Multiplication and Division Number Stories - Everyday Math

Multiplication and
Division Number Stories

Objective To provide practice solving multiplication and division

number stories by using diagrams to organize information.

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Practice Workshop Letters Management Core State Focal Points Teacher’s
Game™ Standards Lesson Guide

Teaching the Lesson Ongoing Learning & Practice Differentiation Options

Key Concepts and Skills Solving Extended Facts READINESS
• Describe the inverse relationship between Math Journal 1, p. 140
Students solve extended Playing Division Arrays
multiplication and division.  multiplication facts. Student Reference Book, p. 240
[Operations and Computation Goal 3] Math Masters, p. 470
Math Boxes 6 1 per partnership: 1 each of number cards
• Solve multiplication and division Math Journal 1, p. 137 6–18 (from the Everything Math Deck, if
number stories.  Students practice and maintain skills available), 1 six-sided die counters
[Operations and Computation Goals 3 and 4] through Math Box problems. Students review the relationship between
multiplication and division using a
• Use repeated addition, skip counting, and Study Link 6 1 concrete model.
arrays to model multiplication.  Math Masters, p. 177
[Operations and Computation Goal 7] Students practice and maintain skills ENRICHMENT
through Study Link activities.
• Write number models to represent Writing Multiplication/Division
multiplication and division number stories.  Number Stories
[Patterns, Functions, and Algebra Goal 2] Students apply their understanding of the
inverse relationship between multiplication
• Write number models containing and division.
grouping symbols. 
[Patterns, Functions, and Algebra Goal 3]

Key Activities
Students use Multiplication/Division Diagrams
to organize information in number stories and
write number models for the problems.

Ongoing Assessment:
Informing Instruction See page 403.

Ongoing Assessment:
Recognizing Student Achievement
Use journal pages 138 and 139. 

[Patterns, Functions, and Algebra Goal 2]

Key Vocabulary

Multiplication/Division Diagram

Materials
Math Journal 1, pp. 138 and 139
Math Masters, p. 437 (optional)
transparency of Math Masters, p. 436
(optional) slate counters (optional)
Multiplication/Division Facts Table (optional)

Advance Preparation

Teacher’s Reference Manual, Grades 4–6 pp. 19, 107–111, 304–306

400 Unit 6 Division; Map Reference Frames; Measures of Angles

Getting Started

Mental Math and Reflexes Math Message

Pose fact pairs to highlight the inverse relationship between There are 6 rows of
multiplication and division. Suggestions: chairs. There are 4 chairs in
each row. How many chairs
2 ∗ 5 = 10 10 / 5 = 2 7 ∗ 4 = 28 28 / 4 = 7 7 ∗ 8 = 56 56 / 7 = 8 are there in all?
3 ∗ 6 = 18 18 / 6 = 3 8 ∗ 3 = 24 24 / 8 = 3 6 ∗ 7 = 42 42 / 6 = 7
5 ∗ 4 = 20 20 / 4 = 5 5 ∗ 9 = 45 45 / 9 = 5 8 ∗ 9 = 72 72 / 8 = 9

1 Teaching the Lesson Interactive whiteboard-ready
ePresentations are available at
Math Message Follow-Up WHOLE-CLASS www.everydaymathonline.com to
ACTIVITY help you teach the lesson.
(Math Masters, pp. 436 and 437)
4 chairs in each row
Algebraic Thinking Have students share their answers and 6 rows
strategies. Such strategies may include using counters; drawing
pictures; making arrays; counting by 1s, 4s, or 6s; repeatedly Array sketched using Math Masters,
adding 4s or 6s; doubling; or knowing that 4 ∗ 6 = 24 or 6 ∗ 4 = 24. page 437
(See margin.)
Lesson 6 1 401
Review the Multiplication/Division Diagram as a way of
keeping track of the information in the number story. Draw the
following diagram on the board (or use a transparency of Math
Masters, page 436), and record the given information. Use a
variable to represent the unknown.

rows chairs per row total chairs

64 t

In this problem, the number of groups and the number of objects
in each group are known. The total number of objects is sought.

Point out that multiplication can be used to find the total number
of objects. Write “Number model with unknown:” below the
diagram. Ask students to give a number model that represents the
number story using a letter for the unknown, and record it on the
board. Number model with unknown: 6 ∗ 4 = t Now erase the
variable in the diagram and replace it with the answer. Write
“Summary number model:” and ask students to give a number
model with the answer in place of the unknown. Record the
summary number model on the board. Summary number model:
6 ∗ 4 = 24

Have students use the relationship between multiplication and
division to check if their answers make sense. There are 6 rows
of chairs with 4 chairs in each row. There are 24 total chairs.
24 ÷ 6 = 4

Tell students that in this lesson they will use Multiplication/
Division Diagrams to help them solve division number stories.

Solving Division WHOLE-CLASS
Number Stories ACTIVITY

(Math Masters, p. 436)

Links to the Future Algebraic Thinking Because the format of the Multiplication/
Division Diagram is identical for multiplication and division, the
In Unit 12 of Fourth Grade Everyday diagram also helps reinforce the inverse relationship between the
Mathematics students solve rate problems. two operations.
Rate problems, like the problems students
will encounter in this lesson, often contain Pose the following problem: Jenna and Salim are setting up chairs
the word per, which means in each or for for the class play. There are 32 chairs. Their teacher asked them
each. to arrange the chairs so that there are 4 rows with the same
number of chairs in each row. How many chairs should Jenna and
rows chairs per row total chairs Salim put in each row?

Explain that this is an equal-sharing problem. In equal-sharing
problems, the number of groups (rows) and the total number of
objects (chairs) are known. The number of objects in each group
(chairs per row) is sought.

Draw a Multiplication/Division Diagram on the board, or use a
transparency of Math Masters, page 436. (See margin.)

Guide students through the following steps:

Step 1: Decide what you need to find out. How many chairs will
be in each row? Write a variable for the unknown in the “chairs
per row” column.

Step 2: Identify the data (information) you need to solve the
problem. The total number of chairs to be shared and the number
of rows that will share them Write the data in the appropriate
columns.

Step 3: Decide what to do to find the answer. Divide. 32 chairs
must be shared (divided) equally by 4 rows. Write a number
model to represent the problem using a variable for the unknown.
Number model with unknown: 32 / 4 = c

Step 4: Do the computation. Encourage students to use counters,
use a Multiplication/Division Facts Table, or draw an array.
Replace the variable in the diagram with the answer to the
computation. 8

rows chairs per row total chairs

4 c8 32

‫ء‬, ր 1 2 3 4 5 6 7 8 9 10

11 10 Adjusting the Activity

24 20 Have students use a Multiplication/Division Facts Table for division.
For example, to find the answer to 32 / 4, move across the 4s row to the cell
39 30 containing 32. Then move up the column containing 32 to find the answer, 8, at
the top. Tell students they can think of the arrow (see margin) from the 4 to the
4 16 32 40 32 as a representation of the question: “How many 4s are in 32?”

AUDITORY KINESTHETIC TACTILE VISUAL

402 Unit 6 Division; Map Reference Frames; Measures of Angles

Step 5: Below the diagram, write the answer with the unit and
write a number model to summarize the number story.

Answer: 8 chairs per row

Summary number model: 32 / 4 = 8

Ask students to use the relationship between multiplication and
division to check if their answers make sense. 32 chairs are
arranged in 4 rows. There are 8 chairs in each row. 4 ∗ 8 = 32

Ongoing Assessment: Informing Instruction

Watch for students who recognize the inverse relationship between multiplication
and division and solve the problem by thinking “4 times what number equals
32?” Help them write a number model that represents their thinking: 4 ∗ c = 32.

Now pose an equal-grouping problem in which the number of
objects per group (chairs per row) and the total number of objects
(chairs) are known and the number of groups (rows) is sought.

The school custodian is setting up chairs for a parent meeting.
There are 43 chairs. The principal asked the custodian to place
7 chairs in each row. How many rows of chairs will there be?
Guide students through Steps 1–5.

rows chairs per row total chairs

r6 7 43

Number model with unknown: 43 / 7 = r Student Page

Answer: 6 rows (1 chair left over) Date Time

Summary number model: 43 / 7 → 6 R1 LESSON Multiplication/Division Number Stories 178A
178B
Have students use the relationship between multiplication and 61
division to check if their answers make sense. 43 chairs are
arranged with 7 chairs in each row. There are 6 rows of 7 chairs Fill in each Multiplication/Division Diagram. Then write a number model with a variable
with 1 chair left over. (6 ∗ 7) + 1 = 43 for the unknown. Be sure to include a unit with your answer. Solve the problem and
write a summary number model.
NOTE When the result of division is expressed as a quotient and non-zero
remainder, the authors prefer to use an arrow rather than an equal sign, as in Sample number models are given.
43 ÷ 7 → 6 R1. The arrow is read as “is,” “yields,” or “results in.” This is because
43 ÷ 7 = 6 R1 is not a proper number sentence. However, this is a subtle usage 1. The profit from the Maple Street lemonade stand was $36. Four girls will
that need not be required of students. share this amount equally. What will each girl’s share be?

girls dollars per girl total dollars

4 d $36

36 ÷ 4 = dNumber model with unknown:

Answer: $9 36 ÷ 4 = 9Summary number model:

2. Sheila has 57 pictures to put in her photo album. She can put 6 pictures on
each page. How many pages will be completely filled up when she is finished?

pages pictures per page total pictures

p 6 57

57 ÷ 6 = pNumber model with unknown:
Answer: 9 pages 57 ÷ 6 ∑ 9 R3Summary number model:

3. Reuben walks a total of 35 blocks going to and from school each week.
He always walks the same route. How many blocks does he walk each day?

days blocks per day total blocks

5 b 35

35 ÷ 5 = bNumber model with unknown:

7 blocks 35 ÷ 5 = 7Answer:
Summary number model:

Math Journal 1, p. 138 1/13/11 3:27 PM

EM3MJ1_G4_U06_137-169.indd 138 403

Lesson 6 1

Student Page

Date Time Solving Number Stories INDEPENDENT
ACTIVITY
LESSON Multiplication/Division Number Stories cont. (Math Journal 1, pp. 138 and 139)
PROBLEM
61 SOLVING

4. Hassan is helping his teacher put 8 centimeter cubes into each paper cup

for a math project. How many paper cups can he fill if there are 79 cubes?

Sample number

paper cups cubes per paper cup total cubes models are given. Students solve Problems 1–4 on their own. Encourage them to
work with a partner, as necessary, to solve Problems 5 and 6.
p 8 79

79 ÷ 8 = pNumber model with unknown:

9 cups 79 ÷ 8 ∑ 9 R7Answer:
Summary number model:

Try This Ongoing Assessment: Journal pages

5. Mr. Henning’s fourth-grade class is planning a field trip to see a play. 138 and 139
The bus will cost $100, and the tickets will cost $125. The 25 students will
share the total cost equally. How much will each student pay for the field trip?

students dollars per student total dollars Recognizing Student Achievement Problems 1–4
25 d 225
Use journal pages 138 and 139, Problems 1–4 to assess students’ ability to
225 ÷ 25 = dNumber model with unknown: write number models to represent number stories. Students are making adequate
progress if the number models accurately reflect the content of the multiplication
Answer: $9 225 ÷ 25 = 9Summary number model: and division number stories and include the correct use of symbols. Some students
may be able to explain the connection between the Multiplication/Division Diagrams
6. Last year, Martina sold 73 boxes of cookies for her club. This year, she sold and the number models.
three times as many. If she collected $876 this year, how much did each
box cost? [Patterns, Functions, and Algebra Goal 2]

boxes dollars per box total dollars
219 d 876

73 ∗ 3 = 219; 876Number model(s) with unknown: 219 = d

Answer: $4

Summary number model(s): 73 ∗ 3 = 219; 876 219 = 4

Math Journal 1, p. 139 2/15/11 5:52 PM 2 Ongoing Learning & Practice

137-169_EMCS_S_MJ1_G4_U06_576361.indd 139

Solving Extended Facts INDEPENDENT
ACTIVITY
(Math Journal 1, p. 140)

Students solve extended multiplication facts. Some problems
require students to find the product; others ask for one of the
factors.

Student Page Math Boxes 6 1 INDEPENDENT
ACTIVITY
(Math Journal 1, p. 137)

Date Time

LESSON Extended Multiplication Facts Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 6-3. The skill in Problem 5
61 previews Unit 7 content.

451. 9 ∗ 5 = 562. 8 ∗ 7 = 17 Writing/Reasoning Have students write a response to the
9 ∗ 50 = 450 8 ∗ 70 = 560 following: Explain how you rounded the number in Problem 2 to
90 ∗ 5 = 450 80 ∗ 7 = 560 the nearest ten million. Sample answer: 409,381,886 is between
90 ∗ 50 = 4,500 80 ∗ 70 = 5,600 400,000,000 and 410,000,000. The number halfway between the
900 ∗ 5 = 4,500 800 ∗ 7 = 5,600 lower number and the higher number is 405,000,000.
90 ∗ 500 = 45,000 80 ∗ 700 = 56,000 409,381,886 is closer to the higher number, so I rounded to
410,000,000.
363. 4 ∗ 9 = 34. 6 ∗ = 18
4 ∗ 90 = 360
40 ∗ 9 = 360 360 ∗ = 180
40 ∗ 90 = 3,600
400 ∗ 9 = 3,600 3060 ∗ = 1,800
40 ∗ 900 = 36,000
3 ∗ 60 = 180

3 ∗ 600 = 1,800

60030 ∗ = 18,000

65. ∗ 8 = 48 36. 8 ∗ = 24
6 ∗ 80 = 480
60 ∗ 80 = 4,800 3008 ∗ = 2,400 Study Link 6 1 INDEPENDENT
ACTIVITY
860 ∗ = 480 3080 ∗ = 2,400 (Math Masters, p. 177)
8006 ∗ = 4,800
6 ∗ 8,000 = 48,000 8 ∗ 30 = 240

80 ∗ 3 = 240

800 ∗ 300 = 240,000 Home Connection Students use Multiplication/Division
Diagrams to help them solve problems.

Math Journal 1, p. 140 1/13/11 2:24 PM

EM3MJ1_G4_U06_137-169.indd 140

404 Unit 6 Division; Map Reference Frames; Measures of Angles

Student Page

Date Time

3 Differentiation Options LESSON Math Boxes

61

1. Measure each line segment to the nearest millimeter.

a.

11R 5 mm S

READINESS PARTNER About cm
ACTIVITY
Playing Division Arrays b.
5–15 Min
(Student Reference Book, p. 240; Math Masters, p. 470) 8C 7 mm S

About cm 128

2. Round 409,381,886 to the nearest 3. Multiply. Use a paper-and-pencil method.

a. hundred. 2,494 = 86 ∗ 29

409,381,900

To explore the relationship between multiplication and division b. ten thousand.
using a concrete model, have students play Division Arrays.
Consider having students record rounds of play on Math Masters, 409,380,000
page 470.
c. ten million.

410,000,000

d. hundred million.

400,000,000

182 183 18 19

ENRICHMENT PARTNER 4. Complete. 5. Circle _5 of the squares. 44
ACTIVITY 6
▶ Writing Multiplication/Division 100a. 102 = 137
5–15 Min
Number Stories ELL 4 1/13/11 2:24 PM

b. 10 = 10 ∗ 10 ∗ 10 ∗ 10

3

c. 1,000 = 10

d. 10 to the ninth power =

1,000,000,000

5

Algebraic Thinking To apply students’ understanding of Math Journal 1, p. 137
the inverse relationship between multiplication and
division, have them write and solve multiplication and EM3MJ1_G4_U06_137-169.indd 137
division number stories. Have them record a number model using
a letter for the unknown and a summary number model. Some
students may be interested in writing and solving multistep
problems and problems that involve distances, intervals of time,
liquid volumes, masses of objects, or money. Stories may look
similar to the following:

Yataiva divided 450 pennies equally among the tellers at a Study Link Master
bank. One roll has 50 pennies, and each teller got 3 rolls.
How many tellers are there? Number model with unknown: Name Date Time
(450 ÷ 50) ÷ 3 = t; Answer: 3 tellers; Summary number model:
(450 ÷ 50) ÷ 3 = 3 STUDY LINK Multiplication/Division Number Stories

To support English language learners, provide an opportunity for 61
students to share and revise their writing. For example:
Fill in each diagram. Write a number model with a variable for the unknown. 178A
Read problems aloud or have students read their own 178B
problems aloud. Then solve and write a summary number model. Sample number
models are given.
Have partners read and comment on each other’s drafts.
1. Trung wants to rearrange his collection of 72 animals on a shelf in his room.
Have students edit problems as a whole-class activity.
How many equal rows of 9 animals can he make?

rows animals per row total animals

r9 72

Number model with unknown: 72 ÷ 9 = r
72 ÷ 9 = 8Summary number model:
Answer: 8 rows

2. An average porcupine has about 30,000 quills. About how many quills
would 4 porcupines have?

porcupines quills per porcupine total quills

4 30,000 t

Number model with unknown: 30,000 ∗ 4 = t

Answer: 120,000 quills

Summary number model: 4 ∗ 30,000 = 120,000

3. There are 168 calculators for the students at Madison School. A box holds
8 calculators. How many boxes are needed to hold all of the calculators?

boxes calculators per box total calculators

b8 168

Number model with unknown: 168 ÷ 8 = b
168 ÷ 8 = 21Summary number model:
Answer: 21 boxes

Practice 7.695. = 12.13 - 4.44

14.944. = 6.17 + 8.77

Math Masters, p. 177

EM3MM_G4_U06_177-202.indd 177 1/13/11 3:07 PM

Lesson 6 1 405

Name Date Time

STUDY LINK Multiplication/Division Number Stories

61

Fill in each diagram. Write a number model with a variable for the unknown. 178A
Then solve and write a summary number model. 178B

1. Trung wants to rearrange his collection of 72 animals on a shelf in his room.
How many equal rows of 9 animals can he make?

rows animals per row total animals

Number model with unknown:

Answer: Summary number model:

2. An average porcupine has about 30,000 quills. About how many quills
would 4 porcupines have?

porcupines quills per porcupine total quills

Copyright © Wright Group/McGraw-Hill Number model with unknown:

Answer:

Summary number model:
3. There are 168 calculators for the students at Madison School. A box holds

8 calculators. How many boxes are needed to hold all of the calculators?

boxes calculators per box total calculators

Number model with unknown:

Answer: Summary number model:

Practice = 6.17 + 8.77 5. = 12.13 - 4.44

4.

177

Name Date Time
in all
Multiplication/Division Diagrams

per

Number model: in all
Answer:
Summary number model:

per

Number model: total
Answer:
Summary number model:

per

Number model: Copyright © Wright Group/McGraw-Hill
Answer:
Summary number model:

436