eToolkitePresentations Interactive Teacher’s

Lesson Guide

Algorithms Practice

EM FactsWorkshop Game™

Family Letters

CurriculumFocal Points

www.everydaymathonline.com

AssessmentManagement

Common Core State Standards

Lesson 6�2 537

Advance Preparation

Teacher’s Reference Manual, Grades 4–6 pp. 144–147, 149–152

Key Concepts and Skills• Apply the concept of a multiple to rename

fractions using a common denominator.  [Number and Numeration Goal 3]

• Use visual models and the Division of Fractions Property to divide fractions and mixed numbers. [Operations and Computation Goal 4]

• Measure line segments to the nearest 1 _ 8 inch. [Measurement and Reference Frames Goal 1]

• Apply the concept of a reciprocal. [Patterns, Functions, and Algebra Goal 4]

Key ActivitiesStudents learn a division algorithm for fractions and use it to divide fractions and mixed numbers.

Ongoing Assessment: Informing Instruction See page 540.

Key VocabularyDivision of Fractions Property

MaterialsMath Journal 2, pp. 208 and 209Student Reference Book, pp. 91 and 92Study Link 6�1inch and centimeter ruler � calculator (optional)

Math Boxes 6�2Math Journal 2, p. 210 Students practice and maintain skillsthrough Math Box problems.

Ongoing Assessment: Recognizing Student Achievement Use Math Boxes, Problems 2a– d.  [Operations and Computation Goal 4]

Study Link 6�2Math Masters, p. 183 Students practice and maintain skillsthrough Study Link activities.

ENRICHMENTSimplifying Complex FractionsMath Masters, p. 184Students use division to simplify complex fractions.

EXTRA PRACTICE Practicing Division of Fractions and Mixed NumbersMath Masters, p. 185Students practice dividing fractions and mixed numbers.

EXTRA PRACTICE 5-Minute Math5-Minute Math™, p. 238Students practice multiplying numbers by unit fractions and reciprocals and explore the relationship between multiplication and division.

Teaching the Lesson Ongoing Learning & Practice Differentiation Options

Division of Fractionsand Mixed Numbers

Objective To introduce an algorithm for division of fractions.

�������

537_EMCS_T_TLG2_G6_U06_L02_576922.indd 537 3/24/11 1:18 PM

Dividing Fractions and Mixed NumbersLESSON

6�2

Date Time

Math Message

1. How many 3-centimeter pieces of string can you cut from a piece that is

12 centimeters long?

2. How many 1 _ 2 -inch pieces of string can you cut from a piece that is

4 inches long?

3. How many 3 _ 4 -inch pieces of string can you cut from a piece that is

3 inches long?

4. How many 3 _ 4 -inch pieces of string can you cut from a piece that is

4 1 _ 2 inches long?

Divide. Show your work. Write your answers in simplest form.

5. 3 _ 8 ÷ 5 _ 6 = 6. 4 _ 7 ÷ 2 _ 3 =

7. 3 _ 10 ÷ 3 _ 5 = 8. 11 _ 12 ÷ 8 _ 5 =

4 pieces

8 pieces

4 pieces

6 pieces

3 _ 8 ∗

6 _ 5 = 18 _ 40 =

9 _ 20 4 _ 7 ∗ 3 _ 2 = 12 _ 14 =

6 _ 7

11 _ 12 ∗ 5 _ 8 =

55 _ 96 3 _ 10 ∗

5 _ 3 = 15 _ 30 = 1 _ 2

0 1 2 3 4 5 6 7 8 9 10 11 12 cm

0 1 2 3 4 in.

0 1 2 3 in.

0 1 2 3 4 5 in.

91 93

Division of Fractions Algorithm

a _ b ÷ c _ d = a _ b ∗ d _ c

205_246_EMCS_S_G6_MJ2_U06_576442.indd 208 3/4/11 10:21 AM

Math Journal 2, p. 208

Student Page

538 Unit 6 Number Systems and Algebra Concepts

Getting Started

1 Teaching the Lesson

▶ Math Message Follow-Up WHOLE-CLASSDISCUSSION

(Math Journal 2, p. 208; Student Reference Book, pp. 91 and 92)

Ask volunteers to describe how they solved the problems. Draw a visual model for each problem by partitioning the whole line segment into equal-length segments to represent the pieces of string. This will be especially helpful for English language learners. As you write each division sentence, say the question that corresponds with that problem. For example:

Problem 3: 3 ÷ 3 _ 4 = 4

34" 3

4" 3

4" 3

4"

0 1 2 3 in.

� Say: How many 3 _ 4 -inch pieces of string can you cut from a piece of string that is 3 inches long?

� Write: 3 ÷ 3 _ 4 .

Show how to solve Problem 3 using the common-denominator method students learned in Fifth Grade Everyday Mathematics.

3 ÷ 3 _ 4 = ?

� Rename 3 as a fraction: 3 _ 1 ÷ 3 _ 4 .

� Rename 3 _ 1 as fourths: 12 _ 4 ÷ 3 _ 4 .

� Divide the numerators: 12 ÷ 3 = 4.

Next, demonstrate problems in which the answer is a mixed number or fraction.

2 _ 3 ÷ 4 _ 9 = ?

� Rename 2 _ 3 as ninths: 6 _ 9 ÷ 4 _ 9 .

� Divide the numerators: 6 ÷ 4 = 6 _ 4 = 1 2 _ 4 , or 1 1 _ 2 .

ELL

Math MessageSolve Problems 1–4 on journal page 208.

Study Link 6�1 Follow-UpBriefly go over the answers.

Mental Math and Reflexes Students use > or < to compare fractions and mixed numbers. Suggestions:

1 _ 6 < 1 _ 3

7 _ 8 < 9 _ 10

14 _ 3 > 7 _ 4

3 _ 5 > 1 _ 4

1 5 _ 8 < 1 6 _ 9

3 4 _ 5 < 3 6 _ 7

Mathematical PracticesSMP1, SMP2, SMP3, SMP4, SMP5, SMP6Content Standards6.NS.1

538-541_EMCS_T_TLG2_G6_U06_L02_576922.indd 538 3/20/12 8:07 AM

Find 10 � �23�. Write 10 � �

23� � .

This problem is equivalent to �23

� * � 10, which means “�

23

� of what number is 10?”

The diagram shows that �23

� of the missing number is 10. Since �

23

� of the missing number is 10, �13

� must be 5. Since �13

� of the missing number is 5, the missing number must be 3 * 5 � 15.

So, �23

� of 15 � 10, which means that �23

� * 15 � 10.

10 � �23� � 15

Division of Fractions

Dividing a number by a fraction often gives a quotient that is larger than the dividend. For example, 4 � �

12� � 8.

To understand why this is, it’s helpful to think about what division means.

Equal GroupsA division problem like a � b � ? is asking “How many bs are there in a?” For example, the problem 6 � 3 � ? asks, “How many 3s are there in 6?” The figure at the right showsthat there are two 3s in 6, so 6 � 3 � 2.

A division problem like 6 � �13� � ? is asking, “How many �

13�s

are there in 6?” The figure at the right shows that there are 18 thirds in 6, so 6 � �

13� � 18.

Missing FactorsA division problem is equivalent to a multiplication problemwith a missing factor.

A problem like 6 � �12� � ■ is equivalent to �

12� * ■ � 6.

�12� * ■ � 6 is the same as asking “�

12� of what number equals 6?”

Since �12� * 12 � 6, you know that 6 � �

12� � 12.

Fractions

Scott has 5 pounds of rice. A cup of rice is about �12� pound.

How many cups of rice does Scott have?

This problem is solved by finding how many �12

�s are in 5, which is the same as 5 � �12

�.

So, Scott has about 10 cups of rice.

6 � 3 � 2

6 � �13� � 18

�23

� of ? � 10

Student Reference Book, p. 91

Student Page

Dividing Fractions and Mixed Numbers cont.LESSON

6�2

Date Time

Divide. Show your work. Write your answers in simplest form.

9. �78� � �

49� � 10. �1

72� � �

13� �

11. �59� � �1

10� � 12. �

34� � �

78� �

13. �53� � �

35� � 14. �1

90� � �

23� �

15. 1�58� � �

46� � 16. �

38� � �

82� �

17. �54� � �

186� � 18. 1�

23� � 2�

14� �

19. �89� � �

89� � 20. 3�

78� � 1�

34� �

21. Explain how you found your answer to Problem 19.

22. How is dividing 5 by �15� different from multiplying 5 by �

15�?

equal to 1.both �

89�, and any number divided by itself is

Sample answer: The dividend and the divisor are

by �15� is the same as finding �

15� of 5, which is 1.

You can think of 5 � �15� as “How many �

15�s are

�78� � �

94� � �

6332� � 1�

33

12� �1

72� � �

31� � �

2112� � 1�

34�

�59� � �

110� � �

590� � 5�

59� �

34� � �

87� � �

2248� � �

67�

�53� � �

53� � �

295� � 2�

79� �1

90� � �

32� � �

2270� � 1�2

70�

1�58� � �

64� � 2�

1342� � 2�1

76� �

38� � �

28� � �6

64� � �3

32�

�54� � �1

86� � �

4604� � �

58� 1�

23� � �

49� � �

2207�

�89� � �

98� � �

7722� � 1 3�

78� � �

47� � 2�

1526� � 2�1

34�

Sample answer:

91 93

in 5?” There are 25 fifths in 5. Multiplying

Math Journal 2, p. 209

Student Page

Adjusting the Activity

Lesson 6�2 539

1 _ 4 ÷ 2 _ 3 = ?

� Rename both fractions with common denominators: 3 _ 12 ÷ 8 _ 12 .

� Divide the numerators: 3 ÷ 8 = 3 _ 8 .

Students may find a different visual model for the division of fractions helpful. For example, ask: Leroy has 3 cups of sugar. One batch of cookies uses 3 _ 4 cup. How many batches of cookies can Leroy make with the sugar he has?

34

34

34

34

Have students compare this model to the line segment model they used in the Math Message. Consider reviewing other visual models for the division of fractions shown on pages 91 and 92 of the Student Reference Book with students.

Pose several problems for students to solve, either by drawing a model or by using the common denominator method. Suggestions:

● 1 1 _ 2 ÷ 3 _ 8 4 ● 2 _ 3 ÷ 3 _ 5 1 1 _ 9

● 5 _ 6 ÷ 2 _ 3 1 1 _ 4 ● 3 _ 4 ÷ 1 _ 3 2 1 _ 4

Some students may wonder why, when using the common-denominator method, the denominators are ignored. Remind them of the original problem: How many 3 _ 4 -inch pieces of string can you cut from a piece of string that is 3 inches long? Rephrase the question as: How many pieces of string of a certain length can you cut from a string 3 inches long? You can cut 4 pieces of a certain length, which is the result of dividing the numerators.

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

▶ Introducing the Division WHOLE-CLASSDISCUSSION

of Fractions(Student Reference Book, p. 91)

Another way to solve a division problem such as 3 ÷ 3 _ 4 is to invert and multiply. Write the following on the board:3 _ 1 ÷ 3 _ 4 = 3 _ 1 ∗ 4 _ 3 = 12 _ 3 = 4.

To help students understand this procedure, walk through the following steps:

6 _ 3 = 2 60 _ 30 = 2 600 _ 300 = 2

538-541_EMCS_T_TLG2_G6_U06_L02_576922.indd 539 3/6/11 10:30 AM

Math Boxes LESSON

6�2

Date Time

4. There are 30.48 centimeters in 1 foot.

Complete each statement.

a. mm � 1 ft

b. cm � 1 yd

c. 304.8 cm � ft

d. cm � 1 in.2.5410

91.44304.8

1. Write the reciprocal.

a. �38�

b. �59�

c. 1�34�

d. 0.68

2. Divide. Simplify if possible.

a. 8 � �45� �

b. 5�15� � �

25� �

c. � �29� � �

13�

d. � �194� � �

37�

1310

3. Lines l and m are parallel. Without using a protractor, find the degree measure ofeach numbered angle. Write eachmeasure on the drawing.

5. Express each decimal as a percent.

a. 0.82 �

b. � 0.4375

c. 0.077 �

d. � 0.0090.9%7.7%

82%43.75%

6. If you randomly pick a date in April, howmany equally likely outcomes are there?

Explain your answer.

30

163

60 150

371

93 91–93

1 2

132�

3 4

7 85

132�

132�48�

48�

48�

132� 48�l

m

�83�, or 2�

23�

�95�, or 1�

45�

�2157�, or 1�1

87�

�47� �

23�

1�12�

�

Math Journal 2, p. 210

Student Page

STUDY LINK

6�2 Fraction Division

93

Name Date Time

pyg

gp

Division of Fractions Algorithm

�ab� � �d

c� � �

ab� º �

dc�

Divide. Show your work.

1. �23� � �

56� � 2. 1�

34� � �

2186� �

3. �2340� � �

45� � 4. �

73� � �

37� �

5. �58� � �

58� � 6. 2 � �

14� �

7. �17� � 2�

45� � 8. 5�

56� � 6 �

Try This

Practice

0 1 2 3 4 5 6 7 8 9 10cm

Round each number to the underlined place.

12. 13.561 13. 589.3552 14. 12.9694 13589.3613.56

9. How many �130�-centimeter segments are in 3 centimeters? segments

10. How many �130�-centimeter segments are in 4�

15� centimeters? segments

11. How many �140�-centimeter segments are in 6�

45� centimeters? segments17

1410

1�34� � �

1268� � 1

�73� � �

73� � �

499� � 5�

49�

2 � �41� � 8

�23� � �

65� � �

1125� � �

45�

�2340� � �

54� � 1

�58� � �

85� � 1

5�56� � �

16� � �

3356��

17� � �1

54� � �9

58�

Math Masters, p. 183

Study Link Master

Adjusting the Activity

540 Unit 6 Number Systems and Algebra Concepts

Draw attention to the fact that when the dividend and divisor are each multiplied by the same number, the quotient remains the same.6 _ 3 ∗ 10 _ 10 = 60 _ 30 = 2 6 _ 3 ∗ 100 _ 100 = 600 _ 300 = 2

In the previous lesson, students learned that x _ x = 1 (x ≠ 0) and x ∗ 1 = x. Therefore, multiplying both dividend and divisor by the same number is the same as multiplying by 1.

When students understand how the value of a quotient is maintained, they can use this knowledge to justify the “invert-and-multiply” algorithm as shown below.

3 ÷ 3 _ 4 = ( 3 _ 1 ∗ 4 _ 3 ) ÷ ( 3 _ 4 ∗ 4 _ 3 )

= ( 3 _ 1 ∗ 4 _ 3 ) ÷ 1

= 12 _ 3 , or 4

Students also maintain the value of a quotient when they move the decimal point in both dividend and divisor and rewrite a decimal division problem such as 7.5 ÷ 0.03 as 750 ÷ 3.

For another example of the Division of Fractions Property, have the class read the top of page 91 of the Student Reference Book.

▶ Dividing Fractions PARTNER ACTIVITY

and Mixed Numbers(Math Journal 2, pp. 208 and 209)

Use the “invert-and-multiply” algorithm to solve Problems 15, 18, and 20 as a class. Have students work in pairs to complete pages 208 and 209. Remind them to write their answers in simplest form. Students should compare their answers to those of their partners and use a fraction calculator to resolve disagreements.

Have students estimate whether each quotient will be less than or greater than 1 before solving. For example, in Problem 9, the dividend ( 7 _ 8 ) is greater than the divisor ( 4 _ 9 ), so the quotient will be greater than 1. In Problem 17, the dividend ( 5 _ 4 ) is less than the divisor ( 16_ 8 ), so the quotient will be less than 1.

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

2 Ongoing Learning & Practice

▶ Math Boxes 6�2

INDEPENDENT ACTIVITY

(Math Journal 2, p. 210)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 6-4. The skills in Problems 5 and 6 preview Unit 7 content.

PROBLEMBBBBBBBBBBBOOOOOOOOOOOBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB MMMMMEEEMMBLEBLLBLEBLLLLBLEBLEBLEBLEBLEBLEBLEBLEEEMMMMMMMMMMMMMMOOOOOOOOOOOBBBBBBLBLBBBLBLLBLLLLPROPROPROPROPROPROPROPROPROPROPRPPROPRPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPROROOROROROROROOPPPPPPP MMMMMMMMMMMMMMMMMMMMMMEEEEEEEEEEEEEEEELELELEEEEEEEEELLLLLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRRRRRRRRPROBLEMSOLVING

BBBBBBBBBBBBBBBBBBBB EEELEMMMMMMMMOOOOOOOOOOBBBBLBLBLBLBBLBBROROOROROROROROROROROROROO LELELELEEEEEELEMMMMMMMMMMMMLEMLLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRGGGLLLLLLLLLLLLLVINVINVINVINVINVINNNVINVINVINNVINVINVINVINV GGGGGGGGGGGOLOOOLOOLOOLOO VINVINVINVLLLLLLLLLLVINVINVINVINVINVINVINVINVINVINVINVINVINV NGGGGGGGGGGOLOLOLOLOLOLOLOOOLO VVVVLLLLLLLLLLVVVVVVVVVOOSOSOSOSOSOSOSOSOSOOSOSOSOOOOOSOSOSOSOSOSOSOSOSOOOSOOSOSOSOSOSOSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS VVVVVVVVVVVVVVVVVVVVVVLLLVVVVVVVVLLVVVVVVVLLLLLLLLVVVVVLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLSSSSSSSSSSSSSSSSSSSSSOOOOOOOOOOOOOOOOOOOO GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGNNNNNNNNNNNNNNNNNNNNNNNNNNIIIIIIIIIIIIIIIIIIISOLVING

Ongoing Assessment: Informing Instruction

Watch for students who find the reciprocal of the dividend instead of the divisor. Encourage students to begin their work by marking or highlighting the divisor of each problem.

538-541_EMCS_T_TLG2_U06_L02_576922.indd 540 1/29/11 8:06 AM

LESSON

6�2

Name Date Time

Complex Fractions

A complex fraction is a fraction whose numerator and/or denominator is also a fraction or a mixed number. Fractions such as

, , and are complex fractions.

To simplify a complex fraction, rewrite it as a division problem and divide.

Simplify each complex fraction. Show your work.

1. 2.

3. 4.

Find each missing divisor.

5. �14� � � 1�

34� 6. 2�

12� � � 1�

14�2�

17�

6�15�

��23�

�34�

����56�

�37�

���63

����12�

22�15�

�

�145�

�16

�

��

�49�

10��23

�

Example 1:

Simplify

� 10 � �23�

� 10 � �32�

� �320�

� 15

10�

�23�

10�

�23�

Example 2:

Simplify

� �16� � �

49�

� �16� � �

94�

� �294�

� �38�

�16�

����49

�

�16�

����49

�

�114�

�190� 9�1

30�

Try This

� 3 � �12�

� 3 � �21�

� 6

3���

�12�

6� �34� � �56�

� �34� � �65�

� �1280� � �1

90�

�34�

����56�

� �37� � 6� �37� � �16�

� �432� � �1

14�

� 6�15� � �23�

� �351� � �32�

� �9130� � 9�1

30�

6�15�

���23�

�37�

���6

Math Masters, p. 184

Teaching Master

Division of Fractions Algorithm

�ab� � �d

c� � �

ab� º �

dc�

Name Date Time

Dividing Fractions and Mixed Numbers

pyg

gp

Divide. Show your work.

1. �78� � �

36� � 2. �

1115� � �

13� � 3. �

76� � �1

52� �

4. 6 � �23� � 5. �

45� � 2 � 6. �1

84� � �1

84� �

7. 1�25� � �1

30� � 8. �

136� � 2�

14� � 9. 2�

34� � �

68� �

10. �57� � 1�

35� � 11. 7 � 5 �

13� � 12. 3 �

45� � 8 �

12� �

Try This

�78� � �

63� � �

4224� � 1�

34� �

1115� � �

31� � �

3135� � 2�

15� �

76� � �

152� � �

8340� � 2�

45�

6 � �32� � �

128� � 9 �

45� � �

12� � �1

40� � �

25�

1�25� � �

130� � 4�

23� 2�

34� � �

86� � 3�

23��

136� � �

49� � �

6247� � 2�

1207�

�184� � �

184� � 1

�57� � �

58� � �

2556� 7 � �1

36� � 1�1

56� 3�

45� � �1

27� � �

3885�

LESSON

6�2

Math Masters, p. 185

Teaching Master

Lesson 6�2 541

Ongoing Assessment: Math Boxes Problems 2a–d �Recognizing Student Achievement

Use Math Boxes, Problem 2 to assess students’ ability to divide fractions. Students are making adequate progress if they can use a reliable algorithm to calculate the quotients in Problems 2a–d. [Operations and Computation Goal 4]

Writing/Reasoning Have students write a response to the following: Explain how you determined the number of feet in Problem 4c. Sample answer: If there are 30.48

centimeters in 1 foot, I know that there are 10 times as many centimeters in 10 feet. 30.48 ∗ 10 = 304.8, so 304.8 cm = 10 ft.

▶ Study Link 6�2

INDEPENDENT ACTIVITY

(Math Masters, p. 183)

Home Connection Students practice dividing fractions and mixed numbers.

3 Differentiation Options

ENRICHMENT

INDEPENDENT ACTIVITY

▶ Simplifying Complex Fractions 15–30 Min

(Math Masters, p. 184)

To extend their knowledge of fraction division, students simplify

complex fractions such as 10 _ 2 _ 3 and

1 _ 6 _

4 _ 9 . They also find the value of a

missing divisor.

EXTRA PRACTICE

INDEPENDENT ACTIVITY

▶ Practicing Division of Fractions 15–30 Min

and Mixed Numbers(Math Masters, p. 185)

Students divide fractions and mixed numbers.

EXTRA PRACTICE

SMALL-GROUP ACTIVITY

▶ 5-Minute Math 5–15 Min

To offer more practice multiplying numbers by their reciprocals, as well as rewriting division problems as multiplication problems, see 5-Minute Math, page 238.

EM3cuG6TLG2_538-541_U06L02.indd 541 1/22/11 1:50 PM