Objectives
To introduce division of decimals by whole numbers;
and to reinforce the partial-quotients division algorithm.
1
materials
Teaching the Lesson
Key Activities
Students use an estimation strategy for dividing decimals. They solve decimal division
problems that offer review and practice of the partial-quotients division algorithm.
Key Concepts and Skills
• Identify place value in decimals through hundredths.
Math Journal 2, pp. 270 and 271
Study Link 9 8
Teaching Aid Master (Math Masters,
p. 404; optional)
slate
[Number and Numeration Goal 1]
• Divide decimals by whole numbers.
[Operations and Computation Goal 4]
• Round decimals and estimate quotients.
[Operations and Computation Goal 6]
• Use conventional notation to write number sentences.
[Patterns, Functions, and Algebra Goal 2]
Ongoing Assessment: Recognizing Student Achievement Use journal page 270.
[Operations and Computation Goal 6]
2
materials
Ongoing Learning & Practice
Students play Polygon Pair-Up to practice identifying properties of polygons.
Students practice and maintain skills through Math Boxes and Study Link activities.
3
Students divide whole
numbers by whole numbers
and estimate quotients.
READINESS
Students use bills and coins
to model division number
stories involving decimals
and whole numbers.
Study Link Master (Math Masters,
p. 298)
Polygon Pair-Up Polygon Deck and
Property Deck (Math Masters,
pp. 496 and 497)
materials
Differentiation Options
READINESS
Math Journal 2, p. 272
Student Reference Book, p. 258
ENRICHMENT
Students write and solve
number stories involving
division of decimals.
Teaching Masters (Math Masters,
pp. 114 and 299)
Teaching Aid Master (Math Masters,
p. 428)
coins
Technology
Assessment Management System
Journal page 270, Problems 5–7
See the iTLG.
768
Unit 9 Fractions, Decimals, and Percents
Getting Started
Mental Math and Reflexes
Pose division problems. Have students estimate the quotient and write a number model to show how they estimated. Discuss
the strategies used. Suggestions:
Sample answers:
61 / 3
37 / 2
59 / 4
86 / 3
60 / 3 20
40 / 2 20
60 / 5 12
90 / 3 30
135 / 7
344 / 5
621 / 8
459 / 9
Math Message
Think of a number story that could be
solved by dividing 4.2 by 7. Be prepared
to discuss your answer.
140 / 7 20
350 / 5 70
640 / 8 80
450 / 9 50
731 / 99
419 / 51
555 / 62
1,946 / 204
700 / 100 7
400 / 50 8
540 / 60 9
2,000 / 200 10
Study Link 9 8
Follow-Up
Have small groups compare answers. Ask volunteers to share
the strategies they used to determine where to place the
decimal point in each product. Students indicate thumbs-up
if they used a similar strategy.
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
Discuss students’ answers. The goal of the Math Message and
follow-up activity is to increase students’ awareness of how
decimal division problems are based on real-life problems.
Suggest examples like the following, which illustrate different
uses for division:
Partitioning into equal parts: A ribbon is 4.2 meters long.
It must be cut into 7 pieces of the same length. 0.6 meters
or 60 cm each
Money and equal sharing: Think of 4.2 as $4.20, which is
to be shared equally among 7 students. $0.60 or 60¢ each
Calculating an average: Tom watched TV for a total of
4.2 hours in one week. His average viewing time per day was
4.2 7 hours. 0.6 hours or 36 minutes
Calculating a fraction or a percent: Alice and Dave took a
4.2
7-hour trip, and Alice drove for 4.2 hours. So 7 is the fraction
of time that Alice drove, and (4.2 7) 100 equals the percent
of the time that she drove. 0.6 100 60 percent
Links to the Future
Use of mental arithmetic, paper-and-pencil algorithms, and calculators to solve
problems involving the division of decimals by whole numbers is a Grade 5 Goal.
Lesson 9 9
769
Estimating Quotients
WHOLE-CLASS
ACTIVITY
of Decimals
As they did with multiplication of decimals, students will use
estimation to place decimal points in the answers to problems
involving division of decimals.
Write the following problems on the board. Write the 4-digit
quotient shown for each problem, but do not include any
decimal point.
13 50
3冄4
苶0
苶.5
苶
3 7 72
4冄1
苶5
苶0
苶.8
苶8
苶
4 2 00
6冄2
苶.5
苶2
苶
2 2 00
7冄1
苶.5
苶4
苶
Tell students that each problem shows the solution, but the
decimal point is missing in the quotient. Ask students to decide
where each decimal point should go by estimating the quotient.
Discuss students’ estimates and answers. Possible responses:
13.50
3冄4
苶0
苶.5
苶
10 threes equals 30; this is a bit less than 40.5,
so the quotient is somewhat larger than 10. The
number 135.0 is too large, so the answer that is
close to 10 is 13.50.
3 7.7 2
4冄1
苶5
苶0
苶.8
苶8
苶
There are at least 30 fours in 150 (30 4 120).
There are not 40 fours in 150 (40 4 160). The
quotient must be between 30 and 40. So 37.72 is
the answer.
.4 200
6冄2
苶.5
苶2
苶
2.52 6 is about 3 6, which is 2, or 0.5. So the
only possible quotient is 0.4200. The other possible
answers are all greater than 1.
.2200
7冄1
苶.5
苶4
苶
There are many ways to round the numbers to
create easy estimates: 2 10, 1 10, 2 5, and
1 5. All of these have answers from 0.1 to 0.4.
Or, think of sharing $1.54 among 7 people: Each
share will be less than $1.00 but more than $0.10.
All estimates are between 0.1 and 1.0, so the answer
must be 0.2200. Because the purpose of this estimate
is to help students place the decimal point, any of
these estimates is satisfactory.
1
Now write the following problem on the board:
Bill paid $5.52 for 8 ballpoint pens. How much did 1 pen cost?
Ask students to estimate the cost of 1 pen. Round $5.52 to $5 or
to $6, and round 8 to 10. $5 10 $0.50, or $6 10 $0.60;
so the cost of 1 pen is about 50 or 60 cents.
Have volunteers come to the board and divide 552 by 8 (ignoring
the decimal point). Remind students about the division algorithm
they have used before. If students are having difficulty getting
started, remind them that either of the estimates they just made
(50 or 60) would be a good starting point.
770
Unit 9 Fractions, Decimals, and Percents
Student Page
8冄5
苶5
苶2
苶
400
152
80
72
72
0
8冄5
苶5
苶2
苶
480
72
72
0
50
10
Date
Time
LESSON
9 9
60
Dividing Decimals
1. Janine is building a bookshelf. She has a board
22 23
233
that is 3.75 meters long. She wants to cut it
into 5 pieces of equal length. What will be
the length of each piece?
9
69
0.75
meters
2. Three sisters set up a lemonade stand.
On Wednesday they made $8.46.
If they shared the money equally,
how much did each girl get?
9
69
2.82
$
3. Alex and his three friends went out to lunch.
Finally, have students use their initial estimates of the total cost to
place the decimal point in the answer. 552 8 69; the estimate
was about $0.50 or $0.60, so place the decimal point before the 6.
The cost for 1 pen is $0.69.
The total bill, including tax and tip, was $42.52.
They decided that each would pay the same
amount. How much did each person pay?
10.63
$
4. Victor divides a 98.4 cm piece of string into
3 equal pieces. What is the length of each piece?
32.8
centimeters
For each problem below, the division has been done correctly, but the decimal point is missing
in the answer. Write a number model to show how to estimate the answer. Use your estimate
to correctly place the decimal point in the answer.
Adjusting the Activity
Have students rename quotients involving amounts in dollars and cents
as cents. Use the amount in cents to estimate and then rename the estimate as
a dollars-and-cents amount. For example, in the problem above, think of $5.52
as 552 cents. Then 1 pen costs between 50 cents and 60 cents, or between
$0.50 and $0.60.
Sample answers for number models:
1 4•6
6.
4 9•8 2
5.
3冄苶4
苶苶.8
3苶
Number model:
7.
45 / 3 15
6冄苶2
苶苶
9苶.9
8苶苶
2
Number model:
1•6 1 5
8.
4冄苶6
苶.4
苶苶
6苶苶
5
Number model: 10 / 5 2
•8
300 / 6 50
7
5冄苶4
苶.3
苶苶
5
Number model:
5/51
270
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 Journal 2, p. 270
Help students summarize the use of estimation to place the
decimal point when dividing decimals.
Example:
3.66 6 ?
1. Estimate the quotient. 3.66 6 is about 3 6, or 0.5.
2. Divide the numbers as though they were whole numbers.
366 6 61
3. Use the estimate to place the decimal point. 0.61 is close to
the estimate of 0.5.
Student Page
Date
Dividing Decimals
PARTNER
ACTIVITY
Time
LESSON
9 9
Dividing Decimals
(Math Journal 2, pp. 270 and 271; Math Masters, p. 404)
Sample answers for number models:
9.
Students complete journal pages 270 and 271. Encourage them to
explain the strategies they used to place the decimal points in the
quotients.
2.36
Journal
page 270
Problems 5–7
9.44 / 4
Number model:
11. 46.8 12 Number model:
Try This
10 / 5 2
3.9
50 / 10 5
22.4
100 / 5 25
253.8 / 6 42.3
Number model: 250 / 5 50
10. 89.6 / 4 Number model:
12.
Sample answers for number models:
13. 2.96 / 8 Ongoing Assessment:
Recognizing Student Achievement
continued
Write a number model to estimate each quotient. Then divide the numbers as though they
were whole numbers. Use the estimate to help you place the decimal point in the answer.
0.37
14.
Number model:
3
9
13 0.33œ
0.73
3.65 5
Number model:
5/51
Use journal page 270, Problems 5–7 to assess students’ ability to estimate the
quotient of a decimal divided by a whole number. Students are making adequate
progress if they are able to correctly place the decimal points and write number
models for Problems 5–7. Some students may be able to solve Problem 8,
which involves a quotient less than 1.
[Operations and Computation Goal 6]
271
Math Journal 2, p. 271
Lesson 9 9
771
Student Page
Date
Time
LESSON
Math Boxes
9 9
1. Calculate.
2. Insert parentheses to make each number
9
7
a. 10% of 90 b. 5% of 140 80
75
c.
d.
(
9
38
Decimal
Percent
0.7
70%
70
100
75
100
)
(
)
Playing Polygon Pair-Up
c. 40 30 60 100 20
(
0.75
75%
3
5
0.6
60%
72
100
0.72
72%
)
d. 56 / 7 3 14
39
3. Complete the table with equivalent names.
Fraction
(
b. 57 24 15 18
% of 48 36
45
)
a. 4 6 3 3 10
% of 30 24
e. 20% of
150
4
22 R4, 22 12,
1
or 22 3
129
62
6. Study the figure. Draw the other half along
Include the correct unit.
Students play Polygon Pair-Up to practice identifying properties of
polygons. See Lesson 1-6 for additional information.
Math Boxes 9 9
INDEPENDENT
ACTIVITY
(Math Journal 2, p. 272)
the vertical line of symmetry.
Mixed Practice Math Boxes in this lesson are linked
with Math Boxes in Lessons 9-5 and 9-7. The skill in
Problem 6 previews Unit 10 content.
2"
8"
8 2 16
16 in2
Area (Student Reference Book, p. 258; Math Masters, pp. 496 and 497)
61
PARTNER
ACTIVITY
4. Divide. Use a paper-and-pencil algorithm.
268 12 5. What is the area of the parallelogram?
Number model:
2 Ongoing Learning & Practice
sentence true.
135
106 109
272
Math Journal 2, p. 272
Writing/Reasoning Have students write a response to
the following: Draw another parallelogram that has the
same area as the parallelogram in Problem 5. Does the
parallelogram you drew have the same perimeter as the one in
Problem 5? Explain your answer. Sample answer: The perimeter
of the parallelogram in Problem 5 is about 20 inches. I cannot tell
exactly because the length of the shorter side is not given. The
perimeter of the parallelogram that I drew is about 16 inches. They
have the same area, but they do not have the same perimeter.
4"
4"
Study Link Master
Name
Date
STUDY LINK
Time
Study Link 9 9
Dividing Decimals
9 9
For each problem below, the division has been done correctly,
but the decimal point is missing in the answer. Correctly place the
decimal point in the answer.
1 4•8
(Math Masters, p. 298)
1.
88.8 / 6 2.
1.35 / 5 • 2
3.
99.84 / 4 2 4•9 6
4.
2.58 / 3 •8
5.
163.8 / 7 2 3•4
6.
233.28 / 4 7.
Explain how you decided where to place the decimal point in Problem 3.
7 0 0
6 0
5 8 •3 2
Sample answer: I rounded 99.84 up and
then divided to get an estimate. 100 4 = 25,
and 24.96 is close to 25.
Try This
Divide. Show your work.
8.
6冄苶
苶5
2
苶苶
.2
9.
4.2
Answer:
4冄1
苶5
苶4
苶.8
苶
Answer:
10.
38.7
9冄苶
苶.8
5
苶5
苶
Answer:
0.65
Practice
11.
7
8
5
8
2
8
5
12. 9
1
3
2
9
13.
9
10
7
10
2
10
9
14. 10
1
2
4
10
Math Masters, p. 298
772
INDEPENDENT
ACTIVITY
Unit 9 Fractions, Decimals, and Percents
Home Connection Students use estimation to place the
decimal point in the quotient of division problems. They
divide decimals by whole numbers.
Teaching Master
Name
3 Differentiation Options
READINESS
Dividing Whole Numbers and
Date
LESSON
Dividing Whole Numbers
99
INDEPENDENT
ACTIVITY
Time
Write a number model to estimate each quotient. Then divide with a
22 23
paper-and-pencil algorithm. Show your work.
1
2
13
R1,
or
13
30
R2,
or
30
6
3
1. 79 / 6 2. 92 / 3 Number model: Sample answer:
Number model: Sample answer:
80 / 8 10
90 / 3 30
15–30 Min
Estimating Quotients
(Math Masters, p. 299)
1
3.
143 R1, or 1434 573 / 4
Sample answer:
600 / 4 150
9
,
To provide experience with whole-number division and estimating
quotients, have students complete Math Masters, page 299.
READINESS
Solving Division Number
PARTNER
ACTIVITY
5–15 Min
1
945 / 18 52 R9, 52 18 or 52 2
Number model: Sample answer:
4.
Number model:
1,000 / 20 50
Try This
5.
The school has $357 to spend on new science books.
If the books cost $9 each, how many books can they buy?
39
books
Explain how to solve this problem without using a paper-and-pencil algorithm.
Stories Involving Money
Sample answer: Round 357 to 360, then divide 360 by 9,
(Math Masters, pp. 114 and 428)
is only enough for 39 books.
which gives 40. I rounded the amount of money up, so there
Math Masters, p. 299
To explore division of decimals by whole numbers using a money
context, have students use the items on Math Masters, page 114
and dollars and cents to model, write, and solve division number
stories. Be sure to have students discuss how they handled any
remainders. For example:
Jen and Carmen bought:
Teaching Master
Name
Date
LESSON
They shared the cost equally. How much did each girl spend?
($3.59 $2.99) 2 $3. 29
ENRICHMENT
Writing and Solving Division
4 4
Time
Items to Purchase
light bulbs
4-pack
VCR tape
$3.25
tissues
$0.73
batteries
toothpaste
$1.39
$1.09
PARTNER
ACTIVITY
15–30 Min
Number Stories with Decimals
To apply students’ understanding of division of decimals,
have them write and solve number stories involving the
division of a decimal by a whole number. Suggest that
students write one of each of the following types of division
number stories:
transparent tape
$0.84
4-pack
$3.59
ballpoint pen
$0.39
tennis balls
can of 3
paperback book
$2.99
$2.59
Partitioning into equal parts
Money and equal sharing
Calculating an average
Calculating a fraction or a percent
Math Masters, p. 114
Lesson 9 9
773