Objective To guide the exploration of a variety of strategies
to solve equal-grouping division number stories.
1
materials
Teaching the Lesson
Key Activities
Students explore a multiples-of-10 strategy as one of many ways to solve equal-grouping
division number stories.
Key Concepts and Skills
• Identify and use multiples of 10.
Math Journal 1, pp. 142 and 143
Study Link 6 1
Teaching Aid Master (Math Masters,
p. 436; optional)
base-10 blocks
slate
Multiplication/Division Facts Table
(optional)
[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]
Key Vocabulary
equal-groups notation • quotient • remainder
2
materials
Ongoing Learning & Practice
Students play High-Number Toss to practice place-value skills and comparing numbers.
Students practice and maintain skills through Math Boxes and Study Link activities.
Ongoing Assessment: Recognizing Student Achievement Use journal page 141.
[Patterns, Functions, and Algebra Goal 2]
3
Students practice extended multiplication/
division facts.
Study Link Master (Math Masters,
p. 178)
Game Master (Math Masters,
p. 487)
1 six-sided die per partnership
materials
Differentiation Options
READINESS
Math Journal 1, p. 141
Student Reference Book, p. 252
EXTRA PRACTICE
Students play Buzz and Bizz-Buzz to practice
naming multiples and common multiples.
Student Reference Book, p. 234
Teaching Master (Math Masters,
p. 179)
Technology
Assessment Management System
Math Boxes, Problems 2a–2c
See the iTLG.
406
Unit 6 Division; Map Reference Frames; Measures of Angles
Getting Started
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
__ in all
2 [6s]? 12
20 [6s]? 120
5 [6s]? 30
50 [6s]? 300
Study Link 6 1
Follow-Up
Ask: How many are
4 [7s]? 28
4 [70s]? 280
9 [3s]? 27
9 [30s]? 270
8 [90s]? 720
80 [90s]? 7,200
7 [60s]? 420
70 [60s]? 4,200
Have students use the
inverse operations to check
their answers to Problems 1–5.
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
ACTIVITY
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?
Remind students that the chocolates cannot be divided into
0 groups of 6. 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冄1
苶3
苶4
苶
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
candies in all
?
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
WHOLE-CLASS
ACTIVITY
Division Problems
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冄1
苶3
苶4
苶 → 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.
Example 1: The school used 336 bottles of water at Parents’
Night. How many 8-packs is that?
Date
LESSON
6 2
Time
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.
in each bag. How many bags of 4 cookies did they make?
10 [4s] 20 [4s] 30 [4s] 40 [4s] 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.
50 [4s] bottles per pack
bottles in all
?
8
336
40
80
120
160
200
64 4 16
Number model:
Answer:
16
bags
2. The community center bought 276 cans of soda for a picnic. How many 6-packs is that?
10 [6s] packs
17
1. José’s class baked 64 cookies for the school bake sale. Students put 4 cookies
20 [6s] 30 [6s] 40 [6s] 50 [6s] 60
120
180
240
300
276 6 46
Number model:
Answer:
46
6-packs
3. Each lunch table at Johnson Elementary School seats 5 people. How many tables
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.
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
are needed to seat 191 people?
10 [5s] 20 [5s] 30 [5s] 40 [5s] 50 [5s] 50
100
150
200
250
191 5 ∑ 38 R1
39 tables
Number model:
Answer:
142
Math Journal 1, p. 142
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
6 2
40 [3s] 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.
50 [3s] people per table
people in all
30
60
90
120
150
Number model:
Answer:
45
135 3 45
tricycles
5. How many 8s are there in 248?
10 [8s] 20 [8s] 30 [8s] 40 [8s] tables
continued
How many tricycles is that?
10 [3s] 30 [3s] Example 2: Each table seats 4 people. How many tables are
needed to seat 195 people?
Solving Division Problems
4. The preschool held a tricycle parade. Trent counted 135 wheels.
20 [3s] 苶3
苶6
苶 42 or 336 8 42.
8冄3
Time
50 [8s] 80
160
240
320
400
Number model:
Answer:
8冄2
苶4
苶8
苶 31
31
6. How many 7s are in 265?
?
4
195
10 [7s] 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 [7s] 40 [7s] 50 [7s] 70
140
210
280
350
7冄2
苶6
苶5
苶 ∑ 37 R6
37 R6
Number model:
Answer:
143
Math Journal 1, p. 143
Lesson 6 2
409
Student Page
Date
Time
LESSON
10 [4s] 40
Math Boxes
6 2
1. There are 32 students in the class. A
2. Solve each open sentence.
yearbook page can show 8 student photos.
How many pages are needed to include all
the student photos?
pages
photos
per page
photos
in all
?
8
32
a
b. 54 / 6 81 / b
b
9
c. (c 4) / 3 7
c
17
m
11.36
d. m 3.87 7.49
4
Answer:
30 [4s] 120
1.46
e. 0.98 4.83 f 4.35 f Number model:
20 [4s] 80
4
a. 24 a (5 1)
32 / 8 = 4
pages
148
3. 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
40 [4s] 160
50 [4s] 200
2. Use the list to make a sequence of estimates for the number
of [4s] in 195. For example:
30 [4s] 120 (too small)
3.05
0.67
2.38
40 [4s] 160 (too small, but closer)
34–37
5. Name a fraction equivalent to
4. Complete.
Circle the best answer.
6.70 m
48 m
4,800 cm 9 m 16
916 cm 18 m 1,800 cm
a. 670 cm b.
c.
d.
A.
cm
50 [4s] 200 (just a bit too much)
40 [4s] is 35 short of 195; 40 [4s] 8 [4s] 192 (only 3 short)
1
.
2
3
4
So there are 40 8, or 48 [4s] in 195, with a remainder of 3.
苶9
苶5
苶 → 48 R3 or 195 4 → 48 R3.
Write this as 4冄1
8
B. 9
5
C. 10
3
D. 5
129
51
Math Journal 1, p.141
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.
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)
Encourage students to use a variety of strategies to solve the
problems on journal pages 142 and 143.
Study Link Master
Name
Date
STUDY LINK
62
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.
20 [6s] 30 [6s] 40 [6s] 50 [6s] 2.
60
120
180
240
300
Number model:
Answer:
228 6 38
38
6-packs
20 [8s] 30 [8s] 40 [8s] 50 [8s] Have copies of Math Masters, page 436 available so students can use
Multiplication/Division Diagrams to organize the information in the problems.
A U D I T O R Y
20 [3s] =
30 [3s] =
40 [3s] =
50 [3s] =
80
160
240
320
400
Number model:
Answer:
K I N E S T H E T I C
T A C T I L E
V I S U A L
184 8 23
23
teams
2 Ongoing Learning & Practice
30
60
90
120
150
3冄1
苶4
苶2
苶 ∑ 47 R1
47
Number model:
Answer:
2,644
661 4
Playing High-Number Toss
PARTNER
ACTIVITY
(Student Reference Book, p. 252; Math Masters, p. 487)
Practice
5.
13 96 1,248
6.
4,838
59 82
Students play High-Number Toss to practice place-value skills and
comparing numbers. See Lesson 2-7 for additional information.
Math Masters, p. 178
410
How many 3s are in 142?
10 [3s] =
4.
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.
There are 8 girls on each basketball team. There are 184 girls in the league. How many
teams are there?
10 [8s] 3.
17
21–24
The community center bought 228 juice boxes for a picnic. How many 6-packs is that?
10 [6s] ELL
Adjusting the Activity
Time
Unit 6 Division; Map Reference Frames; Measures of Angles
Teaching Master
Math Boxes 6 2
INDEPENDENT
ACTIVITY
(Math Journal 1, p. 141)
Name
Date
LESSON
62
Time
Multiples of 10 and 100
17
21
Fill in the missing numbers in the problems below.
1.
50
3 150
150 3 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.
2.
50
•
150
?
7
3.
Ongoing Assessment:
Recognizing Student Achievement
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.
5.
INDEPENDENT
ACTIVITY
40
40
How many [60s] are in 2,400?
700 20
60
•
600
, ?
20
42,000
42,000 700 60
•
42,000
, 60
40
?
40
600
How many [30s] in 600?
6.
2,400
, 30
7
, 20
30 40
•
?
•
320
How many [8s] are in 320?
600 30 60 2,400
2,400 60 7.
, How many [40s] are in 280?
[Patterns, Functions, and Algebra Goal 2]
Study Link 6 2
50
•
280
?
40
8
4.
7
320
3
40 280
280 40 Math Boxes
Problems
2a–2c
320 8 , How many [3s] are in 150?
40
8
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 º and / to solve the
division problems.
Math Masters, p. 179
179
(Math Masters, p. 178)
Home Connection Students solve equal-grouping
division stories.
3 Differentiation Options
READINESS
Finding Multiples of 10 and 100
INDEPENDENT
ACTIVITY
5–15 Min
Student Page
(Math Masters, p. 179)
Games
To explore the relationship between extended multiplication
and division facts, have students complete Fact Triangles on
Math Masters, page 179.
Buzz Games
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.
EXTRA PRACTICE
Playing Buzz and Bizz-Buzz
SMALL-GROUP
ACTIVITY
5–15 Min
(Student Reference Book, p. 234)
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.
Example
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.
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 number is the BUZZ number or a multiple of the
To practice naming multiples and common multiples, have
students play the games Buzz and Bizz-Buzz.
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.
5. Play continues until the STOP number is reached.
Example
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