Download Scott Foresman Addison Wesley, enVision Math

Name
Reteaching
9-1
Divide a fraction by a whole number.
Find _18 4 4.
Use a model to show _18 .
Divide each eighth into 4 equal parts.
1
1.
5 __
Each section shows _____
32
(8 3 4)
1 4 4 5 __
1.
_
8
32
1 1 1 1 1 1 1 1
8 8 8 8 8 8 8 8
1
32
Divide a fraction by a fraction.
Find _34 4 _14.
Use a number line.
Count the number of _14s in _34.
0
1
4
2
4
3
4
Use repeated subtraction.
Subtract _14 from _34 until the difference
is 0.
3 2_
1 5_
2
_
4
4
4
2 2_
1 5_
1
_
1
4
4
4
1 2_
1 50
_
4
4
Count the number of times you
subtracted to find the quotient.
3 4 _
1 53
_
4
4
There are three _14s.
3 4_
1 53
_
4
4
Solve each division sentence. Use a model if you wish.
1. 3 4 _13 5
9
2. _15 4 4 5
__
1
20
Find each quotient. Simplify if possible.
3. 3 4 _12 5
3 4 __
1 5
6. __
16
16
9. 8 4 _14 5
6
3
32
9
5 4 _13 5 15
7 4 __
1 5
__
7
12
12
9 4 __
1 5
4. __
10
10
5. _15 4 3 5
7.
8. _12 4 6 5
10.
11. _67 4 _17 5
__
1
15
__
1
12
6
12. Draw a Picture The square dancing club meets for 3 hours. Every _34 hour, the
dancers change partners. How many different partners will each dancer have in one
meeting? Draw a picture to show your solution.
3 4 _34 5 4; 4 partners. Check students’ drawings.
13. Writing to Explain Explain why the quotient of two fractions less
than 1 is always greater than either fraction.
The quotient is always greater than the fractions because
you are dividing by a number less than one.
26 Topic 9
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Reteaching 9-1
Understanding Division of
Fractions
Name
Practice
9-1
Understanding Division of
Fractions
Solve each division sentence using the models provided.
9
1. 3 ⫼ _13 =
1
2
1
4
3
4. 8 ⫼ _14 ⫽
7. _78 ⫼ _18 ⫽
10. 6 ⫼ _23 ⫽
1
6
0
Find each quotient. Simplify if possible.
32
7
__
1
5. _17 ⫼ 4 ⫽
11 ⫼ __
1 ⫽
8. __
12
12
9
11. 7 ⫼ _13 ⫽
5
3. _56 ⫼ _16 ⫽
24
2
6
3
6
4
6
28
6. 5 ⫼ _12 ⫽
10
11
1 ⫼3⫽
9. __
12
36
21
15 ⫼ __
1 ⫽
12. __
16
16
Practice 9-1
0
__
1
2. _14 ⫼ 6 ⫽
5
6
__
1
15
13. Draw a Picture Olivia has a piece of ribbon _12 yard long. If she cuts it
into 6 equal pieces, what will be the length of each piece, in yards?
1 yard
Check students pictures: __
12
14. Geometry A regular polygon has a perimeter of 12 units. If each
side measures _34 unit, how many sides does the polygon have?
16
15. Which division expression is
shown by this model?
1
2
3
4
5
6
7
8
9
10
0
9 ⫼ __
1
A __
10
10
9
1
B 1⫼ __
10
9
D 10 ⫼ __
10
9 ⫼1
C __
10
© Pearson Education, Inc. 6
16. Writing to Explain When you divide a whole number by a fraction
less than 1, will the quotient be greater than or less than the whole
number? Explain, and give an example.
The quotient will be greater because you are
dividing by a number less than 1. 4 ÷ _13 = 12:
There are more fraction parts than wholes.
1
2
3
4
5
6
1
7
2
1
Each section 3
8
9
10
11
12
3
Topic 9
27
Name
Reteaching
9-2
Dividing a Whole Number by
a Fraction
Number
Reciprocal
Product
3
1
_
1
1
_
1
1
8
2
_
3
3
8
_
1
3
_
2
Find 14 _47.
Step 1
Step 2
Step 3
Rewrite the division as
multiplication using the
reciprocal of the divisor.
Divide out common factors
if possible. Then multiply.
If your answer is an
improper fraction, change it
to a mixed number.
The reciprocal of _47 is _74 .
14
__
7
1
_74
49
__
2
49 24_
1
__
2
2
2
14 ÷ _47 = 14 × _74
Find the reciprocal of each fraction or whole number.
_7
5
1. _57
2. 11
__
1
11
3. _92
_2
9
Find each quotient. Simplify if possible.
4. 12 _45
7. 24 _34
10. 36 _89
15
32
40_12
5. 2 ÷ _14
8. 18 _89
11. 42 _78
8
20_14
48
8
6. 16 ÷ __
10
10
9. 25 ÷ __
11
12. 40 ÷ _45
20
27_12
50
13. Karolyn makes rolls for a friend’s dinner party. She uses 3 lb of
butter. Each stick of butter weighs _14 lb. How many sticks of butter
does Karolyn need to make her rolls?
12 sticks
32
Topic 9
© Pearson Education, Inc. 6
Reteaching 9-2
To divide a whole number by a fraction, you can multiply the whole
number by the reciprocal of the fraction. The reciprocal of a number
has the numerator and the denominator reversed. The product of a
number and its reciprocal is 1.
Name
Practice
9-2
Dividing a Whole Number by
a Fraction
Find the reciprocal of each fraction or whole number.
1. _59
_9
5
2. 8
_1
3. _73
8
Find each quotient. Simplify if possible.
24
42 4 _79 5 54
_
22 4 _56 5 262
5
15 4 _13 5 45
48
10 4 _56 5 12
_
7 4 _23 5 101
2
6 4 _15 5 30
4. 8 4 _25 5
5. 4 4 _16 5
6. 18 4 _38 5
7.
8.
9.
10.
13.
11.
14.
12.
15.
Practice 9-2
20
12
4 _12 5 24
20
4 _34 5 26_23
94 _18 5 72
_37
16. Writing to Explain Will the quotient of 5 4 _78 be greater than or less
than 5? Explain.
Thequotientwillbegreaterthan5becauseyouare
dividingbyanumberthatislessthan1.
17. Reasoning How many times will you need to fill a _12 cup
measuring cup to measure 4 cups of flour?
8
18. Geometry The distance around a circular flower bed is 36 feet.
Jasper wants to put stakes every 8 inches ( _23 of a foot) around the
bed. How many stakes does he need?
54stakes
19. Algebra Which expression is equal to 9 3 _32 ?
A 2 4 _39
B 3 4 _92
C 9 4 _23
© Pearson Education, Inc. 6
D 9 4 _32
Topic 9
33
Name
Reteaching
9-3
Dividing Fractions
To divide by a fraction, you can multiply by its reciprocal. The reciprocal
of a number has the numerator and the denominator reversed.
Step 1
Step 2
Step 3
Rewrite the division as
multiplication using the
reciprocal of the divisor.
Divide out common factors
if possible. Then multiply.
If your answer is an
improper fraction, change it
to a mixed number.
3 is __
10 .
The reciprocal of __
10
3
4
_
2
5
1
3 _
10
4 __
4 × __
_
5
5
10
3
10
___
3
3
Find each quotient. Simplify if possible.
4
1. _12 _14 _12 2
8 =
2. _47 __
21
Reciprocal of _14
3. _13 _12
6. _37 3
9. _59 _12
9 _
4
12. __
5
10
_2
3
4. _25 ÷ _23
7
7. _13 _89
_1
1_19
1_18
10. _35 _34
13. _13 _38
8 2_
2
_
3
3
8
_
_3
_4
7
__
21
8
1_12
8
Reciprocal of __
21
5
7
5. _58 ÷ __
10
8
_4
5
_8
9
8. _56 ÷ _18
_3
11. _34 ÷ _56
14. _47 ÷ _34
__
25
28
6_23
__
9
10
__
16
21
3 -gallon servings
15. Aaron has _78 gallon of bottled water. How many __
16
can he pour?
4_23 servings
16. Draw a Picture Show how Rebecca can divide _34 of a cake into
9 pieces. What fraction of the whole cake will each piece be?
1
Sample answer: Each piece is __
12 of
the cake.
© Pearson Education, Inc. 6
Reteaching 9-3
3.
Find _45 ÷ __
10
38
Topic 9
Name
Practice
9-3
Dividing Fractions
Find each quotient. Simplify if possible.
1. _13 _56 4. _59 5 3 7. _12 __
10
10. _13 _34 _2
5
_1
9
2. _38 _12 5. _67 _34 1_23
5 _
2 8. __
12
3
_4
9
11. _38 4 _3
4
1_17
_5
8
__
3
32
7 3. _78 __
12
1_12
6. _23 _34 9
14 _
2 9. __
5
15
9 _
3 12. __
5
10
_8
2_13
1_12
13. Writing to Explain Serena said that by looking for common factors
5 1__
9 . Do you
4 _
and simplifying the expression, she found that __
10
8
16
agree with Serena? Why or why not?
Practice 9-3
5
10
__
4
5
_
8
4
25
__
16
9
1__
16
No, Serena needs to use the reciprocal of the divisor,
not the dividend, when she rewrites the expression as
5 ⴝ __
8 , which
4 ⴜ_
4 ⴛ_
a multiplication expression: __
5
1016 8
10
can be simplified as _45 ⴛ _45 ⴝ __
.
25
1 yard wide.
14. A _56-yard piece of fencing is made of boards that are __
12
How many boards make up the fence?
10 boards
3 -lb servings does he
15. Nathan has _78 lb of hummus. How many __
10
have?
11 servings
2__
12
16. Algebra Which equation can you use to find the number of _14 −inch
pieces that can be cut from a piece of metal _58 of an inch long?
A _58 _14 n
© Pearson Education, Inc. 6
B _14 _58 n
C _58 _14 n
D _14 _85 n
Topic 9
39
Name
Reteaching
9-4
Estimating Quotients
When you are working with fractions and mixed numbers, you can
estimate using rounding and compatible numbers.
Estimate 23_56 8_37 .
23_56 8_37
Round each mixed
number to the
nearest whole
number.
24 8 3
Divide.
23_56 8_37 ⬇ 3
4. 35_18 5_45
7. 86_34 10_56
9 2__
3
10. 7__
10
10
30 5 6
31_16 4_58 艐 6
Estimate each quotient.
1. 11_12 6_14
31_16 4_58
2
6
8
4
Change 31_16 and
4_58 to the nearest
compatible whole
numbers.
Think: 31_16 and 4_58
are close to 30
and 5.
Sample answers are shown.
2. 19_13 3_23 5
3. 41_79 7_15
5. 61_38 8_59 7
6. 72_29 7_78
9 2_
5 9
8. 26__
9. 11_27 3_35
10
8
6 7__
5 5_
1 7
4
11. 47__
12. 60__
10
12
12
9
6
9
3
10
13. Critical Thinking Which of these two estimates is closer to the actual quotient?
How do you know?
Lisa’s estimate: 55_12 6_34 艐 54 6 9
Hayden’s estimate: 55_12 6_34 艐 56 7 8
Sample answer: Hayden’s estimate is closer because he
used numbers closer to the original numbers.
14. Patrick uses wire to make wreaths. He has 31_12 feet of wire left on
a spool. Estimate how many 3_34 pieces can he cut from the longer
piece of wire.
Sample answer: About 8 pieces
© Pearson Education, Inc. 6
Reteaching 9-4
Estimate 31_16 4_58 .
44
Topic 9
Name
Practice
9-4
Estimating Quotients
Estimate each product.
1. 37_13 ⫼ 5_78 ⫽
4. 12_34 ⫼ 5_59 ⫽
7. 67_15 ⫼ 7_27 ⫽
10. 71_45 ⫼ 7_89 ⫽
13. 26_14 ⫼ 2_38 ⫽
6
2
10
9
13
Sample answers are shown.
2. 25_12 ⫼ 6_14 ⫽ 4
3. 49_45 ⫼ 6_12 ⫽ 7
9 ⫽
5. 43_23 ⫼ 5_25 ⫽ 9
6. 8_13 ⫼ 2__
3
10
8. 55_59 ⫼ 7_16 ⫽ 8
9. 19_67 ⫼ 4_18 ⫽ 5
7 ⫼ 3_
4 ⫽
11. 15__
12. 79_47 ⫼ 8_58 ⫽ 10
4
10
9
14. 40_89 ⫼ 7_35 ⫽ 6
15. 58_13 ⫼ 19_56 ⫽ 3
16. Number Sense Tran wants to cut strips of paper that are 2_14 in. wide.
His sheet of paper is 11_12 in. wide. He estimates that 11_12 ⫼ 2_14 ⫽ 6,
so he can cut 6 strips from each sheet of paper. Is his estimate an
overestimate or an underestimate? Explain.
Practice 9-4
Sample answer: 6 is an overestimate because Tran
rounded the width of the paper up and the width of the
strips down.
17. Writing to Explain Eliza uses 2_78 feet of yarn in each gift basket
she makes. Explain how to estimate how many baskets Eliza can
make if she has 22 feet of yarn.
Sample answer: Divide 22 by 2_78 . Use compatible numbers
to estimate: 21 ⴜ 3 ⴝ 7, so she has enough yarn to make
at least 7 gift baskets.
18. Geometry The area of this rectangle is 257_14 sq in. What is the
best estimate of side length w?
A 66,000 in.
B 50 in.
C 25 in.
257 14 sq in.
1
10 2 in.
w
D 5 in.
© Pearson Education, Inc. 6
19. Critical Thinking What estimation method did you use to find the length of side w in
Problem 18?
Sample answer: I used compatible numbers:
250 ⴜ 10 ⴝ 25.
Topic 9
45
Name
Reteaching
9-5
Dividing Mixed Numbers
You can follow these steps to find 5_13 4 1_13 and 21 4 2_13.
Step 1
Step 2
Step 3
First estimate.
Then write each number
as an improper fraction.
Find the reciprocal of
the divisor. Rewrite
as a multiplication
problem.
Look for common
factors. Simplify,
then multiply.
1
1
1 .
3
3
Estimate 5  1  5.
16
4


3
3
16
3

3
4
16
3


3
4
Find 5
5
1
1
1 
3
3
Find 21  2 1 .
3
1
Estimate 21  2  10 .
2
1
1
1
3
7
21


3
1
21
3

1
7
3
21


7
1
3
9
21
3


9
1
1
7
1
1
9 is close to 10 ,
2
so the answer is
reasonable.
21
7

1
3
Find each quotient. Simplify if possible.
1. 2_23 4 3_14 5
3. 2_15 4 2_13 5
5. 10 4 3_14 5
__
32
39
__
33
30422_T10_10_6RT-1
35
1
3__
13
2. 1_34 4 4_18 5
4. 5_14 4 3 5
6. 7_14 4 2_18 5
__
14
33
1_34
7
3__
17
7. Writing to Explain Paper needs to be cut for voting ballots.
Each piece of paper is 10_12 in. long. Each ballot should be
1_34 in. long. How many ballots can be cut from one piece of paper?
Six ballots can be cut.
© Pearson Education, Inc. 6
Reteaching 9-5
1
4 is close to 5,
so the answer is
reasonable.
16
4

3
3
21  2
4
16
3
4


4
3
4
1
50 Topic 9
Name
Practice
9-5
Dividing Mixed Numbers
Find each quotient. Simplify if possible.
1. 1_12 ⫼ 2_13 ⫽
4. 3_12 ⫼ 2_14 ⫽
7. 8 ⫼ 2_34 ⫽
__
9
14
1_59
10
2__
11
2. 4_14 ⫼ 3_18 ⫽
5. 3_34 ⫼ 2 ⫽
8. 2_12 ⫼ 1_38 ⫽
9
1__
25
_
178
9
1__
11
3. 2_14 ⫼ 5_12 ⫽
6. 1_12 ⫼ 2_14 ⫽
9. 4_23 ⫼ 1_34 ⫽
__
9
22
_
2
3
2_23
10. Reasoning Is it possible to divide 15 by a mixed number and get a
quotient that is greater than 15? Explain.
No, because you are dividing by a
number that is greater than 1.
Room
Gallons of Paint
2_12
Bedroom
3_34
Living room
4_13
Practice 9-5
Kitchen
Max is painting the inside of an apartment complex. The table shows
how many gallons of paint are needed to paint each type of room.
11. How many kitchens can Max paint with 20 gal?
12. How many living rooms can Max paint with 26 gal?
13. How many bedrooms can Max paint with 60 gal?
8 kitchens
6 living rooms
16 bedrooms
14. Find 4_12 ⫼ 2_14 .
A 1
B 2
C 3
D 4
Sample answer:
Write each mixed number as an improper
fraction. Find the reciprocal of the second
term and multiply the two fractions.
Simplify the answer. 1_45
© Pearson Education, Inc. 6
15. Writing to Explain Explain how you would find 4_15 ⫼ 2_13.
Topic 9
51
Name
Reteaching
9-6
Solving Equations
Here is how to solve addition, subtraction, multiplication, and division
equations with fractions.
Addition
Subtraction
4PMWFO
O 9
3
O
5
O 8
0XOWLSOLFDWLRQ
6ROYHy
y y 'LYLVLRQ
Solve a a
a
a
3 .
3
3
( ) ( )
y
Reteaching 9-6
Solve Y 2 6 .
Y 2 6 Y22
x6 2 x8
a
Z 2 Solve each equation and check your answer.
1. z ⫹ 2_13 ⫽ 3_16
3. x ⫺ 1 ⫽ 4_23
5. _38 ⫹ n ⫽ 10
_
5
6
5_23
9_58
2. 6n ⫽ _34
_
1
8
4. y ⫼ _12 ⫽ 2_18
1
1__
16
6. 2_29 ⫼ 5 ⫽ x
9
_
4
9 in. Yesterday it
7. Algebra The rainfall total for this month is 4__
10
9
1
1
in. Use the equation n ⫹ 2__
⫽ 4__
to calculate how
rained 2__
10
10
10
much rainfall was received before yesterday.
© Pearson Education, Inc. 6
2_45 in.
56
Topic 9
Name
Practice
9-6
Solving Equations
Solve each equation and check your answer.
1. y 1 1_14 5 2_38
2. w 2 2 5 3_12
3. z 4 _34 5 4_14
4. _13 5 _78q
5. 6_12 5 _56b
6. 2_14 5 p 2 _38
7. 2_14 5 x 4 _12
y 5 _98 or 1_18
11 or 5_
1
w 5 __
2
2
__
3
z 5 3 16
8
q 5 __
21
b 5 7_45
p 5 2_58
x 5 _98 or 1_18
8. Number Sense Is the solution of m 4 _23 5 9 greater than or less
than the solution of m 4 _14 5 9? Explain.
m 4 _23 5 9 is greater because you are
dividing by a larger number.
10. Alex had a ball of string. He cut the string into 26 equal
pieces. Each piece measured 3_14 in. Use the equation
m 4 26 5 3_14 to find the length of the ball of string.
Practice 9-6
9. The bakery used 42_13 c of flour. There were 10_13 c left in
the flour bin. Use the equation x 2 42_13 5 10_13 to find
out how many cups of flour the bakery had to start with.
Sample
answer:
52_23 c
84_12 in.
11. Solve 12y 5 2_14 .
A 1_12
B 1_18
7
C __
36
3
D __
16
© Pearson Education, Inc. 6
12. Writing to Explain Write the steps you would use to solve the
equation z 1 3_15 5 6_35 . Solve.
Change the mixed numbers to improper
16 5 __
33 . Then subtract
fractions and write z 1 __
5
5
__
16 from both sides of the equation. So,
5
17 , or 3_
2.
z 5 __
5
5
Topic 9
57
Name
Reteaching
9-7
Problem Solving:
Look for a Pattern
Sometimes you can solve a problem by identifying a pattern.
Here are two types of patterns.
Ask yourself:
Are the numbers increasing?
Are they decreasing?
Do they change by the same amount each time?
Do you add, subtract, multiply, or divide to find
the next number?
Patterns in sets of numbers
15 , __
13 , __
9, _
5, _
3
11 , _
7, _
__
4 4 4 4 4 4 4
Ask yourself:
How is the first figure modified to make the
second figure?
How is the second figure modified to make the
third?
Patterns in groups of figures
30422_T10_10-8RT-1
Remember: Once you have identified a possible number pattern,
check at least two other consecutive numbers to make sure that the
pattern is true for all of the numbers.
Find the missing numbers. Describe the pattern.
1_34 , 2 , 2_14 , 2_12 , 2_34
89, 78, 67, 56 , 45 , 34 , 23 , 12, 1
Add _41
Subtract 11
Add _35
2.
10 ,
3. _15, _45, _75, __
5
__
13
5
,
__
16
5
,
__
19
5
,
__
22
5
25
, __
5
4. Draw the next figure in the pattern below.
5. Number Sense The table below shows the number of cells in a
30422_T10_10-8RT-2
culture. How many cells will there be at 4:30?
Time
Number of Cells
128 cells
62
Topic 9
1:00
1:30
2:00
2:30
3:00
1
2
4
8
16
© Pearson Education, Inc. 6
Reteaching 9-7
1. _34, 1, 1_14, 1_12,
Name
Practice
9-7
Problem Solving:
Look for a Pattern
Find the missing numbers. Describe the pattern.
__
21
16 ,
11 , __
1. _18, _68, __
8 8 8
2. _14, _12, 1,
2
,
__
26
,
8
,
,
8
4
__
31
8
,
,
16
__
36
8
41
, __
8
, 32, 64
3. 1.1, 1.1, 2.2, 6.6, 26.4 , 132 , 792 , 5,544
4. 14_12, 12_34 , 11,
5. 27, 9, 3, 1, _13,
6. 3, 5, 9, 15,
9_14
,
9
,
_
1
23
,
7_12
,
27
,
__
1
33
,
5_34
,
81
,
__
1
, 2_14 , _12
4
___
1
243
45 , 59
, 75
7. Number Sense In the figure, the sum of each row
forms a pattern. What is the sum of the seventh row?
49
Add _58
Multiply by 2
Multiply by 1, then
by 2, then by 3, etc.
Subtract 1_34
Multiply by _13 or
divide by 3
Add 2, add 4,
add
6, etc.
1
2
2
3 3 3
4 4 4 4
5 5 5 5 5
8. Which figure completes this pattern?
A
Practice 9-7
B
C
D
© Pearson Education, Inc. 6
9. Writing to Explain How can you find the answer to 7 without
finding the sum of the numbers in a row?
Sample answer: You can square the number
of the row: row 2 ⴝ 22 or 4, row 3 ⴝ 32 or 9,
row 7 ⴝ 72 or 49, and so on.
Topic 9
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