Teaching the Lesson materials

Key ActivitiesStudents use examples of equivalent fractions to develop a rule for finding equivalent fractions.

Key Concepts and Skills• Identify fractional parts of regions. [Number and Numeration Goal 2]• Name equivalent fractions. [Number and Numeration Goal 5]• Use a rule for generating equivalent fractions. [Number and Numeration Goal 5]• Develop a rule for generating equivalent fractions. [Patterns, Functions, and Algebra Goal 1]

Key Vocabularyequivalent fractions • Equivalent Fractions Rule

Ongoing Assessment: Informing Instruction See page 605.

Ongoing Assessment: Recognizing Student Achievement Use Math Masters, page 225.[Number and Numeration Goal 5]

Ongoing Learning & Practice materialsStudents play Fraction Match to practice naming equivalent fractions.

Students practice and maintain skills through Math Boxes and Study Link activities.

Differentiation Options materials

Students use aFraction Number-Line Poster to identify equivalentfractions.

Students investigatehow early Egyptiansrepresented a frac-tion as the sum of unit fractions.

Students complete name-collectionboxes for fractions.

Students practicefinding equivalentfractions.

� Teaching Master (Math Masters, p. 224)

� Teaching Aid Masters (Math Masters,p. 388 or 389 and 397)

� 5-Minute Math, pp. 1, 17, 79, and 165� Fraction Number-Line Poster

(Math Masters, pp. 204 and 205)� straightedge

EXTRA PRACTICEEXTRA PRACTICEENRICHMENTREADINESS

3

� Student Reference Book, p. 243� Math Journal 2, p. 202� Study Link Master (Math Masters,

p. 223)� Game Masters (Math Masters,

pp. 473–476)

2

� Math Journal 2, pp. 201, 342 and 343

� Study Link 7�6 � Teaching Master (Math Masters,

p. 225)� calculator� colored chalk� slate

1

Lesson 7�7 603

Objective To guide the development and use of a rule for

generating equivalent fractions.

Technology Assessment Management System

Math Masters, page 225See the iTLG.

604 Unit 7 Fractions and Their Uses; Chance and Probability

Whole

square

201

Many Names for FractionsLESSON

7�7

Date Time

49Color the squares and write the missing numerators.

1. Color �12� of each large square.

is colored. is colored. is colored.2 4 8

2. Color �14� of each large square.

is colored. is colored. is colored.4 8 16

3. Color �34� of each large square.

is colored. is colored. is colored.4 8 16

1 2 4

1 2 4

3 6 12

Math Journal 2, p. 201

Student Page

� Math Message Follow-Up(Math Journal 2, p. 201)

Ask students to examine the squares they colored on journal page 201. Point out that the three fractions they wrote for eachproblem all name the same fractional part of the square. Suchfractions are called equivalent fractions. To support Englishlanguage learners, have students write equivalent fractions next to the examples in the journals.

Students should notice that whenever the total number of equalparts is doubled (or quadrupled), the number of colored parts isalso doubled (or quadrupled), but the fractional part representedby the colored parts does not change.

Tell students that in this lesson they will develop a rule for findingequivalent fractions.

� Developing a Rule for Finding Equivalent Fractions(Math Journal 2, p. 201)

In each problem on journal page 201, the numerator and denominator of the first fraction are each multiplied by 2 to obtain the second fraction. They are each multiplied by 4 to obtain the third fraction.

To support English language learners, write the following on the board.

Problem 1:�12

��

22� � �

24� �

12

��

44� � �

48�

Problem 2:�14

��

22� � �

28� �

14

��

44� � �1

46�

WHOLE-CLASSDISCUSSION

WHOLE-CLASSDISCUSSION

1 Teaching the Lesson

Getting Started

Mental Math and Reflexes Have students name all the factors for numbers under 100. Suggestions:

6 1, 2, 3, 64 1, 2, 45 1, 5

12 1, 2, 3, 4, 6, 1215 1, 3, 5, 1521 1, 3, 7, 21

50 1, 2, 5, 10, 25, 5052 1, 2, 4, 13, 26, 5272 1, 2, 3, 4, 6, 8, 9,

12, 18, 24, 36, 72

Math MessageComplete journal page 201.

Study Link 7� 6 Follow-Up Have small groups compare answers. Ask volunteers to draw additionalrepresentations of the fractions in Problems 1–4.

Adjusting the Activity

Lesson 7�7 605

342

Equivalent Names for Fractions

Date Time

Fraction Equivalent Fractions Decimal Percent

�20� 0 0%

�12� �24�, �

36�

�22� 1 100%

�13�

�23�

�14�

�34�

�15�

�25�

�35�

�45�

�16�

�56�

�18�

�38�

�58�

�78�

Math Journal 2, p. 342

Student Page

343

Equivalent Names for Fractions continued

Date Time

Fraction Equivalent Fractions Decimal Percent

�19�

�29�

�49�

�59�

�79�

�89�

�110�

�130�

�170�

�190�

�112�

�152�

�172�

�1112�

Math Journal 2, p. 343

Student Page

3 � 1�1

3�

6 � 1�1

3�

Problem 3:�34

��

22� � �

68� �

34

��

44� � �

11

26�

Write �22� and �

44� with colored chalk to emphasize that the numerator

and denominator were multiplied by the same number.

The Equivalent Fractions Rule can be used to rename any fraction: If the numerator and denominator of a fraction are multiplied by the same nonzero number, the result is a fractionthat is equivalent to the original fraction.

Present a more abstract rationale for this rule:� If any number is multiplied by 1, the product is the number you started with.� A fraction with the same numerator and denominator, such as �

44�, is

equivalent to 1.� Multiplying the numerator and denominator of a fraction by the same number

(not 0) is the same as multiplying the fraction by 1. So, the product is equivalent to the original fraction.

A U D I T O R Y � K I N E S T H E T I C � T A C T I L E � V I S U A L

� Generating Equivalent Fractions(Math Journal 2, pp. 342 and 343; Math Masters, p. 225)

Have students turn to the Equivalent Names for Fractions table on journal page 342. Ask them to write 10 fractions that are equivalent to �

13�.

Have students look for patterns in fractions that are equivalent to �

13�. Point out how these patterns relate to the Equivalent

Fractions Rule.

Ongoing Assessment: Informing InstructionWatch for students who note that not every pair of equivalent fractions can befound by multiplying (or dividing) by the same whole number. For example:

� �48�

In this example, the numerator and denominator are both multiplied by the mixed number 1�

13�.

Working in pairs, students use the Equivalent Fractions Rule tofind three equivalent fractions for each of the remaining fractionsin the table.

Ask students to explain how to use a calculator to find equivalentfractions. Sample answer: Enter a fraction. Multiply it by any fraction whose numerator and denominator are the same.

PARTNER

ACTIVITY

606 Unit 7 Fractions and Their Uses; Chance and Probability

Adjusting the Activity

When students complete their work on journal pages 342 and 343ask them to solve the problem on Math Masters, page 225 on their own.

Ongoing Assessment:Recognizing Student AchievementUse Math Masters, page 225 to assess students’ understanding ofequivalent fractions. Students are making adequate progress if they are

able to draw a picture or use the Equivalent Fractions Rule to demonstrate that�14� � (is not equal to) �

36�. Some students may rename the fractions as decimals

and show that 0.25 � 0.5.[Number and Numeration Goal 5]

� Playing Fraction Match(Student Reference Book, p. 243; Math Masters, pp. 473–476)

Students play Fraction Match to practice naming equivalent fractions.

Have a table of equivalent fractions available, such as Math Journal 2,pages 342 and 343 or Student Reference Book, page 51.

A U D I T O R Y � K I N E S T H E T I C � T A C T I L E � V I S U A L

� Math Boxes 7�7(Math Journal 2, p. 202)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 7-5. The skill in Problem 6previews Unit 8 content.

Writing/Reasoning Have students write a response to thefollowing: How did you determine the number of squaresyou needed to circle in Problem 1? Sample answer: There

are 24 total squares. I divided them into 8 equal groups with 3squares in each group. Then I circled 3 of the groups.

INDEPENDENT

ACTIVITY

SMALL-GROUP

ACTIVITY

2 Ongoing Learning & Practice

Math MastersPage 225 �

ExampleExample �23� is the target card. It can be matched with:

♦ an equivalent fraction card such as �46�, �

69�, or �

812�

, or

♦ a like denominator card such as �03�, �

13�, or �

33�, or

♦ a WILD card. The player names any fraction (with a denominatorof 2, 3, 4, 5, 6, 8, 9, 10, or 12) that is equivalent to the target card.The player can match �

23� by saying �

46�, �

69�, or �

812� . The player may not

match �23� by saying �

23�.

Fraction Match

Materials 1 deck of Fraction Match Cards (Math Masters, pp. 473–476)

Players 2 to 4 Skill Recognizing equivalent fractionsObject of the game To match all of your cards and have none left. Directions

1. Shuffle the deck and deal 7 cards to each player. Place theremaining cards facedown on the table. Turn over the top card and place it beside the deck. This is the target card.If a WILD card is drawn, return it to the deck and continuedrawing until the first target card is a fraction.

2. Players take turns trying to match the target card with a card from their hand in one of 3 possible ways:

♦ a card with an equivalent fraction♦ a card with a like denominator ♦ a WILD card.

3. If a match is made, the player’s matching card is placed ontop of the pile and becomes the new target card. It is now thenext player’s turn. When a WILD card is played, the nextplayer uses the fraction just stated for the new target card.

4. If no match can be made, the player takes 1 card from thedeck. If the card drawn matches the target card, it may beplayed. If not, the player keeps the card and the turn ends.

5. The game is over when one of the players runs out of cards,when there are no cards left in the Fraction Match deck, ortime runs out. The player with the fewest cards wins.

Games

2�3

2�3

2�3

1�5

1�5

1�5

WILD

Name anequivalent

fraction with adenominator of

2, 3, 4, 5, 6, 8, 9,10, or 12.

WILD WILD

Student Reference Book, p. 243

Student Page

LESSON

7�7

Name Date Time

An Equivalent Fractions Rule

Margot says the value of a fraction does not change if you do the same thing tothe numerator and denominator. Margot says that she added 2 to the numeratorand the denominator in �

14� and got �

36�.

�14

�

�

22� � �

36�

Therefore, she says that �14� � �

36�. How could you explain or show Margot that she is wrong?

Sample answer: �14� does not equal �

36�, because

�36� equals �

12�. You can multiply or divide the

numerator and denominator by the same number and not change the value of the fraction,but you cannot just add or subtract the same number from the numerator and denominator.

�

Math Masters, page 225

� Study Link 7�7(Math Masters, p. 223)

Home Connection Students identify the missing numerator or denominator of equivalent fractions to complete name-collection boxes.

� Identifying Equivalent Fractions on the Fraction Number-LinePoster(Math Masters, pp. 204, 205, and 388 or 389 )

To explore equivalent fractions using a number-line model, havestudents use a straightedge to vertically line up fractions on theFraction Number-Line Poster (see the optional Readiness activityin Lesson 7-1) that are equivalent to �

14�, �

13�, �

12�, �

23�, and so on. Ask

students to record the results of their exploration in a Math Log oron an Exit Slip.

A straightedge highlights equivalent fractions.

1 Whole

Halves

Fourths

Eighths

Thirds

Sixths

0 1

1212

02

22

24

34

44

14

04

48

38

28

18

08

58

68

78

88

13

23

33

03

36

46

56

66

26

16

06

5–15 Min

SMALL-GROUPACTIVITYREADINESS

3 Differentiation Options

INDEPENDENTACTIVITY

STUDY LINK

7�7 Fraction Name-Collection Boxes

49 50

Name Date Time

In each name-collection box:Write the missing number in each fraction so that the fraction belongs in the box. Write one more fraction that can go in the box.

1. 2. 3.

5. � 95 / 4 6. 57 � 3 � 7. � 882 / 21421923 R3

2

9

10

20

4. Make up your own a. b.

name-collection box problems like the ones above. Ask a friend to solve your problems. Check your friend’s work.

Practice

�12

�

45

10

18

Answers vary.

Answersvary.

6

8

18

30

�23

�

912

20

12

Answersvary.

3

25

20

40

�14

�

125

10

100

Answersvary.

Math Masters, p. 223

Study Link Master

Lesson 7�7 607

Math Boxes LESSON

7 7

Date Time

4. Draw and label a 125° angle.

This angle is an (acute or obtuse) angle.

obtuse

1. Circle 38 of all the squares. Mark Xs on 1

6of all the squares.

2. Insert parentheses to make these number sentences true.

a. 2 (3 10) 26

b. 12 6 (6 4)c. 24 5) 2 38

d. 12 24 3 (6 6)

3. Plot and label each point on thecoordinate grid.

A (0,2)

B (4,0)

C (1,5)

D (5,5)

E (5,3)

5. A bag contains

5 green blocks,6 red blocks,1 blue block, and3 yellow blocks.

You put your hand in the bag and, withoutlooking, pull out a block. About whatfraction of the time would you expect toget a blue block?

115

Sample answer:

59

144

45 145

92 93143

150

1

2

4

3

5

01 2 3 4 50

A

C D

E

B

Sample answer:

O

T

P

6. If 1 inch on a map represents 40 miles,then how many inches represent10 miles? Fill in the circle next to the best answer.

A 2 in.

B 14 in.

C 12 in.

D 4 in.

(

Math Journal 2, p. 202

Student Page

608 Unit 7 Fractions and Their Uses; Chance and Probability

LESSON

7�7

Name Date Time

Egyptian Fractions

Ancient Egyptians only used fractions with 1 in the numerator. These are calledunit fractions. They wrote non-unit fractions, such as �

34� and �

49�, as sums of unit

fractions. They did not use the same unit fraction more than once in a sum.

Examples:

Use drawings and what you know about equivalent fractions to help you find theEgyptian form of each fraction.

1. �38� � 2. �1

52� �

3. �170� � 4. �

56� �

12

13

12

15

�12� � �

13��

12� � �

15�

13

114

18

�13� � �1

12��14� � �

18�

19

13

14

12

5. �35� � 6. �

47� �

12

114

12

110

�12� � �1

14��

12� � �1

10�

�34� � �

12� � �

14� �

49� � �

13� � �

19�

55 57

Math Masters, p. 224

Teaching Master

� Investigating Egyptian Fractions(Math Masters, p. 224)

To apply students’ understanding of fraction addition and equivalent fractions, have students investigate how earlyEgyptians represented a fraction as the sum of unit fractions.

To solve Problems 5 and 6, students need to divide the rectangleinto more regions than indicated by the denominator of the fraction.

NOTE Egyptians also used the fraction �23�.

� Completing Name-Collection Boxes(Math Masters, p. 397)

To provide practice generating equivalent names for fractions,have students complete name-collection boxes. Encourage studentsto complete the boxes with equivalent fractions and mathematicalexpressions that include fractions.

Use Math Masters, page 397 to create problems to meet the needsof individual students or have students create and solve their own problems.

� 5-Minute MathTo offer students more experience with equivalent fractions, see 5-Minute Math, pages 1, 17, 79, and 165.

5–15 Min

SMALL-GROUP

ACTIVITYEXTRA PRACTICE

5–15 Min

INDEPENDENT

ACTIVITYEXTRA PRACTICE

15–30 Min

SMALL-GROUP

ACTIVITYENRICHMENT

Name-Collection Boxes

Name ____________________________

Date _____________________________

Name ____________________________

Date _____________________________

Name ____________________________

Date _____________________________

Name ____________________________

Date _____________________________

Name Date Time

Math Masters, p. 397

Teaching Aid Master