CHAPTE R
3
Model Multiplication and
Division Problems
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The
BIG Idea
Investigate
e
How can I develop
an understanding
of multiplication
and division and
their basic facts?
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Vocabulary
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Virtual
Manipulatives
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Foldables
Practice
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eGames
Worksheets
Make this Foldable to help
you organize information
about multiplication and
division concepts and
facts. Begin with four
1 ˝ × 11˝
sheets of 8 _
2
paper.
of
ning
n
Mea licatio
ip
Mult
Review Vocabulary
etida an addition sentence
repeated addition suma rep
number over and over again
that shows adding the same
4 + 4 + 4 + 4 + 4 = 20
Key Vocabulary
Assessment
English
multiply
product
divide
quotient
inverse operations
116
and
Arrays
ication
Multipl
Español
multiplicar
producto
dividir
cociente
operación inversa
When Will I Use This?
Your Tur n!
You will solve this terr.
problem in the chap
Model Multiplication and Division Problems 117
Are You Ready
You have two options for checking
Prerequisite Skills for this chapter.
for the Chapter?
Text Option
Take the Quick Check below.
Find each sum.
1. 2 + 2 + 2 + 2
2. 4 + 4
3. 5 + 5 + 5
4. 10 + 10 + 10 +10
5. 0 + 0 + 0
6. 1 + 1 + 1 + 1 + 1
Identify a pattern. Then find the missing numbers.
7. 5, 10, 15, , , 30
10. , 8, 12, 16, 8. 12, , 8, 6, , 2
11. 50, , 30, 20, 9. 3, 6, 9, , 15, 12. 6, 12, , 24, Write an addition sentence for each picture.
13.
14.
15.
Solve. Use repeated addition.
16. Larisa has 2 cups with 4 crackers
in each cup. How many crackers
does she have in all?
Online Option
118
17. On Monday and Tuesday, Lance
rode his bike around the block
3 times each day. How many
times in all did he ride his bike
around the block?
Take the Online Readiness Quiz.
Model Multiplication and Division Problems
Multi-Part
Lesson
1
Meaning of Multiplication
PART
A
B
C
D
E
F
G
Model Multiplication
Main Idea
I will use models to
explore the meaning
of multiplication.
Multiplication is an operation on two numbers that can
be thought of as repeated addition. The sign (×) means to
multiply. You can use models to explore multiplication.
Materials
connecting cubes
Find how many are in 4 groups of 5.
Step 1 Model 4 groups of 5.
Get ConnectED
GLE 0306.2.2
Develop understanding of
multiplication and related
division facts through
multiple strategies and
representations. SPI 0306.2.5
Identify various
representations of
multiplication and division.
Also addresses GLE 0306.1.3,
GLE 0306.1.8.
Use connecting cubes
to show 4 groups of
5 cubes.
There are 4 groups.
There are 5 cubes in
each group.
Step 2 Find 4 groups of 5.
Label the groups of
cubes with numbers.
Use repeated addition.
5 + 5 + 5 + 5 = 20
Lesson 1A Meaning of Multiplication
119
Step 3 Record the results.
Copy the table. Record the
number of groups, the number
in each group, and the total.
Number of
Groups
Number in
Each Group
Total
4
5
20
Explore other ways to group
the 20 connecting cubes equally.
About It
1. How can addition help you find the total number
when multiplying?
2. How did you find the total number of cubes in Step 2?
3. What do the numbers stand for in the number sentence
in Step 2?
4. Explain another way to group 20 cubes equally.
and Apply It
Use models to find the total number.
5. 2 groups of 3
6. 3 groups of 4
7. 1 group of 5
8. 8 groups of 2
9. 5 groups of 5
10. 6 groups of 4
Draw a model to find the total number.
11. 6 groups of 2
14.
120
12. 4 groups of 5
E
13. 7 groups of 2
WRITE MATH Explain how addition and multiplication
are similar.
Model Multiplication and Division Problems
Multi-Part
Lesson
1
Meaning of Multiplication
PART
A
Main Idea
I will relate
multiplication and
addition.
B
C
D
E
F
G
Multiplication as
Repeated Addition
Vocabulary
V
multiplication
m
Use Models
multiply
factors
product
Get ConnectED
GLE 0306.2.2
Develop understanding of
multiplication and related
division facts through
multiple strategies and
representations.
GLE 0306.2.4 Solve
multiplication and division
problems using various
representations. Also
addresses GLE 0306.1.4,
SPI 0306.2.5.
FOOD For Gilberto’s party,
his mother made 4 small
pizzas. Each pizza had
6 pieces of pepperoni.
How many pieces of
pepperoni did Gilberto’s
mother use to make
4 small pizzas?
Find how many pieces of pepperoni are in
4 groups of 6.
One Way: Counters
Another Way: Repeated
Addition
Write an addition sentence
to show equal groups.
6 + 6 + 6 + 6 = 24
There are 4 groups. There
are 6 counters in each group.
This is a total of 24 counters.
So, 4 groups of 6 is 24. Gilberto’s mother used
24 pieces of pepperoni.
Lesson 1B Meaning of Multiplication
121
Put equal groups together to multiply . The numbers
multiplied are factors . The result is the product .
Use Models
BEES A honeycomb cell has 6 sides. How many sides
do 5 separate honeycomb cells have altogether?
Find how many sides are in 5 groups of 6.
One Way: Repeated Addition
u
When you multiply, yo
add the same number
multiple times.
+
6
+
6
+
6
+
6
6 = 30
Another Way: Multiplication Sentence
number of
number
cells ( groups)
of sides
5
×
6
factor
total
=
factor
30
product
So, there are 30 sides altogether.
W it an addition
Write
dditi
sentence
t
and
d a multiplication
lti li ti
sentence
t
for each model. See Examples 1 and 2
1.
2.
Multiply. Use models and repeated addition.
3. 2 × 6
4. 4 × 4
7. Marcos gave three friends 4 stickers
each. How many stickers did he
give away?
122
See Examples 1 and 2
5. 5 × 3
8.
Model Multiplication and Division Problems
E
6. 7 × 2
TALK MATH Can you write
2 + 3 + 5 = 10 as a multiplication
sentence? Explain.
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Begins on page EP2.
Write an addition
dd
sentence and
d a multiplication
l l
sentence
for each model. See Examples 1 and 2
9.
10.
11.
12.
13. 6 groups of 6
14. 8 groups of 4
15. 10 groups of 3
16. 7 groups of 5
17. 5 groups of 7
18. 9 groups of 4
Multiply. Use models and repeated addition.
See Examples 1 and 2
19. 3 × 5
20. 5 × 2
21. 3 × 3
22. 6 × 2
23. 9 × 2
24. 10 × 6
25. 5 × 5
26. 4 × 7
27. 6 × 4
28. Adriano bought 3 boxes of paints.
Each box has 8 colors. What is the
total number of paints?
29. Leonora found 4 bags of buttons.
Each bag has 10 buttons. How
many buttons are there altogether?
30. Each boy has 5 balloons and each
girl has 3 balloons. How many
balloons do they have if there are
3 boys and 6 girls?
31. A starfish has 5 legs. There are
5 starfish on the beach. If 4 of the
starfish are each missing 1 leg,
how many legs are there?
32. OPEN ENDED Write a real-world multiplication problem
whose product is greater than 40.
33. CHALLENGE What is 2 more than 5 groups of 3?
34.
E
WRITE MATH Describe a real-world situation where you
would use multiplication to solve a problem.
Lesson 1B Meaning of Multiplication
123
Multi-Part
Lesson
1
PART
Meaning of Multiplication
A
B
C
D
E
F
G
Multiplication with Arrays
Main Idea
I will use arrays to
explore and model
multiplication.
Parker bought 12 fishing hooks. How can he put them in
his tackle box so they are in equal rows and columns?
Materials
color tiles
Arrays and Repeated Addition
Step 1 Arrange 12 tiles in
grid paper and plain
paper
a rectangle. Compare
your rectangle to your
neighbor’s rectangle.
How are they different?
How are they alike?
3
4
3 + 3 + 3 + 3 = 12
Step 2 Write an addition sentence
to show equal rows.
scissors and glue
Arrays and Multiplication
2
Step 1 From the grid paper, cut
Get ConnectED
GLE 0306.2.2
Develop understanding of
multiplication and related
division facts through
multiple strategies and
representations. GLE 0306.3.1
Develop meaning for and
apply the commutative,
associative, and distributive
properties using various
representations.
Also addresses GLE 0306.1.3,
SPI 0306.2.5.
124
an array that has 5 rows of
2 squares.
5
• Glue it on white paper,
and then label it.
2 + 2 + 2 + 2+ 2 = 10
• Write a repeated
addition sentence to represent the array.
Step 2 Cut an array of 2 rows
of 5 squares.
• Glue and label it next
to the first one.
Model Multiplication and Division Problems
5
2
5 + 5 = 10
Step 3 Write a multiplication sentence for each array.
The arrays show the Commutative Property
of Multiplication.
2
5
5
2
number in
rows each row total
5 ×
2
= 10
number in
rows each row total
2 ×
5
= 10
About It
1. What is the connection between repeated addition and
an array?
2. Explain the Commutative Property of Multiplication.
3. How can you use an array to model the Commutative
Property of Multiplication?
and Apply It
Write an addition sentence and a multiplication sentence for
each array.
4.
5.
6.
7.
Make an array to find the total number. Write the
multiplication sentence.
8. 2 × 4
9. 1 × 4
12. Tyrone made a 4 × 6 array using
stones. How many stones did
Tyrone use?
14.
10. 5 × 2
11. 4 × 4
13. If you made an array to find 3 × 5,
how can you change the array to
find 2 × 5?
E
WRITE MATH List and explain the kind of everyday
objects you find in an array.
Lesson 1C Meaning of Multiplication
125
Multi-Part
Lesson
1
PART
Meaning of Multiplication
A
Main Idea
I will use arrays to
multiply.
Vocabulary
V
B
C
D
E
F
G
Arrays and Multiplication
The cups are arranged in equal rows and equal columns.
This arrangement is an array . Arrays can help you multiply.
array
a
Model an Array
Get ConnectED
GLE 0306.2.2
Develop understanding of
multiplication and related
division facts through
multiple strategies and
representations.
GLE 0306.3.1 Develop
meaning for and apply the
commutative, associative, and
distributive properties using
various representations.
Also addresses GLE 0306.1.4,
SPI 0306.2.5.
PARTY CUPS Roberto
places party cups on a
table in 3 rows of 5 cups
each. How many cups
are on the table?
To find the total number of
cups, use counters to model an array.
The array shows 3 rows of 5.
5
3
You can use addition or multiplication to find the total.
One Way: Add
Another Way: Multiply
5 + 5 + 5 = 15
3 × 5 =
factor
3 × 5 = 15
Write a
multiplication
sentence.
So, 3 groups of 5 cups is 15 in all.
126
Model Multiplication and Division Problems
factor
15
product
Commutative Property
of Multiplication
PHOTOS One page of Elsa’s
photo album is shown.
Write two multiplication
sentences to find how many
photos are on each page.
2
4
The models in
Example 2 are also
arrays since they have
columns of equal
number and rows of
equal number.
4
2
number in
each row
rows
4
×
2
total
rows
8
2
=
number in
each row
×
4
total
= 8
Commutative Property
Everyday Use
Commute To go back and
forth.
Math Use
Commutative To change
the order of factors.
Words
The Commutative Property of
Multiplication says the order in which
numbers are multiplied does not change
the product.
Examples
4 ×
factor
3 =
factor
12
product
3
factor
×
4
factor
= 12
product
Write two
t o multiplication
m ltiplication sentences for each array.
arra
See Examples 1 and 2
1.
2.
3. Write two multiplication sentences to
find how many puppies there are if
5 dogs each have 2 puppies.
4.
E
TALK MATH What other
operation uses the Commutative
Property? Explain.
Lesson 1D Meaning of Multiplication
127
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Begins on page EP2.
Write two multiplication
l l
sentences for
f each
h array.
See Examples 1 and 2
5.
6.
7.
8.
9.
10.
Algebra Use the Commutative Property of Multiplication
to find each missing number. See Example 2
11. 5 × 2 = 10
2 × = 10
12. 3 × 5 = 15
× 3 = 15
13. 3 × 9 = 27
9×3= 14. Geometry Hope drew the area
model at the right. Write a multiplication
sentence to represent her model.
Multiply. Use an array if needed.
See Examples 1 and 2
15. Adult tickets to the talent show
cost $8. How much will 4 adult
tickets cost?
16. Tamika gives her dog 2 treats
every day. How many treats does
Tamika’s dog get in one week?
Use the Commutative Property of Multiplication to write two
multiplication sentences for each situation. Then solve.
See Example 2
17. Baily made a 3 by 4 array with
number cards. How many number
cards are there?
128
18. There were 4 students with
5 balloons each. How many
balloons do the students have?
Model Multiplication and Division Problems
19. FIND THE ERROR Alyssa is using the numbers 3, 4, and 12
to show the Commutative Property of Multiplication.
Find and correct her mistake.
3 × 4 = 12
so, 12 × 3 = 4
20.
E
WRITE MATH Describe how an array can help you find the
answer to a multiplication problem.
Test Practice
21. Which multiplication sentence is
modeled below? (Lesson 1D)
A. 5 × 7 = 35
C. 8 × 3 = 24
B. 6 × 6 = 36
D. 4 × 6 = 24
22. Dominic drew 7 lines. Each line is
5 inches long. The total length of
all 7 lines is 35 inches. If Dominic
drew 5 lines and each line was
7 inches long, what is the total
length of all 5 lines? (Lesson 1B)
Multiply. Use models and repeated addition.
F. 30
H. 40
G. 35
I. 45
(Lesson 1B)
23. Jerome saw a group of 8 lizards. Each lizard had 2 stripes on
its back. How many stripes were there in all?
Write an addition sentence and a multiplication sentence
for each model. (Lesson 1B)
24.
25.
Lesson 1D Meaning of Multiplication
129
Multi-Part
Lesson
1
PART
Meaning of Multiplication
A
Main Idea
I will recognize the
comparison of two
groups as another type
of multiplication.
Vocabulary
V
ccomparison problems
bar diagram
B
C
D
E
F
G
Use Multiplication
to Compare
Sometimes you have to look at a problem in a different
way when a phrase like times as many, times more, and
times as much is used. These kinds of problems
are comparison problems .
Get ConnectED
GLE 0306.2.2
Develop understanding of
multiplication and related
division facts through
multiple strategies and
representations. GLE 0306.2.4
Solve multiplication and
division problems using
various representations.
Also addresses GLE 0306.1.4,
SPI 0306.2.5.
Use Models
CAMP Mary attended camp
for 7 days this summer. Tyler
attended 3 times as many
days as Mary. Find the number
of days Tyler attended camp.
Use models to help you compare
the groups of days.
Step 1 Model Mary’s days at camp as 1 group of 7 days.
Step 2 Tyler had 3 times as many days at camp. Model
Tyler’s days at camp as 3 groups of 7.
Step 3
Find the total of 3 groups of 7.
7 + 7 + 7 = 21
or
3 × 7 = 21
So, Tyler attended camp 21 days.
130
Model Multiplication and Division Problems
A type of model drawing is the bar diagram . A bar
diagram can help you understand a problem and plan to
solve it.
Use a Bar Diagram
Multiplication is the
same as repeated
addition.
BEADS Cassady used 5 beads
to make a bracelet. Suki used
3 times as many beads as
Cassady. How many beads
did Suki use?
Step 1
Cassady’s 5 beads are modeled as one part.
Cassady
Step 2
5 beads
Suki has 3 times as many beads as Cassady.
So, the same part is modeled 3 times.
Cassady
5 beads
? beads used
5 beads
Suki
Step 3
5 beads
5 beads
Find the total number of Suki’s beads.
One Way: Repeated
Addition
Another Way: Multiply
15 beads used
15 beads used
5 beads 5 beads 5 beads
5 beads 5 beads 5 beads
5 + 5 + 5 = 15 beads
3 × 5 = 15 beads
So, Suki used 15 beads.
Lesson 1E Meaning of Multiplication
131
Use the models to compare.
compare Then write
rite a multiplication
m ltiplication
sentence. See Example 1
1. 3 times as much
2. 2 times more
3. 4 times as many
Use the bar diagram to compare. Then write a multiplication
sentence. See Example 2
4. twice as many boys
4 boys
5. 6 times as much money
4 pens
$2
? boys
4
6. 3 times more pens
4
? pens
?$
4
$2 $2 $2 $2 $2 $2
Solve. Use a bar diagram if needed.
4
4
See Example 2
7. BAR DIAGRAM While on a trip, Sheri
bought 3 postcards. Willa bought
twice as many. How many postcards
did Willa buy?
8.
E
TALK MATH Do you prefer to
use the bar diagram or models to
help you solve problems? Explain.
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Begins on page EP2.
Use the
h models
d l to compare. Then
h write a multiplication
l l
sentence. See Example 1
9. 3 times as many
10. 5 times more
11. 4 times as much
12. 10 times as much
13. 2 times more
14. twice as many
15. 8 times as many
16. 4 times more
17. 5 times as much
132
Model Multiplication and Division Problems
Use the bar diagram to compare. Then write a multiplication
sentence. See Example 2
18. 5 times as many balls
19. 4 times as many fish
6 fish
4 balls
3 bows
? balls
4
4
6
6
4
21. 3 times more yo-yos
6
6
$6
1 star
? yo-yos
5
3
3
22. 5 times as many stars 23. 3 times as much money
5 yo-yos
5
? bows
? fish
4
4
20. 2 times as many bows
?$
? stars
5
1
Solve. Use a bar diagram if needed.
1
1
1
1
$6
$6
$6
See Example 2
24. BAR DIAGRAM There are 3 times as
many blue balloons as green
balloons. There are 4 green balloons.
How many blue balloons are there?
25. BAR DIAGRAM Nan needs 4 times
as much flour as sugar. She needs
4 cups of sugar. How much flour
does she need?
26. BAR DIAGRAM Devi practiced her
flute 6 days last month. How many
days did she practice this month if
she practiced 3 times as many days?
27. BAR DIAGRAM Perry paid 10¢ for a
rubber snake. Sam paid 3 times as
much for his snake. How much did
Sam pay for his snake?
28. OPEN ENDED Write a real-world comparison problem using
the numbers 2 and 5.
29. WHICH ONE DOESN’T BELONG? Identify the model that
does not represent the number sentence 3 × 4 = 12. Explain.
4 + 4 + 4 = 12
30.
12 - 4 = 8
E
WRITE MATH Explain how a bar diagram can help you
plan and solve a problem.
Lesson 1E Meaning of Multiplication
133
Multi-Part
Lesson
1
PART
Meaning of Multiplication
A
Main Idea
I will use multiplication
to find the total
number of
combinations that can
be made when given
two groups of objects.
B
C
D
F
E
G
Use Multiplication to
Find Combinations
When you make a combination you make a new set that
has one item from each set of items.
Vocabulary
V
Make a Picture
ccombination
tree diagram
Get ConnectED
GLE 0306.2.2
Develop understanding of
multiplication and related
division facts through multiple
strategies and representations.
SPI 0306.1.5 Represent
problems mathematically
using diagrams, numbers, and
symbolic expressions.
Also addresses GLE 0306.1.7,
SPI 0306.2.5.
FOOTBALL Amos’ team has 3 jersey colors—green, red,
and yellow. They can wear black or white socks. Find
all of the jersey and sock combinations for the team.
To find the combinations, match each color of jersey
with each color of socks.
One Way: Make a Picture
green jersey, black socks
green jersey, white socks
2
+
red jersey, black socks
red jersey, white socks
2
+
2
=
yellow jersey, black socks
yellow jersey, white socks
6 combinations
Another Way: Write a Multiplication Sentence
There are 3 jersey colors and 2 sock colors.
Multiply to find the total number of combinations.
3
×
2
=
6
jersey
sock
colors
colors
combinations
There are 6 jersey and sock combinations possible.
134
Model Multiplication and Division Problems
Another way to find combinations is a tree diagram.
A tree diagram uses “branches” to show all the possible
combinations.
Make a Tree Diagram
You can multiply the
number of choices in
each set if you only
need to find the
number of possible
combinations.
ICE-CREAM SUNDAES
What are the possible
sundae combinations if
one ice cream flavor and
one topping is chosen?
Sundaes
Ice Cream
Toppings
Draw a “branch” to
match each ice cream
flavor with each topping.
Flavor
Ice Cream
Toppings
Chocolate
Sprinkles
Chocolate
Sprinkles
Vanilla
Whip Cream
Vanilla
Whipped
Cream
Strawberry
Peanuts
Strawberry
Walnuts
Toppings
Ice-Cream Sundae
sprinkles
whipped cream
peanuts
chocolate, sprinkles
chocolate, whipped cream
chocolate, peanuts
sprinkles
whipped cream
peanuts
vanilla, sprinkles
vanilla, whipped cream
vanilla, peanuts
sprinkles
whipped cream
peanuts
strawberry, sprinkles
strawberry, whipped cream
strawberry, peanuts
chocolate
vanilla
strawberry
Check Multiply to find the number of combinations.
3 flavors × 3 toppings = 9 combinations So, there are 9 possible sundae combinations.
Lesson 1F Meaning of Multiplication
135
Make a pict
picture
re or tree dia
diagram
ram to find all the possible
combinations. Write a multiplication sentence. See Examples 1 and 2
1. The music teacher told her students
to run, walk, or hop while they
clapped or snapped their fingers.
What are the possible combinations
of one hand and one foot motion?
2. There are green, blue, red, and
orange balloons with silver or gold
streamers. Find the possible
combinations of one color balloon
and one streamer.
3. Preston can buy in-line skates or
roller skates in silver or black. Find
Preston’s choices.
4.
E
TALK MATH Explain how a tree
diagram helps you find all the
possible combinations without
repeating any.
EXTRA
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Begins on page EP2.
k a picture or tree diagram
d
f d all
ll the
h possible
bl
Make
to find
combinations. Write a multiplication sentence. See Examples 1 and 2
5. Jackie is playing a card game with
triangles, circles, squares, and
trapezoids. The shapes can be blue,
red, yellow, or green. How many
different cards are there?
6. List all of the 2-digit numbers that
can be made with 3, 4, 2, or 5 as
the tens digit and 1, 6, 7, 8, or 9
as the ones digit.
7. What are the possible color
combinations if both spinners
are spun?
8. The students will choose one piece
of paper and one piece of chalk.
What are the combinations they
may get?
136
Model Multiplication and Division Problems
Use the information to solve the problems.
Pizza Puzzle
Remember, Nate
needs to order
eight different
one-topping
pizzas.
This is quite a puzzle,
... a pizza puzzle.
9. Draw a picture or tree diagram to show the different
one-topping pizza combinations.
10. Is Nate able to make 8 different one-topping pizzas? Write
a multiplication sentence to show the total number
of one-topping pizzas that can be made.
11. OPEN ENDED Write a real-world combination problem. Ask
a neighbor to find all the possible combinations. Provide
the answer.
12. WHICH ONE DOESN’T BELONG? Choose one fruit and
one cheese to make all the possible combinations. Find
the combination that does not belong. Explain.
fruit: peach, pear
pear,
swiss cheese
13.
pear,
cheddar cheese
cheese: cheddar, swiss
pear,
peach
peach,
swiss cheese
E
WRITE MATH Explain a situation when you may need
to know how to find the total number of combinations that
would result when putting two sets of things together.
Lesson 1F Meaning of Multiplication
137
Multi-Part
Lesson
1
Meaning of Multiplication
PART
A
B
C
D
E
G
F
Problem-Solving Strategy:
Make a Table
Main Idea I will use the make a table strategy to solve a problem.
S
Selma
bought 3 shorts and 2 shirts. Her
ssister, Laura, bought 4 shorts and 2 shirts.
How many different shirt and shorts
combinations can each girl make?
Understand
What facts do you know?
• You know what each girl bought.
What do you need to find?
• How many different shirt and shorts
combinations they can each make.
Plan
Organize the information in a table.
Solve
• Make a table for each girl. Make a row for each pair of shorts
and a column for each shirt. List the possible shirt and shorts
combinations.
Selma
Shirt 1
Shirt 2
Laura
Shirt 1
Shirt 2
Shorts A
Shorts B
Shorts C
A1
B1
C1
A2
B2
C2
Shorts A
Shorts B
Shorts C
Shorts D
A1
B1
C1
D1
A2
B2
C2
D2
Selma:
3
×
shorts
Laura:
4
2
shirts
×
2
6
=
combinations
=
8
So, Selma can make 6 combinations, and Laura can make 8.
Check
Since 3 × 2 = 6 and 4 × 2 = 8, you know that the number of
clothing combinations is correct. GLE 0306.1.5 Use mathematical ideas and processes in different settings to formulate patterns, analyze
graphs, set up and solve problems and interpret solutions. SPI 0306.1.8 Express answers clearly in verbal,
numerical, or graphical (bar and picture) form, using units when appropriate. Also addresses GLE 0306.2.2.
138
1
Model Multiplication and Division Problems
Refer to the problem on the previous page.
1. How did the make a table strategy
help you find the answer to the
problem?
3. Look back at your answer for
Exercise 2. How do you know
that the answer is correct? Show
your work.
2. Suppose Laura had 3 shirts instead
of 2. How many combinations would
she have?
4. How are the problems on the
previous page and Exercise 2 alike?
EXTRA
%
)# E
# T4 IC
!C
2A
0R
P
Begins on page EP2.
Solve. Use the make a table strategy.
5. How many lunches can Malia make
if she chooses one main item and
one side item from the menu shown
below?
8. Look at the table. How many pens
do Nestor and Pam have in all?
How many more pens does Carra
have than Pam?
Name
Pens
Pam
Nestor
Carra
7
9
20
9. The students in Mr. Robb’s class
are designing a flag. The flag’s
background can be red or green
with a blue or a purple stripe. How
many flags can they design? Explain
how you solved the problem.
6. Amber has coins in a jar. The sum of
the coins is 13¢. What are the
possible coin combinations Amber
could have?
7. Choose one bread and one meat.
bread: wheat or white
meat: turkey or chicken
What are all the possible
combinations?
10. Geometry Rodrigo is putting up a
fence in the shape of a triangle.
Side A
Side B
Side C
2 times as
long as side B
18 feet
same as
side A
How much fencing does he need?
11.
E
WRITE MATH Write one realworld problem that would involve
making a table to find the answer.
Lesson 1G Meaning of Multiplication
139
LOTS OF
ARMS LEGS
AND
Have you ever wondered why a
cheetah has 4 legs instead of 3?
Or why an octopus has 8 arms
instead of 4? The number of arms
or legs an animal has helps it
hunt for food and escape from
predators.
A cheetah has 4 legs that balance
its body. Its legs help it run as fast
as 70 miles per hour. An octopus
has an unprotected body and no
claws or teeth. So, 8 arms are more
helpful to an octopus than only
4 or 6 arms.
ANIMAL
140
NUMBER OF LEGS
OR ARMS
Model Multiplication and Division Problems
Use the information on the previous page to solve each problem. Write a
multiplication sentence to solve. Then write an addition sentence
to check.
1.
Three ants are on a park bench.
How many legs are there in all?
2.
You see 7 ostriches. How many
legs do you see altogether?
5.
6.
3.
If you see a pack of 3 cheetahs,
how many legs are there in all?
4.
If there are 4 octopuses, how
many octopus arms are there total?
7.
You count 30 sugar star arms in
the aquarium. How many sugar
stars are there? Explain.
There are 3 sea turtles and
2 sugar stars in another
aquarium. How many arms and
legs are there altogether?
How many legs in all do 6 hermit
crabs have?
An octopus
has 240 suction
cups on each of its
8 arms.
Problem Solving in Science
141
Rows and Columns
Use Arrays to Multiply
You will need: 1 set of array cards
for each player; one blank spinner
Get Ready!
Players: 2–3 players
Number the blank spinner 1–10.
Each player cuts a set of array
cards on the dotted lines.
2 by 2
2 by 3
4 by 3
2 by 4
Each player places their set
of cards faceup in an array of
4 rows and 4 columns.
4 by 4
2 by 5
5 by 2
5 by 3
Go!
7 by 2
1 by 3
4 by 2
6 by 3
3 by 4
5 by 4
5 by 1
3 by 3
Get Set!
Player 1 spins the spinner
two times and finds the
product. The player finds an
array card that matches the
product and turns the card
over. If there is no array card
to match, the player’s turn
is over.
The remaining players
repeat the directions in turn.
“Array” is exclaimed when
one player has 1 complete
row or 1 complete column
turned over.
142
Model Multiplication and Division Problems
Mid-Chapter
Check
Find the total number.
(Lesson 1A)
1. 2 groups of 4
2. 4 groups of 5
3. 9 groups of 2
4. 5 groups of 3
Multiply. Use models and repeated
addition. (Lesson 1B)
5. 2 × 6
6. 5 × 2
7. 3 × 3
8. 2 × 8
Write two multiplication sentences
for each array. (Lesson 1D)
9.
14. MULTIPLE CHOICE Nine tigers each
make 4 paw prints. Which of the
following number sentences should
be used to find the total number of
paw prints? (Lesson 1D)
A. 9 + 4 = 13
C. 9 × 4 = 36
B. 9 - 4 = 5
D. 9 × 4 = 40
Use the bar diagram to compare.
Then write a multiplication
sentence. (Lesson 1E)
15. 3 times as many straws
4 straws
? straws
4
10.
4
4
16. 5 times as many pencils
6 pencils
? pencils
Use the models to compare. Then
write a multiplication sentence.
(Lesson 1E)
11. 2 times
as many
12. 5 times
as many
6
6
6
6
6
17. MULTIPLE CHOICE Which of the
following number sentences is
related to this addition sentence?
(Lesson 1B)
5 + 5 + 5 = 15
13. How many possible boy-girl pairs
can be made if Ben, Lauren, Jamal,
Ian, and Angela were chosen? Make
a picture or diagram to find all the
possible combinations. Write a
multiplication sentence. (Lessons 1F, 1G)
18.
F. 3 × 5 = 15
H. 15 - 5 = 10
G. 3 + 5 = 8
I. 5 + 3 = 8
E
WRITE MATH Explain how
multiplication and addition are
related. (Lesson 1B)
Mid-Chapter Check 143
Multi-Part
Lesson
2
Meaning of Division
PART
A
B
C
D
E
F
G
Model Division
Main Idea
I will explore two
meanings of division.
Division is an operation with two numbers. One number
tells you how many things you have. The other tells you
how many equal groups to form or how many to put in
each group.
Materials
counters
paper plates
10 ÷ 5 = 2
Read ÷ as divided by.
10 divided by 5 = 2.
To divide means to separate a number into equal groups, to
find the number of groups, or find the number in each group.
Divide 12 counters into 3 equal groups.
D
Get ConnectED
GLE 0306.2.2
Develop understanding of
multiplication and related
division facts through
multiple strategies and
representations. GLE 0306.2.4
Solve multiplication and
division problems using
various representations.
Also addresses GLE 0306.1.4,
SPI 0306.2.5.
Step 1 Count out
12 counters.
Using paper
plates, show
3 groups.
Step 2 Place one counter
at a time on
each plate
until all of the
unters are
counters
ne.
gone.
Step 3 Twelve
elve counters
re divided
were
o 3 groups.
into
ere are
There
ounters in
4 counters
ch group.
each
So, 12 ÷ 3 = 4.
144
Model Multiplication and Division Problems
roblems
Place 12 counters in groups of 3.
Step 1 Count out 12 counters.
Step 2 Make groups of
3 until all the counters
are gone.
There are 4 groups of 3.
So, 12 ÷ 4 = 3.
About It
1. Explain how you divided 12 counters into equal groups.
2. When you divided the counters into groups of 3, how
did you find the number of equal groups?
and Apply It
3. Make equal groups to find the
number of counters in each group.
4. Find the number of equal groups
of 5.
5. Copy the chart. Then use counters to help complete it.
Number of
Counters
Number of
Equal Groups
Number in
Each Group
Division Sentence
9
3
3
9÷3=3
14
2
15
5
6
6.
3
E
WRITE MATH Can 13 counters be divided equally into
groups of 3? Explain.
Lesson 2A Meaning of Division 145
Multi-Part
Lesson
2
PART
Meaning of Division
A
Main Idea
I will divide by sharing
to make equal groups.
Vocabulary
V
divide
B
C
D
E
F
G
Division as Equal Sharing
Activity 1 in the Explore lesson showed that one way
to divide is to find the number in each group. This can
be done by sharing equally.
Get ConnectED
Share Equally
GLE 0306.2.2
Develop understanding of
multiplication and related
division facts through
multiple strategies and
representations. GLE 0306.2.4
Solve multiplication and
division problems using
various representations.
Also addresses SPI 0306.2.5.
RABBITS Caley has 6 rabbits
that she keeps in 3 hutches.
She has an equal number of
rabbits in each hutch. How
many rabbits are in each hutch?
You can draw a picture. Place one rabbit at a time in
each hutch until there are no more rabbits.
6
rabbits
÷
3
=
hutches
(groups)
2
in each
hutch
So, there are 2 rabbits in each hutch.
146
Model Multiplication and Division Problems
Model an Array
CAMP Fifteen scouts are divided equally to sleep in
3 tents. How many scouts are in each tent?
You can use counters to model an array.
Step 1
Place one counter (scout) in each row (tent).
Tent 1
When you divide you
er
share an equal numb
s.
to all the group
Tent 2
Tent 3
Step 2
Step 3
Continue to place one counter (scout) in each
row (tent) until all of the counters are gone.
Tent 1
5 scouts
Tent 2
5 scouts
Tent 3
5 scouts
Write a number sentence.
15 ÷ 3 = 5
So, 5 scouts are in each tent.
U counters
Use
t
tto model
d l th
the ttotal.
t l Di
Divide
id to
t find
fi d the
th number
b in
i
each group. See Examples 1 and 2
1. 10 counters
2 equal groups
2. 14 counters
7 equal groups
3. 20 counters
5 equal groups
in each group
in each group
in each group
÷
÷
÷
=
4. Dexter had 30 blocks. He stacked
them in equal rows on top of each
other until he ran out of blocks.
Dexter’s towers stood 10 blocks tall.
How many towers did Dexter make?
=
5.
=
E
TALK MATH Explain what it
means to share equally when
dividing.
Lesson 2B Meaning of Division 147
EXTRA
%
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# T4 IC
!C
2A
0R
P
Begins on page EP2.
Use counters to model
d l the
h total.
l Divide
d to find
f d the
h number
b in
each group. See Examples 1 and 2
6. 4 counters
2 equal groups
in each group
÷=
9. 8 counters
4 equal groups
in each group
÷=
7. 12 counters
2 equal groups
in each group
8. 10 counters
5 equal groups
in each group
÷=
÷=
10. 15 counters
5 equal groups
in each group
11. 18 counters
2 equal groups
in each group
÷=
÷=
12. Marla has $30. How
many compact discs
can she buy?
13. There are 6 juice boxes in a
package. How many packages
need to be bought if 24 juice
boxes are needed for the picnic?
14. Mrs. Miller needs 18 feet of fabric.
How many yards of fabric will she
need to buy? (Hint: 1 yard = 3 feet)
15. The bookstore had a sale. Every
time you bought 3 books you got
1 free. Alberto bought 9 books.
How many free books did he get?
Algebra Find each missing number.
16. 8 ÷ = 4
17. ÷ 3 = 3
18. 12 ÷ 3 = 19. ÷ 2 = 6
20. 16 ÷ = 4
21. 20 ÷ 4 = There are many different kinds of saltwater
fish off ocean beaches.
22. Together, 3 snooks weigh about
12 pounds. About how much does each
snook weigh if each weighs about the same?
23. Draw a picture to show two different ways
the fish could be divided equally. Then,
write a number sentence for each picture.
148
Model Multiplication and Division Problems
24. OPEN ENDED Write a real-world division problem in which
5 would be the answer.
25. WHICH ONE DOESN’T BELONG? Identify the number
sentence that does not belong. Explain.
12 ÷ 3 = 4
26.
E
15 ÷ 3 = 5
12 ÷ 6 = 2
12 ÷ 4 = 3
WRITE MATH Explain one meaning of division.
Test Practice
27. Nathan wants to buy a clock.
He can choose one shape and
one color.
Shape
Color
Square
Red
Circle
Blue
28. Alma planted an equal number of
seeds in each pot. How many
seeds did Alma put in each pot?
(Lesson 2B)
Green
How many different combinations
are possible? (Lesson 1F)
A. 3
C. 5
B. 4
D. 6
F. 2
H. 18
G. 9
I. 36
29. The lunch menu choices are carrots or celery and an apple,
orange, or banana. The students must choose one vegetable
and one fruit. Make a table to find all the combinations. (Lesson 1G)
30. Find all the combinations that can be made with one number
and one letter. Write a multiplication sentence. (Lesson 1F)
L, C, T, M 5, 2, 7, 3
Lesson 2B Meaning of Division 149
Multi-Part
Lesson
2
Meaning of Division
PART
A
Main Idea
I will use models to
relate division and
subtraction.
Vocabulary
V
rrepeated subtraction
B
C
D
E
F
G
Relate Division
and Subtraction
Recall that to divide means to separate a number into
equal groups, to find the number of groups, or find the
number in each group.
Get ConnectED
GLE 0306.1.4 Move
flexibly between concrete
and abstract representations
of mathematical ideas in
order to solve problems,
model mathematical ideas,
and communicate solution
strategies. GLE 0306.2.4 Solve
multiplication and division
problems using various
representations.
Also addresses SPI 0306.2.5,
SPI 0306.3.2.
Use a Number
Sentence
PENCILS There are 15 pencils
in a box. Each pencil is either
red, blue, or yellow. There are
the same number of each
color. How many pencils of
each color are there? Use a
number sentence to record
the solution.
Place one counter at a time on each
plate until all 15 counters are gone.
There are
5 counters in
each group.
5
5
5
The number sentence that describes the model is
15 ÷ 3 = 5. So, there are 5 pencils of each color.
You can also divide using repeated subtraction . Subtract
equal groups of 3 repeatedly until you get to zero.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Subtract equal groups
of 3 until you get to 0.
150
15 ÷ 3 = 5
Model Multiplication and Division Problems
For 15 ÷ 3, start at 15.
Repeated
Subtraction
SPORTS Andre wants to put his 10 baseball cards
into equal groups of 2. How many groups can
he make?
Use repeated subtraction to find 10 ÷ 2. Write
a number sentence.
Paper and
One Way: Number Line Another Way:
Pencil
5
4
3
2
1
1
0 1 2 3 4 5 6 7 8 9 10
Start at 10. Count back by
2s until you reach 0. How
many times did you
subtract?
2
3
4
5
8
2
4
10
6
-2 -2 -2 -2 -2
2
8
4
0
6
Subtract groups of 2 until
you reach 0. How many
groups did you subtract?
So, the number sentence 10 ÷ 2 = 5 shows that Andre
will have 5 groups of cards.
Use models
U
d l tto di
divide.
id W
Write
it a number
b sentence.
t
1. There are 16 flowers. Each vase has
4 flowers. How many vases are there?
Use repeated subtraction to divide.
See Example 1
2. There are 14 ears. Each dog has
2 ears. How many dogs are there?
See Example 2
4.
3.
0 1 2 3 4 5 6 7 8
0 1 2 3 4 5 6 7 8 9 10 11 12
12 ÷ 3
5. 6 ÷ 2
8÷2
6. 12 ÷ 6
8. There are 16 mittens. Each student
wears 2 mittens. How many students
are there?
7. 25 ÷ 5
9.
E
TALK MATH Explain how to use
a number line to find 18 ÷ 9.
Lesson 2C Meaning of Division 151
EXTRA
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2A
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P
Begins on page EP2.
Use models
d l to divide.
d d Write a number
b sentence.
See Example 1
10. There are 16 orange slices. Each
orange has 8 slices. How many
oranges are there?
11. Measurement There are 16 miles.
Each trip is 2 miles. How many
trips are there?
12. There are 25 marbles, with
5 marbles in each bag. How many
bags are there?
13. Four friends will share 12 muffins
equally. How many muffins will
each friend get?
Use repeated subtraction to divide.
See Example 2
14.
15.
0
0 1 2 3 4 5 6 7 8 9 10
1
2
10 ÷ 5
3
4
6
6÷3
16.
17.
0
0 1 2 3 4 5 6 7 8 9
9÷3
1
2
3
4
5
6
7
8
8÷4
18. 18 ÷ 3
19. 12 ÷ 2
20. 24 ÷ 6
21. 12 ÷ 3
22. 27 ÷ 3
23. 28 ÷ 7
24. There are 12 erasers. Tobias wants
to share them equally among himself
and his 2 friends. How many erasers
will each person get?
25. Chester has 24 pencils. He kept
4 and shared the others equally
among his 4 brothers. How many
pencils did each brother get?
Social Studies
Chicago’s Ferris wheel is
10 stories tall. Each gondola can seat up to
6 people while they enjoy a 7-minute ride.
26. Suppose someone took a 21-minute ride. How
many rides were taken?
27. If 30 students from a class wanted to ride, how
many gondolas would they need?
28. It costs $24 for 4 people to ride. How much is
each ticket?
152
5
Model Multiplication and Division Problems
29. OPEN ENDED Write a real-world problem that could be
represented by 18 ÷ 6.
30.
E
WRITE MATH How is division related to subtraction?
Test Practice
32. Hally biked 12 miles this week.
She always rode the same 2-mile
path. Which of the following
number sentences shows the
number of days she biked?
31. Mr. Gomez bought the pizzas
shown below. If 3 classes share
the pizzas evenly, how many
pizzas will each class get?
(Lesson 2B)
(Lesson 2C)
F. 6 ÷ 2 = 12
G. 24 ÷ 2 = 12
A. 2
C. 8
H. 12 ÷ 2 = 6
B. 3
D. 16
I. 12 ÷ 4 = 3
Use counters to model the total. Divide to find the number
in each group. (Lesson 2B)
33. 12 counters
4 equal groups
in each group
34. 16 counters
4 equal groups
in each group
35. 18 counters
3 equal groups
in each group
÷=
÷=
÷=
Use counters to help complete the chart.
36.
37.
(Lesson 2A)
Total
counters
Number of
groups
Number in
each group
Division
sentence
4
2
10
5
Lesson 2C Meaning of Division 153
Multi-Part
Lesson
2
PART
Meaning of Division
A
Main Idea
I will explore how
division and
multiplication are
related.
Materials
B
C
D
E
F
G
Relate Division and
Multiplication
You can relate division and multiplication.
Relate Division and Multiplication
counters
Step 1 Find 21 ÷ 3.
Model 21 counters divided into 3 equal groups.
Get ConnectED
GLE 0306.2.3
Relate multiplication and
division as inverse operations.
SPI 0306.3.2 Express
mathematical relationships
using number sentences/
equations. Also addresses
GLE 0306.1.3, SPI 0306.2.5.
There are
7 counters
in each row.
Step 2 Write a division sentence.
number in all
21
number of groups
3
÷
The dividend
is the number
to be divided.
number in each group
7
=
The answer is
the quotient .
The divisor is the
number the dividend
is divided by.
Step 3 Write a multiplication sentence.
number of groups
3
154
Model Multiplication and Division Problems
number in each group
number in all
7
21
×
=
About It
1. Explain how you used models to show 21 ÷ 3.
2. Explain how the array shows that 21 ÷ 3 = 7 is related to
3 × 7 = 21.
3. What pattern do you notice in the number sentences?
4. How can multiplication facts be used to divide?
and Apply It
Use counters to model each problem. Then write related
division and multiplication sentences to help find the answer.
5. 12 ÷ 6
6. 18 ÷ 3
7. 25 ÷ 5
8. 15 ÷ 3
9. 16 ÷ 2
10. 24 ÷ 8
Write a related division and multiplication sentence
for each picture.
11.
12.
13.
14.
15.
E
WRITE MATH How do you know what multiplication
sentence to use to find 28 ÷ 4?
Lesson 2D Meaning of Division 155
Multi-Part
M
Mu
ult
ltilti
i-Pa
Part
rt
Lesson
L
Le
esson
n
2
Meaning of
Meaning
of Division
Division
PART
A
Main Idea
I will divide using
related multiplication
facts.
Vocabulary
V
iinverse operations
dividend
B
C
E
D
F
Inverse Operations
In the Explore Activity, you used arrays to help you
understand how division and multiplication are related.
Operations that are related are inverse operations ;, they
undo each other.
divisor
quotient
related facts
Use an Array
fact family
Get ConnectED
GLE 0306.1.4 Move
flexibly between concrete
and abstract representations
of mathematical ideas in
order to solve problems,
model mathematical ideas,
and communicate solution
strategies. GLE 0306.2.3 Relate
multiplication and division as
inverse operations.
Also addresses SPI 0306.2.8.
MUFFINS The pan of
blueberry muffins
represents an array. The
array shows 3 rows of
muffins with 4 muffins
in each row. Use the
array to write a related
multiplication and
division sentence.
Multiplication
number number number
of rows in each row in all
3
factor
×
4
= 12
factor
product
Division
number
in all
12
number
number
of rows in each row
÷
3
=
4
dividend divisor quotient
The related multiplication and division sentences show
how multiplication and division are inverse operations.
3 × 4 = 12 is the inverse of 12 ÷ 3 = 4.
156
Model Multiplication and Division Problems
A group of related facts using the same numbers is
a fact family . Each fact family follows a pattern by using
the same numbers.
When a number is
divided in half, it is the
same as dividing a
number by two.
Fact Family
3, 4, and 12
Fact Family
7 and 49
3 × 4 = 12
4 × 3 = 12
12 ÷ 3 = 4
12 ÷ 4 = 3
7 × 7 = 49
49 ÷ 7 = 7
Write a Fact Family
Use the fact family 3, 6, and 18 to write four related
multiplication and division sentences.
3 × 6 = 18
6
6 × 3 = 18
3
18 ÷ 3 = 6
18 ÷ 6 = 3
The pattern shows that 3, 6, and 18 are used in
each number sentence.
Use the array
arra to complete each pair of number
n mber sentences.
sentences
1. × 5 = 15
÷3=5
2. 4 × = 24
24 ÷ = 6
Write the fact family for each set of numbers.
3. 2, 6, 12
SSee Example
E
l 1
See Example 2
4. 4, 5, 20
6. Isabella will divide 18 marbles
equally into 2 bags. Show this with a
number sentence.
5. 3, 9, 27
7.
E
TALK MATH Why are the
product and the dividend the same
in 3 × 7 = 21 and 21 ÷ 3 = 7?
Lesson 2E Meaning of Division 157
EXTRA
%
)# E
# T4 IC
!C
2A
0R
P
Begins on page EP2.
Use the
h array to complete
l
each
h pair off number
b sentences.
See Example 1
8. × 2 = 8
÷4=2
9. 2 × = 4
4÷=2
10. × 2 = 14
÷2=7
11. 4 × = 20
20 ÷ = 4
Write the fact family for each set of numbers.
See Example 2
12. 2, 3, 6
13. 2, 7, 14
14. 4, 16
15. 4, 8, 32
16. 4, 3, 12
17. 4, 7, 28
Identify the pattern by writing the set of numbers for each
fact family.
18. 5 × 9 = 45
9 × 5 = 45
45 ÷ 5 = 9
45 ÷ 9 = 5
19. 7 × 2 = 14
2 × 7 = 14
14 ÷ 2 = 7
14 ÷ 7 = 2
20. 3 × 3 = 9
9÷3=3
Solve. Write a number sentence.
21. All 5 members of the Malone family
went to the movies. Their tickets cost
a total of $30. How much was each
ticket?
22. The petting zoo has 21 animals.
There are an equal number of
goats, ponies, and cows. How
many of each animal are there?
23. Measurement Mr. Thomas travels
20 miles each week to and from
work. If he works 5 days a week,
how many miles does Mr. Thomas
travel each day to go to work?
24. Stacia and her friend are each
making a bracelet. They have
18 beads to share. If they use the
same number of beads, how many
beads will each bracelet have?
158
Model Multiplication and Division Problems
25. NUMBER SENSE What multiplication fact will help
you find 27 ÷ 9?
26. WHICH ONE DOESN'T BELONG? Identify the number
sentence that does not belong. Explain.
3 × 6 = 18
27.
18 ÷ 2 = 9
18 ÷ 6 = 3
6 × 3 = 18
E
WRITE MATH Explain how multiplication facts can
help you with division facts. Give an example.
Test Practice
28. The figure below is a model for
4 × 6 = 24.
29. Which number sentence is
modeled by repeated subtraction
on the number line? (Lesson 2C)
0 1 2 3 4 5 6 7 8
Which number sentence is in the
same fact family? (Lesson 2E)
A. 4 ÷ 6 = 24
C. 24 ÷ 4 = 6
B. 24 ÷ 3 = 8
D. 24 ÷ 6 = 6
Use repeated subtraction to divide.
30. 12 ÷ 4
F. 4 ÷ 2 = 8
G. 16 ÷ 2 = 8
H. 8 ÷ 2 = 4
I. 24 ÷ 8 = 3
(Lesson 2C)
31. 18 ÷ 3
32. 28 ÷ 7
33. 25 ÷ 5
34. One frog sat on a log for 18 minutes. A second frog sat
for half as long. How long did the second frog sit
on the log? (Lesson 2B)
Multiply. Use repeated addition.
35. 8 × 3
36. 2 × 9
(Lesson 1B)
37. 3 × 10
38. 10 × 5
Lesson 2E Meaning of Division 159
Multi-Part
Lesson
2
PART
Meaning of Division
A
B
C
D
E
F
Problem-Solving Investigation
Main Idea I will choose the best strategy to solve a problem.
DENZELL: Our third-grade class will
make 3 holiday baskets to give away.
We have a total of 21 food items to
equally fill the baskets with.
YOUR MISSION: Find how many items will
fill each basket.
Understand
• You know the class will make 3 baskets.
• There are 21 items to equally fill the baskets with.
• Find the number of items that will fill each basket.
Plan
You can use models to solve the math problem.
Solve
Use counters to model
the situation.
The model shows that
21 ÷ 3 = 7.
So, the third-grade class will fill each basket with 7 items.
Check
Check by using repeated addition.
7 + 7 + 7 = 21
The answer is 21, so you know your answer is correct
and reasonable. GLE 0306
0306.1.2
1 2 Apply and adapt a variety of appropriate strategies to problem solving
solving, including estimation
estimation,
and reasonableness of the solution. Also addresses GLE 0306.2.4.
160
1
Model Multiplication and Division Problems
EXTRA
%
)# E
# T4 IC
!C
2A
0R
P
Begins on page EP2.
• Make a table.
• Draw a picture.
• Make a model.
6. Geometry Blaine built a cube
staircase. How many cubes does he
need to build 6 steps?
Use any strategy to solve each
problem.
1. Solana buys the following items. She
gives the cashier $20. How much
change will she receive?
7. Marjorie made 48 pancakes for the
school breakfast. Elian ate some of
the pancakes, and now Marjorie
2. Dasan planted 30 tomato seeds in
his garden. Three out of every
5 seeds grew into plants. How many
tomato plants did he have?
3. Would it cost more to send 2 letters
or 3 postcards? Explain.
4. Claudia and Danielle bought paint
for their project. They chose 5 colors.
Each bottle of paint costs $3. Find
the total cost.
only has 43 pancakes. How many
pancakes did Elian eat?
8. There are 3 children in line. Cami is
right after Brock. Branden is third.
What place is each child in line?
9. One campsite has 3 tents with
5 people in each tent. Another
campsite has 3 tents with 4 people
in each. How many campers are
there in all?
10. Rachel sold 4 glasses of lemonade
for 25¢ each. How much money
did she make?
11.
5. Measurement Alfonso exercised
20 minutes yesterday. Today he is
going to exercise twice as long.
How long does Alfonso plan to
exercise today?
E
WRITE MATH Mrs. Felps read
her students one book each day for
2 weeks. If there are 5 days in each
school week, how many books did
she read in all? Explain your
reasoning.
To assess mastery of SPI 0306.2.5, see your Tennessee Assessment Book.
161
Chapter Study
Guide and Review
Be sure the following Key
Concepts
p are noted in
your Foldable.
Fo
Key Vocabulary
array
divisor
fact family
factors
Mea
nin
Mult
l ipli g of
catio
n
product
Arr
Mu ays an
ltip
lica d
tion
quotient
Vocabulary Check
Key Concepts
• Multiplication can be thought of as
repeated addition. (Lesson 1)
4 × 5 = 20
is the same as
5 + 5 + 5 + 5 = 20
• The Commutative Property of
Multiplication states that the order in
which numbers are multiplied does
not change the product. (Lesson 1)
3×2=6
2×3=6
• In the operation of division , one
number tells you how many things you
have. The other number tells you how
many equal groups to form or how
many to put in each group.
Choose the vocabulary word that
completes each sentence.
?
1. An
is an arrangement
of equal rows and equal
columns.
2. The answer to a division
?
.
problem is the
3. In the multiplication sentence
?
2 × 6 = 12, 12 is the
.
?
is a group of
4. A
related facts using the same
numbers.
(Lesson 2)
8
÷
2
=
4
• Multiplication and division are inverse
operations . They “undo” each other.
(Lesson 2)
162
Model Multiplication and Division Problems
5. In the division sentence
?
12 ÷ 3 = 4, 3 is the
?
6. Two
are multiplied
together to get a product.
.
Multi-Part Lesson Review
Lesson 1
Meaning of Multiplication
Multiplication as Repeated Addition
(Lesson 1B)
Write an addition and a multiplication
sentence for each model.
EXAMPLE 1
8.
7.
Write
i an addition
dd
sentence and a
multiplication sentence.
Multiply. Use repeated addition.
9. 4 × 6
6 + 6 + 6 = 18
10. 2 × 7
Arrays and Multiplication
(Lesson 1D)
Write two multiplication sentences
for each array.
12.
11.
EXAMPLE 2
There
h
are 3 rows of 4 muffins. How
many muffins altogether? Write two
multiplication sentences.
3 × 4 = 12
Use Multiplication to Compare
4 × 3 = 12
(Lesson 1E)
Use the model and bar diagram to
compare. Then write a multiplication
sentence.
13. 4 times as many
3 × 6 = 18
EXAMPLE 3
Timmy
downloaded
5 songs. His
i
d
sister downloaded three times as
many. How many songs did she
download?
3 times as many songs
14. twice as much money
5 songs
$5
? songs
5
?$
$5
$5
5
5
3 × 5 = 15 songs
Chapter Study Guide and Review 163
Chapter Study Guide and Review
Use Multiplication to Find Combinations
Make a picture or tree diagram to
find all the possible combinations.
Write a multiplication sentence.
15. Find the possible combinations of
one yogurt and one topping.
Yogurt
(Lesson 1F)
EXAMPLE 4
Jane is
i buying
b i a bike. List all of
Jane’s bike and color choices.
Bike Choices
Toppings
Strawberry
Granola
Peach
Strawberries
Road Bike
Mountain Bike
Vanilla
16. Ian can buy a skateboard or a
surfboard. Both come in yellow,
orange, green, or red. Find Ian’s
choices.
green silver red
Color Choices
Jane has 6 choices of bikes.
Check
2
×
types of
bikes
Problem-Solving Strategy: Make a Table
3
colors
=
6
combinations
(Lesson 1G)
Solve. Use the make a table strategy.
EXAMPLE 5
17. Toya finishes reading a book every
3 days. How many books had she
read after 21 days?
For the
h first
fi
day of school, Maggie
can wear a yellow or a blue shirt.
With her shirt she can wear a skirt,
shorts, pants, or capris. How many
possible outfits does Maggie have
to wear?
18. Algebra Polly is putting balloons
in bunches. If Polly keeps her
pattern going, how many balloons
will be in the sixth bunch?
Organize the information in a table.
Yellow shirt
Blue shirt
Skirt
yellow shirt, skirt
blue shirt, skirt
Balloon Bunches
Bunch
Number
First
3
Shorts
yellow shirt, shorts
blue shirt, shorts
Second
5
Pants
yellow shirt, pants
blue shirt, pants
Third
7
Capris
yellow shirt, capris
blue shirt, capris
Maggie has 8 possible outfits to wear.
164
Model Multiplication and Division Problems
Lesson 2
Meaning of Division
Division as Equal Sharing
(Lesson 2B)
Use counters to model the total.
Divide to find the number in
each group.
19. 12 counters
3 equal groups
÷
EXAMPLE 6
Coach
divided 27 players
h Shelton
h l
into 3 equal-sized teams. How many
players are on each team?
=
20. 15 counters
5 equal groups
÷
=
21. 8 counters
2 equal groups
÷
Team 1
players
Team 2
players
Team 3
players
27 ÷ 3 = 9 players on each team.
=
Relate Division and Subtraction
(Lesson 2C)
Use repeated subtraction to divide.
EXAMPLE 7
22.
Find 8 ÷ 2.
0
2
4
6
8
10 12
12 ÷ 4
One Way: Number Line
4
3
2
1
23.
0 1 2 3 4 5 6 7 8
0 2 4 6 8 10 12 14 16
16 ÷ 8
24. 6 ÷ 2
25. 27 ÷ 3
26. 14 ÷ 2
27. 4 ÷ 2
28. Chang has 15 frogs in his pond.
If he catches 3 a day, how many
days will it take him to catch all of
the frogs?
Start at 8. Count back by 2s until you
reach 0. Count how many times you
subtracted. So, 8 ÷ 2 = 4.
Another Way: Repeated Subtraction
1
2
8
6
2
2
−−− −−−
4
6
So, 8 ÷ 2 = 4.
3
4
4
2
2
2
−−− −−−
2
0
Chapter Study Guide and Review 165
Chapter Study Guide and Review
Inverse Operations
(Lesson 2E)
Use the array to complete each pair
of number sentences.
29.
30.
×2=8
÷4=2
EXAMPLE 8
Show
division
and multiplication as
h
di
i i
inverse operations. Write related
multiplication and division facts.
× 6 = 24
÷4=6
3 × 7 = 21
Write the fact family for each set
Write the fact family for the array.
of numbers.
31. 6, 7, 42
32. 8, 4, 2
33. 5, 4, 20
34. 4, 9, 36
21 ÷ 3 = 7
3 × 4 = 12
4 × 3 = 12
12 ÷ 3 = 4
12 ÷ 4 = 3
Problem-Solving Investigation: Choose a Strategy
(Lesson 2F)
Use any strategy to solve.
EXAMPLE 9
35. Algebra One day, Juana
received 2 gifts. The next day
she received 4 gifts. The third day
she received 6 gifts. If the pattern
continues, how many gifts will she
receive on the 6th day? How many
gifts did she receive altogether?
Miriam
ii
bought
b
3 toys. Jamil bought
2 more toys than Miriam. How many
toys did they buy?
36. You need to read 5 books a month
during the school year. The school
year is from August to May. How
many books will you read in
a year?
166
You can model the problem with
counters.
Miriam
3
Jamil
+
3
+
2 = 8
So, Miriam and Jamil bought 8 toys.
Model Multiplication and Division Problems
Practice
Chapter Test
Tell whether each statement is true
or false.
1. The Commutative Property of
Multiplication says the order in
which numbers are multiplied can
change the product.
10. If Kathryn chooses 1 item from
each menu, how many different
combinations could she make?
Make a table to explain.
Toast
Bagel
Muffin
2. Repeated subtraction can help you
solve a division problem.
Multiply. Use repeated addition.
3. 3 × 6
4. 3 × 9
5. 5 × 5
6. 6 × 4
Yogurt
Bacon
Cereal
Use the bar diagram to compare.
Then write a multiplication sentence.
11. 4 times as many stars
5 stars
? stars
5
Write two multiplication sentences
for each array.
8.
7.
5
5
5
12. 5 times as many spoons
2 spoons
? spoons
2
9. MULTIPLE CHOICE A cook boils
16 potatoes in 2 pots. Each pot
has the same number of potatoes.
Which number sentence shows how
many potatoes are in each pot?
2
2
2
2
13. MULTIPLE CHOICE Benita did this
division problem.
15 ÷ 5 = 3
Which problem could she do to
check her answer?
F. 5 + 3
H. 5 × 3
G. 3 - 5
I. 3 ÷ 5
A. 16 + 2 = 18
B. 16 - 2 = 18
C. 16 × 2 = 32
D. 16 ÷ 2 = 8
14.
E
WRITE MATH Can 6 roses
be divided equally between
2 vases? Explain.
Practice Chapter Test
167
Test Practice
Tylerr rides his bicycle 2 miles a day. He rides
4 days a week. How many miles does Tyler
ride in a week?
A. 4 miles
C. 8 miles
B. 6 miles
D. 10 miles
Make an array to help
you find the product to a
multiplication problem.
Read the Question
You need to find how many miles Tyler
rides his bike in a week.
2
Solve the Question
You can draw an array to find 4 × 2.
4
So, Tyler rides 8 miles a week.
The answer is C.
4×2=8
Read each question. Then fill in the correct answer on the answer
sheet provided by your teacher or on a separate sheet of paper.
1. The model shows 12 ÷ 3 = 4.
2. Which number sentence is modeled
by the figure below?
Which number sentence below is from
the same family of facts?
A. 3 + 4 = 7
C. 7 - 4 = 3
F. 5 × 8 = 40
H. 8 + 8 + 8 = 24
B. 3 × 4 = 12
D. 6 ÷ 3 = 2
G. 4 × 8 = 32
I. 3 × 8 = 24
168
Model Multiplication and Division Problems
3. Bella arranged 24 shells in
6 equal-size groups. How many
were in each group?
6. Evita swims 5 times a week for
2 hours. How many hours does
Evita swim in a week?
F. 7
G. 10
H. 15
A. 3
C. 6
B. 4
D. 8
I. 25
4. Marquez has 16 baseball cards. He
puts the cards in piles of 8. How
many piles does he make?
F. 2
H. 6
G. 4
I. 8
7. Marcy ran 2 miles today. Brett ran
4 times as far as Marcy. How many
miles did Brett run?
5. Ming-Su equally divides 6 fish into
3 fish tanks. Which picture shows
Ming-Su’s fish?
A. 4
C. 8
B. 6
D. 10
8. Seth arranged a group of buttons in
rows and columns as shown.
A.
B.
C.
What operation best shows how he
arranged them?
D.
F. 6 + 4
H. 4 - 6
G. 6 ÷ 4
I. 4 × 6
NEED EXTRA HELP?
If You Missed Question . . .
1
2
3
4
5
6
7
8
Go to Chapter-Lesson . . .
3-2E
3-1D
3-2B
3-2C
3-2B
3-1B
3-1E
3-1D
For help with . . .
SPI 2.8
SPI 2.5
SPI 2.5
SPI 2.5
SPI 2.5
SPI 2.5
SPI 2.5
SPI 2.5
Test Practice 169