Division of Fractions
and Mixed Numbers
Objective To introduce an algorithm for division of fractions.
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Ongoing Learning & Practice
Key Concepts and Skills
Math Boxes 6 2
• Apply the concept of a multiple to rename
fractions using a common denominator. Math Journal 2, p. 210
Students practice and maintain skills
through Math Box problems.
[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. Ongoing Assessment:
Recognizing Student Achievement
Use Math Boxes, Problems 2a– d. [Operations and Computation Goal 4]
Study Link 6 2
Math Masters, p. 183
Students practice and maintain skills
through Study Link activities.
[Patterns, Functions, and Algebra Goal 4]
Key Activities
Students learn a division algorithm for
fractions and use it to divide fractions and
mixed numbers.
Ongoing Assessment:
Informing Instruction See page 540.
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
ENRICHMENT
Simplifying Complex Fractions
Math Masters, p. 184
Students use division to simplify
complex fractions.
EXTRA PRACTICE
Practicing Division of Fractions
and Mixed Numbers
Math Masters, p. 185
Students practice dividing fractions
and mixed numbers.
EXTRA PRACTICE
5-Minute Math
5-Minute Math™, p. 238
Students practice multiplying numbers by
unit fractions and reciprocals and explore
the relationship between multiplication
and division.
Key Vocabulary
Division of Fractions Property
Materials
Math Journal 2, pp. 208 and 209
Student Reference Book, pp. 91 and 92
Study Link 61
inch and centimeter ruler calculator
(optional)
Advance Preparation
Teacher’s Reference Manual, Grades 4–6 pp. 144–147, 149–152
Lesson 6 2
537
Mathematical Practices
SMP1, SMP2, SMP3, SMP4, SMP5, SMP6
Content Standards
Getting Started
6.NS.1
Mental Math and Reflexes
Math Message
Students use > or < to compare fractions and mixed numbers. Suggestions:
1
_
6
7
_
8
14
_
3
<
1
_
3
_
3
5
<
9
_
5
1_
10
7
> _
4
8
4
3_
5
Solve Problems 1–4 on
journal page 208.
1
_
>
4
<
1_
9
<
6
3_
6
Study Link 6 1
Follow-Up
7
Briefly go over the answers.
1 Teaching the Lesson
▶ Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
ELL
(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:
3 =4
Problem 3: 3 ÷ _
4
3"
4
Student Page
Date
0
Time
LESSON
Math Message
1.
91 93
How many 3-centimeter pieces of string can you cut from a piece that is
1
2
3
4
5
6
7
8
9
10
11
0
3.
1
2
4.
3
4 in.
3
-inch pieces of string can you cut from a piece that is
How many _
4
0
4 pieces
1
4
2
3 ÷_
3.
Rename 3 as a fraction: _
1
4
3
3
12
_
_
_
Rename as fourths:
÷ .
3 in.
3
-inch pieces of string can you cut from a piece that is
How many _
4
1
inches long?
4_
6 pieces
2
0
1
2
3
4
1
5 in.
b
d
b
c
3
_
8
∗
6
_
5
=
18
_
40
=
9
_
20
4
_
6. 7
2 =
÷_
3
4
_
7
∗
3
_
2
=
12
_
14
=
6
_
7
3
3
_
7. 10
3
÷_
5 =
3
_
10
∗
5
_
3
=
15
_
30
=
1
_
2
11
_
8. 12
8
÷_
5 =
11
_
12
∗
5
_
8
=
538
Unit 6
9
6 ÷_
2 as ninths: _
4.
Rename _
3
9
9
55
_
96
6 = 1_
2 , or 1_
1.
Divide the numerators: 6 ÷ 4 = _
4
4
2
Math Journal 2, p. 208
205_246_EMCS_S_G6_MJ2_U06_576442.indd 208
4
Next, demonstrate problems in which the answer is a mixed
number or fraction.
2 ÷_
4 =?
_
Divide. Show your work. Write your answers in simplest form.
5
÷_
6 =
4
Divide the numerators: 12 ÷ 3 = 4.
Division of Fractions Algorithm
a ÷_
c =_
a ∗_
d
_
3
_
5. 8
3 in.
Show how to solve Problem 3 using the common-denominator
method students learned in Fifth Grade Everyday Mathematics.
3 =?
3÷_
8 pieces
3 inches long?
2
3.
Write: 3 ÷ _
4
12 cm
1
_
2. How many 2 -inch pieces of string can you cut from a piece that is
4 inches long?
1
3"
4
4 pieces
12 centimeters long?
0
3"
4
3 -inch pieces of string can you cut from a piece
Say: How many _
4
of string that is 3 inches long?
Dividing Fractions and Mixed Numbers
62
3"
4
3/4/11 10:21 AM
Number Systems and Algebra Concepts
Student Page
2 =?
1 ÷_
_
4
Fractions
3
Division of Fractions
3 ÷_
8.
Rename both fractions with common denominators: _
12
12
3.
Divide the numerators: 3 ÷ 8 = _
8
Students may find a different visual model for the division of
fractions helpful. For example, ask: Leroy has 3 cups of sugar.
3 cup. How many batches of cookies can
One batch of cookies uses _
4
Leroy make with the sugar he has?
3
4
3
4
3
4
Dividing a number by a fraction often gives a quotient
1
that is larger than the dividend. For example, 4 2 8.
To understand why this is, it’s helpful to think about what
division means.
Equal Groups
A 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 shows
that there are two 3s in 6, so 6 3 2.
1
3
632
1
s
3
A division problem like 6 ? is asking, “How many
are there in 6?” The figure at the right shows that there are
1
18 thirds in 6, so 6 3 18.
Scott has 5 pounds of rice. A cup of rice is about
How many cups of rice does Scott have?
3
4
1
3
6
1
2
18
pound.
1
1
This problem is solved by finding how many 2s are in 5, which is the same as 5 2.
So, Scott has about 10 cups of rice.
Missing Factors
A division problem is equivalent to a multiplication problem
with a missing factor.
A problem like 6 1
2
1
2
Since
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.
●
3 4
1 ÷_
1_
2
8
5
● _
6
2
● _
3 1_
1
÷_
5 9
3
● _
1 2_
1
÷_
3 4
3
2 1_
1
÷_
3 4
4
1
1
* 12 6, you know that 6 2 12.
2
2
3
Find 10 3. Write 10 2
3
This problem is equivalent to
2
“3 of what number is 10?”
*
.
10, which means
2
The diagram shows that 3 of the missing number is 10.
2
1
1
Since 3 of the missing number is 10, 3 must be 5. Since 3 of
the missing number is 5, the missing number must be 3 * 5 15.
So,
2
3
10 Pose several problems for students to solve, either by drawing a
model or by using the common denominator method. Suggestions:
1
■ is equivalent to * ■ 6.
2
2
1
* ■ 6 is the same as asking “2 of what number equals 6?”
of 15 10, which means that
2
3
2
3
2
3
of ? 10
* 15 10.
15
Student Reference Book, p. 91
Adjusting the Activity
Some students may wonder why, when using the common-denominator
method, the denominators are ignored. Remind them of the original problem:
3
How many _
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
Student Page
V I S U A L
Date
Time
LESSON
62
䉬
▶ Introducing the Division
WHOLE-CLASS
DISCUSSION
of Fractions
(Student Reference Book, p. 91)
3 is to invert
Another way to solve a division problem such as 3 ÷ _
4
and multiply. Write the following on the board:
3 ÷_
3 =_
3 ∗_
4 =_
12 = 4.
_
1
4
1
3
5
11. 9
5
1
9
1
0 ⴱ
1
0
1
5
13. 3
3
5 5
3
ⴱ
5
3
15.
5
4
18 6 5
17. 4
16
8 8
19. 9
8
9 3
To help students understand this procedure, walk through the
following steps:
6 =2
60 = 2
600 = 2
_
_
_
3
Dividing Fractions and Mixed Numbers
Divide. Show your work. Write your answers in simplest form.
63
3
1
7
7 ⴱ 9 7
4
1
7
12
4
32 132
9. 8 9 8
10. 12 3 30
21.
5
18 ⴱ
559
50
9
2
5
9
2 79
21342 2 176
6
4
5
4
ⴱ
8
16
8
9
ⴱ
9
8
40
64
72
72
1
5
8
3
12. 4
3
16. 8
3
1
21
12
3
14
3
4
ⴱ
8
7
24
28
9
2
3 10
ⴱ
3
2
27
20
120
7
8 9
14. 10
ⴱ
cont.
8
2 2
1
7
3
3
8
18.
13 24 20.
38 14 ⴱ
2
8
2
13 ⴱ
7
38 ⴱ
4
7
6
64
4
9
91 93
6
7
7
3
32
20
27
2 1526 2 134
Explain how you found your answer to Problem 19.
Sample answer: The dividend and the divisor are
8
both 9, and any number divided by itself is
equal to 1.
300
22.
1
1
Sample answer:
1
You can think of 5 as “How many 5s are
in 5?” There are 25 fifths in 5. Multiplying
1
1
by 5 is the same as finding 5 of 5, which is 1.
How is dividing 5 by 5 different from multiplying 5 by 5?
1
5
Math Journal 2, p. 209
Lesson 6 2
539
Student Page
Date
Time
LESSON
62
Write the reciprocal.
1.
8
,
3
9
,
5
3
a. 8
5
b. 9
夹
25
0.68 17 ,
Divide. Simplify if possible.
2.
2
or 23
4
or 15
a.
or
2
5
d.
13
2
3
1
12
c.
8
117
10
4
5
8 1
b. 5
5
4
7
3
c. 1
4
d.
Draw attention to the fact that when the dividend and divisor are
each multiplied by the same number, the quotient remains
the same.
6 ∗_
10 = _
60 = 2
6 ∗_
100 = _
600 = 2
_
_
Math Boxes
䉬
2
9
3
1
3
9
14
91–93
There are 30.48 centimeters in 1 foot.
4.
304.8
91.44
b.
132ⴗ
4
48ⴗ
2
3
48ⴗ 5
132ⴗ
l
132ⴗ
7
c.
304.8 cm d.
2.54
cm 1 yd
10
ft
cm 1 in.
48ⴗ
163
6.
82%
b. 43.75%
0.4375
c. 0.077 7.7%
d. 0.9% 0.009
0.82 a.
4
371
Express each decimal as a percent.
5.
100
300
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 ∗_
3 ∗_
4 ) ÷ (_
4)
3÷_
mm 1 ft
m
8
3
Complete each statement.
a.
48ⴗ 1
132ⴗ
30
In the previous lesson, students learned that _xx = 1 (x ≠ 0) and
x ∗ 1 = x. Therefore, multiplying both dividend and divisor by the
same number is the same as multiplying by 1.
3
7
93
Lines l and m are parallel. Without using
a protractor, find the degree measure of
each numbered angle. Write each
measure on the drawing.
3.
10
=
If you randomly pick a date in April, how
many equally likely outcomes are there?
30
=
Explain your answer.
60
1
3
4
3
4
_
_
(1 ∗ 3) ÷ 1
12 , or 4
_
3
3
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.
150
For another example of the Division of Fractions Property, have
the class read the top of page 91 of the Student Reference Book.
Math Journal 2, p. 210
▶ Dividing Fractions
Ongoing Assessment:
Informing Instruction
PARTNER
ACTIVITY
and Mixed Numbers
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.
PROBLEM
PRO
PR
P
RO
R
OB
BLE
BL
LE
L
LEM
EM
SO
S
SOLVING
OL
O
LV
VIN
IN
NG
G
(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.
Study Link Master
Name
Date
STUDY LINK
Time
62
䉬
Division of Fractions Algorithm
a
b
Divide. Show your work.
2
6
12
ⴱ 2
5
5
15
1. 3
3
Adjusting the Activity
Fraction Division
c
d
a
b
3
45
6
º
2.
Have students estimate whether each quotient will be less than or
7
greater than 1 before solving. For example, in Problem 9, the dividend (_
8 ) is
4
_
greater than the divisor ( 9 ), so the quotient will be greater than 1. In Problem 17,
5
16
_
the dividend (_
4 ) is less than the divisor ( 8 ), so the quotient will be less than 1.
93
d
c
3
4
28
16
1 16
14 ⴱ 28 1
A U D I T O R Y
24
3. 30
4
5
5
8
24
30
5
8
5
5. 8
1
7. 7
1
4
2 7
5
5
ⴱ 4 1
7
4. 3
3
7
7
3
8
ⴱ 5 1
5
1
4
7
49
2 8.
5 6 5
6
556 ⴱ 16 36
3
10
1
5
4
10
4
5
How many -centimeter segments are in 4 centimeters?
11.
How many -centimeter segments are in 6 centimeters?
g
p
10
14
17
3
10
How many -centimeter segments are in 3 centimeters?
10.
py g
2
13.56
3
13.
4
5
6
7
8
9
segments
▶ Math Boxes 6 2
Unit 6
V I S U A L
589.3552
589.36
14.
12.9694
INDEPENDENT
ACTIVITY
(Math Journal 2, p. 210)
10
13
Math Masters, p. 183
540
segments
segments
Practice
13.561
T A C T I L E
2 Ongoing Learning & Practice
Round each number to the underlined place.
12.
35
5
ⴱ 14 98
0 1
cm
K I N E S T H E T I C
4
2 ⴱ 1 8
6.
Try This
9.
4
ⴱ 3 9 59
Number Systems and Algebra Concepts
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.
Teaching Master
Ongoing Assessment:
Recognizing Student Achievement
Math Boxes
Problems
2a–d
Name
Date
LESSON
Complex Fractions
62
䉬
A complex fraction is a fraction whose numerator and/or
denominator is also a fraction or a mixed number. Fractions such as
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.
1
10 , 6 ,
4
2
9
3
1
22 5
and are complex fractions.
15
4
To simplify a complex fraction, rewrite it as a division problem and divide.
[Operations and Computation Goal 4]
Example 1:
10
2
Simplify Example 2:
10
2
2 10 1
6
4
9
1
6
Simplify 4
9
3
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.
10 ⴱ
3
2
1
6
4
9
1
6
9
4
ⴱ 30
2
9
24
15
3
8
Simplify each complex fraction. Show your work.
1.
3 12
3 ⴱ 21
6
6
3
3
5
4
4
6
5
6
34 ⴱ 65
9
1280 190 10
3
7
6
6
2
(Math Masters, p. 183)
37 6
37 ⴱ 16
1
432 114 14
1
65 1 2
65
3
2
3
351 ⴱ 32
3
9130 9130 910
3
7
2. 3
3
1
1
3
4
3. 5
6
INDEPENDENT
ACTIVITY
3
3
2
▶ Study Link 6 2
Time
1
4.
6
5
2
3
Try This
Home Connection Students practice dividing fractions
and mixed numbers.
Find each missing divisor.
1
1
3
7
5. 1
4
6.
4
2
1
2
2 1
4
1
Math Masters, p. 184
3 Differentiation Options
ENRICHMENT
▶ Simplifying Complex Fractions
INDEPENDENT
ACTIVITY
15–30 Min
(Math Masters, p. 184)
To extend their knowledge of fraction division, students simplify
1
_
10 and _
6
. They also find the value of a
complex fractions such as _
2
_
missing divisor.
3
4
_
9
Teaching Master
Name
EXTRA PRACTICE
▶ Practicing Division of Fractions
Date
LESSON
INDEPENDENT
ACTIVITY
62
䉬
Division of Fractions Algorithm
15–30 Min
a
b
and Mixed Numbers
7
1. 8
3
6
4.
11
2. 15
ⴱ 63 4224 134
7
8
Students divide fractions and mixed numbers.
a
b
d
c
2
3
4
5. 5
6 SMALL-GROUP
ACTIVITY
7.
5–15 Min
10
3
ⴱ
16
8. 3
423
16
3
1
2
4
10
2
5
8
14
1
4
4
9
ⴱ 152 8340 245
8
6. 14
2 ⴱ
7
6
2
4
5
5
12
7
3. 6
ⴱ 31 3135 215
9.
64
27
21207
3
4
8
14
ⴱ
1
14
8
6
8
2 234 ⴱ
8
6
323
g
p
Try This
5
10. 7
5
7
3
5
1 ⴱ
5
8
11.
25
56
1
3
7 5 7ⴱ
3
16
1156
12.
4
5
1
2
3 8 3 45 ⴱ
2
17
38
85
py g
To offer more practice multiplying numbers by their reciprocals, as
well as rewriting division problems as multiplication problems, see
5-Minute Math, page 238.
3
10
2
5
1 125 ⴱ
1
3
11
15
6 ⴱ 32 128 9
▶ 5-Minute Math
c
d
º Divide. Show your work.
(Math Masters, p. 185)
EXTRA PRACTICE
Time
Dividing Fractions and Mixed Numbers
Math Masters, p. 185
Lesson 6 2
541