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Division of
Whole Numbers
Objectives To estimate quotients; and to use a paper-and-pencil
division
algorithm to divide whole numbers.
d
www.everydaymathonline.com
ePresentations
eToolkit
Algorithms
Practice
EM Facts
Workshop
Game™
Teaching the Lesson
Key Concepts and Skills
• Find differences of partial quotients. [Operations and Computation Goal 1]
• Use multiplication and extended facts to
compute partial quotients. [Operations and Computation Goal 2]
• Divide multidigit whole numbers. [Operations and Computation Goal 2]
Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
1 2
4 3
Playing Division Top-It
(Advanced Version)
Student Reference Book, p. 336
Math Masters, p. 478
per partnership: 4 each of number
cards 1–9 (from the Everything Math
Deck, if available)
Students practice whole-number
division.
Key Activities
Math Boxes 2 7
Students estimate quotients and practice
the partial-quotients division algorithm for
whole numbers.
Math Journal 1, p. 65
Students practice and maintain skills
through Math Box problems.
Ongoing Assessment:
Informing Instruction See page 138.
Study Link 2 7
Ongoing Assessment:
Recognizing Student Achievement
Math Masters, p. 57; p. 414 (optional)
Students practice and maintain skills
through Study Link activities.
Use journal page 66. Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
ENRICHMENT
Divisibility by 7
Students use a little-known rule to test
for divisibility by 7.
ENRICHMENT
Exploring an Alternative
Division Algorithm
Student Reference Book, p. 24
Students explore the column division
algorithm.
ELL SUPPORT
Building a Math Word Bank
Differentiation Handbook, p. 130
Students add the terms divisor, dividend,
quotient, and remainder to their Math
Word Banks.
[Operations and Computation Goal 2]
Key Vocabulary
partial-quotients division algorithm dividend divisor quotient remainder
Materials
Math Journal 1, pp. 66 and 67
Student Reference Book, pp. 22 and 23
Study Link 26
Math Masters, p. 414
Advance Preparation
Make one or two copies of the computation grid (Math Masters, p. 414) for each student.
Teacher’s Reference Manual, Grades 4–6 pp. 132–140
Lesson 2 7
135
Mathematical Practices
SMP5, SMP6
Getting Started
Content Standards
6.NS.2, 6.EE.2b, 6.SP.4
Mental Math and Reflexes
Math Message
Pose problems such as the following:
Josie has 327 photographs. She can put 12 photos
on each page of her scrapbook. Estimate the number
of scrapbook pages she will need.
How many 5s are in 45? 9
Which number multiplied by 9 equals 27? 3
Multiply 3 by 120. 360
How many 4s are in 32? 8
Which number multiplied by 8 equals 40? 5
Multiply 5 by 80. 400
Study Link 2 6 Follow-Up
Review answers. Ask students to explain
their strategies for placing the decimal point in
Problems 1–4.
Which number multiplied by 50 equals 600? 12
How many 12s are in 132? 11
Multiply 3 by 55. 165
1 Teaching the Lesson
NOTE When expressing the result of a
division problem in terms of a quotient and a
remainder, Everyday Mathematics uses an
arrow rather than an equal sign. For example,
157 / 12 = 13 R1 is not a proper number
sentence because the right side (13 R1) is
not an actual number. To write the solution as
a number sentence, express the remainder
1
as a fraction: 157 / 12 = 13_
12 . Students
learned to express remainders as fractions in
Fifth Grade Everyday Mathematics.
▶ Math Message
Follow-Up
WHOLE-CLASS
DISCUSSION
SOLVING
This is an equal-grouping problem. The total number of photos is
divided into groups of 12. The number of pages or groups Josie
needs can be found by dividing the number of photographs (327)
by 12.
Discuss students’ strategies for estimating the quotient.
For example:
10 [12s] = 120; 20 [12s] = 240; 30 [12s] = 360
Josie would need 30 pages for 360 photos, so she will
need fewer than 30 pages for 327 photos.
Algorithm Project To teach U.S.
traditional long division of whole numbers, see
Part A of Algorithm Project 4 on page A18.
Use close numbers that are easy to divide. 327 is close to 300,
and 12 is close to 10. Because 300 divided by 10 is 30, Josie
needs about 30 pages.
▶ Reviewing the Partial-
WHOLE-CLASS
ACTIVITY
Quotients Division Algorithm
(Math Masters, p. 414)
Continue with the Math Message problem. To answer this
question exactly, students need to figure out how many 12s
are in 327.
Model the partial-quotients division algorithm while students
follow along using a computation grid (Math Masters, p. 414).
136
Unit 2
Operations with Whole Numbers and Decimals
Adjusting the Activity
ELL
Write a division problem on the board. Label the dividend, divisor,
quotient, and remainder.
remainder
↓
27R3 ← quotient
Example:
__
12 327 ← dividend
↑
divisor
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
__
Write the problem in this form: 12 327
1. A good strategy is to start with multiples of 10 because they are
“easy” numbers with which to work. Ask: Are there at least 20
[12s]? Yes, because 20 ∗ 12 = 240 Are there more than 30 [12s]?
No, because 30 ∗ 12 = 360 So, the answer is at least 20 but not
more than 30. Try 20, because 20 is the first partial quotient.
Partial quotients will be used to build up to the final quotient.
__
12 327
240
20
← (The
first partial quotient) 20 ∗ 12 = 240
2. The next step is to find out how much is left to divide.
Subtract 240 from 327.
__
12 327
- 240
87
20
← Estimate.
← Subtract.
Write 20 ∗ 12.
3. Now find the number of 12s in 87. There are two ways to
do this:
Use a fact family to find the number of 12s in 87.
There are 7, since 7 ∗ 12 = 84. Record as follows:
__
12 327
- 240
87
- 84
3
20
7
first partial quotient) 20 ∗ 12 = 240
← Subtract. 87 is left to divide.
← (The second partial quotient) 7 ∗ 12 = 84
← Subtract.
← (The
Use at least/not more than estimates with numbers that are
easy to multiply. Ask: Are there at least 10 [12s] in 87?
No, because 10 ∗ 12 = 120 Are there at least 5 [12s]? Yes,
because 5 ∗ 12 = 60 Next, subtract 60 from 87 and
continue by asking, How many 12s are in 27?
__
12 327
- 240
87
- 60
27
- 24
3
20
5
2
← (The first partial quotient) 20 ∗ 12 = 240
← Subtract. 87 is left to divide.
← (The second partial quotient) 5 ∗ 12 = 60
← Subtract. 27 is left to divide.
← (The third partial quotient) 2 ∗ 12 = 24
← Subtract.
Lesson 2 7
137
Student Page
Date
Ongoing Assessment: Informing Instruction
Time
LESSON
2 7
䉬
Practicing Division
3 Ways to Write a Division Problem
22–24
246 12 ∑ 20 R6
122
4
6
∑ 20 R6
246 / 12 ∑ 20 R6
2 Ways to Express a Remainder
6
1
122
4
6
20, or 20
12
2
122
4
6
∑ 20 R6
Divide.
夹
夹
16
∑ 32 R16, or 32 23
1.
752 / 23
3.
2,436 28
87
Encourage students to use numbers that are easy for them to work with. Using
easy numbers to make partial quotients may require more steps, but it makes
the work go faster.
夹
2.
839 58
4.
1501
,3
5
0
27
∑ 14 R27, or 14 58
4. With either strategy, the division is complete when the
subtraction results in a number less than 12 (the divisor,
or the number by which you are dividing). The final step is
to add the partial quotients.
9
__
__
12 327
- 240
87
- 84
3
12 327
- 240
87
- 60
27
- 24
3
20
+ 7
27
20
5
+ 2
27
5. Because the work shows that there are 27 [12s] in 327,
27 is the quotient. The work also shows 3 left, which is the
remainder. Josie needs 27 full pages, as well as part of
another page, to include all 327 photos. Josie needs a total
of 28 pages.
Math Journal 1, p. 66
▶ Practicing the Partial-
WHOLE-CLASS
ACTIVITY
Quotients Division Algorithm
(Student Reference Book, pp. 22 and 23)
For additional whole-class practice, pose problems similar to the
ones below. Have students estimate the quotients using “close”
numbers. They can use the estimates to check answers.
866 / 27 Sample answer: 900 / 30 = 30; solution: → 32 R2,
2
or = 32_
27
Student Page
Date
LESSON
2 7
䉬
Time
Practicing Division
791 / 33 Sample answer: 750 / 25 = 30; solution: → 23 R32,
32
or = 23_
33
continued
Solve the following problems mentally or use a division algorithm.
The Petronas Twin Towers is an 88-story
building in Malaysia. It costs about $60,000 to rent
2,500 square feet of office space in the towers
for 1 year. What is the cost per month?
About
5.
$5,000 per month
(unit)
A professional hockey stick costs about $60.
Lucero’s team has $546 to use for equipment.
How many sticks can the team buy?
6.
9 hockey sticks
(unit)
In 1650, it took about 50 days to sail from London,
England, to Boston, Massachusetts, which
is a distance of about 3,700 miles. On average,
about how many miles were sailed each day?
About
7.
74 miles
Adjusting the Activity
Have students use doubles and halves to construct a list of easy
multiples. For example, if the divisor is 36, students would make the
following list:
200 ∗
100 ∗
50 ∗
25 ∗
20 ∗
10 ∗
5∗
2∗
(unit)
Tutunendo, Colombia has the greatest annual
rainfall in the world—about 464 inches per year. On
average, about how many inches is that per month
About
(to the nearest whole number)?
8.
Tour buses at the zoo leave when every seat is
occupied. Each bus holds 29 people. On Saturday,
1,827 people took tour buses. How many tour buses
were filled?
9.
39 inches
(unit)
63 buses
(unit)
36 = 7,200
36 = 3,600
36 = 1,800
36 = 900
36 = 720
36 = 360
36 = 180
36 = 72
Try This
10.
The diameter of the planet Neptune is about 30,600 miles.
1
Pluto’s diameter is about 21 that of Neptune. About how
many miles is the diameter of Pluto?
About
A U D I T O R Y
1,457 miles
(unit)
Math Journal 1, p. 67
138
Unit 2
Operations with Whole Numbers and Decimals
K I N E S T H E T I C
T A C T I L E
V I S U A L
Student Page
INDEPENDENT
ACTIVITY
Division Algorithm
Time
LESSON
Multiply mentally.
(Math Journal 1, pp. 66 and 67)
-4
d.
35
Times to Run 1 Mile (min)
8.5
9.3
9.9
10.5
11.2
8.6
9.4
8.7
10.5
7.5
9.2
9.6
10.2
8.7
9.0
14.7 ∗ 0.65 =
36
The members of the Smith School
cross-country team were timed on a
one-mile run. Their times were rounded
to the nearest 0.1 minute and recorded
in the table below. Use the data to
construct a bar graph.
Journal
Page 66
Problems 1–3
Use journal page 66, Problems 1–3 to assess students’ abilities to estimate
quotients and to use an algorithm to divide whole numbers by 2-digit divisors.
Students are making adequate progress if they give reasonable estimates for
partial quotients and choose a reliable algorithm to solve Problems 1–3. Some
students may be able to divide by a 3-digit divisor to solve Problem 4.
Sample estimate: 8
Multiply 14.7 ∗ 0.65. Show your work.
5
c.
3.
Estimate the product 14.7 ∗ 0.65.
Estimate:
-3
b.
Ongoing Assessment:
Recognizing Student Achievement
2.
7.5
2.8476
2,847.6 ∗ 10 =
3,590.0
0.0359 ∗ 10 =
0.0919
919 ∗ 10 =
0.075 ∗ 102 =
a.
Have students solve the problems on journal pages 66 and 67.
After students have completed the pages, ask volunteers to discuss
their solutions. Encourage those students who used different
estimates to share their work.
Math Boxes
27
1.
9.555
37
Times to Run 1 Mile
Number of Students
▶ Using the Partial-Quotients
Date
6
5
4
3
2
1
0
7.5
8.0
8.5
9.0
9.5
10.0 10.5 11.0 11.5
Time (minutes)
138
Write the rule for the table in words.
4.
in
out
$47.99
$479.90
$12.10
$121
Match each description with the point
on the number line that represents it.
D
C
-3
-2
A
-1
0
B
1
2
3
$0.59
$5.90
$0.08
$0.80
a.
The opposite of 2
C
Multiply the in
b.
The sum of any number
and its opposite
A
Rule:
[Operations and Computation Goal 2]
5.
number by 10.
35
99 100
253
Math Journal 1, p. 65
EM3MJ1_G6_U02_45_81.indd 65
12/29/10 1:29 PM
2 Ongoing Learning & Practice
▶ Playing Division Top-It
PARTNER
ACTIVITY
(Advanced Version)
(Student Reference Book, p. 336; Math Masters, p. 478)
Divide the class into pairs and distribute 4 each of number cards
1–9 to each partnership, as well as a game record sheet. Students
may need to play a practice game.
▶ Math Boxes 2 7
Study Link Master
INDEPENDENT
ACTIVITY
Name
STUDY LINK
27
䉬
Date
Time
Dividing Numbers
(Math Journal 1, p. 65)
3 Ways to Write a Division Problem
42–43
4
6
∑ 20 R6
122
246 12 ∑ 20 R6
Mixed Practice Math Boxes in this lesson are paired with
Math Boxes in Lesson 2-5. The skills in Problems 4 and 5
preview Unit 3 content.
Writing/Reasoning Have students write a response to the
following: Explain whether the bar graph you created for
Problem 3 is a histogram. Sample answer: The bar graph
is a histogram because the data come from a continuous set of
numbers and are grouped into intervals on the graph. The
horizontal axis is a number line, and the bars share sides.
246 / 12 ∑ 20 R6
2 Ways to Express a Remainder
122
4
6
20162, or 2012
4
6
∑ 20 R6
122
When estimating quotients, use “close” numbers that are easy to divide.
Sample estimates given.
Example: 346 / 12
Estimate
1.
234 / 6
Estimate
2.
659 / 12
Estimate
3.
512 / 9
Estimate
4.
1,270 / 7 Estimate
5.
728 / 34
Estimate
35
40
50
60
200
20
How I estimated: 350 / 10 = 35
How I estimated:
How I estimated:
How I estimated:
How I estimated:
How I estimated:
Solve using a division algorithm. Show your work on a separate sheet of
paper or a computation grid.
6
6. 85
3
4
∑66 R6, or 66 8
7. 976 / 15 ∑65 R1,
8.
980 20
10.
6,024 / 38
1
or 65 15
15
49
9.
468
4
3
∑18 R15, or 18 46
20
∑158 R20, or 158 38
11.
5,586 44
∑126 R42, or 126 4424
Practice
Multiply mentally.
$3.98
$11.84
3 books at $24.98 each $74.94
5 gifts at $99.99 each $499.95
12.
2 notebooks at $1.99 each 13.
4 pens at $2.96 each 14.
15.
Math Masters, p. 57
Lesson 2 7
139
▶ Study Link 2 7
Divisibility Rules For 7
Example 1:
INDEPENDENT
ACTIVITY
Is 2,758 divisible by 7?
(Math Masters, p. 57; p. 414 optional)
Step 1: Isolate the ones digit. Multiply by 2.
(8 ∗ 2 = 16)
Home Connection Students use close numbers to
estimate quotients. They use a paper-and-pencil
division algorithm to solve division problems.
2,75 8
- 16
Step 2: Subtract the product (16) from
the remaining digits.
25 9
-18
Repeat Steps 1 and 2 until you get 0 or
another multiple of 7. If the result is 0 or
7
another number divisible by 7, then the
original number is divisible by 7.
7 is divisible by 7, so 2,758 is
also divisible by 7.
3 Differentiation Options
INDEPENDENT
ACTIVITY
ENRICHMENT
Example 2:
▶ Divisibility by 7
Is 1,667 divisible by 7?
Step 1: Isolate the ones digit. Multiply by 2.
(7 ∗ 2 = 14)
1,66 7
- 14
Step 2: Subtract the product (14) from
the remaining digits.
15 2
- 4
11 is not divisible by 7, so 1,667 is not
11
5–15 Min
ELL
None of the simple divisibility rules apply to the prime
number 7. To extend students’ knowledge of divisibility rules,
consider introducing them to the divisibility rule for 7.
(See margin.)
To support English language learners, discuss the meaning of the
terms divisibility rule and isolate. Have students test the following
numbers for divisibility by 7. Suggestions:
divisible by 7.
343 Yes
1,372 Yes
5,527 No
ENRICHMENT
▶ Exploring an Alternative
17,276 Yes
INDEPENDENT
ACTIVITY
15–30 Min
Division Algorithm
(Student Reference Book, p. 24)
Student Page
Students begin by reviewing the column division algorithm on
page 24 of the Student Reference Book. They extend and explore
this
algorithm to solve a problem with a 2-digit divisor, such as
__
15 2,589 .
Whole Numbers
Column-Division Method
The best way to understand column division is to think of a
division problem as a money-sharing problem. In the example
below, think of sharing $863 equally among 5 people.
58
6
3
?
1. Draw lines to separate the digits in the dividend
(the number being divided).
Work left to right. Begin in the left column.
2. Think of the 8 in the hundreds column as
8 $100 bills to be shared by 5 people.
Each person gets 1 $100 bill. There are
3 $100 bills remaining.
3. Trade the 3 $100 bills for 30 $10 bills.
Think of the 6 in the tens column as
6 $10 bills. That makes 30 6 36 $10 bills.
5 8
6
3
6
3
5 8
6
3
5
36
1
5 8
5
3
1
ELL SUPPORT
▶ Building a Math Word Bank
INDEPENDENT
ACTIVITY
5–15 Min
(Differentiation Handbook, p. 130)
3
4. If 5 people share 36 $10 bills, each person
gets 7 $10 bills. There is 1 $10 bill remaining.
1
7
5 8
6
5
36
3
3 35
1
5. Trade the 1 $10 bill for 10 $1 bills.
Think of the 3 in the ones column as
3 $1 bills. That makes 10 3 13 $1 bills.
1
7
5 8
6
3
5
36
13
3 35
1
6. If 5 people share 13 $1 bills, each person
gets 2 $1 bills. There are 3 $1 bills remaining.
1
7
2
5 8
6
3
5
36
13
To provide language support for division, have students use
the Word Bank template found on Differentiation Handbook,
page 130. Ask them to write the terms divisor, dividend, quotient,
and remainder, record examples representing each term, and
write other related words. See the Differentiation Handbook for
more information.
3 35 10
Record the answer as 172 R3.
Each person receives $172 and $3 are left over.
1
3
Student Reference Book, p. 24
140
Unit 2
Operations with Whole Numbers and Decimals