Strategies for Division
Objective To guide the exploration of a variety of strategies
tto solve equal-grouping division number stories.
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ePresentations
eToolkit
Algorithms
Practice
EM Facts
Workshop
Game™
Teaching the Lesson
Key Concepts and Skills
• Identify and use multiples of 10. [Number and Numeration Goal 3]
• Add multiples of 10. [Operations and Computation Goal 1]
• Apply extended multiplication facts to
long-division situations. [Operations and Computation Goal 3]
• Solve equal-grouping division
number stories. [Operations and Computation Goal 4]
• Write number models to represent
multiplication and division number stories. [Patterns, Functions, and Algebra Goal 2]
Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
1 2
4 3
Playing High-Number Toss
Differentiation Options
READINESS
Finding Multiples of 10 and 100
Math Boxes 6 2
Playing Buzz and Bizz-Buzz
Math Journal 1, p. 141
Students practice and maintain skills
through Math Box problems.
Ongoing Assessment:
Recognizing Student Achievement
Use Math Boxes, Problems 2a–2c. [Patterns, Functions, and Algebra Goal 2]
Study Link 6 2
Students explore a multiples-of-10
strategy as one of many ways to solve
equal-grouping division number stories.
Math Masters, p. 178
Students practice and maintain skills
through Study Link activities.
Key Vocabulary
equal-groups notation quotient remainder
Materials
Math Journal 1, pp. 142 and 143
Study Link 61
Math Masters, p. 436
base-10 blocks (optional) slate Multiplication/Division Facts Table (optional)
Advance Preparation
Teacher’s Reference Manual, Grades 4–6 pp. 19, 107–111, 304–306
Unit 6
Interactive
Teacher’s
Lesson Guide
Student Reference Book, p. 252
Math Masters, p. 487
per partnership: 1 six-sided die
Students practice place-value skills
and comparing numbers.
Key Activities
406
Curriculum
Focal Points
Division; Map Reference Frames; Measures of Angles
Math Masters, p. 179
Students practice extended multiplication/
division facts.
EXTRA PRACTICE
Student Reference Book, p. 234
Students practice naming multiples and
common multiples.
Mathematical Practices
SMP1, SMP2, SMP3, SMP4, SMP5, SMP6, SMP7, SMP8
Content Standards
Getting Started
4.OA.3, 4.OA.4, 4.NBT.2, 4.NBT.6
Mental Math and Reflexes
Math Message
Display a Multiplication/Division Diagram. Explain that you are
thinking of things that are packaged in equal groups.
A box holds 6 chocolate
candies. How many boxes are
needed to hold 134 chocolate candies?
packages
__ per package
total
__
Study Link 6 1
Follow-Up
Ask: How many are
2 [6s]? 12
20 [6s]? 120
5 [6s]? 30
50 [6s]? 300
4 [7s]? 28
4 [70s]? 280
9 [3s]? 27
8 [90s]? 720
80 [90s]? 7,200
7 [60s]? 420
9 [30s]? 270
70 [60s]? 4,200
Have students use the
inverse operations to check
their answers to Problems 1–5.
1 Teaching the Lesson
Math Message
WHOLE-CLASS
ACTIVITY
Follow-Up
SOLVING
Algebraic Thinking Remind students that by packing 134
chocolate candies into the boxes, they are dividing 134 chocolates
into groups of 6. The problem is a division problem: How many
[6s] are in 134?
Discuss with students that the 134 chocolates could be divided into
groups of other sizes besides 6. For example, they could be divided
into 13 groups of 10, with 4 chocolates left over: 134 / 10 → 13 R4.
But the 134 chocolates cannot be divided into boxes with 0
chocolates in each box. That is, division by 0 is not possible.
On the board, write the problem in four of the ways that division
problems can be written:
134 ÷ 6
6 134
134 / 6
134
_
6
Ask several students to give their solutions to the Math Message
problem and to describe their strategies. Four possible strategies:
Use a Multiplication/Division Diagram to organize the
information in the problem. Some students may think “What
number times 6 equals 134?” while others may reason “134
divided by 6 equals what number?”
boxes
candies per box
total candies
b
6
134
Lesson 6 2
407
Take 134 cubes. Divide the cubes into as many groups of
6 cubes as possible. 22 groups, 2 cubes left over
Draw a picture.
6
6
6
134 candies
How many boxes?
6 in each box
How many 6s in 134?
Break 134 into smaller “friendly numbers.” For example:
Links to the Future
This method for solving equal-grouping
division number stories is formalized in the
following lesson. In Lesson 6-10, after
students have had opportunities to practice
division with 1-digit divisors, students work
with 2-digit divisors.
●
120 + 14 = 134. There are 20 [6s] in 120 and 2 [6s] in 14
with 2 left over. 20 + 2 = 22, so there are 22 [6s] in 134
with 2 left over.
●
60 + 60 + 14 = 134. There are 10 [6s] in 60, 10 [6s] in
60, and 2 [6s] in 14 with 2 left over. 10 + 10 + 2 = 22, so
there are 22 [6s] in 134 with 2 left over.
Tell students that there are many ways to solve equal-grouping
division problems. One strategy, multiples-of-10, is introduced in
this lesson.
Using Multiples to Solve
Division Problems
WHOLE-CLASS
ACTIVITY
ELL
Explain the following multiples strategy as one way to solve the
Math Message problem:
Ask if there are at least 10 [6s] in 134. Yes, because 10 ∗ 6 = 60.
Write this on the board as shown: 10 [6s] = 60.
NOTE When students confront a division
problem such as 134 ÷ 6, they should ask
themselves, “How many 6s are there in 134?”
A good way to keep track of the number of
6s is to use equal-groups notation: 10 [6s]
are 60, 20 [6s] are 120, 21 [6s] are 126, and
so forth.
If you choose not to use the equal-groups
notation, you can write 10 ∗ 6 in place of
10 [6s], and 20 ∗ 6 in place of 20 [6s]. But
continue to read 10 ∗ 6 as “10 sixes,” and
20 ∗ 6 as “20 sixes.”
Then ask if there are at least 20 [6s] in 134. Yes, because
20 ∗ 6 = 120. Ask if there are at least 30 [6s] in 134. No, because
30 ∗ 6 = 180. Record these multiples on the board.
20 [6s] = 120
30 [6s] = 180
10 [6s] is far less than the required total of 134. 30 [6s] is far
greater than 134. 20 [6s] is just 14 short of 134. You need
20 boxes to hold 120 candies. Ask: How many boxes do you need to
hold the remaining 14 candies? 2 boxes will hold 12 candies. There
are 2 candies left over. Therefore, there is not an exact number of
[6s] in 134; 22 is the quotient, and the 2 left over is the
remainder. Write the answer on the board in two ways. To
support English language learners, label the quotient and
remainder in each of the example problems.
6 134 → 22 R2 and 134 ÷ 6 → 22 R2
408
Unit 6 Division; Map Reference Frames; Measures of Angles
Student Page
Pose and solve equal-grouping division problems that are similar
to the Math Message problem. Show how multiples can be used for
each problem. Encourage students to use the relationship between
multiplication and division to check their answers.
Date
62
Solving Division Problems
For Problems 1–6, fill in the multiples-of-10 list if it is helpful. If you prefer to solve
the division problems in another way, show your work.
Sample number models are given.
40
80
120
160
200
10 [4s] =
20 [4s] =
30 [4s] =
40 [4s] =
50 [4s] =
bottles per pack
total bottles
p
8
336
1 [8] = 8
10 [8s] = 80
2 [8s] = 16
20 [8s] = 160
3 [8s] = 24
30 [8s] = 240
4 [8s] = 32
40 [8s] = 320
5 [8s] = 40
50 [8s] = 400
Answer:
16
bags
64 ÷ 4 = 16
Summary number model:
60
120
180
240
300
20 [6s] =
30 [6s] =
40 [6s] =
50 [6s] =
1. Make a list of the number of bottles in 1, 2, 3, 4, and 5 groups
of 8. Also make a list of the number of bottles in 10, 20, 30,
40, and 50 groups of 8.
64 ÷ 4 = b
Number model with unknown:
The community center bought 276 cans of soda for a picnic. How many 6-packs is that?
2.
10 [6s] =
packs
17 178A
178B
José’s class baked 64 cookies for the school bake sale. Students put 4 cookies
in each bag. How many bags of 4 cookies did they make?
1.
Example 1: The school used 336 bottles of water at Parents’
Night. How many 8-packs is that?
This is a division problem. The 336 bottles are divided into groups
of 8. The problem is to find how many [8s] there are in 336.
Time
LESSON
276 ÷ 6 = p
Number model with unknown:
Answer:
46
6-packs
276 ÷ 6 = 46
Summary number model:
Each lunch table at Johnson Elementary School seats 5 people. How many tables
are needed to seat 191 people?
3.
50
100
150
200
250
10 [5s] =
20 [5s] =
30 [5s] =
40 [5s] =
50 [5s] =
191 ÷ 5 = t
Number model with unknown:
Answer:
39
tables
Summary number model:
191 ÷ 5 ∑ 38 R1
Math Journal 1, p. 142
EM3MJ1_G4_U06_137-169.indd 142
1/13/11 3:27 PM
2. Use the list to make a sequence of estimates for the number
of [8s] in 336. For example:
10 [8s] = 80 (far too small)
20 [8s] = 160 (too small)
30 [8s] = 240 (too small, but close)
40 [8s] = 320 (too small, but closer)
50 [8s] = 400 (too much)
40 [8s] is 16 short of 336; 40 [8s] + 2 [8s] = 336 (exact agreement)
So there are 40 + 2, or 42 [8s] in 336, with no remainder. We now
know that 42 packs of water, with 8 in each pack, were used for
Parents’ Night. Write this as
Student Page
Date
LESSON
62
4.
10 [3s] =
30 [3s] =
40 [3s] =
Example 2: Each table seats 4 people. How many tables are
needed to seat 195 people?
50 [3s] =
5.
20 [8s] =
30 [8s] =
40 [8s] =
tables
people per table
total people
t
4
195
50 [8s] =
6.
continued
80
160
240
320
400
10 [7s] =
30 [7s] =
40 [7s] =
50 [7s] =
70
140
210
280
350
135 ÷ 3 = t
Number model with unknown:
Answer:
45
tricycles
Summary number model:
135 ÷ 3 = 45
Number model with unknown:
Answer:
31
Summary number model:
How many 7s are in 265?
20 [7s] =
1. Begin by making a list of the number of people in 10, 20, 30,
40, and 50 groups of 4. Show students that it is not necessary
to also make a list of the number in 1, 2, 3, 4, and 5 groups of 4.
For example, since 30 [4s] = 120, it is clear that 3 [4s] = 12.
30
60
90
120
150
How many 8s are there in 248?
10 [8s] =
This is a division problem. The 195 people are divided into groups
of 4. The problem is to find how many [4s] there are in 195.
Solving Division Problems
The preschool held a tricycle parade. Trent counted 135 wheels.
How many tricycles is that?
20 [3s] =
336 = 42 or 336 ÷ 8 = 42.
8 Time
248 ÷ 8 = y
_____
8 248 = 31
265 ÷ 7 = y
_____
7 265 ∑ 37 R6
Number model with unknown:
Answer:
37 R6
Summary number model:
Sample number models are given.
Math Journal 1, p. 143
EM3MJ1_G4_U06_137-169.indd 143
1/13/11 3:27 PM
Lesson 6 2
409
Student Page
Date
Math Boxes
62
There are 32 students in the class. A
yearbook page can show 8 student photos.
How many pages are needed to include all
the student photos?
1.
10 [4s] = 40
Time
LESSON
pages
photos
per page
total
photos
p
8
32
2.
Number model with unknown:
20 [4s] = 80
Solve each open sentence.
4
a.
24 = a ∗ (5 + 1)
a=
b.
54 / 6 = 81 / b
b=
9
c.
(c + 4) / 3 = 7
c=
17
d.
m - 3.87 = 7.49
m=
11.36
e.
0.98 + 4.83 = f + 4.35 f =
30 [4s] = 120
40 [4s] = 160
50 [4s] = 200
1.46
32 8 = p
4 pages
Answer:
2. Use the list to make a sequence of estimates for the number
of [4s] in 195. For example:
Summary number model:
32 8 = 4
3.
178A
178B
148
30 [4s] = 120 (too small)
Use a paper-and-pencil algorithm to add or subtract.
a.
0.85
+ 0.53
b.
1.38
0.64
+ 1.73
c.
2.37
12.38
- 1.09
d.
11.29
3.05
- 0.67
40 [4s] = 160 (too small, but closer)
2.38
50 [4s] = 200 (just a bit too much)
34–37
4.
5.
Complete.
6.70 m
48 m
9 m 16
916 cm =
18 m = 1,800 cm
670 cm =
a.
b.
c.
d.
A.
4,800 cm =
40 [4s] is 35 short of 195; 40 [4s] + 8 [4s] = 192 (only 3 short)
1
Name a fraction equivalent to _
2.
Circle the best answer.
3
_
B.
4
5
_
C. 10
D.
So there are 40 + 8, or 48 [4s] in 195, with a remainder of 3.
195 → 48 R3 or 195 ÷ 4 → 48 R3.
Write this as 4 8
_
9
3
_
5
cm
129
3. Point out the answer and ask students whether 48 R3 tables
is the solution to the problem. No. The remainder indicates
that 3 people would be left over. Therefore, 49 tables are
actually needed to seat 195 people with 4 per table.
51
Math Journal 1, p.141
EM3MJ1_G4_U06_137-169.indd 141
1/13/11 3:27 PM
Lead students through several more problems on the board,
asking, How many [ns] are there in m? Each n should be a
1-digit number; each m should be a 2- or 3-digit number.
Practicing Division Strategies
INDEPENDENT
ACTIVITY
(Math Journal 1, pp. 142 and 143; Math Masters, p. 436)
Encourage students to use a variety of strategies to solve the
problems on journal pages 142 and 143. Have copies of Math
Masters, page 436 available so students can use Multiplication/
Division Diagrams to organize the information in the problems.
Study Link Master
Name
Date
STUDY LINK
Time
Equal-Grouping Division Problems
62
For Problems 1–3, fill in the multiples-of-10 list if it is helpful. If you prefer to
solve the division problems in another way, show your work.
1.
The community center bought 228 juice boxes for a picnic. How many 6-packs is that?
10 [6s] =
20 [6s] =
30 [6s] =
40 [6s] =
50 [6s] =
2.
60
120
180
240
300
Number model with unknown:
Answer:
38
228 ÷ 6 = p
Have students use a Multiplication/Division Facts table, and begin by
listing the numbers in 1, 2, 3, 4, and 5 groups of n before listing the numbers in
10, 20, 30, 40, and 50 groups of n.
6-packs
228 ÷ 6 = 38
Summary number model:
A U D I T O R Y
10 [8s] =
30 [8s] =
40 [8s] =
50 [8s] =
80
160
240
320
400
20 [3s] =
30 [3s] =
40 [3s] =
50 [3s] =
23
teams
Summary number model:
T A C T I L E
V I S U A L
184 ÷ 8 = 23
2 Ongoing Learning & Practice
30
60
90
120
150
Number model with unknown:
Answer:
47
Summary number model:
142 ÷ 3 = c
____
3142 ∑ 47 R1
Playing High-Number Toss
2,644
= 661 ∗ 4 5. 13 ∗ 96 =
1,248
6.
4,838
Students play High-Number Toss to practice place-value skills and
comparing numbers. See Lesson 2-7 for additional information.
= 59 ∗ 82
Math Masters, p. 178
EM3MM_G4_U06_177-202.indd 178
PARTNER
ACTIVITY
(Student Reference Book, p. 252; Math Masters, p. 487)
Practice
410
K I N E S T H E T I C
184 ÷ 8 = t
Number model with unknown:
Answer:
How many 3s are in 142?
10 [3s] =
4.
There are 8 girls on each basketball team. There are 184 girls in the league. How many
teams are there?
20 [8s] =
3.
ELL
Adjusting the Activity
17 178A
178B
Sample number models are given.
1/13/11 3:07 PM
Unit 6 Division; Map Reference Frames; Measures of Angles
Teaching Master
Math Boxes 6 2
Name
INDEPENDENT
ACTIVITY
(Math Journal 1, p. 141)
Date
LESSON
62
Time
Multiples of 10 and 100
17
21
Fill in the missing numbers in the problems below.
50
1.
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 6-4. The skill in Problem 5
previews Unit 7 content.
7
40
= 320
40
•
320
∗, ?
•
280
20
30 ∗
600 ÷ 30 =
= 600
20
•
600
∗, ?
40
5.
∗, 7
∗ 60 = 2,400
2,400 ÷ 60 =
40
30
40
How many [40s] are in 280?
700 ∗
60
= 42,000
42,000 ÷ 700 =
60
2,400
•
42,000
∗, ∗, 60
How many [60s] are in 2,400?
?
20
How many [30s] in 600?
6.
•
?
40
How many [8s] are in 320?
4.
7
8
50
∗ 40 = 280
280 ÷ 40 =
7.
8∗
320 ÷ 8 =
3
How many [3s] are in 150?
[Patterns, Functions, and Algebra Goal 2]
Study Link 6 2
•
150
∗, Math Boxes
Problems
2a–2c
Use Math Boxes, Problems 2a–2c to assess students’ ability to solve open
sentences. Students are making adequate progress if they are able to solve
the open sentences involving multiplication and division facts. Some students
may be able to solve Problems 2d and 2e, which involve addition and subtraction
of decimals.
2.
50
?
3.
Ongoing Assessment:
Recognizing Student Achievement
∗ 3 = 150
150 ÷ 3 =
40
700
How many [700s] in 42,000?
?
60
Explain how solving one problem in each set helps you solve the other two problems.
Sample answer: I solved the multiplication problem and
then used the relationship between ∗ and to solve the
division problems.
INDEPENDENT
ACTIVITY
Math Masters, p. 179
EM3MM_G4_U06_177-202.indd 179
(Math Masters, p. 178)
1/13/11 2:16 PM
Home Connection Students solve equal-grouping
division stories.
3 Differentiation Options
READINESS
Finding Multiples of 10
INDEPENDENT
ACTIVITY
5–15 Min
and 100
Games
(Math Masters, p. 179)
Buzz Games
To explore the relationship between extended multiplication
and division facts, have students complete Fact Triangles on
Math Masters, page 179.
EXTRA PRACTICE
Playing Buzz and Bizz-Buzz
Student Page
SMALL-GROUP
ACTIVITY
5–15 Min
(Student Reference Book, p. 234)
Buzz
Materials none
Players
5–10
Skill
Finding multiples of a number and common
multiples of two numbers
Object of the game To correctly say either “BUZZ” or the
next number when it is your turn.
Directions
1. Players sit in a circle and choose a leader. The leader
names any whole number from 3 to 9. This number is the
BUZZ number. The leader also chooses the STOP number.
The STOP number should be at least 30.
2. The player to the left of the leader begins the game by
saying “one.” Play continues clockwise with each player
saying either the next whole number or “BUZZ.”
3. A player must say “BUZZ” instead of the next number if:
The BUZZ
number is 4. Play
should proceed as
follows: 1, 2, 3,
BUZZ, 5, 6, 7,
BUZZ, 9, 10, 11,
BUZZ, 13, BUZZ,
15, and so on.
♦ The number is the BUZZ number or a multiple of the
BUZZ number; or
♦ The number contains the BUZZ number as one of its digits.
4. If a player makes an error, the next player starts with 1.
To practice naming multiples and common multiples, have
students play the games Buzz and Bizz-Buzz. Before playing the
game, remind students that a whole number is a multiple of each
of its factors.
5. Play continues until the STOP number is reached.
6. For the next round, the player to the right of the leader
becomes the new leader.
Bizz-Buzz
Bizz-Buzz is played like Buzz, except the leader names
2 numbers: a BUZZ number and a BIZZ number.
Players say:
1. “BUZZ” if the number is a multiple of the BUZZ number.
2. “BIZZ” if the number is a multiple of the BIZZ number.
3. “BIZZ–BUZZ” if the number is a multiple of both the BUZZ
number and the BIZZ number.
The BUZZ
number is 6, and
the BIZZ number
is 3. Play should
proceed as follows:
1, 2, BIZZ, 4, 5,
BIZZ-BUZZ, 7, 8,
BIZZ, 10, 11,
BIZZ-BUZZ, 13,
14, BIZZ, 16, and
so on. The numbers
6 and 12 are
replaced by “BIZZBUZZ” since 6 and
12 are multiples
of both 6 and 3.
Student Reference Book, p. 234
Lesson 6 2
411
Name
STUDY LINK
62
Date
Time
Equal-Grouping Division Problems
For Problems 1–3, fill in the multiples-of-10 list if it is helpful. If you prefer to
solve the division problems in another way, show your work.
1.
17 178A
178B
The community center bought 228 juice boxes for a picnic. How many 6-packs is that?
10 [6s] =
Number model with unknown:
20 [6s] =
Answer:
30 [6s] =
Summary number model:
6-packs
40 [6s] =
50 [6s] =
2.
There are 8 girls on each basketball team. There are 184 girls in the league. How many
teams are there?
10 [8s] =
Number model with unknown:
20 [8s] =
Answer:
30 [8s] =
Summary number model:
teams
40 [8s] =
50 [8s] =
Copyright © Wright Group/McGraw-Hill
3.
How many 3s are in 142?
10 [3s] =
Number model with unknown:
20 [3s] =
Answer:
30 [3s] =
Summary number model:
40 [3s] =
50 [3s] =
Practice
4.
178
= 661 ∗ 4 5. 13 ∗ 96 =
6.
= 59 ∗ 82