About the MathProfessional Development

539A Chapter 9

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LESSON AT A GLANCE

Professional Development Videos

Modeling Equivalent FractionsEquivalent fractions are fractions that represent the same area or number. The concept of equivalent fractions can be presented by having students compare models with different denominators that have the same areas shaded. Students learn that equal areas show equivalent fractions. The number of equal parts can change as long as the total area remains the same.

Students also explore equivalent fractions on a number line. They learn that two fractions are equivalent if the fractions are at the same point on a number line. The three models all show that 3 _ 4 is equivalent to 6 _ 8 .

Investigate • Model Equivalent Fractions

LESSON 9.6

Learning ObjectiveModel equivalent fractions by folding paper, using area models, and using number lines.

Language ObjectiveStudent pairs illustrate and write captions showing how you can use models to find equivalent fractions.

MaterialsMathBoard, sheet of paper, crayon or color pencil, Fraction Strips (see eTeacher Resources)

F C R Focus:Common Core State Standards

3.NF.A.3a Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.

Also 3.NF.A.1, 3.NF.A.2a, 3.NF.A.2b, 3.NF.A.3, 3.NF.A.3b, 3.NF.A.3c, 3.G.A.2

MATHEMATICAL PRACTICES (See Mathematical Practices in GO Math! in the Planning Guide for full text.)MP4 Model with mathematics. MP5 Use appropriate tools strategically.MP7 Look for and make use of structure.

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

Grade 33.NF.A.3a

After4.NF.A.1

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

ENGAGE1Daily Routines

Common Core

Lesson 9.6 539B

How can you use models to f ind

equivalent fractions?

with the Interactive Student Edition

Essential QuestionHow can you use models to find equivalent fractions?

Making ConnectionsInvite students to tell you what they know about creatures that hop.

Can you name some insects that hop or jump? grasshoppers, crickets, fleas

Can you name some animals that hop or jump? frogs, rabbits, kangaroos, flying squirrels, dolphins

Learning ActivityAsk students to restate the problem in their own words.

• What is going on in this problem? Gable is hopping to his friend’s house but stops before he gets there.

• What fraction represents how far Gable has hopped? 2 __ 3

• What is the question you have to answer? What fraction is equivalent to 2 __

3 ?

Literacy and MathematicsHave students work together in pairs.

1. Have one student in each pair use a ruler to draw a number line scaled from 0 to 1, divided into thirds.

2. Have that student draw a point that shows 2 _ 3 .

3. Have the other student draw a circle and divide it into thirds, shading 2 _ 3 of it.

4. Then, have the partners explain why both models show 2 _ 3 .

Vocabulary Builder

Materials Word Web (see eTeacher Resources)

Have students make a word web for the term equivalent. Students should generate words that relate to each other structurally by writing words with the same root or the same suffix or prefix on the arms of the graphic.

Problem of the Day 9.6Juanita has 56 stuffed animals. She wants to display them on 8 shelves. Write a multiplication number sentence that can help Juanita divide the animals equally.

Vocabulary equivalent, equivalent fractions

8 × 7 = 56 or 7 × 8 = 56

Interactive Student EditionMultimedia Glossary e

EXPLORE2

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InvestigateInvestigate

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Chapter 9 539

Model Equivalent FractionsEssential Question How can you use models to find equivalent fractions?

Lesson 9.6

Materials ■ sheet of paper ■ crayon or color pencil

Two or more fractions that name the same amount are called equivalent fractions. You can use a sheet of paper to model fractions equivalent to 1 _ 2 .

A. First, fold a sheet of paper into two equal parts. Open the paper and count the parts.

There are _ equal parts. Each part is _ of the paper.

Shade one of the halves. Write 1 _ 2 on each of the halves.

B. Next, fold the paper in half two times. Open the paper.

Now there are _ equal parts. Each part is

_ of the paper.

Write 1 _ 4 on each of the fourths.

Look at the shaded parts. 1 _ 2 = ___ 4

C. Last, fold the paper in half three times.

Now there are _ equal parts. Each part is

_ of the paper.

Write 1 _ 8 on each of the eighths.

Find the fractions equivalent to 1 _ 2 on your paper.

So, 1 _ 2 , ___ , and ___ are equivalent.

Number and Operations—Fractions—3.NF.A.3a Also 3.NF.A.1, 3.NF.A.2a, 3.NF.A.2b, 3.NF.A.3, 3.NF.A.3b, 3.NF.A.3c, 3.G.A.2

MATHEMATICAL PRACTICESMP4, MP5, MP7

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2 __ 3 5 ___ 6 3 __ 4 5 ____ 8

1 __ 2 5 ____ 8

Shade the model. Then divide the pieces to find the equivalent fraction.

1. 2.

3. 4.

Model Equivalent Fractions

Equivalent fractions are two or more fractions that name the same amount.

You can use fraction circles to model equivalent fractions.

Find a fraction that is equivalent to 1 _ 2 . 1 _ 2 5 __

4

Step 1 Look at the first circle. It is divided into 2 equal parts. Shade one part to show 1 _ 2 .

Step 2 Draw a line to divide the circle into 4 equal parts because 4 is the denominator in the second fraction.

Step 3 Count the number of parts shaded now. There are 2 parts out of 4 parts shaded.

1 _ 2 5 2 _

4 So, 1 _

2 is equivalent to 2 _

4 .

Lesson 9.6Reteach

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9-15 ReteachChapter Resources© Houghton Mifflin Harcourt Publishing Company

Name

Name Equivalent Fractions

For each of the following shapes, shade some of the parts. Write the fraction that represents the parts you shaded. Then use the shape to write an equivalent fraction for the parts you shaded.

1.

Fraction:

Equivalent Fraction:

2.

Fraction:

Equivalent Fraction:

3.

Fraction:

Equivalent Fraction:

4.

Fraction:

Equivalent Fraction:

5.

Fraction:

Equivalent Fraction:

6.

Fraction:

Equivalent Fraction:

7. Stretch Your Thinking Draw a model that shows 3 __ 3 shaded.

Then use your drawing to find two equivalent fractions.

Lesson 9.6Enrich

Check students’ drawings. Possible answer: 1 _ 1 , 6 _ 6

2 _ 8

1 _ 4

1 _ 3

2 _ 6

6 _ 6

2 _ 2

1 _ 4

2 _ 8

6 _ 8

3 _ 4

Possible answers are given.

1 _ 2

2 _ 4

9-16 EnrichChapter Resources© Houghton Mifflin Harcourt Publishing Company

DifferentiatedInstruction1

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539 Chapter 9

3.NF.A.3a Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.

Enrich 9.6Reteach 9.6

Investigate MATHEMATICAL PRACTICES

MP5 Use appropriate tools strategically.Work together with students to complete the steps of the activity in Part A. Be sure students fold their papers in two equal parts and label each half.• What do the two halves equal? one whole Have students follow the steps to fold and label the paper as directed in Part B.• Describe how one half of the paper is now

labeled. One half is divided into two equal parts. Each part is labeled one fourth.

Have students follow the steps to fold and label the paper as directed in Part C. Students can label the eighths on the back side of the folded paper if the paper is too crowded. You may also suggest that students label the eighths by writing the fractions smaller.• Describe how one fourth of the paper is

now labeled. One fourth is divided into two equal parts. Each part is labeled one eighth.

• How many one-eighth parts are in one half? 4

Guide students to name the equivalent fractions and to complete the statement.MP7 Look for and make use of structure.• Look at the fractions 1 _ 2 , 2 _ 4 , and 4 _ 8 . What do

you notice about the relationship between the numerator and denominator. Possible answer: For each fraction, the denominator is twice the numerator.

ELL Strategy: Scaffold LanguageStudents can use sentence frames to support them in describing how to solve a problem.• As students work through problems to find

equivalent fractions, have them describe the process to a partner.

• Write the sentence frame on the board for students to use. First, I drew a and divided it into pieces. Then, I shaded

pieces. That’s how I know that and are equivalent.

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LESSON 9.6

COMMON ERRORS

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MathTalk MATHEMATICAL PRACTICES 2

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N1. Explain how many 1 _ 8 parts are equivalent to

one 1 _ 4 part on your paper.

2. SMARTER What do you notice about how the numerators changed for the shaded part as

you folded the paper? _____

What does this tell you about the change in the

number of parts? ______

How did the denominators change for the shaded

part as you folded? ______

What does this tell you about the change in the

size of the parts? ______

N

You can use a number line to find equivalent fractions.

Find a fraction equivalent to 2 _ 3 .

Materials ■ fraction strips

STEP 1 Draw a point on the number line to represent the distance 2 _ 3 .

STEP 2 Use fraction strips to divide the number line into sixths. At the end of each strip, draw a mark on the number line and label the marks to show sixths.

STEP 3 Identify the fraction that names the same point as 2 _ 3 . _

So, 2 __ 3

= ___ 6

.

Two or more numbers that have the same value or name the same amount are equivalent.

Make ConnectionsMake Connections

Use Reasoning Explain how the number of sixths in a distance on the number line is related to the number of thirds in the same distance.

Possible explanation: there are two 1 _ 8 parts in one 1 _ 4 part,

so 2 _ 8 = 1 _ 4 .

The numerators got larger.

The number of parts increased.

They doubled.

The size of the parts got smaller.

The number of sixths is double the number of thirds.

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4 _ 6

NDraw ConclusionsDraw Conclusions

Advanced Learners

Lesson 9.6 540

Error Students may think that fractions with the same numerator are equivalent.

Example In Exercise 1, students think that 1 _ 8 = 1 _ 4 .Springboard to Learning Have students outline 1 _ 4 part on their paper. Then, instruct students to shade the two 1 _ 8 parts. Have students point to each 1 _ 8 part and count “1 eighth, two eighths.”

Draw ConclusionsHave students read the Math Idea before completing the exercises. Discuss the definition for the term equivalent and have students provide examples of equivalent numbers. Have students refer to their folded papers to explain the answer for Exercise 1 and to extend their thinking to complete Exercise 2.

SMARTER

• How do you know that 2 _ 8 equals 1 _ 4 ? Explain. Possible explanation: the two eighths parts are the same size as the 1 _ 4 part.

• 3 _ 6 and 1 _ 2 are equivalent fractions. Compare the number of parts, and the sizes of the parts, in each fraction. 3 _ 6 will have more parts and they will be smaller than the part for 1 _ 2 .

Make Connections Hands

On

Review with students how they used fraction strips in Chapter 8 to divide a number line into equal parts. • How can you identify equivalent fractions

on a number line? Equivalent fractions name the same point on the number line.

• Use your number line to find another pair of equivalent fractions. Explain how you found your answer. Possible answer: 1 _ 3 and 2 _ 6 ; I used thirds and sixths to name the same distance from 0 on the number line.

MathTalk Use Math Talk to focus on

students’ understanding of equivalent fractions on a number line.

• What number do you multiply by 3 to get 6? 2

• How many sixths equal one-third? 2

• Present this problem to students:

There were three sandwiches of the same size. Mark ate 1 _ 2 of the first one. The second sandwich was cut into 8 equal pieces. Amy ate 4 pieces. The third sandwich was cut into 6 equal pieces. Lola ate 3 pieces. How could you cut another sandwich into equal pieces so that David could eat an equivalent amount?

• Ask students to draw a picture to solve the problem.

• Have students write to explain how they solved the problem and label their drawings to illustrate their explanations. Check students’ work.

Logical / MathematicalIndividual / Partners

EXPLAIN3

Quick Check

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Rt I RR1

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Share and ShShare and ShShare and Show MATHBOARDMATHBOARD

Personal Math Trainer

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Chapter 9 • Lesson 6 541

Shade the model. Then divide the pieces to find the equivalent fraction.

Use the number line to find the equivalent fraction.

5. MATHEMATICALPRACTICE 6 Explain why 2 _ 2 = 1.

Write another fraction that is equal to 1. Draw to justify your answer.

1. 2.

3.

1 _ 2 = ___ 6

4.

3 _ 4 = ___ 8

1 _ 4 = ___ 8 2 _ 3 = ___ 6

6. SMARTER For numbers 6a–6d, select True or False to tell whether the fractions are equivalent.

6a. 6 _ 6 and 3 _ 3 True False

6b. 4 _ 6 and 1 _ 3 True False

6c. 2 _ 3 and 3 _ 6 True False

6d. 1 _ 3 and 2 _ 6 True False

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Possible explanation: 2 _ 2 = 1 because

the numerator names both parts in the

whole; possible answer: 8 _ 8 .

Possible shadings are shown.

Drawings may vary.

2 __ 2 1 8 __ 8

ELABORATE4

541 Chapter 9

MP6 Attend to precision. Exercise 5 focuses on fractions equal to 1.• Describe how all fractions that name

a whole will be alike. The numerator and denominator will be the same number, because the numerator will name all the parts in the whole.

Personal Math Trainer SMARTER

Be sure to assign this problem to students in the Personal Math Trainer. It features a video to help them model and answer the problem. Students need to be able to identify equivalent fractions on the number line. They should understand that each mark on the number line that has a fraction both above and below it represents an equivalence between those two fractions. Students who do not correctly select the first choice may not understand that both of the fractions represent one whole. Other errors may be clarified if students discuss how the number line shows thirds as well as sixths.

Share and ShowThe first problem connects to the learning model. Have students use the MathBoard to explain their thinking. Discuss with students how drawing lines in Exercises 1 and 2 is like tracing the folds from the paper-folding investigation.Use the checked exercises for Quick Check. Students should show their answers for the Quick Check on the MathBoard.

a student misses the checked exercises

Differentiate Instruction with • Reteach 9.6 • Personal Math Trainer 3.NF.A.3a • Rtl Tier 1 Activity (online)

HandsOnMATH

BOARDMATHBOARDMBMMMBBBMATHABOARDMMMAAATHATHTHHAAAAAAAAATTAAAABOARDBOARDBOARD

Problem Solving • Applications

Differentiated Centers Kit

DIFFERENTIATED INSTRUCTION INDEPENDENT ACTIVITIESD

EVALUATE5 Formative Assessment

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Are all the fractions equivalent? _

Which fractions are equivalent? ___ = ___

So, __ and __ ate the same amount of a sandwich.

SummarizeYou can summarize the information in a problem by underlining it or writing the information needed to answer a question.

Read the problem. Underline the important information.

7. SMARTER Mrs. Akers bought three sandwiches that were the same size. She cut the first one into thirds. She cut the second one into fourths and the third one into sixths. Marian ate 2 pieces of the first sandwich. Jason ate 2 pieces of the second sandwich. Marcos ate 3 pieces of the third sandwich. Which children ate the same amount of a sandwich? Explain.

The first sandwich was cut

into __.

The second sandwich was

cut into __.

The third sandwich was

cut into __.

Marian ate _ pieces of the sandwich. Shade the part Marian ate.

Jason ate _ pieces of the sandwich. Shade the part Jason ate.

Marcos ate _ pieces of the sandwich. Shade the part Marcos ate.

Marian ate ___ of the

first sandwich.

Jason ate ___ of the second

sandwich.

Marcos ate ___ of the third

sandwich.

thirds

Possible shading is shown.

fourths sixths

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no

Jason Marcos

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

Lesson 9.6 542

Connect to Reading

SummarizeSummarizing is a strategy that can help students with reading comprehension. Have students read the text and complete the graphic organizer to summarize information from the problem. Then have students use the graphic organizer to find the equivalent fractions and solve the problem.

SMARTER

• How do you know which fractions are equivalent? Possible answer: they name the same amount in the sandwich models.

Essential QuestionUsing the Language ObjectiveReflect Have student pairs illustrate and write captions to answer the Essential Question.How can you use models to find equivalent fractions? Possible answer: I can fold my paper in half and label one part 1 _ 2 . Then I can fold the paper again to make more equal parts. I can label each of the folded parts and compare the parts to find equivalent fractions.

Math Journal WRITE MathDraw a number line that shows two equivalent fractions. Label your number line and explain how you know the fractions are equivalent.

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

Students complete blue Activity Card 11 by finding

fractional parts of a group of pattern blocks.

ActivitiesFraction Action

LiteraturePizza Parts!

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

Extend the Math Activity

Problem SolvingProblem Solving

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Model Equivalent Fractions

Shade the model. Then divide the pieces to find the equivalent fraction.

Use the number line to find the equivalent fraction.

1.

2 _ 4

= ___ 8

2.

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3 _ 4

= ___ 8

3. Mike says that 3 _ 3 of his fraction model is shaded blue. Ryan says that 6 _ 6 of the same model is shaded blue. Are the two fractions equivalent? If so, what is another equivalent fraction?

4. Brett shaded 4 _ 8 of a sheet of notebook paper. Aisha says he shaded 1 _ 2 of the paper. Are the two fractions equivalent? If so, what is another equivalent fraction?

Chapter 9 543

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Lesson 9.6Practice and Homework

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

5. WRITE Math Draw a number line that shows two equivalent fractions. Label your number line and explain how you know the fractions are equivalent.

Check students’ shadings.

Yes; possible answer: 4 _ 4 Yes; possible answer: 2 _ 4

6

Check students work.

543 Chapter 9

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.

Use Equivalent Fractions to Compare FractionsMaterials Fraction Circles (see eTeacher Resources)

Investigate Have students compare 3 _ 4 and 5 _ 8 using equivalent fractions. Have students start with a fourths fraction circle and ask them to shade 3 _ 4 . Then have them draw 2 lines to divide fourths into eighths. Discuss how many 1 _ 8 pieces are equivalent to 3 _ 4 .

Six 1 _ 8 pieces 5 3 _ 4 , so 6 _ 8 5 3 _ 4 .

Explain that now students can compare fractions with the same denominator to solve the problem.

• Which is greater, 6 _ 8 or 5 _ 8 ? 6 _ 8

• So, which fraction is greater, 3 _ 4 or 5 _ 8 ? Explain. 3 _ 4 is greater because 6 _ 8 5 3 _ 4 and 6 _ 8 is greater than 5 _ 8 .

Summarize How can you use equivalent fractions to compare two fractions with different denominators? Possible answer: I can find an equivalent fraction for one of the fractions that has the same denominator as the other fraction. I can then compare the equivalent fraction and the other fraction using numerators.

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Personal Math Trainer

FOR MORE PRACTICE GO TO THE

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

Spiral Review (3.OA.A.3, 3.OA.C.7, 3.NF.A.1)

5. There are 56 students going to the game. The coach puts 7 students in each van. How many vans are needed to take the students to the game?

6. Write a division equation for the picture.

3. Eric practiced piano and guitar for a total of 8 hours this week. He practiced the piano for 1 _

4 of that

time. How many hours did Eric practice the piano this week?

4. Kylee bought a pack of 12 cookies. One-third of the cookies are peanut butter. How many of the cookies in the pack are peanut butter?

1. Name a fraction equivalent to 2 _ 3

.

2. Find the fraction equivalent to 1 _ 4

.

Possible answer: 4 _ 6

2 hours

8 vans

2 _ 8

4 cookies

8 ÷ 2 = 4 or 8 ÷ 4 = 2

Lesson 9.6 544

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.