problem solving • compare fractions · 2018-12-23 · if students have difﬁculty solving a...

507A Chapter 9A Chapter 9

About the MathProfessional Development

Ch 9

LESSON AT A GLANCE

Interactive Student Edition

Personal Math Trainer

Math on the Spot

Animated Math Models

iTools: Fractions

HMH Mega Math

Why Teach ThisIf students have difficulty solving a problem, making visual representations helps students understand the structure of the problem. Acting out problems using manipulatives is especially helpful when presenting new concepts, because students are able to visualize how the information given in the problem is related.

Using representations to compare fractions helps students develop their number sense about fraction size. Using representations also helps students recognize that comparisons are valid only when the two fractions refer to the same whole. Help students understand that the processes they used to compare whole numbers do not necessarily apply to fractions.

In this lesson, students use fraction strips and fraction circles to compare fractions. Guide students to align one end of the fraction strips when comparing lengths of models.

Problem Solving • Compare Fractions

LESSON 9.1

Learning ObjectiveSolve comparison problems by using the strategy act it out.

Language ObjectiveStudent pairs role play the strategy act it out to solve comparison problems.

MaterialsMathBoard, Fraction Circles, Fraction Strips (see eTeacher Resources)

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. MP3 Construct viable arguments and critique the reasoning of others. MP4 Model with mathematics. MP5 Use appropriate tools strategically.

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

Professional Development Videos

ENGAGE1

Lesson 9.1 507B

Daily RoutinesCommon CoreDaily RoutinesCommon Core

How can you use the strategy act it out to solve

comparison problems?

with the Interactive Student Edition

Essential QuestionHow can you use the strategy act it out to solve comparison problems?

Making ConnectionsHave students sit with a partner, compare physical characteristics of an object, and make a list. Examples include size, color, and common use. Invite students to discuss their comparisons. What does it mean to compare? to see how things are alike and how they are different What are some words you used when you compared? Possible answers: greater than, less than, bigger, smaller, longer, taller, shorter, the same, equal

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

• How far is the crosswalk from Lucia’s burrow? 2 _ 4 mile

• How far is the crosswalk from Finn’s den? 2 _ 8 mile

• What does Lucia want to know? if Finn lives farther from the crosswalk than she does

• What do you need to do to find out? compare fractions

Literacy and MathematicsView the lesson opener with the students. Then, choose one or more of the following activities:

• Have students write a story using the settings of Lucia’s burrow and Finn’s den. Students should incorporate details from the lesson opener problem in their stories.

• Have students work in pairs to read a recipe that includes different fractions of ingredients. Have the partners share the recipes with their classmates.

The Whole Picture

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

Students read the book and model fractional parts.

Vocabulary BuilderCompare Discuss the word compare. Help students recall that when you compare two numbers, you find which is greater or lesser. Review the symbols for equal to (=), greater than (>), and less than (<).

List pairs of whole numbers on the board, and have volunteers write the correct symbol between the numbers.

23 < 32 8 > 2

45 > 35 88 = 88

16 > 7 74 < 91

Problem of the Day 9.1Laurel and her class are collecting bottles and cans to recycle. She collected 129 bottles and 92 cans. About how many items did Laurel collect in all?

______

Vocabulary• Interactive Student Edition• Multimedia Glossary e

about 200

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EXPLORE2

Unlock the ProblemUnlock the Problem

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Lesson 9.1Problem Solving • Compare FractionsEssential Question How can you use the strategy act it out to solve comparison problems?

Chapter 9 507

Mary and Vincent climbed up a rock wall at the park. Mary climbed 3 _ 4 of the way up the wall. Vincent climbed 3 _ 8 of the way up the wall. Who climbed higher?

You can act out the problem by using manipulatives to help you compare fractions.

What do I need to find?

What information do I need to use?

Mary climbed _ of the way.

Vincent climbed _ of the way.

How will I use the information?

I will use _____

and ___ the lengths of

the models to find who climbed

__.

Read the Problem Solve the Problem

< is less than

> is greater than

= is equal to

Compare the lengths.

_ ● _The length of the 3 _ 4 model is __

than the length of the 3 _ 8 model.

So, __ climbed higher on the rock wall.

Record the steps you used to solve the problem.

MathTalk MATHEMATICAL PRACTICES 4

Use Models When comparing fractions using fraction strips, how do you know which fraction is the lesser fraction?

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

MATHEMATICAL PRACTICESMP2, MP4, MP5, MP6

whether Mary or Vincent climbed higher

3 _ 4

3 _ 8

fraction strips

compare

Possible answer: when you place the fraction strips above each other, there will be one longer and one shorter strip. The shorter strip in length will be the lesser fraction.

3 _ 4 3 _

8

>

Maryhigher

greater

or 3 _ 8 < 3 _

4

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3_MNLESE342156_C09L01.indd 507 13/10/14 10:47 AM

Name

1 __ 2 1 __ 4

2 __ 6 5 __ 6

3 __ 8 7 __ 8

4 __ 8 5 __ 8

4. Draw a set of 8 counters and color 4 __ 8 of the counters red. Draw another set of 8 counters and color 5 __ 8 red. Write , or . to compare the fraction of red counters in the two sets.

Fraction Frenzy

Use the model to help you compare the fractions. Write , or . .

1. Compare 3 __ 8 and 7 __ 8 .

2. Compare 2 __ 6 and 5 __ 6 .

3. Compare 1 __ 2 and 1 __ 4 .

Lesson 9.1Enrich

.

,

,

,

Check students’ models.

Check students’ drawings.

DO NOT EDIT--Changes must be made through "File info"CorrectionKey=B

9-6 EnrichChapter Resources© Houghton Mifflin Harcourt Publishing Company

3_MNLEAN343023_C09E01.indd 6 11/20/14 11:59 AM

Name

1 __ 2 1 __ 4

2 __ 6 5 __ 6

3 __ 8 7 __ 8

4 __ 8 5 __ 8

4. Draw a set of 8 counters and color 4 __ 8 of the counters red. Draw another set of 8 counters and color 5 __ 8 red. Write , or . to compare the fraction of red counters in the two sets.

Fraction Frenzy

Use the model to help you compare the fractions. Write , or . .

1. Compare 3 __ 8 and 7 __ 8 .

2. Compare 2 __ 6 and 5 __ 6 .

3. Compare 1 __ 2 and 1 __ 4 .

Lesson 9.1Enrich

.

,

,

,

Check students’ models.

Check students’ drawings.

DO NOT EDIT--Changes must be made through "File info"CorrectionKey=B

9-6 EnrichChapter Resources© Houghton Mifflin Harcourt Publishing Company

3_MNLEAN343023_C09E01.indd 6 11/20/14 11:59 AM

Reteach 9.11

2

3 DifferentiatedInstruction

507 Chapter 9

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.

Enrich 9.1

Unlock the Problem MATHEMATICAL PRACTICES

How can you use the strategy act it out to help you compare fractions? Read the problem. You can use fraction strips to act it out.MP4 Model with mathematics. Check that students understand how to use fraction strips to represent the fractions. Point out that each piece of the fraction strip is labeled with the fraction name for one part and that they are comparing parts of the same whole. Make sure students identify the length of a fraction strip by counting the total number of fraction pieces. For example, the length of 3 _ 8 is the total length of three 1 _ 8 -size pieces.Have students read the problem and complete the graphic organizer.

• How can you compare 3 _ 4 and 3 _ 8 to find which is greater? Possible answer: model each fraction with a fraction strip and line up the left edges. Then compare their lengths to find which fraction is greater.

MathTalk Use Math Talk to provide students

a chance to explain how to know which fraction is the least.

• How do the fraction strips show that neither Mary nor Vincent climbed all the way to the top? Possible answer: both fraction strip models are shorter than the whole fraction strip. So, the fractions are less than 1.

MP6 Attend to precision.• To find who climbed higher, you looked

for the longer fraction model. What other words, like higher and longer, can mean to look for the greater fraction? Possible answers: farther, more, bigger, taller

ELL Strategy: Develop Meaning

Students develop meaning by connecting visuals to vocabulary.•Give students fraction circles.

•Name two fractions and ask students to tell their partner which is greater than or less than the other.

•Use sentence frames such as, of a circle is greater than of a circle. Possible answer: 5 _ 6 of a circle is greater than 3 _ 8 of a circle.

LESSON 9.1

HandsOn


Meeting Individual Needs

DifferentiatedInstruction

COMMON ERRORS

COMMON ERRORS

Try Another Problem Try Another Problem

MathTalk MATHEMATICAL PRACTICES 2

508

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Students at day camp are decorating paper circles for placemats. Tracy finished 3 _ 6 of her placemat. Kim finished 5 _ 6 of her placemat. Who finished more of her placemat?

What do I need to find?

What information do I need to use?

How will I use the information?

Read the Problem Solve the Problem

1. How did your model help you solve the problem?

2. Tracy and Kim each had a carton of milk with lunch. Tracy drank 5 _ 8 of her milk. Kim drank 7 _ 8 of her milk. Who drank more of her milk? Explain.

Record the steps you used to solve the problem.

Use Reasoning How do you know that 5 _ 6 is greater than 3 _ 6 without using models?

I need to � nd who � nished more of her placemat.

Tracy � nished 3 _ 6 of her placemat. Kim � nished 5 _ 6 of her placemat.

Possible answer: I will use fraction circles to model the fractions. Then I will compare the models to � nd who � nished more.

Possible answer: � rst, I modeled 3 _ 6 with sixth-size pieces for Tracy’s placemat. Then, I modeled 5 _ 6 with sixth-size pieces for Kim’s placemat.

Last, I compared the models to � nd the greater fraction. 5_6 > 3_

6 . So, Kim � nished more of her placemat.

Possible answer: I could compare

the size of the models for each placemat and see that 5 _ 6 is greater than 3 _ 6 .

Math Talk: Possible answer: I know that the sixth-size pieces are all the same size from the same-size whole. 5 is greater than 3, so 5 _ 6 is greater than 3 _ 6 .

Kim; possible explanation: a model with 7 eighth-size pieces put together is bigger than a

model with 5 eighth-size pieces put together.

Tracy Kim



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Advanced LearnersAdvanced LearnersAdvanced Learners

Lesson 9.1 508

Error Students might confuse the numerator and denominator when comparing fractions with the same denominator.

Example For Problem 2, they may be unable to compare 5 _ 8 and 7 _ 8 .Springboard to Learning Guide students to use and focus on models to see that the fraction with the greater number of same-size pieces is the greater fraction.

Try Another Problem Have students answer the questions in the graphic organizer to help them solve the problem. Students should be able to describe how they used fraction circles to model and compare the fractions to solve the problem.

• How are the fractions in this problem different from the fractions in the last problem? Possible answer: these fractions have the same denominator.

• If two fractions have the same denominator, what does that tell you about the size of the fraction pieces in each model? They are the same.

• How do you know how many sixth-size pieces you need to model each fraction? Looking at the numerator, I need 3 sixth-size pieces to model the part Tracy finished. I need 5 sixth-size pieces to model the part Kim finished.

• Why might you choose fraction strips or fraction circles to model a problem? Possible answer: so I can see the amounts. Looking at the models makes it easy to compare amounts and choose the fraction that is greater or lesser.

MathTalk Use Math Talk to focus on

students’ understanding of comparing fractions with like denominators without modeling them.

• What does the 6 in the denominator of the fractions represent? The 6 represents the total number of equal parts.

You may suggest that students place completed Try Another Problem graphic organizers in their portfolios.

Materials Fraction Strips (see eTeacher Resources)

•Present this problem:

Chris rode his bike 3 _ 8 mile to the park. Kyra rode her bike 5 _ 8 mile to meet Chris at the park. Which friend rode a shorter distance? Chris

•Have students use fraction strips. Ask them to compare the fractions using <, >, or = to solve the problem. Possible answers: 3 _ 8 < 5 _ 8 , or 5 _ 8 > 3 _ 8

•Ask students to write their own problems that require comparing fractions. Have students exchange problems and use fractions strips to model the problems. Ask them to compare the fractions using <, >, or = to solve the problems. Check students’ work.

Kinesthetic / Visual Individual / Partners


DifferentiatedInstruction

EXPLAIN3

Quick Check

If

Rt I 1

2

3

Quick Check

If

Rt I 1

2

3

Then

Share and ShowShare and ShowShare and ShowShare and ShowShare and ShowShare and ShowShare and ShowShare and Show MATHBOARDMATHBOARDMATHBOARDMATHBOARDMATHMATHMATHMATHBOARDBOARDBOARDBOARD

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√ Circle the question.

√ Underline important facts.

√ Act out the problem using manipulatives.

Chapter 9 • Lesson 1 509

1. At the park, people can climb a rope ladder to its top. Rosa climbed 2 _ 8 of the way up the ladder. Justin climbed 2 _ 6 of the way up the ladder. Who climbed higher on the rope ladder?

First, what are you asked to find?

Then, model and compare the fractions.

Last, find the greater fraction.

_ ● _

So, __ climbed higher on the rope ladder.

2. What if Cara also tried the rope ladder and climbed 2 _ 4 of the way up? Who climbed highest on the rope ladder: Rosa, Justin, or Cara? Explain how you know.

3. MATHEMATICALPRACTICE 5 Use a Concrete Model Ted walked 2 _ 3 mile

to his soccer game. Then he walked 1 _ 3 mile to his friend’s house. Which distance is shorter? Explain how you know.

On Your OwnOn Your Own

Unlock the Problem

Think: Compare 2 _ 8 and 2 _ 6 .

Cara; possible explanation: I know that Justin climbed

higher than Rosa since 2 _ 6 > 2 _ 8 . So, I modeled 2 _ 6 and 2 _ 4 and

compared them. 2 _ 4 > 2 _ 6 So, Cara climbed the highest.

1 _ 3 mile; possible explanation: the third-size pieces are all

the same size. 1 < 2, so 1 _ 3 < 2 _ 3 .

whether Rosa or Justin climbed higher

Justin

or 2 _ 8 < 2 _

6

> 2 _ 6 2 _

8

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509 Chapter 9

MP6 Attend to precision. Have students use fractions to describe how far Rosa, Justin, and Cara were from the top of the ladder. Ask them to explain their answer. Rosa: 6 _ 8 ; Justin: 4 _ 6 ; Cara: 2 _ 4 ; possible explanation: I used my model to find how many pieces were needed to complete each whole.

On Your Own If students complete the checked exercises correctly, they may continue with the On Your Own section.MP5 Use appropriate tools strategically. Have students draw fraction strips to model the problem.

a student misses the checked exercises

Differentiate Instruction with • Reteach 9.1

• Personal Math Trainer 3.NF.A.3d

• Rtl Tier 1 Activity (online)

Share and ShowThe first problem connects to the learning model. Have students use the MathBoard to explain their thinking. Remind students that they can only compare fractions that refer to the same whole. In Exercise 1, they are comparing distances up a rope ladder.Use the checked exercises for Quick Check.

MATHBOARDMATHBOARD


ELABORATE4

Games

Differentiated Centers Kit

DIFFERENTIATED INSTRUCTION INDEPENDENT ACTIVITIES

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

ELABORATE4

EVALUATE5 Formative Assessment

Suri’s Biscuits

Raspberry

Strawberry

Peach 384818

JamFlavor

Fraction ofBiscuits

WRITE Math • Show Your Work

MATHEMATICAL PRACTICES ANALYZE • LOOK FOR STRUCTURE • PRECISION

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8. SMARTER Rick lives 4 _ 6 mile from school. Noah lives 3 _ 6 mile from school.

Use the fractions and symbols to show which distance is longer.

36

46 <, >, and ●

Use the table for 4–5.

4. DEEPER Suri is spreading jam on 8 biscuits for breakfast. The table shows the fraction of biscuits spread with each jam flavor. Which flavor did Suri use on the most biscuits? Hint: Use 8 counters to model the biscuits.

5. WRITE Math What’s the Question? The answer is strawberry.

6. SMARTER Suppose Suri had also used plum jam on the biscuits. She frosted 1 _ 2 of the biscuits with peach jam, 1 _ 4 with raspberry jam, 1 _ 8 with strawberry jam, and 1 _ 8 with plum jam. Which flavor of jam did Suri use on the most biscuits?

7. Ms. Gordon has many snack bar recipes. One recipe uses 1 _ 3 cup oatmeal, 1 _ 4 cup of milk, and 1 _ 2 cup flour. Which ingredient will Ms. Gordon use the most of?

raspberry

Possible question: Which jam avor did Suri use

on the fewest biscuits?

peach

our

or 3 _ 6 < 4 _

6

3 _ 6 4 _

6 >


3_MNLESE342156_C09L01.indd 510 13/10/14 10:59 AM

Lesson 9.1 510

Students practice comparing fractions.

GamesFraction Action

LiteratureThe Whole Picture

Students read the book and model fractional parts.

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 student pairs role play to answer the Essential Question.How can you use the strategy act it out to solve comparison problems? Possible answer: to compare fractions, I can use fraction strips or fraction circles to represent each fraction and compare the models.

Math Journal WRITE Math

Explain how you can find whether 5 _ 6 or 5 _ 8 is greater.

MATHEMATICAL PRACTICES

DEEPER

MP4 Model with mathematics.• How do you determine the total number

of counters to model a problem? The denominator tells the total number of counters.

SMARTER

Exercise 6 requires students to analyze a multi-step problem and compare four fractions.

SMARTER

Students should recognize that they can compare fractions to solve the problem. Students may use models or compare the numerators. Students also are required to make correct use of the < and > signs. Have students check their answers by making sure the symbol points toward the lesser value or by drawing a picture to compare the fractions.

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


Meeting Individual Needs

Cross-Curricular

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Problem Solving • Compare Fractions

Solve.

1. Luis skates 2 _ 3

mile from his home to school.

Isabella skates 2 _ 4

mile to get to school. Who

skates farther?

Think: Use fraction strips to act it out.______

2. Sandra makes a pizza. She puts

mushrooms on 2 _ 8

of the pizza. She adds

green peppers to 5 _ 8

of the pizza. Which

topping covers more of the pizza?______

3. The jars of paint in the art room have different amounts of paint. The green

paint jar is 4 _ 8

full. The purple paint jar

is 4 _ 6

full. Which paint jar is less full?______

4. Jan has a recipe for bread. She uses 2 _ 3

cup of flour and 1 _ 3

cup of chopped onion.

Which ingredient does she use more of, flour or onion?

______

Chapter 9 511

Lesson 9.1

Luis

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

5. WRITE Math Explain how you can find whether 5 _ 6

or 5 _ 8

is greater.

Practice and Homework

green peppers

Check students’ work.

the green paint jar

fl our

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.

511 Chapter 9

SOCIAL STUDIESSCIENCE

Materials Fraction Strips (see eTeacher Resources)• Limestone is a sedimentary rock

often formed in clear, shallow seawater. Suppose a box of limestone rocks has a total weight of 24 ounces.

• You choose two rocks and find that one is 1 _ 4 the total weight and the other is 1 _ 6 the total weight. Use fraction strips to find which rock weighs more. The rock that is 1 _ 4 the total weight weighs more.

• Ask students to justify their reasoning. Possible answer: compare the fractions: 1 _ 4 . 1 _ 6 . The greater fraction represents the

rock with the greater weight.

Materials map of the United States• Display a map of the United States that shows the

names of all 50 states. • Have students identify the states that begin with the

letter A. Alabama, Alaska, Arizona, and Arkansas Then ask students what fraction of the state names begin with the letter A. 4 __ 50

• Tell students that 8 __ 50 of the state names begin with the letter M. Ask students whether more state names begin with the letter A or the letter M. M Students should explain how they know they are correct. Possible answer: the denominators are the same, so I compared the numerators. 8 . 4 Have students find all the states that begin with the letter M. Maine, Maryland, Massachusetts, Michigan, Minnesota, Mississippi, Missouri, and Montana

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Personal Math Trainer

FOR MORE PRACTICE GO TO THE

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512

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

1. Ali and Jonah collect seashells in identical buckets. When they are

finished, Ali’s bucket is 2 _ 6

full and

Jonah’s bucket is 3 _ 6

full. Compare the

fractions using >, < or =.

3 _ 6

2 _ 6

2. Rosa paints a wall in her bedroom.

She puts green paint on 5 _ 8

of the wall

and blue paint on 3 _ 8

of the

wall. Compare the fractions

using >, < or =.

5 _ 8

3 _ 8

Spiral Review (3.OA.B.6, 3.OA.D.9, 3.NF.A.1)

5. Charles places 30 pictures on his bulletin board in 6 equal rows. How many pictures are in each row?

6. Describe a pattern in the table.

Tables 1 2 3 4 5

Chairs 5 10 15 20 25

3. Dan divides a pie into eighths. How many equal parts are there?

4. Draw lines to divide the circle into 4 equal parts.

> >

8 equal parts

5 picturesPossible answers: add 5 chairs for each table; multiply the number of tables by 5.


3_MNLESE342156_C09P01.indd 512 13/10/14 11:09 AM

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 9.1 512


problem solving • compare fractions · 2018-12-23 · if students have difﬁculty solving a...

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