467A Chapter 8

About the MathProfessional Development

LESSON AT A GLANCE

Professional Development Videos

Teaching for DepthIn this lesson, students will be introduced to using a number line to represent fractions.

A number line is a measurement model. Fractions on a number line represent the distance from 0 to the location marked on the number line. So, if the distance from 0 to 1 is partitioned into 4 equal increments, the third mark represents 3 _ 4 because it is 3 _ 4 of the distance from 0 to 1. It may help if students visualize the distance between 0 and 1 as the whole and see the length between the tick marks as fractional parts.

Fractions on a Number Line

LESSON 8.5

The point represents the fraction 3 _ 4 .

Learning ObjectiveRepresent and locate fractions on a number line.

Language ObjectiveStudents use the sequence words first, next, then, and finally to explain how you can represent and locate fractions on a number line.

MaterialsMathBoard, Fraction Strips (see eTeacher Resources), color pencils

F C R Focus:Common Core State Standards3.NF.A.2a Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into

b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.

3.NF.A.2b Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

Also 3.NF.A.2

MATHEMATICAL PRACTICESMP1 Make sense of problems and persevere in solving them. MP4 Model with mathematics.MP7 Look for and make use of structure.

F C R Coherence:Standards Across the GradesBefore2.MD.B.6

Grade 33.NF.A.2a3.NF.A.2b

After4.NF.B.3

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 441J.

FOCUS COHERENCE RIGOR

ENGAGE1Daily Routines

Common Core

Lesson 8.5 467B

How can you represent and locate fractions on

a number line?

Fluency BuilderDivision Facts Write the following division problems on the board. Have students practice their division facts by solving each problem. Then have students check each answer by writing a related multiplication fact.

27 ÷ 3 9; 3 × 9 = 27 35 ÷ 7 5; 7 × 5 = 35

40 ÷ 4 10; 4 × 10 = 40 18 ÷ 3 6; 3 × 6 = 18

16 ÷ 2 8; 2 × 8 = 16 24 ÷ 6 4; 4 × 6 = 24

56 ÷ 8 7; 8 × 7 = 56 18 ÷ 9 2; 9 × 2 = 18

Problem of the Day 8.5Dan ordered a pizza. It was divided into 8 equal slices. His brother ate 1 slice. How much of the pizza did his brother eat?

–––––––

Vocabulary

with the Interactive Student Edition

Essential QuestionHow can you represent and locate fractions on a number line?

Making ConnectionsInvite students to talk about farms. Pose the following questions.

Have you ever been to a farm where they grow a crop? What types of food did they grow on the farm? Where do farmers sell their crops?

Learning ActivityConnect the story to the problem.

• What information are you given in this story? The potatoes are 1 _ 4 of the way to the vegetable stand. The onions are 3 _ 4 of the way to the vegetable stand.

• When might you use a fraction to describe a distance? on a family trip

Literacy and MathematicsYou may choose one or both of the following activities.

• Ask students to describe how displaying the fractions on a number line helps them determine the distances between the stops.

• Have students use rulers to draw number lines that represent day trips they would like to take. Have them mark and label six places of interest they’d like to visit or see at equal distances apart on their number lines. Invite students to share their number lines with the class. Then have the students write a short story about what they would do on one of those day trips.

One eighth; 1 _ 8

Common Core Fluency Standard 3.OA.C.7

Interactive Student EditionMultimedia Glossary e

EXPLORE2

HandsOn

06

26

36

56

66

A

0 1

B

16

16

16

16

16

16

0 of 3equal lengths

0 1

33

23

13

03

13

13

13

1 of 3equal lengths

2 of 3equal lengths

3 of 3equal lengths

0 114

14

14

14

0 444

1

4

2

4

3

4

Name

2. point A 3. point B

Write the fraction that names the point.

1. Complete the number line. Draw a point to show 2 __ 4 .

Fractions on a Number Line

Use the fraction strips to help name the points on the number line.

Draw a point to show 1 _ 3 .

Step 1 The denominator is 3, so use fraction strips for thirds. Place

the fraction strips above the number line. Use the fraction

strips to divide the number line into three equal lengths.

Step 2 Label each mark on the number line.

Think: The distance between each mark is 1 _ 3 of the total

distance, so count the number of 1 _ 3

lengths.

Step 3 Draw a point on the number line to show 1 _ 3 .

Lesson 8.5Reteach

4 _ 6 1 _

6

8-13 ReteachChapter Resources© Houghton Mifflin Harcourt Publishing Company

08

48

78

AEC B FD

0 1

Name

Fraction Find

Write a fraction that names the shaded part of

each whole. Then locate the fraction on the

number line below. Write the letter of the model

that represents the fraction.

A.

B.

C.

D.

E.

F.

Lesson 8.5Enrich

8 _ 8

2 _ 8

5 _ 8 1 _

8

6 _ 8

3 _ 8

8-14 EnrichChapter Resources© Houghton Mifflin Harcourt Publishing Company

1

G1 G2

4

2

4

3

4

04

44

14

14

14

14

Billy’s House0

Grandma’s House1

Unlock the ProblemUnlock the Problem

Name

© H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

Chapter 8 467

Activity Locate fractions on a number line.

Materials ■ fraction strips

Billy’s family is traveling from his house to his grandma’s house. They stop at gas stations when they are 1 _ 4 and 3 _ 4 of the way there. How can you represent those distances on a number line?

You can use a number line to show fractions. The length from one whole number to the next whole number represents one whole. The line can be divided into any number of equal parts, or lengths.

Lesson 8.5

STEP 1 Divide the line into four equal lengths, or fourths. Place four 1 _ 4 -fraction strips end-to-end above the line to help.

STEP 2 At the end of each strip, draw a mark on the line.

STEP 3 Count the fourths from zero to 1 to label the distances from zero.

STEP 4 Think: 1 _ 4 is 1 out of 4 equal lengths. Draw a point at 1 _ 4 to represent the distance from 0 to 1 _ 4 . Label the point G1.

STEP 5 Think: 3 _ 4 is 3 out of 4 equal lengths. Draw a point at 3 _ 4 to represent the distance from 0 to 3 _ 4 . Label the point G2.

A point on a number line shows the endpoint of a length, or distance, from zero. A number or fraction can name the distance.

Fractions on a Number LineEssential Question How can you represent and locate fractions on a number line?

HandsOn

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

MATHEMATICAL PRACTICESMP1, MP4, MP7

DifferentiatedInstruction1

2

3

467 Chapter 8

3.NF.A.2a Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line. 3.NF.A.2b Represent a fraction a /b on a number line diagram by marking off a lengths 1/b from 0. Recognize the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

Enrich 8.5Reteach 8.5

LESSON 8.5

Unlock the Problem MATHEMATICAL PRACTICES

MP1 Make sense of problems and persevere in solving them. Make sure that students understand that the number line represents the complete trip in the problem. • What does way mean? the total distance to Billy’s

grandma’s house

• What does of the mean? part of the total distance

• One gas station is 1 __ 4 of the way. What part of this fraction tells the number of equal parts? the denominator, 4

• What does the numerator tell you? The number of equal lengths that have been traveled so far.

ActivityMP4 Model with mathematics.Direct students’ attention to the number line.• Why is the number line being divided into

4 equal lengths? Possible answer: because 4 is the denominator of the fractions, which means there will be 4 equal lengths in the whole.

• How do you know where to draw a point for 3 __ 4 ? The numerator is 3, so I counted from zero 3 equal lengths, or three fourths.

Have students describe how they would make a number line for this situation if one were not provided for them.MP2 Reason abstractly and quantitatively. • Billy’s family stops for lunch when they are 1 __ 2

of the way to his grandma’s house. How would you show this on the number line? How many fourths is this? Place a point halfway between 0 and 4 __ 4 , or at 2 __ 4 .

ELL Strategy: Rephrase

Students rephrase modeled steps to show understanding of how fractions can equal one whole.• Have students fold a strip of paper into

fourths.

• Have them color one fourth after the teacher models and explains why it is 1 __ 4 .

• Continue to color each fourth until all parts are colored. Explain 4 __ 4 equals one whole.

• Ask students to rephrase in their own words to a partner why 4 _ 4 equals one whole.

COMMON ERRORS

COMMON ERRORS

Quick Check

If

Rt I 1

2

3

Quick Check

If

Rt I 1

2

3

Then

HandsOn

0

B CA

1

08

18

28

48

58

88

18

18

18

18

18

18

18

18

3

3

2

3

1

3

0

3

0 1

13

13

13

16

26

36

46

0 1

66

5

606

16

16

16

16

16

16

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

MathTalk MATHEMATICAL PRACTICES 4

© H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

© H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

468

2. point A _ 3. point B _ 4. point C _

1. Complete the number line. Draw a point to show 2 _ 3 .

Shade the fraction strips to show the location of the point.

There are _ out of _ equal lengths shaded. The shaded length shows 5 _ 6 .

So, _ names the point.

Example Complete the number line to name the point.

Write the fraction that names the point.

Materials ■ color pencils

Write the fraction that names the point on the number line.

Think: This number line is divided into six equal lengths, or sixths.

The length of one equal part is _.

HandsOn

Use Models What does the length between each mark on this number line represent?

1 _ 6

5 6

3 _ 8

Possible explanation: 1 _ 3 of the whole distance, or the length, from zero to 1

6 _ 8 7 _

8

5 _ 6

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

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

3_MNLESE342149_C08L05.indd 468 15/10/14 11:18 PM

Advanced LearnersAdvanced LearnersAdvanced Learners

Lesson 8.5 468

Error  Students do not label a point on the number line correctly.

Example  For Exercise 1, a student may label the point at 1 _ 3 instead of 2 _ 3 .Springboard to Learning  Have students use a shading method to correct the error. Students shade from 0 __

3 to 1 __

3 , and then 1 __

3 to 2 __

3

to understand that 2 __ 3 represents the distance

from 0 to that point.

ExampleGuide students to complete the number line. Make sure that they understand that the denominators of the fractions match the number of equal lengths between two whole numbers on the number line. •  What fraction names the point? 5 __ 6 What

does the numerator show? the number of equal lengths that are shaded

a student misses the checked exercises

Differentiate Instruction with  • Reteach 8.5

  • Personal Math Trainer 3.NF.A.2a, 3.NF.A.2b

  • RtI Tier 1 Activity (online)

EXPLAIN3

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

students’ understanding of how the number line connects to fractions.

Use the checked exercises for Quick Check. Students should show their answers for the Quick Check on the MathBoard.

Materials Fraction Strips (see eTeacher Resources)

•Write the following problem on the board.

Mr. Smith is driving to work. He stops for breakfast when he is 1 __ 2 of the way to work.How can you show this distance on the number line?

•Have one student choose the fraction strips that will represent the problem. Then the partner draws a number line and uses the fraction strips to show the distance on the number line.

Visual / AuditoryPartners

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

DifferentiatedInstruction

0

0

C D E

1

444

1

62

63

64

65

60

0 1

6

16

66

1

32

30

0 1

333

13

On Your OwnOn Your Own

Name

© H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

Chapter 8 • Lesson 5 469

Write the fraction that names the point.

Use fraction strips to help you complete the number line. Then locate and draw a point for the fraction.

5. 2 _ 6

6. 2 _ 3

7. point C _ 8. point D _ 9. point E _

10. There is a walking trail at the park. Four laps around the trail is a distance of 1 mile. How many laps does it take to walk 3 _ 4 mile?

11. DEEPER A recipe for pasta makes enough for eight servings. How many servings can be made using 4 _ 8 of each ingredient in the recipe?

2 _ 4 1 _

4 3 _

4

3 laps 4 servings

469 Chapter 8

On Your OwnIf students complete the checked exercises correctly, they may continue with the On Your Own section.• How many fraction strips do you need for

the number line in Exercise 5? How do you know? Possible answer: I see that the number line shows sixths, so I need 6 sixth strips.

• When looking at the number lines, how do you know what numbers to write for each fraction? Possible answer: I look to see how many equal parts are shown between 0 and 1 and I write that number for the denominator. Then I look to see how many parts away from the 0 it is and write that number as the numerator.

MP4 Model with mathematics. To extend their thinking, have students draw a number line on notebook paper to represent the fraction 5 __ 8 . Students could present their number lines to the class.

ELABORATE4

Differentiated Centers Kit

DIFFERENTIATED INSTRUCTION INDEPENDENT ACTIVITIESD

EVALUATE5 Formative Assessment

0 1

38

08

88

18

18

18

18

18

18

18

18

Unlock the ProblemUnlock the Problem

0 F 1

Personal Math Trainer

MATHEMATICAL PRACTICES ANALYZE

© H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

470

c. MATHEMATICALPRACTICE 4 Use Models Make a model to solve the problem.

d. Complete the sentences.

There are _ laps in 1 mile.

Each lap represents _ of a mile.

_ laps represent the distance of three eighths of a mile.

So, Javia will need to run _ laps to run 3 _ 8 of a mile.

12. SMARTER Javia ran 8 laps around a track to run a total of 1 mile on Monday. How many laps will she need to run on Tuesday to run 3 _ 8 of a mile?

a. What do you need to find?

____

____

____

b. How will you use what you know about number lines to help you solve the problem?

____

____

____

____

13. SMARTER Locate and draw point F on the number line to represent the fraction

2

_ 4 .

Possible answer: I will

use the number line to

count the number of laps

in three eighths of a mile.

Check students’ models.

Three

8

3

1 _ 8

the number of laps

Javia will need to run

on Tuesday

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

Lesson 8.5 470

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

Students complete orange Activity Card 11 by

playing a Go Fish game in which they match fraction symbols, words, and pictures.

LiteraturePizza Parts!

ActivitiesFish for Fractions

Essential QuestionUsing the Language ObjectiveReflect Have students use the sequence words first, next, then, and finally to answer the Essential Question. How can you represent and locate fractions on a number line? Possible answer: I divide the number line into equal lengths and draw a mark on the line at the end of each equal length. To locate a fraction, I use the numerator to count the number of equal lengths.

Math Journal WRITE MathExplain how showing fractions with models and a number line are alike and different.

Unlock the Problem MATHEMATICAL PRACTICES

SMARTER

Have students read Exercise 12. Ask them to describe how they will solve the problem.MP4 Model with mathematics.• How do you know how many equal

lengths to show on your number line? The total distance is 8 laps, so the number line will have 8 equal lengths.

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 a student’s ability to locate a fraction on a number line. Students should understand that each equal part shows 1 __ 4 . An incorrectly plotted point at 3 __ 4 may show that students did not start counting from zero. A point at 4 __ 4 may indicate confusion between numerator and denominator meanings.

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

Mathematical Practices in Your Classroom PROFESSIONALDEVELOPMENT

08

1

0 1

838

48

68

88

A B C

0 1

13

33

23

13

03

Problem SolvingProblem Solving

Name

© H

ou

gh

ton

Miff

lin H

arc

ou

rt P

ublis

hin

g C

om

pa

ny

Chapter 8 471

Fractions on a Number Line

Use fraction strips to help you complete the number line. Then locate and draw a point for the fraction.

Write the fraction that names the point.

1. 1 _ 3

5. Jade ran 6 times around her neighborhood to complete a total of 1 mile. How many times will she need to run to complete 5 _ 6 of a mile?

6. A missing fraction on a number line is located exactly halfway between 3 _ 6 and 5 _ 6 . What is the missing fraction?

2. point A __ 3. point B __ 4. point C __

Lesson 8.5Practice and Homework

COMMON CORE STANDARDS—3.NF.A.2a, 3.NF.A.2b Develop understanding of fractions as numbers.

7. WRITE Math Explain how showing fractions with models and a number line are alike and different.

2 _ 8 5 _

8 7 _

8

4 _ 6 5 times

Check students' work.

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.

471 Chapter 8

CCSS.Math.Practice.MP4 Model with mathematics.Students should recognize that while each fraction model compares equal parts to the whole, one model may be more useful than another in a given problem situation. Students should master selecting the most useful model for the problem situation.

The area model shows parts that have equal areas and is best used for partitioning whole objects. The linear model shows lengths that are compared instead of areas. While both fraction strips and number lines are linear models, the number line model is more sophisticated and is best used for problems involving distances from zero. In this lesson, students use fraction strips to transition into using a number line model for fractions that describe lengths and distances.

Ask questions such as the following to help students connect distance to fractions on a number line:

• How do you show the entire distance on the number line? From 0 to 1 shows the entire distance; it shows one whole.

• If a distance was 1 __ 4 of a complete trip, where would you write the fraction on the number line? I would count one equal part on the number line and write the fraction at that point.

• What type of real-world situations can be modeled using a fraction circle? using a number line? Explain. Possible answer: I can use a fraction circle to model dividing a pie into equal slices. I can use a number line to model the distance the bus travels from my house to school, and the stops it makes along the way.

0 1

44

34

24

04

G

0 1

33

13

03

R

Personal Math Trainer

FOR MORE PRACTICE GO TO THE

© H

oughto

n M

ifflin

Harc

ourt

Publis

hin

g C

om

pany

472

Lesson Check (3.NF.A.2a, 3.NF.A.2b)

1. What fraction names point G on the number line?

2. What fraction names point R on the number line?

Spiral Review (3.OA.B.5, 3.OA.C.7, 3.NF.A.1)

3. Each table in the cafeteria can seat 10 students. How many tables are needed to seat 40 students?

4. Use the Commutative Property of Multiplication to write a related number sentence.

4 × 9 = 36

5. Pedro shaded part of a circle. What fraction names the shaded part?

6. Find the quotient.

8 ÷ 1 =

1 _ 4

4 tables

7 _ 8

2 _ 3

9 × 4 = 36

8

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 8.5 472