Investigate • Model Equivalent Fractions

LESSON 9.6

Investigate ? Model Equivalent Fractions

FOCUS COHERENCE RIGOR

LESSON AT A GLANCE

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 Grades Before Grade 3 After 2.NBT.A.4 3.NF.A.3a 4.NF.A.1

Learning Objective

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

Language Objective

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

Materials

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

F C R Rigor:

Level 1: Understand Concepts....................Share and Show ( Checked Items) Level 2: Procedural Skills and Fluency.......On Your Own, Practice and Homework Level 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.

About the Math

Professional Development

Modeling Equivalent Fractions

Equivalent 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

.

Professional Development Videos

0 1 2345678 888888888

0

1

2

3

4

4

4

4

4

4

3__ 4

=

6__ 8

539A Chapter 9

Daily Routines

Common Core

Problem of the Day 9.6

Juanita 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.

8 ? 7 = 56 or 7 ? 8 = 56

Vocabulary equivalent, equivalent fractions

Interactive Student Edition Multimedia eGlossary

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.

1 ENGAGE

with the Interactive Student Edition

Essential Question

How can you use models to find equivalent fractions?

Making Connections

Invite 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 Activity

Ask 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 Mathematics

Have 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

.

How can you use models to find

equivalent fractions?

Lesson 9.6 539B

LESSON 9.6

2 EXPLORE

Investigate HOannds

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 Language

Students 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.

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.

Name

Model Equivalent Fractions

Essential Question How can you use models to find equivalent fractions?

IInnvveessttiiggaattee

HOannds

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.

1 _

There are _ 2 equal parts. Each part is _ 2 of the paper.

Lesson 9.6

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 PRACTICES MP4, MP5, MP7

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 _ 4 equal parts. Each part is

1 _

_ 4 of the paper.

Write

1 _ 4

on

each

of

the

fourths.

Look

at

the

shaded

parts.

1 _ 2

=

_2__ 4

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

Now there are _ 8 equal parts. Each part is

1 _

_8 of the paper.

Write

1 _ 8

on

each

of

the

eighths.

Find

the

fractions

equivalent

to

1 _ 2

on

your

paper.

So,

1 _ 2

,

_2__ 4

,

and

_4__ 8

are

equivalent.

Chapter 9 539

? Houghton Mifflin Harcourt Publishing Company 1 4

Reteach 9.6

Name

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 _21.

_ 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.

1 4

1 4

3 2 1 Lesson 9.6 Reteach

11 22

1 4

Shade the model. Then divide the pieces to find the

equivalent fraction. Possible shading is shown.

1.

2.

2 1__

4

5

____ 8

3.

4 1__

2

5

____ 8

4.

4 2__

3

5

___ 6

Chapter Resources ? Houghton Mifflin Harcourt Publishing Company

6 3__

4

5

____ 8

9-15

Reteach

Enrich 9.6

Differentiated Instruction

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.

2.

Lesson 9.6 Enrich

Possible answers are given.

2 _

Fraction:

4

1 _

Equivalent Fraction:

2

3.

3 _

Fraction:

4

6 _

Equivalent Fraction:

8

4.

2 _

Fraction:

8

1 _

Equivalent Fraction:

4

5.

2 _

Fraction:

2

6 _

Equivalent Fraction:

6

6.

2 _

Fraction:

6

1 _

Equivalent Fraction:

3

1 _

Fraction:

4

2 _

Equivalent Fraction:

8

7.

Stretch Your Thinking

Draw

a

model

that

shows

_3_ 3

shaded.

Then use your drawing to find two equivalent fractions.

Check students' drawings. Possible answer:

1_1,

_ 6 6

Chapter Resources ? Houghton Mifflin Harcourt Publishing Company

9-16

Enrich

N Draw Conclusions

1. oEnxpel1_4aipnahrtoownmyoaunryp1_8appaerrt.s are equivalent to

Possible

explanation:

there

are

two

1 _ 8

parts

in

one

1 _ 4

part,

= so

2 _ 8

1 _ 4

.

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

2.

SMARTER What do you notice about how

the numerators changed for the shaded part as

you folded the paper? _ The_ numer_ ators g_ ot lar_ ger.

What does this tell you about the change in the

number of parts? _ The _ numb_ er of p_ arts in_ creas_ ed.

How did the denominators change for the shaded

part as you folded? __Th_ ey do_ ubled_ . _

What does this tell you about the change in the size of the parts? _ The s_ ize of_ the p_ arts go_ t sma_ ller.

N MMaakkee CCoonnnneeccttiioonnss

HOannds

You can use a number line to find equivalent fractions.

Find a fraction equivalent to 2_3.

Materials fraction strips

MTaatlkh

MATHEMATICAL PRACTICES 2

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.

0

1

6

6

2 6

3 6

4 6

5 6

6 6

The number of sixths is

double the number of thirds.

0

1

2

3

3

3

3

3

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. _ 4_6

So,

_2_ 3

=

_46__.

540

Advanced Learners

Logical / Mathematical Individual / Partners

? Present this problem to students:

There were three sandwiches of the same size.

Mwaasrkcuattein1_2toof8tehqeufiarlsptioencees..TAhme yseactoen4d

sandwich 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.

? Houghton Mifflin Harcourt Publishing Company

Draw Conclusions

Have 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.

? t3_6haenndu1_2mabreereoqfupivaarltesn, at nfrdacthtieonsisz.eCsoomfptharee

parts, in each fraction.

they will be smaller than the

p3_6awrtilflohra1_2v.e

more

parts

and

Make Connections HOannds

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.

MTaatlkh

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

COMMON ERRORS

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

Example In Exercise 1, students think

that

_ 1

8

=

_41 .

Springboard to Learning Have students

outline

1 _

4

students

students

part on their paper. Then, instruct tpooisnhtatdoeetahcehtw1_8 opa1_8rtpaanrtds.cHouanvet

"1 eighth, two eighths."

Lesson 9.6 540

3 EXPLAIN

Share and Show

MATH

BOARD

HOannds

The 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.

Quick Check

3 2

RtI

1

If

a student misses the checked

exercises

Then Differentiate Instruction with ? Reteach 9.6

? Personal Math Trainer 3.NF.A.3a

? Rtl Tier 1 Activity (online)

4 ELABORATE

Problem Solving ? Applications

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.

SMARTER

Personal Math Trainer 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.

541 Chapter 9

? Houghton Mifflin Harcourt Publishing Company

Name

SShhaarree aanndd SShhow

MATH

BOARD

Shade the model. Then divide the pieces to find the

equivalent fraction. Possible shadings are shown.

1.

2.

1 _ 4

=

_2__ 8

HOannds

2 _ 3

=

_4__ 6

Use the number line to find the equivalent fraction.

3. 0 1 2 3 4 5 6 6666666

0

1

2

2

2

2

1 _ 2

=

_3__ 6

4. 0 1 2 3 4 5 6 7 8 888888888 0 1 234 44444

3 _ 4

=

_6__ 8

OOqqnnaakkddllRRnnkkuuhhmmff??@@ooookkhhbb``sshhnnmmrr

5. MATHEMATICAL PRACTICE

6

Explain

why

2 _ 2

=

1.

Write another fraction that is equal

to 1. Draw to justify your answer.

Possible

explanation:

2 _ 2

=

1

because

the numerator names both parts in the

whole;

possible

answer:

8 _ 8

.

Drawings may vary.

2__ 2

1

8__ 8

Personal Math Trainer

6.

SMARTER For numbers 6a?6d, select True or False

to tell whether the fractions are equivalent.

6a.

6 _ 6

and

3 _ 3

6b.

4 _ 6

and

1 _ 3

6c.

2 _ 3

and

3 _ 6

6d.

1 _ 3

and

2 _ 6

True True True True

False False False False

01 23456 6666666

0

1

2

3

3

3

3

3

Chapter 9 ? Lesson 6 541

 ................
................

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