning to conclude that 3 _ 4 is the greatest fraction.MP2 Reason abstractly and quantitatively.• When fractions with the same numerators

are ordered from least to greatest, what do you notice about the denominators? The numbers go from greatest to least.

• Based on this, would the fraction 3 __ 10 be greater than or less than 3 _ 8 ? less than

ELL Strategy: Model Concepts

Students solidify the concept that the larger the denominator, the smaller the size of the fraction.• Have students use 3 paper strips of equal

size. Model how to fold into fourths, sixths, and eighths. Label each strip with the fraction.

• Place the strips side by side and have pairs ofstudents discuss the pattern using a sentence frame. As the denominator gets ,the size of the fraction gets . For example, is a larger piece than .

3.NF.A.3d Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

HandsOn

COMMON ERRORS

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MathTalk MATHEMATICAL PRACTICES 8

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Shade fraction strips to order 5 _ 8 , 8 _ 8 , and 3 _ 8 from least to greatest.

Try This! Order 2 _ 6 , 2 _ 3 , and 2 _ 4 from greatest to least.

Activity 2 Order fractions with the same denominator.

Materials ■ color pencil

Order the fractions 2 _ 6 , 2 _

3 , and 2 _ 4 by thinking about the

length of the unit fraction strip. Then label the fractions

shortest, between, or longest.

Fraction Unit Fraction Length

2 _ 6

2 _ 3

2 _ 4

• When the numerators are the same, think about the

_ of the pieces to compare and order fractions.

So, the order from greatest to least is _ , _ , _ .

• When the denominators are the same, the size of the pieces is the __.

So, think about the __ of pieces to compare and order fractions.

_ is the shortest. _ is the longest.

_ is between the other two fractions.

So, the order from least to greatest is _ , _ , _ .

Shade 5 _ 8 .

Shade 8 _ 8 .

Shade 3 _ 8 .

Generalize When ordering three fractions, what do you know about the third fraction when you know which fraction is the shortest and which fraction is the longest? Explain your answer.

number

same

Possible explanation: the length of the third fraction is somewhere between the shortest fraction and the longest fraction. It must be longer than the shortest fraction and shorter than the longest fraction.

size

1 _ 6

1 _ 3

2 _ 3

2 _ 4

2 _ 6

1 _ 4

5 _ 8

5 _ 8

8 _ 8

8 _ 8

3 _ 8

3 _ 8

shortest

longest

between

Advanced Learners

Lesson 9.5 534

Try This!Work through the problem with students. If students have difficulty thinking about the size of each unit fraction, encourage them to use the reasoning skills they have learned with the same numerator strategy. If students continue to struggle, allow them to use fraction strips to visualize each unit fraction.Review with students that the longest fraction strip represents the greatest number and the shortest fraction strip represents the least number.

MathTalk Use Math Talk to focus on

students’ understanding of ordering three fractions.

• When you order three fractions, where do you list the fraction that is shown by the middle-length fraction strip? between the least fraction and the greatest fraction

Activity 2MP4 Model with mathematics.Remind students that they are comparing the three fractions to the same-size whole.• Which is greater, 5 _ 8 or 8 _ 8 ? 8 _ 8

• Which is greater, 8 _ 8 or 3 _ 8 ? 8 _ 8

• How do those comparisons help you find the greatest fraction? Since 8 _ 8 is greater than 5 _ 8 and 3 _ 8 , 8 _ 8 is the greatest fraction.

• How can you find the least fraction? Compare 5 _ 8 and 3 _ 8 .

Error Students might order fractions by comparing numbers for the denominators rather than the size of pieces.

Example Students might order 2 _ 4 , 2 _ 8 , and 2 _ 6 from least to greatest as 2 _ 4 , 2 _ 6 , and 2 _ 8 .Springboard to Learning Have students use models to represent each fraction so they can see how the size of the equal parts differs when the denominators differ.

HandsOn

Materials index cards

• Challenge students to order several fractions.

• Write the fractions 2 _ 3 , 1 _ 3 , 1 _ 2 , 1 _ 8 , 5 _ 6 , 7 _ 8 , 3 _ 4 , and 1 _ 6 on separate index cards.

• Have students work in small groups. Give one set of cards to each group. Students should shuffle the fraction cards and place the pile facedown.

• One member of the group draws three cards and turns them over.

• Then the students order the fractions by comparing pairs of fractions.

• Have them repeat the process until all the cards have been drawn and ordered. Check students’ work.

Kinesthetic / VisualSmall Group

EXPLAIN3

Quick Check

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On Your OwnOn Your Own

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Share and ShShare and ShShare and Show MATHBOARDMATHBOARD

MathTalk MATHEMATICAL PRACTICES 5

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Chapter 9 • Lesson 5 535

Write the fractions in order from greatest to least.

Write the fractions in order from least to greatest.

Write the fractions in order from least to greatest.

2. 1 __ 2

, 0 __ 2

, 2 __ 2

_ , _ , _ 3. 1 __

6 , 1 __

2 , 1 __

3 _ , _ , _

4. 6 __ 6

, 2 __ 6

, 5 __ 6

_ , _ , _ 5. 1 __

8 , 1 __

4 , 1 __

2 _ , _ , _

6. SMARTER

6__3

, 6 __ 2

, 6 __ 8

_ , _ , _

7. SMARTER

4 __ 2

, 2 __ 2

, 8 __ 2

_ , _ , _

1. Shade the fraction strips to order 4 _ 6 , 4 _ 4 , and 4_

8 from

least to greatest.

_ is the shortest. _ is the longest.

_ is between the other two lengths. _ , _ , _

8. MATHEMATICALPRACTICE 6 Compare Pam is making biscuits.

She needs 2 _ 6 cup of oil, 2 _

3 cup of water, and 2 _ 4 cup of milk.

Write the ingredients from greatest to least amount.

_ , _ , _

Use a Concrete Model Why does using fraction strips help you order fractions with unlike denominators?

Possible answer: I can line up the fraction strips above each other. The longest set of fraction strips is the greatest fraction, and the shortest length fraction strip is the least fraction.

0 _ 2

1 _ 2

2 _ 2

1 _ 6

1 _ 3

1 _ 2

6 _ 6

5 _ 6

2 _ 6

1 _ 2

1 _ 4

1 _ 8

6 _ 8

6 _ 3

6 _ 2

2 _ 2

4 _ 2

8 _ 2

4 _ 8

4 _ 4

4 _ 4

4 _ 6

4 _ 6

4 _ 8

2 _ 3

2 _ 4

2 _ 6

535 Chapter 9

a student misses the checked exercises

Differentiate Instruction with • Reteach 9.5

• Personal Math Trainer 3.NF.A.3d

• Rtl Tier 1 Activity (online)

On Your OwnIf students complete the checked exercises correctly, they may continue with the On Your Own section.Show students another way to complete Exercise 4 by replacing the commas with the greater than symbol. Then have them use symbols to order the fractions in Exercises 5–8. Students should be able to determine which symbol to use based on key words in the direction lines.

SMARTER

Exercises 6 and 7 require students to analyze a set of numbers to select an appropriate way to order them.MP6 Attend to precision. For Exercise 8, have students underline the last sentence to reinforce they are to order the fractions from greatest to least.

Share and Show

MathTalk Use Math Talk to focus on ordering

fractions with like denominators.• If two fractions have the same denominator,

how can you quickly tell which fraction is greater? Compare numerators. The fraction with the greater numerator is the greater fraction

Use the checked exercises for Quick Check.

HandsOn

ELABORATE4

Differentiated Centers Kit

DIFFERENTIATED INSTRUCTION INDEPENDENT ACTIVITIESD

EVALUATE5 Formative Assessment

24

23, 2

8and

WRITE Math Show Your Work

MATHEMATICAL PRACTICES M

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9. In fifteen minutes, Greg’s sailboat went 3 _ 6 mile,

Gina’s sailboat went 6 _ 6 mile, and Stuart’s sailboat

went 4 _ 6 mile. Whose sailboat went the longest

distance in fifteen minutes?

Whose sailboat went the shortest distance?

10. DEEPER Look back at Problem 9. Write a similar

problem by changing the fraction of a mile each

sailboat traveled, so the answers are different from

Problem 9. Then solve the problem.

11. SMARTER Tom has three pieces of wood.

The length of the longest piece is 3 _ 4 foot. The

length of the shortest piece is 3 _ 8 foot. What

might be the length of the third piece of wood?

12. SMARTER Jesse ran 2 _ 4 mile on Monday,

2 _ 3 mile on Tuesday, and 2 _

8 mile on Wednesday.

Order the fractions from least to greatest.

2 _ 8

2 _ 4

2 _ 3

Gina’s sailboat

Possible problem: In fi fteen minutes, Greg’s sailboat

went 4 _ 4 mile, Stuart’s sailboat went 4 _

8 mile, and Gina’s

sailboat went 4 _ 6 mile. Whose sailboat went the longest

distance in fi fteen minutes? Whose sailboat went the

shortest distance? Greg’s sailboat; Stuart’s sailboat

Answers will vary.

Greg’s sailboat

3 _ 6

footMath on the Spot videos are in the Interactive Student Edition and at www.thinkcentral.com.

Lesson 9.5 536

GamesFraction Action

LiteraturePizza Parts!

Students practice comparing fractions.

Students read about how to find equal parts to write fractions.

Students complete purple Activity Card 11 by using fraction tiles to compare and order fractions.

ActivitiesWho’s the Greatest?

Essential QuestionUsing the Language ObjectiveReflect Have students write a description in their Math Journal to answer the Essential Question.How can you compare and order fractions? Possible answer: if the denominators are the same, you can compare the numerators to order the number of pieces. If the numerators are the same, you can compare the denominators to order the size of the pieces.

Math Journal WRITE MathDescribe how fraction strips can help you order fractions.

MATHEMATICAL PRACTICES

Exercise 9 requires that students identify the greatest fraction and the least fraction.

DEEPER

Exercise 10 asks students to write a similar problem that yields different answers. Guide students to change only the fractions and keep the scenario and questions the same.

SMARTER

SMARTER

Exercise 12 requires that students must be able to order three fractions with like numerators. Students should recognize that they can compare the fractions by comparing the denominators. Students may need a reminder that when numerators are the same, the fraction with the greater denominator has the lesser value. Use fraction strips or drawings to reinforce the comparisons.

Math on the Spot Video TutorUse this video to help students model and solve this type of Think Smarter problem.

Problem Solving • Applications

Extend the Math Activity

Problem SolvingProblem Solving

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Compare and Order FractionsWrite the fractions in order from greatest to least.

Write the fractions in order from least to greatest.

1. 4 _ 4

, 1 _ 4

, 3 _ 4

_, _, _ 2. 2 _ 8

, 5 _ 8

, 1 _ 8

_, _, _

3. 1 _ 3

, 1 _ 6

, 1 _ 2

_, _, _ 4. 2 _ 3

, 2 _ 6

, 2 _ 8

_, _, _

7. Mr. Jackson ran 7 _ 8

mile on Monday.

He ran 3 _ 8

mile on Wednesday and 5 _ 8

mile on Friday. On which day did

Mr. Jackson run the shortest distance?

8. Delia has three pieces of ribbon.

Her red ribbon is 2 _ 4

foot long. Her

green ribbon is 2 _ 3

foot long. Her yellow

ribbon is 2 _ 6

foot long. She wants to use

the longest piece for a project. Which

color ribbon should Delia use?

Chapter 9 537

5. 2 _ 4

, 4 _ 4

, 3 _ 4

_, _, _ 6. 4 _ 6

, 5 _ 6

, 2 _ 6

_, _, _

4 __ 4

3 __ 4

1 __ 4

Think: The denominators

are the same, so compare the

numerators: 4 . 3 . 1.

Lesson 9.5Practice and Homework

COMMON CORE STANDARD—3.NF.A.3d Develop understanding of fractions as numbers.

9. WRITE Math Describe how fraction strips can help you

order fractions.

1 _ 2 1 _

6 1 _

3

5 _ 8 2 _

8 1 _

8

2 _ 3 2 _

6 2 _

8

2 _ 6 4 _

6 5 _

6

Wednesday green

2 _ 4 4 _

4 3 _

4

Possible answer: by placing fraction strips above and below each other, I can quickly see

which strips, therefore which fractions, are smaller and which are larger. I can then place

the strips in order based on their lengths.

Practice and HomeworkUse the Practice and Homework pages to provide students with more practice of the concepts and skills presented in this lesson. Students master their understanding as they complete practice items and then challenge their critical thinking skills with Problem Solving. Use the Write Math section to determine student’s understanding of content for this lesson. Encourage students to use their Math Journals to record their answers.

537 Chapter 9

Compare Fractions Using Number LinesMaterials scissors, Number Lines (see eTeacher Resources)

Investigate Have students use number lines to compare 2 _ 3 , 1 _ 4 , and 3 _ 8 . Provide students with number lines divided into thirds, fourths, and eighths. Show them how to mark the location of 2 _ 3 on the number line divided into thirds. Repeat the process for a number line divided into fourths, plotting 1 _ 4 , and a number line divided into eighths, plotting 3 _ 8 . Students should cut out each number line and vertically align them.

• Which is greater, 2 _ 3 or 1 _ 4 ? 2 _ 3 How do you know? 2 _ 3 is farther from 0 on the number line than 1 _ 4 .

• Which is greater, 2 _ 3 or 3 _ 8 ? 2 _ 3 What can you conclude about 2 _ 3 ? It is the greater fraction.

• Which is less, 1 _ 4 or 3 _ 8 ? 1 _ 4 What is the lesser fraction, and how can you tell? 1 _ 4 ; Possible explanation: 1 _ 4 is closer to 0 on the number line.

• What is the order of the fractions from greatest to least? 2 _ 3 , 3 _ 8 , 1 _ 4

Summarize Suppose you are comparing and ordering 3 fractions with the same denominator. Do you still need to use 3 different number lines? Explain. Possible explanation: no; I can represent all 3 fractions on the same number line if all 3 fractions are divided into the same number of equal parts.

Personal Math Trainer

FOR MORE PRACTICE GO TO THE

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Lesson Check (3.NF.A.3d)

1. Write the fractions in order from least

to greatest.

1 _ 8

, 1 _ 3

, 1 _ 6

2. Write the fractions in order from

greatest to least.

3 _ 6

, 3 _ 4

, 3 _ 8

Spiral Review (3.OA.B.5, 3.NF.A.1, 3.MD.B.3)

3. What fraction of the group of cars

is shaded?

4. Wendy has 6 pieces of fruit. Of these,

2 pieces are bananas. What fraction

of Wendy’s fruit is bananas?

5. Toby collects data and makes a bar

graph about his classmates’ pets.

He finds that 9 classmates have dogs,

2 classmates have fish, 6 classmates

have cats, and 3 classmates have

gerbils. What pet will have the longest

bar on the bar graph?

6. The number sentence is an example

of which multiplication property?

6 × 7 = (6 × 5) + (6 × 2)

1 _ 8 , 1 _

6 , 1 _

3

3 _ 8

dog

3 _ 4 , 3 _

6 , 3 _

8

2 _ 6

Distributive

Continue concepts and skills practice with Lesson Check. Use Spiral Review to engage students in previously taught concepts and to promote content retention. Common Core standards are correlated to each section.

Lesson 9.5 538