compare fractions with the same numerator · 3.nf.a.3d compare two fractions with the same...

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519A Chapter 9

About the MathProfessional Development

LESSON AT A GLANCE

Interactive Student Edition

Personal Math Trainer

Math on the Spot

Animated Math Models

iTools: Fractions

HMH Mega Math

Professional Development Videos

Teaching for DepthIn this lesson, students explore the relationship of the number of fractional parts and the size of each part. Students will apply what they learned about comparing fractions to compare fractions with the same numerator.

Students use their prior knowledge that the number in the denominator tells the number of parts in the whole. When a whole is divided into more parts, the parts are smaller. So, fractions with greater denominators have smaller parts. For example, for two wholes that are the same size, 1 _ 2 is greater than 1 _ 3 because a piece that is 1 _ 2 of the whole is larger than a piece that is 1 _ 3 of the whole.

To compare fractions with a numerator of 1, guide students to visualize the number of parts in each whole. It may help to have them think about sharing with more or fewer friends.

Compare Fractions with the Same Numerator

LESSON 9.3

Learning ObjectiveCompare fractions with the same numerator by using models and reasoning strategies.

Language ObjectiveStudent pairs develop step-by-step directions on how to compare fractions with the same numerator.

MaterialsMathBoard

F C R Focus:Common Core State Standards3.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 PRACTICESMP1 Make sense of problems and persevere in solving them. MP2 Reason abstractly and quantitatively. MP4 Model with mathematics. MP7 Look for and make use of structure.

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 OwnLevel 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 9.3 519B

Daily RoutinesCommon Core

Daily RoutinesCommon Core

How can you compare fractions with the same

numerator?

with the Interactive Student Edition

Essential QuestionHow can you compare fractions with the same numerator?

Making ConnectionsInvite students to talk about what they know about fractions. Ask students the following questions.

Name the parts of a fraction. numerator and denominator What does each part represent? The numerator tells how many equal parts are being counted; the denominator tells how many equal parts are in the whole.

Learning ActivityConnect the story to the problem.

• Describe the problem in your own words. We have to figure out which ant went farther—the one who went 4 _ 6 of the way or the one who went 4 _ 8 of the way.

Literacy and Mathematics• Have students write a paragraph describing how to use models to

compare distances.

Problem of the Day 9.3Todd’s collection of stickers is 3 _ 4 triangles. Draw a model using shapes to represent Todd’s sticker collection.

possible answer:

Vocabulary• Interactive Student Edition• Multimedia Glossary e

Fluency BuilderMental Math Have students practice multiplying. Present problems similar to the following:

• 3 × 2

•30 × 2

• 4 × 3

•40 × 3

• 6 × 2

•60 × 2

• 5 × 3

•50 × 3

• 3 × 3

•30 × 3

• 5 × 5

•50 × 5

• 4 × 5

•40 × 5

• 2 × 2

•20 × 2

Common Core Fluency Standard 3.OA.C.7

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DifferentiatedInstruction1

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MathTalk MATHEMATICAL PRACTICES 1

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Chapter 9 519

Compare Fractions with the Same NumeratorEssential Question How can you compare fractions with the same numerator?

Markos is at Athena’s Cafe. He can sit at a table with 5 of his friends or at a different table with 7 of his friends. The same-size spinach pie is shared equally among the people at each table. At which table should Markos sit to get more pie?

Model the problem.

There will be 6 friends sharing Pie A or 8 friends sharing Pie B.

So, Markos will get either 1 _ 6 or 1 _ 8 of a pie.

• Shade 1 _ 6 of Pie A.

• Shade 1 _ 8 of Pie B.

• Which piece of pie is larger?

• Compare 1 _ 6 and 1 _ 8 .

1 __ 6 1 __

8

So, Markos should sit at the table with _ friends to get more pie.

• Including Markos, how many friends will be sharing pie at each table?

• What will you compare?

1. Which pie has more pieces? _The more pieces a whole is divided into,

the ___ the pieces are.

2. Which pie has fewer pieces? _The fewer pieces a whole is divided into,


Pie BPie A

HandsOn

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

MATHEMATICAL PRACTICESMP1, MP2, MP4, MP7

Make Sense of Problems Suppose Markos wants two pieces of one of the pies above. Is 2 _ 6 or 2 _ 8 of the pie a greater amount? Explain how you know.

6 friends or 8 friends

Possible shading is shown.

the size of a piece of pie at

the two tables

>

5

smaller

larger

B

A

2 _ 6 ; Possible explanation: the

sixths are larger than the eighths,

so 2 _ 6 is greater than 2 _

8 .

1 3

13

13

Dog Owners Cat Owners

Name

Compare. Write ,, ., or 5.

1. 3 __ 4

3 __ 6

2. 1 __ 8

1 __ 6

3. 2 __ 4

2 __ 6

4. 2 __ 3

2 __ 6

5. 4 __ 6

4 __ 8

6. 2 __ 8

2 __ 4

7. 5 __ 6

5 __ 8

8. 1 __ 3

1 __ 4

9. 3 __ 6

3 __ 4

10. 1 __ 3

1 __ 3

11. 3 __ 3

3 __ 4

12. 2 __ 8

2 __ 6


Ryan takes a survey of his class. 1 _ 8

of the class has dogs,

and 1 _ 3 of the class has cats. Are there more dog owners or cat

owners in Ryan’s class?

Compare the fractions. 1 _ 8 1 _

3

Step 1 Divide the first circle into 8 equal parts. Shade 1 _

8 of the circle to show dog owners.

Step 2 Divide the second circle into 3 equal parts. Shade 1 _

3 of the circle to

show cat owners.

Step 3 Compare the shaded parts of the circles. Which shaded part is larger?

1 _ 3 is larger than 1 _

8 . 1 _

8 1 _

3

So, there are more cat owners than dog owners in Ryan’s class.

Lesson 9.3Reteach

,

,

,

.5

..

,..

.,.

9-9 ReteachChapter Resources© Houghton Mifflin Harcourt Publishing Company

Spinner A Spinner B Spinner C

Name

Spin the Wheel of Fractions

Use the spinners for 1–6.

1. Use fractions to compare the white section on Spinner A to the white section on Spinner B.

2. Use fractions to compare the striped sections on Spinner B to the striped sections on Spinner C.

3. Use fractions to compare the gray sections on Spinner B to the gray sections on Spinner A.

4. Use fractions to compare the gray sections on Spinner B to the white sections on Spinner C.

5. Use fractions to compare the striped section and white section combined on Spinner A to the gray sections on Spinner A.

6. Use fractions to compare the white sections on Spinner C to the gray sections on Spinner A.

7. Stretch Your Thinking Draw two spinners that are the same size. Divide each spinner into a different number of equal parts. Color two parts on each spinner red. Then use fractions to compare the red parts on your spinners.

Lesson 9.3Enrich

2 _ 8 , 2 _

4 2 _

4 5 2 _

4

2 _ 6 , 2 _

4 2 _

6 . 2 _

8

3 _ 6 . 3 _

8 1 _

4 . 1 _

6

Check students’ work.

9-10 EnrichChapter Resources© Houghton Mifflin Harcourt Publishing Company

11

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3333

1

2

3 DifferentiatedInstruction

519 Chapter 9

Enrich 9.3Reteach 9.3

LESSON 9.33.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.

Unlock the Problem MATHEMATICAL PRACTICES

Read the problem. You can solve the problem by thinking about the size of the pieces.MP1 Make sense of problems and persevere in solving them. Check that students understand that they need to find the table at which the slices of pie will be larger.Have students follow the instructions and shade the two fraction circles to make representations of the fractional amount of each pie Markos would get.

• Why are the pies divided into sixths and eighths? Possible answer: they represent Markos’ choices. If Markos sits at the table with 5 friends, there will be 6 friends sharing a pie. If Markos sits at the table with 7 friends, there will be 8 friends sharing a pie.

Make sure students understand that Markos will get 1 piece of either pie, and they will use the denominators to compare the size of the pieces.

• If Markos wants a larger piece, should he sit at the table with friends sharing Pie A or Pie B? Explain. Pie A; possible explanation: it is divided into fewer pieces since there are fewer friends sharing.

MP2 Reason abstractly and quantitatively. • How many more people will have to sit at

the table with Pie A so that the pieces of pie will be the same size? 2 more people.

It is important that students use their own words to generalize the concepts in Exercises 1 and 2.

MathTalk Use Math Talk to focus on students’

understanding of how the number of pieces in a whole affects the size of the pieces.

• Three of Markos’ friends sit at another table in the restaurant. If Markos joined that table would his slice of pie be larger or smaller than his slice of Pie A? The pie at the new table would be divided into 4 equal slices. The pieces at the new table would be larger than the sixths served of Pie A.

HandsOn

COMMON ERRORS

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Use fraction strips.

On Saturday, the campers paddled 2 _ 8 of their planned route down the river. On Sunday, they paddled 2 _ 3 of their route down the river. On which day did the campers paddle farther?

Compare 2 _ 8 and 2 _ 3 .

• Place a ✓ next to the fraction strips that show more parts in the whole.

• Shade 2 _ 8 . Then shade 2 _ 3 . Compare the shaded parts.

• 2 __ 8 2 __

3

So, the campers paddled farther on ___.

Use reasoning.

For her class party, Felicia baked two trays of snacks that were the same size. After the party, she had 3 _ 4 of the carrot snack and 3 _ 6 of the apple snack left over. Was more carrot snack or more apple snack left over?

Compare 3 _ 4 and 3 _ 6 .

• Since the numerators are the same, look at the denominators to compare the size of the pieces.

• 1 __ 4 is ___ than 1 __

6 because there are

___ pieces.

• 3 __ 4 3 __

6

So, there was more of the ___ snack left over.

Think: 1 _ 8 is less than 1 _ 3 , so 2 _ 8 is less than 2 _ 3 .

3 __ 4 ● 3 __

6

• The more pieces a whole is divided into,


• The fewer pieces a whole is divided into,

the ___ the pieces are.When comparing fractions with the same numerator, be sure the symbol shows that the fraction with fewer pieces in the whole is the greater fraction.

HandsOn

<

Sunday

smaller

larger

carrot

✓

larger

fewer

>

Lesson 9.3 520

Error Students think the fraction with the smaller denominator has smaller pieces, so it is the lesser fraction.

Example Students write 3 _ 6 < 3 _ 8 Springboard to Learning Provide students with same numerator problems and same denominator problems side-by-side. This will enable them to develop a better understanding of numerators and denominators and to correctly reason about the relative size of fractions.

HandsOn The first problem uses fraction strips to

represent fractions.

• Using fraction strips, how can you compare the fractions to solve the problem? Possible answer: shade the fraction strips to represent the fractions in the problem and compare amounts to find the greater fraction.

• If the numerators are the same, why isn’t the greater fraction the one with the greater number in the denominator? Possible answer: the greater number in the denominator means that the whole is divided into more parts. So, each part is smaller, not larger.

In the Use Reasoning example, students look at the denominators to compare fractions.

• How do you compare the fractions to find which snack had more left over? Possible answer: the numerators are the same, so look at the denominators. The fraction with the greater denominator is 3 _ 6 . It has more parts, so the parts are smaller. The fraction with the lesser denominator is 3 __ 4 . It has fewer parts, so the parts are larger. So, there was more carrot snack left over.

ELL Strategy: Develop Meaning

Students understand vocabulary by studying drawings that illustrate concepts and unfamiliar words.• Have students make flash cards that

show same-sized wholes to represent unit fractions.

• Have students play a game where they hold up two fractions, say the fraction, and explain which is greater. Make sure students compare the sizes of the equal parts.

Advanced LearnersAdvanced LearnersAdvanced Learners Visual / KinestheticPartners

• Have students use their understanding of fractions to write a problem that requires comparing fractions with the same numerator. Share the example below.

Joe has a piece of pizza that is 4 _ 6 of a whole pizza. Jorge has a piece of pizza that is 4 _ 8 of a whole pizza. What could their pieces of pizza look like? Explain.

• Have students exchange problems and draw pictures to solve and explain their answers. Check students’ work.

EXPLAIN3

Quick Check

If

Rt I 1

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Quick Check

If

Rt I 1

2

3

Then

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MathTalk MATHEMATICAL PRACTICES 1

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Chapter 9 • Lesson 3 521

Compare. Write <, >, or =.

2. 1 _ 8

1 _ 3

3. 3 _ 4 3 _ 8

4. 2 _ 6

2 _ 3

Compare. Write <, >, or =.

1. Shade the models to show 1 _ 6 and 1 _ 4 .

Then compare the fractions.

1 _ 6 1 _ 4

5. 4 _ 8

4 _ 4 6. 3 _ 6

3 _ 6

7. 8 _ 4 8 _ 8

8. 1 _ 3

1 _ 4 9. 2 _ 3

2 _ 6

10. 4 _ 8

4 _ 2

11. 6 _ 8

6 _ 6

12. 1 _ 6

1 _ 2

13. 7 _ 8

7 _ 8

James David

Evaluate Why is 1 _ 2 greater than 1 _ 4 ?

14. DEEPER James ate 3 _ 4 of his quesadilla. David ate 2 _ 3 of his quesadilla. Both are the same size. Who ate more of his quesadilla?

James said he knows he ate more because he looked at the amounts left. Does his answer make sense? Shade the models. Explain.

<

<

<

=

<

>

=

>

>

Possible explanation: since halves are larger pieces than fourths and there are the same number of pieces in each fraction, 1 _ 2 is greater than 1 _ 4 .

Possible shading is shown.

<

<

Yes; possible explanation: the amount James has left is smaller than the

amount David has left. 1 _ 4 < 1 _ 3 , so James ate more.

<

>

On Your OwnOn Your Own

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521 Chapter 9

On Your Own If students complete the checked exercises correctly, they may continue with the On Your Own section.MP6 Attend to precision. To extend their thinking, have students represent each comparison in Exercises 8–13 another way. For example, they might use words or switch the fractions and use the opposite symbol.

DEEPER

Exercise 14 requires students to evaluate a statement and explain why it makes sense. By looking at the amounts left, students compare fractions with the same numerator.

Share and Show MATHBOARDMATHBOARD

The first problem connects to the learning model. Have students use the MathBoard to explain their thinking.

MathTalk Use Math Talk to focus on the size

of the fraction pieces when comparing fractions.

• What does the denominator of a fraction tell you? The denominator tells you how many pieces the whole is divided into or the size of the pieces.

Use the checked exercises for Quick Check.

a student misses the checked exercises

Differentiate Instruction with • Reteach 9.3

• Personal Math Trainer 3.NF.A.3d

• Reteach Tier 1 Activity (online)


ELABORATE4

Differentiated Centers Kit

DIFFERENTIATED INSTRUCTION INDEPENDENT ACTIVITIESD

EVALUATE5 Formative Assessment

WRITE Math Show Your Work

Unlock the ProblemUnlock the Problem


MATHEMATICAL PRACTICES COMMUNICA E CONSTRUCT ARGUMENTS

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15. MATHEMATICALPRACTICE 1 Make Sense of Problems Quinton and

Hunter are biking on trails in Katy Trail State Park. They

biked 5 _ 6 mile in the morning and 5 _ 8 mile in the afternoon.

Did they bike a greater distance in the morning or in

the afternoon?

c. How can you solve the problem? d. Complete the sentences.

In the morning, the boys biked

__ mile. In the afternoon, they biked __ mile.

So, the boys biked a greater distance

in the __. 5 _ 6

5 _ 8

a. What do you need to know?

b. The numerator is 5 in both fractions, so compare 1 _ 6 and 1 _ 8 . Explain.

16. SMARTER Zach has a piece of pie that is 1 _ 4 of a pie. Max has a piece of pie that is 1 _ 2 of a pie. Max’s piece is smaller than Zach’s piece. Explain how this could happen. Draw a picture to show your answer.

17. SMARTER Before taking a hike, Kate and Dylan each ate part of their same-size granola bars. Kate ate 1 _ 3 of her bar. Dylan ate 1 _ 2 of his bar. Who ate more of the granola bar? Explain how you solved the problem.

which distance is greater, 5 _ 6 mile or 5 _

8 mile

Possible

Possible answer: since the

numerators are the same, I can look

at the denominators to compare the

size of the pieces. Sixth-size pieces

are larger than eighth-size pieces.

So, 5 _ 6 > 5 _

8 .

5 _ 6

5 _ 8

morning >

Possible explanation: Max’s piece is smaller

because the two whole pies are different sizes. Max’s pie is smaller.

Dylan; possible explanation: I drew 2 squares the same size and divided one in half

Check students’ drawings.

Possible drawing:

explanation: the more pieces a whole is divided into, the smaller the pieces are, so 1 _ 6 > 1 _

8 .

and the other in thirds. Thirds have more, smaller parts than halves, so 1 _ 2 > 1 _

3 .

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

Lesson 9.3 522

Students practice comparing fractions.

GamesFraction Action

ActivitiesWho’s the Greatest?

Students complete purple Activity Card 11

by using fraction tiles to compare and order fractions.

Essential QuestionUsing the Language ObjectiveReflect Have student pairs develop step-by-step directions to answer the Essential Question.How can you compare fractions with the same numerator? Possible answer: I can look at the denominators and compare the size of the pieces. The fraction with the greater denominator has smaller pieces, so it is the lesser fraction.

Math Journal WRITE MathExplain how the number of pieces in a whole relates to the size of each piece.

Unlock the Problem MATHEMATICAL PRACTICES

SMARTER

Exercise 16 requires students to conclude that the two pies have to be different sizes for a 1 _ 2 piece to be smaller than a 1 _ 4 piece.

Personal Math Trainer SMARTER

Be sure to assign this problem to students in the Personal Math Trainer. It features an animation to help them model and answer the problem. This item assesses students’ ability to compare fractions having the same numerator. Students should recognize that they can compare unit fractions by comparing the denominators, and remember that the fraction with the greater denominator is the lesser fraction. Help students check their own work by naming the symbol they used or by comparing fraction strips.

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

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Problem SolvingProblem Solving

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Compare. Write <, >, or =. 1. 1 _

8 1 _

2 2. 3 _

8 3 _

6 3. 2 _

3 2 _

4

10. Javier is buying food in the lunch line.

The tray of salad plates is 3 _ 8 full.

The tray of fruit plates is 3 _ 4 full.

Which tray is more full?

11. Rachel bought some buttons.

Of the buttons, 2 _ 4 are yellow and 2 _ 8 are red. Rachel bought more

of which color buttons?

Chapter 9 523

4. 2 _ 8

2 _ 3

7. 5 _ 6

5 _ 8

5. 3 _ 6

3 _ 4

8. 4 _ 8

4 _ 8

6. 1 _ 2

1 _ 6

9. 6 _ 8

6 _ 6

<

Lesson 9.3Practice and Homework

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

12. WRITE Math Explain how the number of pieces in a whole relates to the size of each piece.

the fruit plate tray yellow

<

<

<

<

>

>

> =

Check students' work.

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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.

523 Chapter 9


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

1. What symbol makes the statement true? Write <, >, or =.

3 _ 4

3 _ 8

2. What symbol makes the statement true? Write <, >, or =.

2 _ 4

2 _ 3

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

3. Anita divided a circle into 6 equal parts and shaded 1 of the parts. What fraction names the part she shaded?

4. What fraction names the shaded part of the rectangle?

5. Chip worked at the animal shelter for 6 hours each week for several weeks. He worked for a total of 42 hours. How many weeks did Chip work at the animal shelter?

6. Mr. Jackson has 20 quarters. If he gives 4 quarters to each of his children, how many children does Mr. Jackson have?

1 _ 6

7 weeks

2 _ 8

5 children

<>

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 9.3 524

compare fractions with the same numerator · 3.nf.a.3d compare two fractions with the same...

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