U.S. Traditional
Long Division
Algorithm
Objective To review and practice U.S. traditional long division
Project
Projject
with whole numbers.
w
www.everydaymathonline.com
eToolkit
Algorithms
Practice
EM Facts
Workshop
Game™
Family
Letters
Assessment
Management
Doing the Project
Recommended Use After Lesson 42
Common
Core State
Standards
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Mathematical Practices
SMP1, SMP2, SMP3, SMP4, SMP5, SMP6, SMP8
Content Standards
5.NBT.6
Key Concepts and Skills
• Apply multiplication and division facts with long division. [Operations and Computation Goal 2]
• Use long division to divide whole numbers by single-digit and multidigit
whole numbers. [Operations and Computation Goal 3]
Materials
Math Journal 1 or 2, pp. 25P–28P
Student Reference Book, pp. 24E–24J
play money (optional)
• Use estimation to carry out long division efficiently. [Operations and Computation Goal 6]
Key Activities
Students review, practice, and extend U.S. traditional long division with whole numbers.
Key Vocabulary
U.S. traditional long division dividend divisor quotient remainder
Extending the Project
Ex
Students create and solve long division puzzles.
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. 28P
Math Masters, p. 415
Online Additional Practice, pp. 28A–28D
Student Reference Book, pp. 22–24,
and 24E–24J
A32
Algorithm Project 7
U.S. Traditional Long Division
1 Doing the Project
► Reviewing Long Division with
WHOLE-CLASS
DISCUSSION
Single-Digit Divisors
(Math Journal 1 or 2, p. 25P)
Have students solve Problem 1 on journal page 25P. Tell them they
may use paper and pencil, or any manipulatives or tools that they
wish, except calculators.
Have volunteers explain their solutions. $2,365 / 6 → $394 R$1
Expect that students will use several different methods, including
partial-quotients division, various informal paper-and-pencil
approaches, dealing with manipulatives, and drawing pictures.
Some students may also use U.S. traditional long division, which
was introduced and practiced in fourth-grade algorithm projects.
For example:
Using partial-quotients division
6 2365
- 1200
1165
- 600
565
- 300
265
- 240
25
- 24
1
394
6 2365
- 18
56
- 54
25
- 24
1
200
100
50
$2,365 ÷ 6 = $394
remainder = $1
40
Long division
4
394
$2,365 ÷ 6 → $394 R$1
Student Page
Date
Time
PROJECT
Using an informal paper-and-pencil method
$2365
- 600
1765
- 600
1165
- 600
565
- 300
265
- 240
25
- 24
1
U.S. Traditional Long Division
7
Algorithm Project 7
Use any strategy to solve the problem.
← $100 for each worker
1.
A local newspaper decided to give bonuses to its delivery
workers. The paper gave $2,365 in bonuses, which was
shared evenly by 6 workers. How much did each worker get?
$394, with $1 left over
← $100 for each worker
Use U.S. traditional long division to solve each problem.
2.
← $100 for each worker
3.
$837 / 3
Answer:
$279
$2,257 / 5
Answer:
$451 R$2
← $50 for each worker
← $40 for each worker
4.
5.
$8,091 / 9
Answer:
$899
782 / 4
Answer:
195 R2
← $4 for each worker
$100 + $100 + $100 + $50 + $40 + $4 → $394 remainder $1
Using U.S. traditional long division (See margin.)
Math Journal, p. 25P
25P-28P_EMCS_S_MJ2_G5_P07_576434.indd 25
3/8/11 11:38 AM
Algorithm Project 7
A33
NOTE Sharing money is a useful context for teaching and learning long
division. The process of sharing bills of various denominations ($100 bills,
$10 bills, and $1 bills) and coins (dimes and pennies), starting with the largest
denomination and trading for smaller denominations as necessary, is broadly
parallel to the steps in long division.
NOTE Although the largest note in
circulation today is the $100 bill, in the past,
limited numbers of much larger bills have
been issued, including $1,000, $10,000, and
$100,000 bills. These notes were withdrawn
from circulation by President Nixon because
they were rarely used and were attractive to
counterfeiters.
Review how to solve the problem using U.S. traditional
long division. Illustrate each step with pictures and, if
possible, model the steps using play money. Emphasize the
connections between the steps in long division and the process
of sharing money.
Step 1:
Step 1
Write the problem $2,365 / 6 in the long division format on the
board or a transparency.
7dcjhid
WZH]VgZY
6 2 3 6 5 ← $2,365 is to be shared: $2,365 is the dividend.
↑
' &!%%% h
( &%% h
+
&%
h
*
&
h
The money is to be
shared by 6 workers:
6 is the divisor.
Step 2:
Step 2
Encourage students to think about sharing actual bills:
2 [$1,000]s, 3 [$100]s, 6 [$10]s, and 5 [$1]s. There are not enough
[$1,000]s for 6 equal shares, so trade the 2 [$1,000]s for 20
[$100]s. There were 3 [$100]s already, so there are 20 + 3 = 23
[$100]s after the trade.
7dcjhid
WZH]VgZY
'( &%% h
+
&%
h
*
&
h
6 2 3 6 5 ← 20 [$100]s from the 2 [$1,000]s +
3 [$100]s = 23 [$100]s
Step 3:
Step 3
7dcjhid
WZH]VgZY
Share the 23 [$100]s. Each worker gets 3 [$100]s; 5 [$100]s are
left over.
L&
* &%% h
+
&%
h
*
&
h
L'
L(
L)
L*
L+
( &%% h ( &%% h ( &%% h ( &%% h ( &%% h ( &%% h
← Each worker gets 3 [$100]s.
3
6 2365
← 3 [$100]s each for 6 workers = 18 [$100]s
18
−−−−
← 5 [$100]s are left.
5
Step 4:
Step 4
7dcjhid
WZH]VgZY
*+
&%
h
*
&
h
Trade the 5 [$100]s for 50 [$10]s. There were 6 [$10]s already,
so there are 50 + 6 = 56 [$10]s.
L&
L'
L(
L)
L*
L+
( &%% h ( &%% h ( &%% h ( &%% h ( &%% h ( &%% h
3
6 2365
18
−−−−
56
A34
Algorithm Project 7
U.S. Traditional Long Division
← 50 [$10]s from the 5 [$100]s + 6 [$10]s = 56 [$10]s
Step 5:
Step 5
Share the 56 [$10]s. Each worker gets 9 [$10]s; there are 2 [$10]s
left over.
← Each worker gets 9 [$10]s.
39
6 2365
18
−−−−
56
4
−−−−
2
-5
7dcjhid
WZH]VgZY
L&
L'
L(
L)
L*
L+
'
&%
h
( &%% h ( &%% h ( &%% h ( &%% h ( &%% h ( &%% h
*
&
h
. &% h . &% h . &% h . &% h . &% h . &% h
← 9 [$10]s each for 6 workers = 54 [$10]s
← 2 [$10]s are left.
Step 6
Step 6:
Trade the 2 [$10]s for 20 [$1]s. There were 5 [$1]s already, so there
are 20 + 5 = 25 [$1]s.
7dcjhid
WZH]VgZY
'*
&
h
L&
L'
L(
L)
L*
L+
( &%% h ( &%% h ( &%% h ( &%% h ( &%% h ( &%% h
. &% h . &% h . &% h . &% h . &% h . &% h
39
6 2365
18
−−−−
56
-5 4
−−−−
2 5 ← 20 [$1]s from the 2 [$10]s + 5 [$1]s = 25 [$1]s
Step 7
Step 8:
Each worker gets $394: $394 is the quotient. The $1 that is left
over is the remainder. A number model is a good way to show
the answer:
&
h
L&
L'
Suggestions:
$4,967 / 6 $827 R$5
$2,850 / 8 $356 R$2
$705 / 5 $141
$1,008 / 9 $112
$2,048 / 4 $512
$402 / 4 $100 R$2
$4,290 / 7 $612 R$6
$2,040 / 8 $255
L)
L*
L+
. &% h . &% h . &% h . &% h . &% h . &% h
)
&
h )
&
h )
&
h )
&
h )
&
h )
&
$2,365 shared by 6 workers
NOTE Some students may benefit from
seeing long division on a computation grid
that shows place-value column names.
$2,365 / 6 → $394 R$1
U.S. traditional long division is complicated, so consider working
one or two more examples with the whole class. Continue to use
sharing money as a context and drawing pictures to show actions
with money, as needed.
L(
( &%% h ( &%% h ( &%% h ( &%% h ( &%% h ( &%% h
6
-
Ones
56
-5 4
−−−−
25
2 4 ← 4 [$1]s each for 6 workers = 24 [$1]s
−−−−
1 ← 1 [$1] is left.
&
Tens
3 9 4 ← Each worker gets 4 [$1]s.
6 2365
18
−−−−
7dcjhid
WZH]VgZY
Hundreds
Share the 25 [$1]s. Each worker gets 4 [$1]s; 1 [$1] is left over.
Thousands
Step 7:
3
9
4
2
3
6
5
1
8
-
5
6
5
4
-
2
5
2
4
1
Algorithm Project 7
A35
h
Student Page
Date
► Solving Long Division Problems
Time
PROJECT
U.S. Traditional Long Division
7
cont.
with One-Digit Divisors
Algorithm Project 7
6.
PARTNER
ACTIVITY
The fifth graders at Freemont School washed cars to
raise money for instruments for their music classes.
They worked for two Saturdays and raised $632. If they
charged $8 for each car they washed, how many cars
did they wash?
(Math Journal 1 or 2, pp. 25P and 26P; Student Reference Book,
pp. 24E and 24F)
79 cars
7.
Have students use U.S. traditional long division to solve
Problems 2–9 on journal pages 25P and 26P. They might find the
examples on Student Reference Book, pages 24E and 24F helpful.
Write a number story that fits $750 / 8. Then solve your
number story.
$93 R$6; Number stories vary.
► Introducing Long Division
Fill in the missing digits in these long division problems.
8.
9.
18 5
4 7 4 2
–4
34
–3 2
2 2
–2 0
2
5 9 2
with Multidigit Divisors
7
4, 1 4 5
–3 5
6
4
(Math Journal 1 or 2, p. 27P)
–6 3
1 5
–1 4
Have students solve Problem 1 on journal page 27P. Circulate
and assist.
1
Math Journal, p. 26P
25P-28P_EMCS_S_MJ2_G5_P07_576434.indd 26
3/8/11 11:38 AM
Time
PROJECT
Long Division with Multidigit Divisors
7
Algorithm Project 7
Use any strategy to solve the problem.
1.
When most students have finished, discuss students’ solutions.
$6,714 / 27 → $248 R$18 Expect that students will use various
methods.
27 6714
- 2700
100
4014
- 2700
100
1314
- 540
774
- 540
234
- 135
99
- 54
45
- 27
18
Student Page
Date
WHOLE-CLASS
DISCUSSION
Holmes School had a carnival to raise money for
classroom supplies. The carnival raised $6,714, which
was to be shared equally by all 27 teachers at Holmes.
How much should each teacher have received?
6714 ÷ 27
$248, with $18 left over
20
20
5
2
1
248
248 R18
Partial quotients
27 ∗ 1 = 27
27 ∗ 2 = 54
27 ∗ 3 = 81
27 ∗ 4 = 108
27 ∗ 5 = 135
27 ∗ 6 = 162
27 ∗ 8 = 216
27 ∗ 10 = 270
248
27 6714
- 54
131
- 108
234
- 216
18
6714 ÷ 27
248 R18
Long division
Use U.S. traditional long division to solve each problem.
2.
$4,864 / 32
Answer:
4.
$152
3.
Answer:
5.
5,238 / 27
194
Answer:
$17,187 / 43
Demonstrate how to solve the problem using long division.
Illustrate each step on the board or a transparency.
$399 R$30
Carrying out long division with multidigit divisors can be
challenging, particularly for students whose estimation skills
are not well developed. Have students make a table of simple
multiples of the divisor to use in deciding how many to share at
each step.
4,785 / 87
Answer:
55
Math Journal, p. 27P
25P-28P_EMCS_S_MJ2_G5_P07_576434.indd 27
A36
Algorithm Project 7
3/8/11 11:38 AM
U.S. Traditional Long Division
Student Page
Example: Solve: $6,714 / 27
$6,714 has only 6 [$1,000]s, which are not enough to share among
27 teachers. Trade the 6 [$1,000]s for 60 [$100]s, which makes
67 [$100]s in all. Share the 67 [$100]s among the 27 teachers. Use the
table of multiples to help decide how many each teacher should get.
← Each teacher gets 2 [$100]s.
2
27 6 7 1 4 ← Trade 6 [$1,000]s for 60 [$100]s.
There are 67 [$100]s to share
54 ←
−−−−
2 [$100]s * 27 teachers
13 ←
13 [$100]s are left.
Whole Numbers
U.S. Traditional Long Division
U.S. traditional long division is a method you can use to divide.
Share $845 among 3 people.
Step 1: Share the
2
3
845
-6
2
131
108
−−−−−−
23
← 13 [$100]s + 1 [$10] = 131 [$10]s
← 4 [$10]s * 27 teachers
s.
← 2 $100 s each for 3 people
← 2 $100 s are left.
$100
Step 2: Trade 2
s for 20 $10 s.
That makes 24 $10 s in all.
2
845
3
-6
← 24
24
Step 3: Share the
$10
$10
s are to be shared.
s.
← Each person gets 8 $10 s.
28
845
3
-6
24
-24
← 8 $10 s each for 3 people
0
← 0 $10 s are left.
Trade the 13 [$100]s for 130 [$10]s. There was 1 [$10] already, so
there are 131 [$10]s after the trade. Share the 131 [$10]s among the
27 teachers.
2 4 ← Each teacher gets 4 [$10]s.
27 6714
54
−−−−
$100
← Each person gets 2 $100 s.
Student Reference Book, p. 24E
024A-024J_EMCS_S_SRB_G5_576515.indd 24E
3/8/11 4:50 PM
← 23 [$10]s are left.
Trade the 23 [$10]s for 230 [$1]s. There were 4 [$1]s already, so there
are 234 [$1]s altogether. Share the 234 [$1]s among the 27 teachers.
2 4 8 ← Each teacher gets 8 [$1]s.
27 6714
54
−−−−
131
108
−−−−−−
2 3 4 ← 23 [$10]s + 4 [$1] = 234 [$1]s
← 8 [$1] * 27 teachers
1 8 ← 18 [$1]s are left.
Student Page
216
−−−−−−
$6,714 / 27 → $248 R$18
Whole Numbers
Step 4: Share the
$1
s.
281 ← Each person gets 1
3
845
-6
.
24
Each teacher gets $248. $18 are left over.
Discuss what the remainder means in this problem. It’s the part of
the original $6,714 raised by the carnival that is left over after each
of the 27 teachers receives $248. Ask students what they think should
be done with the leftover money. Some students may want to continue
dividing. The $18 left over could be traded for 180 dimes, shared 27
ways, and so on. The quotient, even carried to pennies, won’t come
out evenly. There will be 18 pennies left over. However, by continuing
the division, a bit more of the carnival proceeds could be distributed.
Project 8 will extend long division to decimal dividends. Some students
might want to attempt to extend the method on their own.
$1
-24
05 ← 5
$1
-3
←1
$1
2 ←2
$1
s are to be shared.
each for 3 people
s are left.
Each person gets $281; $2 is left over.
$845 3 → $281 R$2
Divide.
1. $780 6
2. $973 4
3. $729 6
4. $828 8
Check your answers on page 442.
Student Reference Book, p. 24F
024A-024J_EMCS_S_SRB_G5_576515.indd 24F
3/8/11 4:50 PM
Algorithm Project 7
A37
Work more examples as necessary until students understand the
procedure.
NOTE Everyday Mathematics uses →
rather than = in number models showing
division with remainders because = is
technically incorrect in such situations.
The simplest way to see that a number
model such as 22 / 5 = 4 R2 is incorrect
is to notice that the left-hand side (22 / 5)
is a number, whereas the right-hand side
(4 R2) is not a number — numbers, after all,
don’t usually include letters such as R.
But something that is a number cannot
possibly equal something that is not a
number, so the use of = in such number
models is improper.
Suggestions:
7.
Write a number story that fits 8,792 / 24. Then solve
your number story.
cont.
366 R8; Number stories vary.
597 R5
9.
59,241 / 31
Answer:
1,911
Long Division Puzzles
Follow these steps to make a long division puzzle:
1.
Make up a division problem and use long division to solve it.
2.
Check that your answer is correct.
3.
Copy your long division work on a separate piece of paper, but leave
blanks for some of the digits.
4.
Trade your puzzle with a partner. Can you solve your partner’s puzzle?
Math Journal, p. 28P
25P-28P_EMCS_S_MJ2_G5_P07_576434.indd 28
Algorithm Project 7
1
6 8 8 → 889 R2
7 6225
42
−−−−
20
14
−−−−
62
56
−−−−
65
56
−−−−
9
7
−−−
2
39 buses
8,960 / 15
PARTNER
ACTIVITY
One of the hardest steps in carrying out U.S. traditional long
division is accurately estimating the quotient at each step. As
traditionally performed, any wrong estimate, either too high or
too low, will cause the algorithm to fail—the incorrect estimate
must be erased and replaced. Using a table of easy multiples is
one way to address this problem.
2
All 1,237 students at John Muir Elementary School are
going on a field trip to the museum. They will take school
buses that can each hold 32 students. How many buses
will be needed?
A38
$3,749 / 25 $149 R$24
Another approach, which combines features of Partial Quotients
Division with U.S. Traditional Long Division, is illustrated below.
The stacked digits in the place-value positions of the quotient
represent partial quotients that must be added to yield the final
quotient. For example, the 6 in the hundreds place is from
6 ∗ 7 = 42 (actually 600 ∗ 7 = 4,200). The 2 above the 6 is from
2 ∗ 7 = 14 (actually 200 ∗ 7 = 1,400). The 6 and 2 must be added
to determine the hundreds digit in the final quotient. Similarly,
the 8 and the 1 must be added to determine the ones digit in the
final quotient. So, the final quotient for 6,225 / 7 is 889 R2.
Long Division with Multidigit Divisors
6.
Answer:
$7,089 / 47 $150 R$39
Have students use U.S. traditional long division to solve
Problems 2–9 on journal pages 27P and 28P. They may find the
examples on Student Reference Book, pages 24I and 24J helpful.
Algorithm Project 7
8.
$5,535 / 12 $461 R$3
(Math Journal 1 or 2, pp. 27P and 28P; Student Reference Book,
pp. 24I and 24J)
Time
7
$16,248 / 13 $1,249 R$11
with Multidigit Divisors
Student Page
PROJECT
$19,240 / 33 $583 R$1
► Solving Long Division Problems
Another way to see that using = can cause
problems is to consider 22 / 5 = 4 R2 and
202 / 50 = 4 R2. If both 22 / 5 and 202 / 50
were equal to 4 R2, one would have to
conclude that 22 / 5 and 202 / 50 are equal
to each other (since equality is a transitive
relation). But 22 / 5 and 202 / 50 are not
equal to each other: 22 / 5 > 202 / 50. This
shows that using = in such number models
can cause problems. Accordingly, in
Everyday Mathematics, we use the →
notation when there are remainders.
Date
$9,475 / 26 $364 R$11
3/8/11 11:38 AM
U.S. Traditional Long Division
Student Page
Whole Numbers
2 Extending the Project
U.S. traditional long division is not limited to dividing money.
9,427 7 → ?
► Long Division Puzzles
PARTNER
ACTIVITY
Think about the problem as dividing 9,427 into 7 equal shares.
1
9427
7
-7
2
Step 1: Start with the thousands.
(Math Journal 1 or 2, p. 28P; Math Masters, p. 415)
Have partners follow the directions at the bottom of journal
page 28P. Provide copies of the computation grid (Math Masters,
page 415). Ask volunteers to explain how they checked their
answers in Step 2. Consider having students make long division
puzzles for classroom display.
► Solving Division Problems
Step 2: Trade 2 thousands
for 20 hundreds.
Share the hundreds.
Step 3: Trade 3 hundreds
for 30 tens.
Share the tens.
← Each share gets 1 thousand.
← 1 thousand ∗ 7 shares
← 2 thousands are left.
13
9427
7
-7
24
-21
3
← Each share gets 3 hundreds.
134
7
9427
-7
24
-21
32
-28
4
← Each share gets 4 tens.
← 20 hundreds + 4 hundreds
← 3 hundreds ∗ 7 shares
← 3 hundreds are left.
← 30 tens + 2 tens
← 4 tens ∗ 7 shares
← 4 tens are left.
INDEPENDENT
ACTIVITY
(Online Additional Practice, pp. 28A–28D; Student Reference Book,
pp. 22–24 and 24E–24J)
Online practice pages 28A–28D 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 28A. Invite them
to use any algorithm they wish to solve the problems on the
remaining pages. Students may find the examples on Student
Reference Book, pages 22–24 and 24E–24J helpful.
Student Reference Book, p. 24G
024A-024J_EMCS_S_SRB_G5_576515.indd 24G
3/8/11 4:50 PM
Go to www.everydaymathonline.com
to access the additional practice
pages.
Online Master
Name
Date
PROJECT
Student Page
Time
Whole Numbers
Online
Additional
Practice
Partial-Quotients Division
7
continued
Algorithm Project 7
Step 4: Trade 4 tens for
40 ones.
Share the ones.
Use partial-quotients division to solve each problem.
1.
Five families collected $3,540 in a garage sale.
They plan to split the money equally. How much
should each family get?
$708
2.
990 / 6
Answer:
165
3.
← Each share gets 6 ones.
← 40 ones + 7 ones
← 6 ones ∗ 7 shares
← 5 ones are left.
9,427 7 → 1,346 R5
$8,562 / 7
Answer:
1346
9427
7
-7
24
-21
32
-28
47
-42
5
$1,223 R$1
Divide.
1. 3,896 8
2. 3 6,789
6,725
3. 5 4. 4,304 4
Check your answers on page 442.
4.
5.
$7,623 / 9
Answer:
$847
641 / 4
Answer:
160 R1
Online Additional Practice, p. 28A
EM3cuG5OP_28A-28D_P07.indd 28A
Student Reference Book, p. 24H
4/1/10 5:36 PM
024A-024J_EMCS_S_SRB_G5_576515.indd 24H
3/8/11 4:50 PM
Algorithm Project 7
A39