Division Ties to
Multiplication
Objective To provide opportunities to model division number
stories with arrays, multiplication/division diagrams, and
s
number models.
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Practice
EM Facts
Workshop
Game™
Teaching the Lesson
Key Concepts and Skills
• Use multiplication facts to solve division
problems. [Operations and Computation Goal 3]
• Use arrays and diagrams to model equalsharing and equal-grouping number stories. [Operations and Computation Goal 6]
• Identify the quotient, dividend, divisor,
and remainder. [Operations and Computation Goal 6]
• Write number sentences to model
number stories. [Patterns, Functions, and Algebra Goal 2]
Key Activities
Children draw arrays, fill in multiplication/
division diagrams, and write number models
to solve division number stories.
Family
Letters
Assessment
Management
Common
Core State
Standards
Curriculum
Focal Points
Ongoing Learning & Practice
1 2
4 3
Playing Division Arrays
Student Reference Book, p. 282
per group: 1 each of number cards
6 –18 (from the Everything Math Deck,
if available), 18 counters, 1 six-sided die
Children practice modeling equal
sharing by making arrays.
Math Boxes 4 4
Math Journal 1, p. 87
Children practice and maintain skills
through Math Box problems.
Home Link 4 4
Differentiation Options
READINESS
Making Equal Groups on a Number Line
Math Masters, p. 93
Children use number lines to model equal
groups.
ENRICHMENT
Finding the Mystery Number
Math Masters, p. 94
counters calculator
Children find mystery numbers using
multiplication and division.
Math Masters, p. 92
Children practice and maintain skills
through Home Link activities.
ELL SUPPORT
Building a Math Word Bank
Differentiation Handbook, p. 132
Children add the term quotient to their Math
Word Banks.
Ongoing Assessment:
Recognizing Student Achievement
Use the Math Message. [Operations and Computation Goal 6]
Ongoing Assessment:
Informing Instruction See page 263.
Key Vocabulary
quotient dividend divisor remainder
Materials
Math Journal 1, p. 86
Student Reference Book, p. 250 (optional)
Home Link 43
Math Masters, p. 406 (optional); pp. 407
and 419
pennies or other counters calculator
(optional)
Advance Preparation
Post the Guide to Solving Number Stories (Math Masters, page 406). Make multiple copies of Math
Masters, page 419 for each child to use during Part 1.
Teacher’s Reference Manual, Grades 1–3 pp. 84, 90–92
260
Unit 4
Multiplication and Division
Interactive
Teacher’s
Lesson Guide
Mathematical Practices
SMP1, SMP2, SMP4, SMP5, SMP6, SMP7
Content Standards
Getting Started
3.OA.2, 3.OA.3, 3.OA.4, 3.OA.6, 3.OA.7
Mental Math and Reflexes
Math
Message
Children count from 1 to 30, clapping at the intervals indicated below.
Every 2nd number 1, 2 (clap); 3, 4 (clap); 5, 6 (clap); 7, 8 (clap); 9, 10
(clap); 11, 12 (clap); and so on to 30.
Every 4th number 1, 2, 3, 4 (clap); 5, 6, 7, 8 (clap); 9, 10, 11, 12 (clap);
and so on to 30
Every 6th number 1, 2, 3, 4, 5, 6 (clap); 7, 8, 9, 10, 11, 12 (clap); and so
on to 30
12 pennies are shared equally by 4 children.
How many pennies per child is that? If you
wish, you may use your tool-kit pennies to
act out the story.
Home Link 4 3
Follow-Up
Have children share their
strategies for solving Problem 6.
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
Check whether any children solved the story by drawing arrays.
If so, have them draw their solutions on the board. If no one used
arrays, demonstrate how they can be used for division by
distributing the 12 pennies into 4 rows as you count from 1 to 12.
Each child gets 3 pennies.
4 × 3 array
You might also have children experiment with a calculator to find
a key sequence to solve the problem. Ask partners to work
together. Emphasize the question: How many pennies per child?
After a few minutes, ask children to share their key sequences.
12, ÷, 4, =, 3 Point out that the ÷ symbol means divided by.
Ongoing Assessment:
Recognizing Student Achievement
Math
Message
Use the Math Message to assess children’s progress toward using equal
sharing to demonstrate the meaning of division. Children are making adequate
progress if they solve the number story using pennies. Some children may be
able to solve the story using other strategies (arrays, pictures, multiplication).
[Operations and Computation Goal 6]
Lesson 4 4
261
Teaching Aid Master
Name
Date
Using Number Models
Time
Multiplication/Division Diagram
in all
per
WHOLE-CLASS
ACTIVITY
and Diagrams for
Division Stories
PROBLEM
PRO
P
RO
R
OB
BLE
BL
LE
L
LEM
EM
SO
S
SOLVING
OL
O
LV
LV
VIN
IIN
NG
(Math Masters, p. 419)
in all
per
Algebraic Thinking With the 4-by-3 array from the Math Message
left on the board, display a multiplication/division diagram (see
below). Ask children to fill in the known numbers and identify the
missing numbers in a diagram on Math Masters, page 419. The
diagram is the same for division and multiplication—the diagram
reinforces the inverse relationship between the two operations.
in all
per
●
Math Masters, p. 419
children
pennies
per child
pennies
in all
4
?
12
Ask whether a volunteer can write a number model to match
12
= ?, or
the story. 12 ÷ 4 = ?, _
4
?
_
412
Any of these notations can be read or thought of in a variety of
ways: What is 12 divided by 4? How many 4s in 12? 4 goes into
12 how many times? Some children might think in terms
of multiplication and suggest the number model 4 × ? = 12. This
is also correct. Accept this answer and take the opportunity to
discuss the relationship between multiplication and division.
Adjusting
the Activity
ELL
Label the dividend, divisor, and quotient of
the number model on the board.
12
dividend
AUDITORY
÷4=3
KINESTHETIC
Ask: How many pennies per child? 3 pennies
●
Ask: Does the answer make the number model true? yes Write a
summary number model for the story. 12 ÷ 4 = 3
The answer in each of the division number models, or the result
of the division, is called the quotient.
The dividend is the total before sharing.
divisor
●
The divisor is the number of equal parts or the number in
each equal part.
quotient
TACTILE
VISUAL
It is beneficial to use these words when discussing division
number stories, but do not insist that children use them.
Pose the following problem: There are 15 pennies. Each child
receives 4 pennies. How many children are there?
Display or draw a multiplication/division diagram on the board.
Ask children to fill in the known numbers and identify the
missing number.
NOTE Some children may think in terms of
multiplication and suggest more than one
number model to solve the problem. They
might first use 3 × 4 = 12 and 4 × 4 = 16.
Then 12 + 3 = 15 (or 15 - 12 = 3).
262
Unit 4 Multiplication and Division
●
children
pennies
per child
pennies
in all
?
4
15
Label the dividend, divisor, quotient, and
remainder of the number model on the
board.
Ask a volunteer to write a number model to match the story.
15 →
15 ÷ 4 → ?, _
?, _
?
4
415
15
remainder
dividend
Ask: How many children are there? 3 children How many
pennies are left over? 3 pennies are left over
●
Does the answer make the number model true? yes Write a
summary number model. 15 ÷ 4 → 3 R3
●
For problems with remainders, use the same notations, but
replace the = sign with an arrow. This arrow is read as leads
to, gives, results in, or a similar expression.
●
The remainder is the quantity left over when a set of objects
is shared equally or separated into equal groups. You can
expect children to use the word remainder when describing the
result of a division problem.
÷ 4 ∑ 3 R3
}
●
ELL
Adjusting
the Activity
AUDITORY
divisor
quotient
KINESTHETIC
TACTILE
VISUAL
NOTE Explain that there are several symbols
for division. The symbol ÷ is used for division
in most textbooks and on calculators. The
fraction notation symbol / is commonly used
on computer keyboards, in newspapers, and
in most other non-textbook situations. The
symbol is used mostly in textbooks when
teaching the traditional long-division algorithm.
Ongoing Assessment: Informing Instruction
Watch for children who are confused by the → symbol. Have them use the
= symbol in their work. Continue to model problems using the → symbol so
that children will become accustomed to the notation.
Pose additional problems as necessary. For each problem, fill in a
multiplication/division diagram and write a number model.
●
●
●
23 candles are arranged with 3 in each row. How many rows
are there? 7 rows How many candles are left over? 2 candles;
23 ÷ 3 = ? or 23 ÷ 3 → 7 R2
21 puppies are placed equally in 3 pens. How many puppies
are there per pen? 7 puppies How many puppies are left over?
none; 21 ÷ 3 = ? or 21 ÷ 3 = 7
18 cards are dealt so that each player gets 4 cards. How many
players are there? 4 players How many cards are left over?
2 cards; 18 ÷ 4 = ? or 18 ÷ 4 → 4 R2
Student Page
Date
44
●
17 markers are shared equally among 6 children. How many
markers does each child get? 2 markers How many markers are
left over? 5 markers; 17 ÷ 6 = ? or 17 ÷ 6 → 2 R5
A piece of cloth 15 yards long is cut into smaller pieces for
dresses. Each dress needs 2 yards. How many dresses can be
made? 7 dresses How many yards of cloth are left over? 1 yard;
15 ÷ 2 = ? or 15 ÷ 2 → 7 R1
Solving Multiplication and Division Number Stories
Solve each number story. Use counters or draw an array to help you.
Fill in the diagrams and write number models.
1.
253
259 260
Roberto has 3 packages of pencils. There are 12 pencils in each
package. How many pencils does Roberto have in all?
3 × 12 = ?
36 pencils
Number model:
Answer:
packages
pencils per
package
pencils
in all
3
12
?
(unit)
2.
●
Time
LESSON
The sewing club has 36 feet of cloth. They need 3 feet of cloth to
make 1 book bag. How many book bags can they make?
36 ÷ 3 = ?
12 book bags
Number model:
Answer:
(unit)
book
bags
feet per
book bag
feet
in all
?
3
36
3a. A class of 30 children wants to play ball. How many teams can be
made with exactly 6 children on each team?
30 ÷ 6 = ?
5 teams
Number model:
Answer:
(unit)
teams
children
per team
children
in all
?
6
30
3b. For another game, the same class of 30 children wants to have
exactly 4 children on each team. How many teams can they make?
30 ÷ 4 = ?
7 teams
Number model:
Answer:
(unit)
teams
children
per team
children
in all
?
4
30
Math Journal 1, p. 86
79-101_EMCS_S_SMJ_G3_U04_576353.indd 86
1/26/11 9:11 AM
Lesson 4 4
263
Student Page
Date
Math Boxes
4 4
䉬
Complete the bar graph.
1.
Solving Multiplication and
Time
LESSON
Show an array and complete a
number model to match the diagram.
2.
Book Club Totals
Books Read
5
Joe read
3 books.
4
3
Maya read
2 books.
Fran read
4 books.
2
1
0
Joe
Maya
packs
cards per
pack
cards
in all
3
6
?
Number model:
Fran
3 6 18
Children
Median books read:
3 books
Add. Make a ballpark
estimate.
3.
64 65
259 260
80
86 87
(unit)
Solve.
4.
Unit
239
6 dimes 47 192
80
80
estimate:
439
147 292
estimate:
3
¢
8 10
¢ 8 dimes
PROBLEM
PRO
P
RO
R
OB
BLE
BL
LE
L
LEM
EM
SO
S
SOLVING
OL
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LV
VIN
ING
(Math Journal 1, p. 86; Math Masters, pp. 406 and 407;
Student Reference Book, p. 250)
Algebraic Thinking Refer children to the Guide to Solving
Number Stories. Partners or small groups work cooperatively to
solve each story on journal page 86. Encourage children to write
their own division number stories on Math Masters, page 407.
Have partners use the inverse relationship between multiplication
and division to check their answers.
52 53
56
57–59
192
Fill in the empty frames.
5.
60
60
6 10 Estimates vary.
Division Number Stories
PARTNER
ACTIVITY
6. What number has
18
6
9
2
6
3
2 hundreds
5 ones
4 thousands
6 ten-thousands
3 tens
Fill in the circle for the best answer
and read it to a partner.
3
A
24,536
B 42,356
C 63,542
D 64,235
200 201
18 19
Math Journal 1, p. 87
2 Ongoing Learning & Practice
Playing Division Arrays
SMALL-GROUP
ACTIVITY
(Student Reference Book, p. 282)
Children practice modeling equal sharing by making arrays in
Division Arrays. For detailed instructions, see page 282 in the
Student Reference Book.
Math Boxes 4 4
INDEPENDENT
ACTIVITY
(Math Journal 1, p. 87)
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 4-2. The skill in Problem 6
previews Unit 5 content.
Home Link Master
Name
Date
HOME LINK
44
䉬
Home Link 4 4
Time
Division Number Stories
(Math Masters, p. 92)
Family
Note
Help your child solve the division number stories by using counters such as pennies or
uncooked macaroni to model the problems. Refer to pages 73, 74, 259, and 260 in the
Student Reference Book. Your child is not expected to know division facts at this time.
73 74
259 260
Please return this Home Link to school tomorrow.
Use counters or draw pictures to show someone at home how you can
use division to solve number stories. Fill in the diagrams.
1. Jamal gave 24 marbles to 4 friends.
marbles
marbles
per friend
in all
?
24
cookies
cookies
per bag
in all
?
29
complete
stamps
stamps
rows
per row
in all
?
10
46
Each friend got the same number of
marbles. How many marbles did
each friend get?
6
friends
4
marbles
0
How many marbles were left over?
marble(s)
2. Ellie had 29 cookies to put in
14 lunch bags. She put the same
number in each bag. How many
cookies did she put in each bag?
2
bags
14
cookies
How many cookies were left over?
1
3. A sheet of stamps has 46 stamps.
A complete row has 10 stamps. How
many complete rows are there?
4
complete rows
How many stamps were left over?
6
cookie(s)
stamp(s)
Math Masters, p. 92
264
Unit 4 Multiplication and Division
INDEPENDENT
ACTIVITY
PROBLEM
PRO
PR
P
RO
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OB
BLE
BL
LE
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LEM
EM
SO
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SOLVING
OL
O
LV
VIN
ING
Home Connection Children use division to solve
number stories.
Teaching Master
Name
To apply children’s understanding of the relationship
between multiplication and division, have them find
mystery numbers and identify patterns on Math Masters,
page 94. When children have finished the page, have them share
their ideas on why the Mystery Number is the same as the first
number. Sample answer: First you multiply a number, then you
subtract that number one time and divide by one less than you
multiplied by.
Time
4
20 5 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
8
7
6
5
4
3
1
Start at
2
0 . Show hops of 5 . Stop at 20 . How many hops? 4 hops
3
18 6 18 . How many hops? 3 hops
0
2
1
4. 20 5 ?
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
8
. Stop at
7
6
6
3
4
5
Start at 0. Show hops of
4
16 4 0
8
7
6
5
4 hops
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
5
4
3
2
8
7
6
5
4
3
2
3. 18 6 ?
(Math Masters, p. 94)
1
15–30 Min
1
Finding the Mystery Number
0
INDEPENDENT
ACTIVITY
ENRICHMENT
1. 15 3 ?
To provide experience with equal-grouping situations using a
number-line model, have children solve equal-grouping problems
by marking and counting hops on a number line.
Follow the directions to solve each problem.
(Math Masters, p. 93)
Start at 0. Show hops of 4. Stop at 16. How many hops?
on a Number Line
0
5–15 Min
15 3 Making Equal Groups
5 hops
INDEPENDENT
ACTIVITY
2. 16 4 ?
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
READINESS
44
䉬
Start at 0. Show hops of 3. Stop at 15. How many hops?
3 Differentiation Options
Date
Equal Groups
LESSON
Math Masters, p. 93
Step 3
Step 2
Step 1
Step 4
Teaching Master
First number is 4 and second number is 8.
Name
Step 5: 8 - 1 = 7
Date
LESSON
44
Step 6: 28 ÷ 7 = 4
Time
Equal Sharing Mystery Number
1. Find the Mystery Number for three different Start Numbers.
In Trial #1, the Start Number and 2nd Number should be less than 10.
In Trial #3, the Start Number and 2nd Number should each have
three digits. Use your calculator.
Step
Sample answer for Start Number 4
and 2nd number 8
ELL SUPPORT
Building a Math Word Bank
SMALL-GROUP
ACTIVITY
5–15 Min
(Differentiation Handbook, p. 132)
What to Do
1
Start Number—write a
number greater than 1.
2
2nd Number—write a number
greater than 1.
3
Multiply the Start Number
times your 2nd Number.
4
Subtract the Start Number
from the number in Step 3.
5
Subtract 1 from the 2nd
Number.
6
Divide the number in Step 4
by the number in Step 5.
Mystery
Number!
Trial #1
Trial #2
Trial #3
2. Look for patterns in the table and discuss the patterns with a partner.
3. Predict what the Mystery Number will be if the Start Number is 4.
To provide language support for division, have children use
the Word Bank template found on Differentiation Handbook,
page 132. Ask children to write quotient and remainder, draw a
picture representing each word, and write other related words.
See the Differentiation Handbook for more information.
The mystery number will be 4.
4. Use counters to show what happened when you followed the steps in Trial
#1. Draw a picture on the back of this page.
Math Masters, p. 94
Lesson 4 4
265