Objective
To review and practice U.S. traditional long division
with whole numbers and decimals.
1
Doing the Project
Recommended Use: Part A: After Lesson 2-7; Part B: After Lesson 2-8; Part C:
After Lesson 8-2
materials
Math Journal, pp. 13P–15P
Student Reference Book,
pp. 24E–24H and 60E–60I
Key Activities
Students review long division for whole numbers (Part A), decimal dividends (Part B),
and decimal divisors (Part C).
Key Concepts and Skills
• Use long division to rename common fractions as decimals.
[Number and Numeration Goal 5]
• Use the Multiplication Rule to find equivalent fractions.
[Number and Numeration Goal 5]
• Use long division to divide whole numbers and decimals.
[Operations and Computation Goal 2]
• Multiply numbers by powers of 10.
[Operations and Computation Goal 2]
Key Vocabulary
U.S. traditional long division • divisor • dividend • short division
2
Extending the Project
Students learn short division for single-digit divisors.
For additional practice, students solve division problems, first using the focus algorithm (partialquotients division) and then using any algorithm they choose.
materials
Math Journal, p. 16P
Online Additional Practice,
pp. 16A–16D
Student Reference Book, pp. 22, 23,
24E–24H, 42– 44, and 60E– 60I
Additional Information
This project has three parts, each of which is structured as follows:
1. Students work individually to solve a problem using whatever methods they choose.
2. Solutions to the problem are examined in whole-class discussion, including solutions using long division.
3. As necessary, the class works together to use long division to solve one or more similar problems.
4. Students work in partnerships to solve problems with long division.
U.S. traditional long division with whole numbers and decimals is introduced and practiced in a series of algorithm projects in Fourth and Fifth
Grade Everyday Mathematics. If students completed those projects, then the work of this project will be review (except for the extension on
short division) and students may be able to work with minimal direction from you. If your students did not complete the long division projects in
fourth and fifth grades, then you should expect that they will need more support and instruction as they work on this project.
In Everyday Mathematics, U.S. traditional long division is introduced in situations that involve sharing money equally. There are two reasons for
this. One is that U.S. traditional long division fits most naturally with what Everyday Mathematics calls equal-sharing situations—situations in
which a given amount is shared equally in a known number of shares. In applying long division to such problems, one can think about sharing
the larger amounts—those in the left-most places in the dividend—first, and then sharing progressively smaller and smaller amounts as the
algorithm moves to places further to the right in the dividend. (continued)
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Student Page
Date
The other reason for using sharing money problems in early work with long
division is that money naturally models place value, including decimal place
value through hundredths, so using money problems emphasizes important
place-value aspects of long division. Of course, long division is not limited to
problems involving the equal sharing of money, so after initial work with such
situations, students use the algorithm to solve all sorts of division problems. Still,
you will notice that the opening problems in Parts A and B of this project involve
sharing money. If your students have little prior experience with long division, be
sure to continue to use sharing money as a primary context until they understand
how and why the algorithm works.
Time
PROJECT
4
Long Division
Algorithm Project 4
Use any strategy to solve the problem.
1.
The four sixth-grade classes at Linda Vista Elementary
School held a book sale to raise money for their
classroom libraries. The sale raised $464. How much
should each class get?
$116
Use U.S. traditional long division to solve each problem.
2.
3.
$395 / 5 = ?
$79
$908 / 22 = ?
Student Reference Book pages 24E–24H and 60E–60I are important resources for
this project. If your students have significant prior experience with long division,
they may be able to understand these pages well enough to do several parts of
this project on their own. If your students have less experience with long division,
you may want to refer to these pages as background.
$41.27
The directions provide an outline for each part of this project; you will need to
adjust your approach depending on your students’ experience with long division.
4.
5.
837 / 3 = ?
279
975 / 75 = ?
13
1 Doing the Project
Math Journal, p. 13P
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► Part A: Whole Number U.S.
PARTNER
ACTIVITY
Traditional Long Division
(Math Journal, p. 13P; Student Reference Book, pp. 24E–24H)
Ask students to solve Problem 1 on journal page 13P. Once
students have solved the problem individually, they should check
their work with a partner’s work and check the reasonableness
of the calculated results. As students work, circulate to help and
note what methods they are using.
Student Page
Student Page
Whole Numbers
U.S. Traditional Long Division: Single-Digit Divisors
Whole Numbers
U.S. traditional long division is not limited to
dividing money.
U.S. traditional long division is another method you can use to divide.
Note
The “leading” 0 in the quotient
is shown in the problem to help
you understand the long division
method. It should not be
included in the answer.
Share $957 among 5 people.
Step 1: Share the
$100
Step 2: Trade 4
s.
$100
s for 40 $10 s.
That makes 45 $10 s in all.
1
苶5
苶7
苶
5冄9
⫺5
4
1
苶5
苶7
苶
5冄9
⫺5
45
Ò Each person gets 1 $100 .
Ò 1 $100 each for 5 people
Ò 4 $100 s are left.
Step 3: Share the
$10
191
苶5
苶7
苶
5冄9
⫺5
45
⫺45
07
⫺5
19 Ò Each person gets 9 $10 s.
5冄9
苶5
苶7
苶
⫺5
45
⫺45 Ò 9 $10 s each for 5 people
0 Ò 0 $10 s are left.
2
Step 1: Start with the thousands.
Step 2: So trade 3 thousands for 30 hundreds.
Ò There are not enough thousands
0
to share 5 ways.
苶6
苶2
苶8
苶
5冄3
07
苶6
苶2
苶8
苶
5冄3
⫺35
1
Step 3: Trade 1 hundred for 10 tens.
Step 4: Trade 2 tens for 20 ones.
Ò 45 $10 s are to be shared.
Step 4: Share the
s.
3,628 / 5 ⫽ ?
Think about the problem as dividing 3,628 into 5 equal shares.
$1
Share the hundreds.
Ò
Ò
Ò
Ò
Each share gets 7 hundreds.
36 hundreds
7 hundreds 5 shares
1 hundred is left.
s.
Ò Each person gets 1
$1
.
Share the tens.
Ò 7 s are to be shared.
Ò1
$1
Ò2
$1
each for 5 people
s are left.
$957 / 5 ∑ $191 R$2
Each person gets $191; $2 are left over.
072
苶6
苶2
苶8
苶
5冄3
⫺35
12
⫺10
2
Share the ones.
Ò Each share gets 2 tens.
Ò 10 tens ⫹ 2 tens
Ò 2 tens 5 shares
Ò 2 tens are left.
0725
苶6
苶2
苶8
苶
5冄3
⫺35
12
⫺10
28
⫺25
3
Ò Each share gets 5 ones.
Ò 20 ones ⫹ 8 ones
Ò 5 ones 5 shares
Ò 3 ones are left.
3,628 / 5 ∑ 725 R3
Divide.
1. 840 / 7 ⫽ ?
2. 6冄9
苶8
苶4
苶
3. 4冄5
苶3
苶9
苶
4. 5,280 / 6 ⫽ ?
Check your answers on page 424.
1. 5,376 / 6 = ?
2. 6冄8
苶,5
苶8
苶6
苶
3. 4冄6
苶,9
苶2
苶3
苶
4. 8,029 / 3 = ?
Check your answers on page 424.
Student Reference Book, p. 24E
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Student Reference Book, p. 24F
Long Division
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Student Page
Discuss solutions as a class. Expect that students will use several
different methods, including partial-quotients division, informal
paper-and-pencil approaches, and U.S. traditional long
division. Discuss methods other than long division first, and then
work through the long division solution step-by-step. Highlight the
connections between the steps in U.S. traditional long division
and the process of sharing money. See Student Reference Book,
page 24E for an example of the connections.
Whole Numbers
U.S. Traditional Long Division: Multidigit Divisors
You can use U.S. traditional long division to divide by larger numbers.
Share $681 among 21 people.
Make a table of easy multiples of the divisor.
This can help you decide how many to share at each step.
* 21
* 21
* 21
4 * 21
5 * 21
6 * 21
8 * 21
10 * 21
As necessary, use long division to solve one or more similar
problems with the whole class, including problems with multidigit
divisors. (See Student Reference Book, pages 24G and 24H for a
discussion of U.S. traditional long division with multidigit divisors.)
As students solve the problems related to money, ask them to
evaluate the reasonableness of their solutions in the context of the
original situation.
1
21
2
42
Double 21.
3
63
Add 2 21 and 1 21.
84
Double 2 21.
105
Halve 10 21.
126
Double 3 21.
168
Double 4 21.
210
Move decimal point one place to the right.
Step 1: There are not enough [$100]s to
Step 2: Trade the 5 [$10]s for 50 [$1]s.
share 21 ways, so trade 6 [$100]s
for 60 [$10]s.
Share the 51 [$1]s.
Share the 68 [$10]s.
3
苶8
苶1
苶
21冄6
⫺63
5
32
苶8
苶1
苶
21冄6
⫺63
51
⫺42
9
Ò Each person gets 3 [$10]s.
Ò There are 68 [$10]s to share.
Ò 3 [$10]s 21
Ò 5 [$10]s are left.
Ò Each person gets 2 [$1]s.
Ò 50 [$1]s ⫹ 1 [$1]
Ò 2 [$1]s 21
Ò 9 [$1]s are left.
$681 / 21 ∑ $32 R$9
Suggestions:
Share $359 among 6 people. $59 R$5 or $59.83
Share $8,295 among 5 people. $1,659
Student Reference Book, p. 24G
Share $2,859 among 25 people. $114 R9 or $114.36
When students are ready, ask them to solve Problems 2–5 on
journal page 13P. Encourage students to share their solutions
and the strategies they utilized.
► Part B: U.S. Traditional Long
PARTNER
ACTIVITY
Division with Decimal Dividends
(Math Journal, p. 14P; Student Reference Book, pp. 60E, 60F, and 60I)
Ask students to solve Problem 1 on journal page 14P. Once
students have solved the problem individually, they should check
their work with a partner’s work and check the reasonableness of
the calculated results. As students work, circulate to help and note
what methods they are using.
Discuss solutions as a class, again starting with methods other
than long division and then working through the U.S. traditional
long division solution step-by-step. See Student Reference Book,
page 60E for a step-by-step solution of a similar problem.
Student Page
Date
PROJECT
4
Time
Long Division with Decimal Dividends
Algorithm Project 4
Use any strategy to solve the problem.
1. Three friends bought some supplies for a school project
for $14.07. How much should each friend pay?
$4.69
Use U.S. traditional long division to solve each problem.
2. $25.86 / 6 = ?
3. 1.071 / 7 = ?
$4.31
0.153
4. 7.0000 / 11 = ?
5. Rename
−−
0.63
3
_
as a decimal.
8
0.375
Math Journal, p. 14P
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Decimals and Percents
U.S. Traditional Long Division: Decimal Dividends
You can use U.S. traditional long division to divide money in
dollars-and-cents notation.
Share $5.29 among 3 people.
Step 1: Share the dollars.
1
3冄$
苶5
苶.2
苶9
苶
⫺3
2
Ò Each person gets 1 dollar.
Ò 1 dollar each for 3 people
Ò 2 dollars are left.
Step 2: Trade the dollars for dimes. Share the dimes.
1.7
3冄$
苶5
苶.2
苶9
苶
⫺3
22
⫺2 1
1
Ò Each person gets 7 dimes. Write a decimal point
to show amounts less than a dollar.
Suggestions:
Ò 20 dimes ⫹ 2 dimes
Ò 7 dimes each for 3 people
Ò 1 dime is left.
Share $5.79 among 3 people. $1.93
Step 3: Trade the dime for pennies. Share the pennies.
1.76
3冄$
苶5
苶.2
苶9
苶
⫺3
22
⫺2 1
19
⫺18
1
As necessary, use long division to solve similar problems with the
whole class, including problems in non-money contexts, problems
that involve extending the division into decimal places not present
in the original dividend, and renaming-fractions-as-decimals
problems. See Student Reference Book, pages 60E, 60F, and
60I. As students solve the problems, ask them to evaluate the
reasonableness of their solutions in the context of the original
situation.
Ò Each person gets 6 pennies.
A ribbon that is 7.5 m long is to be cut into 6 pieces.
How long should each piece be? 1.25 m
−
7
Rename _
as a decimal. 0.77 or 0.78
Ò 10 pennies ⫹ 9 pennies
Ò 6 pennies each for 3 people
Ò 1 penny is left.
9
Each person gets $1.76. There is 1¢ left.
$5.29 / 3 ∑ $1.76 R1¢
Divide.
苶8
苶.6
苶1
苶
2. 7冄$
1. $7.26 / 6 = ?
3. 7冄$
苶5
苶.6
苶2
苶
4. $8.04 / 3 = ?
When students are ready, ask them to solve Problems 2–5 on
journal page 14P. Encourage them to share their solutions and
the strategies they utilized.
Check your answers on page 424A.
Student Reference Book, p. 60E
► Part C: U.S. Traditional Long
PARTNER
ACTIVITY
Division with Decimal Divisors
(Math Journal, p. 15P; Student Reference Book, pp. 60G and 60H)
Ask students to solve Problem 1 on journal page 15P. Once
students have solved the problem individually, they should check
their work with a partner’s work. As students work, circulate to
help and note what methods they are using.
Discuss solutions as a class, starting with methods other than
long division and then working through the U.S. traditional
long division solution step-by-step. See Student Reference Book,
pages 60G and 60H for step-by-step solutions of similar problems.
Student Page
Solve one or more similar problems with the whole class.
Decimals and Percents
Suggestions:
You can use U.S. long division to divide decimals that do not represent money.
732 / 0.6 = ? 1,220
3.97 / 5 ⫽ ?
Step 1: Trade the ones for tenths and share the tenths.
.7
5冄3
苶.9
苶7
苶
⫺35
4
Ò
Ò
Ò
Ò
Each share gets 7 tenths. Write a decimal point in the quotient.
3 ones ⫹ 9 tenths ⫽ 39 tenths
7 tenths ⴱ 5 ⫽ 35 tenths
4 tenths are left.
45.05 / 0.5 = ? 90.1
603 / 0.0009 = ? 670,000
Step 2: Trade the remaining tenths for hundredths. Share the hundredths.
.79
苶.9
苶7
苶
5冄3
⫺35
47
⫺ 45
2
When students are ready, ask them to solve Problems 2–5 on
journal page 15P.
Ò Each share gets 9 hundredths.
Ò 4 tenths ⫹ 7 hundredths ⫽ 47 hundredths
Ò 9 hundredths ⴱ 5 ⫽ 45 hundredths
Ò 2 hundredths are left.
At this point, you can either round 0.79 to 0.8 and write 3.97 / 5 ≈ 0.8,
or you can continue dividing into the thousandths.
Step 3: Continue dividing into the thousandths. Add a 0 to the end of 3.97.
(Adding 0s or “padding” a decimal with 0s doesn’t change its value.)
.794
苶.9
苶7
苶0
苶
5冄3
⫺35
47
⫺ 45
20
⫺ 20
0
Ò Each share gets 4 thousandths.
Ò 3.97 ⫽ 3.970
Ò 2 hundredths ⫹ 0 thousandths ⫽ 20 thousandths
Ò 4 thousandths ⴱ 5 ⫽ 20 thousandths
Ò No thousandths are left.
3.97 / 5 ⫽ 0.794
Divide.
1. 8.28 / 4 ⫽ ?
苶.6
苶4
苶
2. 4冄9
3. 6冄8
苶.6
苶7
苶
4. 38.65 / 5 = ?
Check your answers on page 424A.
Student Reference Book, p. 60F
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2 Extending the Project
Time
PROJECT
Long Division with Decimal Divisors
4
Algorithm Project 4
Use any strategy to solve the problem.
► Exploring Short Division
PARTNER
ACTIVITY
1. Donuts cost $0.89 each at the Farmers’ Market.
How many donuts can be bought with $11.50?
12
(Math Journal, p. 16P)
Students may enjoy learning short division, which is an efficient
paper-and-pencil method for solving problems with single-digit
divisors. The method can be used with multidigit divisors, but the
mental arithmetic involved is complicated, so short division is
normally used only with single-digit divisors.
Students should study the examples on journal page 16P. Once
they understand the method, they use it to solve Problems 1–6.
Ask students to discuss their solution strategies.
► Solving Division Problems
For Problems 2–5, find equivalent problems with no decimals in the divisors.
Then solve the equivalent problems.
2. 24 / 0.8 =
3. 27.090 / 0.06 =
240 / 8 = 30
2,709 / 6 = 451.5
equivalent problem
equivalent problem
4. 28.8 / 1.8 =
5. 0.0084 / 0.3 =
288 / 18 = 16
0.084 / 3 = 0.028
equivalent problem
equivalent problem
INDEPENDENT
ACTIVITY
(Online Additional Practice, pp. 16A–16D; Student Reference Book,
pp. 22, 23, 24E–24H, 42–44, and 60E–60I)
Math Journal, p. 15P
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Online practice pages 16A–16D provide students with additional
practice solving division problems. Use these pages as necessary.
Encourage students to use the focus algorithm (partial-quotients
division) to solve the problems on practice page 16A. Invite them
to use any algorithm to solve the problems on the remaining
pages. Students may find the examples on Student Reference Book,
pages 22, 23, 24E–24H, 42– 44, and 60E–60I helpful.
NOTE Go to www.everydaymathonline.com to access the additional
practice pages.
Online Master
Name
Student Page
Date
PROJECT
4
Date
Time
Time
PROJECT
Online
Additional
Practice
Partial-Quotients Division
Short Division
4
Algorithm Project 4
Algorithm Project 4
Use partial-quotients division to solve each problem.
Short division is a fast way to divide with paper and pencil. It’s like long division,
but all the multiplying and subtracting is done mentally. Short division works
best with single-digit divisors.
1. Rita bought 18 pounds of grapes for the class picnic.
She spent $54. What was the price per pound?
Study the examples below. Then use short division to solve Problems 1–6.
$3 per pound
Example 1:
Example 2:
Long Division
2. 423 / 9 = ?
47
3. 985 / 5 = ?
Short Division
197
Long Division
3 5 8 R2
3 5 8
5 1 7
5 1 7 9 2
2
9 2
2 1
4 0
2 1
0
2
7
7
8 R2
3
8
39
42
2
2
9
6 R2
6
8
29
20
1
4
2
5 R2
5
17
10
22
5
2.
7
1
2
2
5 R4
8
15
17
39
56
3.
3
5.
4
Online Additional Practice, p. 16A
16A
1
0
4 2
1.
EM3cuOP1_G6_P04_16A-16D.indd
2
0 2
2 5
5. 784 / 14 = ?
3 9 2
9
2 9
$46
3 0 7
3 9 2 1
1 5
4. $138 / 3 = ?
Short Division
3 0 7
4
4.
6
5
6.
9
3
9
4
6 R5
6
28
41
4
2
2 R3
8
20
21
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