Multiplication
and Division
Objectives To guide exploration of the relationship between
multiplication and division; and to provide practice with
division facts.
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eToolkit
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Practice
EM Facts
Workshop
Game™
Teaching the Lesson
Key Concepts and Skills
• Solve multiplication facts. [Operations and Computation Goal 3]
• Use multiplication facts to generate related
division facts. [Operations and Computation Goal 3]
• Apply multiplication and division facts and
extended facts to solve problems. [Operations and Computation Goal 3]
• Write multiplication and division number
sentences. [Patterns, Functions, and Algebra Goal 2]
Key Activities
Students explore the relationships between
multiplication and division. They use the
Multiplication/Division Facts Table to solve
division facts, and they use Multiplication/
Division Fact Triangles to generate fact
families and to practice division facts.
Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
1 2
4 3
Playing Beat the Calculator
Student Reference Book, p. 233
Math Masters, p. 461
calculator 4 each of number cards
1–10 (from the Everything Math Deck,
if available)
Students practice multiplication facts.
Math Boxes 3 5
Math Journal 1, p. 62
Students practice and maintain skills
through Math Box problems.
Study Link 3 5
Math Masters, p. 85
Students practice and maintain skills
through Study Link activities.
Ongoing Assessment:
Recognizing Student Achievement
Use journal page 61. [Patterns, Functions, and Algebra Goal 2]
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Playing Division Arrays
Student Reference Book, p. 240
Math Masters, p. 470
number cards 6–18 (from the Everything
Math Deck, if available) 1 six-sided die 18 counters grid paper (optional)
Students explore the connections between
multiplication and division using a concrete
model.
ENRICHMENT
Exploring the Relationship between
Division and Fractions
Math Masters, p. 86
Students use division facts to solve a
fraction problem.
EXTRA PRACTICE
Practicing with Fact Families
Math Masters, p. 418
Students complete Fact Triangles and write
the related fact families.
ELL SUPPORT
Key Vocabulary
Creating a Poster
dividend divisor quotient remainder fact family
colored pencils, markers, or crayons
Students create a poster for the terms
dividend, divisor, quotient, and remainder.
Materials
Math Journal 1, p. 61
Student Reference Book, pp. 178A and 178B
Study Link 34
transparencies of Math Masters, pp. 408 and
409 (optional) º, / Fact Triangles slate
180
Unit 3
Multiplication and Division; Number Sentences and Algebra
Mathematical Practices
SMP1, SMP2, SMP3, SMP4, SMP5, SMP6, SMP7
Content Standards
Getting Started
4.OA.1, 4.NBT.6, 4.MD.2
Mental Math and Reflexes
Have students name the next three numbers in each pattern. Suggestions:
10, 12, 14,
12, 16, 20,
30, 25, 20,
18, 15, 12,
16 ,
24 ,
15 ,
9 ,
18 ,
28 ,
10 ,
6 ,
18, 27, 36, 45 , 54 , 63
24, 30, 36, 42 , 48 , 54
49, 42, 35, 28 , 21 , 14
40, 32, 24, 16 , 8 , 0
20
32
5
3
24, 28, 32,
24, 36, 48,
63, 56, 49,
72, 64, 56,
36 , 40 ,
60 , 72 ,
42 , 35 ,
48 , 40 ,
44
84
28
32
Math Message
Study Link 3 4 Follow-Up
How many bags of oranges, with 5 oranges in
each bag, can be made with 35 oranges? How
many can be made with 42 oranges?
Have students represent the multiplication
equations in Problems 6–11 with multiplicative
comparison statements. For example, “20 is 4
times as many as 5 and 20 is 5 times as many as 4.”
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
(Student Reference Book, pp. 178A and 178B)
As you and the students discuss the first problem, use alternative
statements such as these:
How many 5s in 35? This is an equal-groups interpretation of
the problem.
35 is 5 times as many as what number? This is a comparison
interpretation of the problem.
What number multiplied by 5 gives 35? This suggests an area
interpretation and points out the inverse relationship between
multiplication and division.
These interpretations are described on Student Reference Book,
pages 178A and 178B.
Point out that in the division fact 35 / 5 = 7, the number 35 is the
dividend, 5 is the divisor, and 7 is the quotient.
In the second problem, there will still be 2 oranges left over after
making 8 bags with 5 oranges in each bag. Remind students that
2 is called the remainder.
Adjusting the Activity
ELL
To help students keep the terms dividend and divisor straight, point
out that the suffix -or indicates someone or something doing the action—doctor,
senator, tailor, generator, and so on. So the divisor is the number that is doing
the dividing. The dividend is the number that is being divided.
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
Lesson 3 5
181
Student Page
Date
Exploring the Relationship
Time
LESSON
Multiplication and Division
35
16 20
between Multiplication
and Division
Equivalents
3 ∗ 4 12 / 3
1.
3×4
3
_
3 12
3<5
(< means “is less than”)
5>3
(> means “is greater than”)
Answers vary.
Choose 3 Fact Triangles. Write the fact family for each.
∗
=
×
=
(Math Journal 1, p. 61; Math Masters, p. 409)
=
∗
=
×
=
/
=
/
=
÷
=
/
=
/
=
÷
=
∗
2.
12
_
12 ÷ 3
=
∗
On the board, draw the ∗, / Fact Triangle shown below, or use a
transparency of Math Masters, page 409.
Solve each division fact.
a.
b.
27 / 3 =
9
9
36 ÷ 6 =
d.
24 / 8 =
Remind students of the following symbols: ∗ or × for
multiplication, and ÷ or / for division. Then show them how
to use the Fact Triangle to generate a multiplication/division
fact family:
Think: How many 3s in 27?
= 45 / 5
c.
Think: 45 is 5 times as many as what number?
6
3
Think: 6 times what number equals 36?
Think: 24 is 8 times as many as what number?
Try This
3.
A cashier has 5 rolls of quarters and 6 rolls of dimes in the cash register.
Each roll of quarters is worth $10, and each roll of dimes is worth $5.
a.
How much are the rolls of quarters and dimes worth in all? $
b.
How many quarters are in 1 roll?
c.
How many quarters are in the 5 rolls?
d.
How many dimes are in 1 roll?
e.
How many dimes are in the 6 rolls?
f.
There is also $7.50 worth of half-dollars in
the cash register. How many half-dollars is that?
6 ∗ 7 = 42
•
42
80
40
200
50
300
15
WHOLE-CLASS
DISCUSSION
7 ∗ 6 = 42
∗
quarters
,/
quarters
dimes
42 / 6 = 7
7
6
dimes
42 / 7 = 6
half-dollars
61
Math Journal 1, p. 61
EM3MJ1_G4_U03_53-77.indd 61
11/10/10 3:15 PM
Do several more examples with the class. Then have students
choose three of their Fact Triangles and record the fact families
for those triangles in Problem 1 on journal page 61.
Ongoing Assessment:
Recognizing Student Achievement
Adjusting the Activity
Use a piece of paper to track
across and up. For example, to solve 56 / 7,
line up the bottom of a piece of paper along
the top of the 7s row. Scan across the 7s
row to find 56. Line up the right edge of the
paper with the left side of the column
containing 56. Then scan up that column to
find the answer, 8, at the top.
1
1
2
3
4
5
6
7
1
2
4
3
9
4
16
5
25
6
36
8
10
20
24
27
30
32
36
40
40
45
50
48
54
60
14
21
28
35
42
49
56
63
70
8
8
16
24
32
40
48
56
64
72
80
90
9
9
18
27
36
45
54
63
72
81
10
10
20
30
40
50
60
70
80
90 100
KINESTHETIC
TACTILE
WHOLE-CLASS
ACTIVITY
(Math Journal 1, p. 61; Math Masters, p. 408)
9
18
7
Using the Multiplication/
Division Facts Table for Division
7
AUDITORY
[Patterns, Functions, and Algebra Goal 2]
10
8
Use journal page 61, Problem 1 to assess students’ ability to use conventional
notation to write multiplication and division number sentences. Students are
making adequate progress if they are able to record the appropriate fact families.
Some students may use factors greater than 10.
9
16
Journal
page 61
Problem 1
You may want to use an overhead transparency of the
Multiplication/Division Facts Table (Math Masters, page 408) to
remind students how to use it for division. For example, to find the
answer to 24 / 3, move across the 3s row to the cell containing 24.
Then move up the column containing 24 to find the answer, 8, at
the top. Tell students that the arrow (see below) from the 3 to the
24 can represent the question “How many 3s in 24?”
VISUAL
ⴱ, 1
1
1
2
3
4
2
3
4
5
6
7
8
9
10
10
20
4
24
9
16
30
40
Have students complete Problems 2 and 3 on journal page 61 on
their own or with a partner.
182
Unit 3 Multiplication and Division; Number Sentences and Algebra
Student Page
Practicing Division Facts
Date
PARTNER
ACTIVITY
with Fact Triangles
Time
LESSON
3 5
Write , , or to make each number
sentence true.
1.
Demonstrate the following procedure:
2. Partners help each other. If Student B is not sure of the
answer, Student A gives a clue. For example, if the fact is 42
divided by 6, Student A can help Student B by asking, “What
number times 6 is equal to 42?” or “How many 6s in 42?” or “42
is 6 times as many as what number?” If Student B still is not
sure, Student A reads the answer.
The number of glasses of milk drunk
by 10 students in a week:
2.
3,389
70,699
6 million 6,000,000
8,000,032 8 million, 32 thousand
400 30 5 4,000 30 5
a.
5,389
b.
70,642
c.
1. Student A takes the top triangle from his or her OK pile
and asks Student B a division fact from the triangle, such as
“42 divided by 6 is equal to what number?”
Math Boxes
䉬
d.
e.
16, 13, 15, 20, 8, 10, 15, 12, 10, 18
What is the range? Circle the best answer.
6 149
Make a ballpark estimate. Write a number
model to show your strategy.
3.
A
8
B
20
C
12
D
14
Sample answers:
a.
31 in. b.
17 ft 13,685 8,379
a.
14,000 8,000
6,000
7,600
42
84
c.
7,602 3,213
b.
3,200
4,400
73
Complete.
4.
d.
2
5
in.
ft
ft 14 yd
in. 2 yd 1 ft
1
4
2 miles e.
7
2
ft
yd
11,880
ft
181
129
Complete.
5.
3. Partners take turns asking and answering until they have
gone through the triangles in both their OK and Try Again
piles, or as long as time permits.
Solve mentally or with a paper-and-pencil
algorithm.
6.
19 , 23 , 27
4
Rule:
13 , 16 , 19 , 22, 25, 28
3
Rule:
7 , 14, 21 , 28, 35 , 42
7
Rule:
7, 11, 15,
a.
b.
c.
a.
$2.27
$4.96
b.
$5.00
$3.64
$1.36
$7.23
160 161
34–37
Math Journal 1, p. 62
2 Ongoing Learning & Practice
Playing Beat the Calculator
SMALL-GROUP
ACTIVITY
(Student Reference Book, p. 233;
Math Masters, p. 461)
NOTE For practice with facts
through 12 º 12, have students
use the Online Master available
at www.everydaymathonline.com
in place of Study Link 35.
Students play Beat the Calculator to maintain automaticity with
multiplication facts.
Adjusting the Activity
Study Link Master
Have students play the extended-facts version of the game described
on Student Reference Book, page 233.
Name
Date
STUDY LINK
35
䉬
A U D I T O R Y
K I N E S T H E T I C
Math Boxes 3 5
T A C T I L E
V I S U A L
1.
30 / 6 3.
9
5
72 8
32 / 4 8
42 /
7.
9.
(Math Journal 1, p. 62)
Missing Numbers
Complete each fact by filling in the missing numbers.
Use the Multiplication/Division Facts Table to help you.
5.
INDEPENDENT
ACTIVITY
4
11.
Study Link 3 5
13.
5º
14.
54 2.
21 /
4.
100 /
6.
25 8.
8/
/
10.
/2
12.
Try This
Mixed Practice Math Boxes in this lesson are linked with
Math Boxes in Lessons 3-1 and 3-3. The skill in Problem 6
previews Unit 4 content.
Time
16 20
7
10
3
10
Answers
vary.
1
10 º
Sample answers.
3
3
º
º
2
3
30
º
6
INDEPENDENT
ACTIVITY
(Math Masters, p. 85)
º, / 1
2
3
4
5
6
7
8
9 10
1
1
2
3
4
5
6
7
8
9 10
2
2
4
6
8 10 12 14 16 18 20
3
3
6
9 12 15 18 21 24 27 30
4
4
8 12 16 20 24 28 32 36 40
5
5 10 15 20 25 30 35 40 45 50
6
6 12 18 24 30 36 42 48 54 60
7
7 14 21 28 35 42 49 56 63 70
8
8 16 24 32 40 48 56 64 72 80
9
9 18 27 36 45 54 63 72 81 90
10 10 20 30 40 50 60 70 80 90 100
Practice
Home Connection Students fill in the missing numbers
in multiplication and division facts.
15.
17.
1,646 989 657
289
887 598 16.
18.
5,033
1,288 2,004 716
314 4,719 Math Masters, p. 85
Lesson 3 5
183
Student Page
Games
3 Differentiation Options
Division Arrays
Materials □ number cards 6–18 (1 of each)
□ 1 six-sided die
□ 18 counters
Players
2 to 4
Skill
Division and equal shares
PARTNER
ACTIVITY
READINESS
Object of the game To have the highest total score.
Playing Division Arrays
Directions
1. Shuffle the cards. Place the deck number-side down
on the table.
2. Players take turns. When it is your turn, draw a card
and take the number of counters shown on the card.
You will use the counters to make an array.
15–30 Min
(Student Reference Book, p. 240; Math Masters, p. 470)
♦ Roll the die. The number on the die is the number
of equal rows you must have in your array.
♦ Make an array with the counters.
♦ Your score is the number of counters in 1 row. If there
To explore the connections between multiplication and division
using a concrete model, have students play Division Arrays and
record their work on Math Masters, page 470. Some students may
find it helpful to organize the counters on grid paper.
are no leftover counters, your score is double the number
of counters in 1 row.
3. Keep track of your scores. The player with the highest total
score at the end of 5 rounds wins.
Dave draws a 14-card and takes 14 counters. He rolls
a 3 and makes an array with 3 rows by putting 4 counters in each
row. Two counters are left over.
Dave scores 4 because there are 4 counters in each row.
PARTNER
ACTIVITY
ENRICHMENT
Exploring the Relationship
Marsha draws a 15-card and takes 15 counters. She rolls
a 3 and makes an array with 3 rows by putting 5 counters
in each row.
Her score is 5
* 2 = 10 because there are 5 counters in each
5–15 Min
between Division and Fractions
(Math Masters, p. 86)
Student Reference Book, p. 240
To apply students’ understanding of division, have them use
division facts to solve a fraction problem. Both fractions and
division involve sharing something equally or forming equal
groups. Fraction notation is one way to write a division problem.
Some students may solve the pizza problem by dividing each pizza
into 4 equal pieces. Each person gets 1 piece, or _14 , of each pizza.
So one person’s share of the 3 pizzas is three one-fourths, or _34 of
one pizza.
Name
Game Master
LESSON
35
䉬
Name
Date
Time
Division Arrays Record Sheet
240
1 2
4 3
Your score is the number of counters per row. If there are 0 leftover counters,
your score is double the number of counters per row.
Round
Rows
Counters
per Row
Counters
in All
Leftover
Counters
Score
Sample
3
5
15
0
10
Date
Time
Fractions and Division
Four friends want to share 3 pizzas evenly. Each person cannot have a whole
pizza, so the pizzas need to be divided.
1.
How many slices of pizza
should each person get?
slices.
Use a drawing to explain your answer.
2.
Marisa says these expressions all mean
the same thing:
3
1
4
of 3
3
4
34
4冄3
苶
Do you agree or disagree? Explain your answer.
1
Sample answer: Agree. They are all different names for the
1
3
1
3
same thing. 4 of 3 pizzas is 4 of a pizza, so 4 of 3 4. The
fraction bar, , and 冄苶 all mean division.
2
3
4
5
Total Score
Name
Math Masters, page 86
Date
Time
Division Arrays Record Sheet
240
1 2
4 3
Your score is the number of counters per row. If there are 0 leftover counters,
your score is double the number of counters per row.
Round
Rows
Counters
per Row
Counters
in All
Leftover
Counters
Score
Sample
3
5
15
0
10
1
2
3
4
5
Total Score
Math Masters, p. 470
184
46
Unit 3 Multiplication and Division; Number Sentences and Algebra
Teaching Aid Master
EXTRA PRACTICE
Practicing with Fact Families
INDEPENDENT
ACTIVITY
Name
Date
Time
Fact Families
Complete each Fact Triangle. Write the fact family for each Fact Triangle.
5–15 Min
2.
1.
(Math Masters, p. 418)
•
•
ⴱ, ⴱ, To practice the connection between multiplication and division,
have students complete Fact Triangles and related fact families.
ELL SUPPORT
Creating a Poster
SMALL-GROUP
ACTIVITY
5–15 Min
3.
4.
•
•
ⴱ, ⴱ, To provide language support for division, have students create a
poster representing the terms dividend, divisor, quotient, and
remainder. Ask them to write a division number sentence and
label all of the parts. Students can color code the number sentence
so that each part is drawn in the same color as the label.
Math Masters, p. 418
Planning Ahead
Lesson 3-6 provides a detailed introduction to the World Tour.
Read through the lesson well in advance, along with the related
pages in the World Tour section of the Student Reference Book. In
addition to a globe and a large map of the world, try to collect as
many reference materials as possible for a World Tour Corner.
The authors recommend several copies of an almanac, such as the
World Almanac, a full-sized atlas, and several student atlases
similar to the National Geographic World Atlas for Young
Explorers (National Geographic Society, 2007).
Ask students to bring country guide books to school. Begin a
collection of travel information, including airline schedules and
fares, travel articles, weather reports, and currency exchange
rates from the local newspaper or the Internet.
Lesson 3 5
185