Algorithm
U.S. Traditional
Long Division: Decimals
Objective To extend long division to problems in which
Project
Projject
both the divisor and the dividend are decimals.
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Teacher’s
Lesson Guide
Doing the Project
Recommended Use After Lesson 45 and Algorithm Project 8
Materials
Key Concepts and Skills
Math Journal 1 or 2, p. 32P
• Use the Multiplication Rule to find equivalent fractions. Student Reference Book, pp. 37, 54G, 54H,
and 60
[Number and Numeration Goal 5]
• Use long division to solve division problems with decimal divisors. [Operations and Computation Goal 3]
• Multiply numbers by powers of 10. [Operations and Computation Goal 3]
• Explore the meaning of division by a decimal. [Operations and Computation Goal 7]
Key Activities
Students explore the meaning of division by a decimal and extend long division
to decimal divisors.
Key Vocabulary
decimal divisors
Extending the Project
Ex
Students express the remainder in a division problem as a whole number, a fraction,
an exact decimal, and a decimal rounded to the nearest hundredth.
For additional practice, students solve division problems, first using the focus algorithm
(partial-quotients division) and then using any algorithm they choose.
Materials
Math Journal 1 or 2, p. 33P
Student Reference Book, pp. 54E–54J
Online Additional Practice, pp. 33A–33B
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Student Page
Date
Time
PROJECT
1 Doing the Project
Long Division with Decimal Divisors
9
Algorithm Project 9
1.
Draw lines to connect each number model with the number story that fits it best.
Number Model
► Exploring Meanings for
Number Story
What is the area of a rectangle
1.75 m by 50 cm?
2 ∗ 0.10
1.75 ∗ 0.50
2 / 0.10
Sales tax is 10%. What is the sales
tax on a $2 purchase?
Decimal Division
How many dimes are there in $2?
(Math Journal 1 or 2, p. 32P)
1.75 / 0.50
A rectangle’s area is 8 cm 2 and its
width is 0.25 cm. What is its length?
8 / 0.25
How many 50-cm pieces can be cut
from 1.75 m of string?
Have partners solve Problem 1 on journal page 32P. When most
students have finished, have volunteers explain their solutions.
Use student responses to emphasize the following ideas:
Find equivalent problems with no decimals in the divisors and solve.
2.
380 / 5
38 / 0.5 =
3.
84 / 7
12
0.84 / 0.07 =
equivalent problem
Solution:
4.
equivalent problem
76
Solution:
501 / 0.03 =
50,100 / 3
Solution:
16,700
5.
0.465 / 1.5 =
equivalent problem
6.
Solution:
WHOLE-CLASS
DISCUSSION
One way to think about division is as “How many ___
4.65 / 15
equivalent problem
0.31
are in ___?”
Think of missing factors. Given the area and one dimension of
Indy cars use about 1.3 gallons of fuel for each
2.5-mile lap. About how many miles per gallon is that?
a rectangle, for example, we can use division to find the other
dimension.
1.92 miles/gallon
(unit)
Numbers in problems can be represented one way in the
problem’s number story and another way in the matching
number model. For example, 10% in one of the number stories
corresponds to 0.10 in the matching number model.
Math Journal, p. 32P
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The idea that multiplication makes bigger and division makes
smaller, which many students have formed from their work
with whole numbers, does not apply to multiplication and
division by numbers less than 1.
► Dividing with Decimal
WHOLE-CLASS
DISCUSSION
Divisors
(Math Journal 1 or 2, p. 32P)
Remind students of two key facts that may be used to solve
division problems with decimal divisors:
Student Page
Date
Time
PROJECT
9
Representing Remainders as Decimals
A fraction can be interpreted as a division problem, and
vice versa.
Multiplying the numerator and denominator of a fraction by
the same nonzero number results in an equivalent fraction.
Algorithm Project 9
You might write the answer to a problem such as 17 / 6 in two ways: 17 / 6 → 2 R5,
5
or by rewriting the remainder as a fraction: 17 / 6 = 2 _6 .
−
With decimal long division, you can show the quotient as a decimal: 17 / 6 = 2.83.
The repeat bar means that the 3s repeat forever. But, in most situations, having
infinitely many 3s is not practical, so answers are often rounded to some reasonable
number of decimal places, usually two or three: 17 / 6 = 2.83, or 17 / 6 = 2.833.
Write 27 / 0.3 = ? on the board or a transparency. Using
division methods such as partial quotients and long division
is cumbersome with decimal divisors. Ask students how they
might use multiplication to rename the problem to an equivalent
problem that is easier to solve.
Notice that 17 / 6 = 2.83 is not actually true. You can check this by multiplying
2.83 by 6. You won’t get 17. But, for most practical purposes, 2.83 or 2.833 is
close enough to 17 / 6 that most people aren’t bothered by a rounded number model.
Complete the table.
Answer as
Problem
Quotient and
Remainder
Mixed
Number
5
Exact
Decimal
Decimal
Rounded to
Hundredths
−
2.83
2.83
3.75
1.
17 / 6
2 R5
2 _6
2.
15 / 4
3 R3
3_4
3.75
5
_
8
2
5_3
2
6_9
0.625
−
5.66
−
6.22
3.
5/8
0 R5
4.
17 / 3
5 R2
5.
56 / 9
6 R2
3
27
27 / 0.3 can be thought of as ___
.
0.3
0.63
27
27 10
270
___
= ___ * ___ = ___
0.3
0.3 10
3
5.67
270
But _
can be thought of as 270 / 3.
6.22
So, 27 / 0.3 = 270 / 3.
3
The resulting problem, 270 / 3, is easier to solve than the original
problem, 27 / 0.3, but has the same answer.
Math Journal, p. 33P
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U.S. Traditional Long Division: Decimals
3/9/11 9:52 AM
Student Page
Work through similar problems until students understand
the principle:
Decimals and Percents
U.S. Traditional Long Division:
Renaming Fractions as Decimals
Any whole number can
be written as a decimal
by attaching a decimal
point and one or more
0s; the value of the
number remains the
same: 5 = 5.0.
U.S. traditional long division can be used to rename
fractions as decimals.
Multiplying the dividend and the divisor in a division problem by
the same nonzero number does not change the quotient.
With all decimal numbers,
attaching one or more
zeros to the right of the
digit that is furthest to the
right will not change the
value of the number:
8.3 = 8.3000.
Suggestions:
45 ___
10
450
= ___ = 450 / 9 = 50
45 / 0.9 45 / 0.9 = ___
0.9 * 10
9
Use U.S. traditional long division to rename __5 as a decimal.
1,200
100
12
___
= _____ = 1,200 / 3 = 400
12 / 0.03 12 / 0.03 = ____
0.03 * 100
3
1,000
105,000
105
_____
= _______ =
105 / 0.015 105 / 0.015 = _____
0.015 * 1,000
15
105,000 / 15 = 7,000
8
Step 1: Write __58 as a division problem. Write 5 with several 0s after the
decimal point: 5.000. (You can always add more 0s if you need them.)
8 5.000
Step 2: Solve the division problem. Stop when the remainder is 0, or when you have
enough precision for your purposes, or when you notice a repeating pattern.
.625
8 5.000
-4 8
20
-16
40
-40
0
Have students complete journal page 32P. Circulate and assist.
This division problem divided evenly in three decimal places.
5
__
= 0.625
8
2 Extending the Project
Student Reference Book, p. 54I
► Using Long Division to Rename
PARTNER
ACTIVITY
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Fractions as Decimals
(Student Reference Book, pp. 54I and 54J; Math Journal, p. 33P)
Have partners read Student Reference Book, pages 54I and 54J
and then complete journal page 33P. When students have finished,
discuss any difficulties or curiosities they encountered.
Go to www.everydaymathonline.com
to access the additional practice
pages.
NOTE Student Reference Book, pages 54G and 54H provide a detailed step-bystep explanation of how to “clear decimals” in the divisor so that the long division
algorithm can be applied. Other relevant Student Reference Book pages include
page 37 (multiplication by powers of 10) and 60 (using multiplication to find
equivalent fractions).
Online Master
► Solving Division Problems
INDEPENDENT
ACTIVITY
(Online Additional Practice, pp. 33A and 33B; Student Reference Book,
pp. 42, 43, and 54E–54J)
Name
Date
PROJECT
Time
Online
Additional
Practice
Partial-Quotients Division
9
Algorithm Project 9
Use partial-quotients division to solve each problem.
1.
Online practice pages 33A and 33B provide students with
additional practice solving division problems. Use these pages
as necessary.
Linda is wrapping gifts for a charity giveaway. She has
a roll of wrapping paper that is 4.5 m long. If she needs
0.75 m to wrap each gift, how many gifts can Linda wrap
with the roll she has?
6 gifts
2.
Encourage students to use the focus algorithm (partial-quotients
division) to solve the problems on practice page 33A. Invite them
to use any algorithm they wish to solve the problems on the
remaining page. Students may find the examples on Student
Reference Book, pages 42, 43, and 54E–54J helpful.
42 / 0.25
168
Answer:
4.
580 / 0.05
Answer:
11,600
3.
0.68 / 0.04
Answer:
5.
0.48 / 0.15
Answer:
17
3.2
Online Additional Practice, p. 33A
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