6.5
Commutative Property of
Multiplication
?
Essential Question How can you use the Commutative Property of
Multiplication to find products?
Texas Essential
Knowledge and Skills
Number and Operations—3.4.E
Represent multiplication facts by using a variety of approaches
such as equal-sized groups and arrays
How can you use the
Commutative Proper t y
of Multiplication to f ind
products?
3.4.K Solve one-step and two-step problems involving
multiplication within 100 using strategies based on objects;
pictorial models, including arrays and equal groups;
properties of operations; or recall of facts
Algebraic Reasoning—3.5.B Represent and solve oneand two-step multiplication problems within 100 using arrays
and equations Also 3.4.D
MATHEMATICAL PROCESSES
3.1.C Select tools, technology, and techniques
3.1.E Create and use representations
Are You Ready?
Access Prior Knowledge
Lesson Opener
Making Connections
Engage students to tell you what they know about arrays and multiplication.
What are factors? (numbers that are multiplied) What is a product? (the answer to a
multiplication problem) What is an array? (Possible answer: objects arranged in rows
with the same number of objects in each row) What multiplication sentence can you
write for an array? (Possible answer: the number of rows times the number of objects
in each row is the total number of objects in the array.)
Use the Are You Ready? 6.5 in the
Assessment Guide to assess students’
understanding of the prerequisite skills
for this lesson.
Vocabulary
Commutative Property of
Multiplication
Go to Multimedia eGlossary at
thinkcentral.com
Using the Digital Lesson
Materials
Have students share what they know about cactus plants. Where do they usually live?
What are the characteristics of a cactus? Does it have leaves? Flowers?
square tiles, MathBoard
Learning Task
What is the problem the students are trying to solve? Connect the story to the problem.
Ask the following questions.
• How many prickly pear cacti plants are in the garden? (18)
• How are the 18 plants arranged? (in 3 equal rows of 6)
• What addition sentence is equivalent to 3 × 6 = 18? (6 + 6 + 6)
• What are other pairs of factors of 18 besides 3 and 6? (1 and 18, 2 and 9)
Literacy and Mathematics
• Have students work in pairs. Give each pair a set of 18 tiles. Have them model the
array in the problem to show 3 rows of 6 plants. Have them explain the relationship
between the number of rows and objects in each row and the multiplication
sentence 3 × 6 = 18, using the proper vocabulary: factors, product, and array.
• Have students work in pairs and arrange 18 tiles in 2 alternating rows of 5 and 4 tiles.
Then, ask them to explain to each other why this arrangement is not an array.
Resources
For the student
For the teacher
Interactive
Student Edition
provides students
with an interactive learning
environment!
Digital Management
Center organizes program
resources by TEKS!
eTeacher
Edition
Math on the Spot
Video Tutor
Online Assessment
System
iTools Virtual
Manipulatives
Soar to Success Math
Online Intervention
Lesson 6.5 197A
Name
6.5
ALGEBRA
Unlock the Problem
?
Be sure students understand they are finding
different arrangements for 15 boxes. Students should
also note that there needs to be the same number of
boxes in each row.
Essential Question
Algebraic Reasoning—3.5.B
Also 3.4.D
MATHEMATICAL PROCESSES
3.1.C, 3.1.E
How can you use the Commutative Property of
Multiplication to find products?
Hands
On
Unlock
Unlock the
the Problem
Problem
Dave works at the Bird Store. He arranges 15 boxes
of birdseed in rows on the shelf. What are two
ways he can arrange the boxes in equal rows?
• Why did you only circle the 15 in the problem and
not the two? Possible answer: the number two tells me
how many ways I need to arrange the tiles. The number
15 tells the total number of tiles, so it will be the product.
Activity
Number and
Operations—3.4.E, 3.4.K
Commutative Property
of Multiplication
• Circle the number that
is the product.
Activity Make an array.
Materials ■ square tiles ■ MathBoard
Arrange 15 tiles in 5 equal rows.
Draw a quick picture of your array. Check students’ drawings.
Hands
On
Help students work through the problem.
• How can you arrange the tiles in 5 rows? Possible
answer: I can put 1 tile in each row and keep adding one
more tile to each row until there are no tiles left.
3 tiles
How many tiles are in each row? __
• If you turn the array so the rows become the
columns, what multiplication sentence would you
write to represent it? 3 × 5 = 15
5 × 3 = 15
What multiplication sentence does your array show? ___
Suppose Dave arranges the boxes in 3 equal rows.
Draw a quick picture of your array.
Check students’ drawings.
• Is there any other way you can arrange the
15 tiles in an array that has more than 1 row?
Explain. No; possible explanation: the only numbers that
make equal rows for 15 are 3 and 5. If you put any other
number of tiles in a row, the rows are not equal.
Mathematical Processes
Use Math Talk to focus on students’ understanding
that changing the order of the factors does not
change the product.
English Language Learners
3 × 5 = 15
What multiplication sentence does your array show? ___
© Houghton Mifflin Harcourt Publishing Company
Math Talk
5 tiles
How many tiles are in each row? __
So, two ways Dave can arrange the 15 boxes are
5 .
5 rows of 3 or in 3 rows of _
in _
Possible answer: it is like turning the array around.
You do not change the total when you turn it around.
ELL Language Support
Leveled Activities
ELPS
Beginning: Activity 20
1.A.1, 3.G.2, 4.C.3
Intermediate: Activity 40
4.F.6, 4.G.2, 4.G.4
Strategy: Rephrase
Advanced: Activity 41
4.F.3, 4.F.8
Materials: square tiles, paper, color pencils
Advanced High: Activity 59
1.F, 3.E, 3.H.3, 4.C.3
thinkcentral.com for the ELL Activity
Guide containing these leveled activities.
197 Module 6
Math Talk
Mathematical Processes
Why do 5 rows of 3 and
3 rows of 5 both equal
the same number?
Module 6 197
Visual / Linguistic
Small Group
ELPS 2.C.4, 2.E.3, 3.D.2
• Students can demonstrate an understanding of the Commutative
Property of Multiplication by rephrasing the definition in their own
words.
• Read aloud the definition of Commutative Property of Multiplication.
• Have students rephrase it in their own words or represent it in
drawings or models.
• If students need help verbalizing the definition,
provide this sentence starter: The Commutative
Property of Multiplication says that if 3 × 2 = 6,
then _. 2 × 3 = 6
Multiplication Property The Commutative Property
of Multiplication states that when you change the
order of the factors, the product stays the same. You can
think of it as the Order Property of Multiplication.
6
3 =_
2×_
Multiplication Property
Have a volunteer read the statement about the
Commutative Property of Multiplication at the top of
page 198. Then work with students through the steps
that show 2 × 3 = 3 × 2.
2 =_
6
3×_
• How could you use a fact like 2 × 9 = 18 and the
Commutative Property of Multiplication to find
9 × 2? I know that 2 × 9 = 9 × 2, so I know that
9 × 2 = 18.
Math Idea
Facts that show the Commutative
Property of Multiplication have the
same factors in a different order.
Share
Share and
and Show
Show
1.
Discuss the Math Idea and be sure students
understand the meaning of the Commutative
Property of Multiplication, or Order Property of
Multiplication.
2 × 3 = 6 and 3 × 2 = 6
2 .
3 =3×_
So, 2 × _
Write a multiplication sentence for the array.
• What is the related fact for 5 × 4? 4 × 5
In 2 × 6, the 2 tells the number
of rows. In 6 × 2, the 2 tells
the number in each row.
• How many different multiplication sentences can
be made from the numbers 4, 4, and 16? Explain.
Math Talk
One; the factors are the same, so switching the order of
the factors gives the same sentence: 4 × 4 = 16.
Mathematical Processes
Explain what the
factor 2 means in each
multiplication sentence.
2 × 6 = 12
__
Share and Show
6 × 2 = 12
__
The first problem connects to the learning model.
Have students use the MathBoard to explain their
thinking.
Write a multiplication sentence for the model. Then
use the Commutative Property of Multiplication to
write a related multiplication sentence.
4.
2 ×_
4 =_
8
_
5 ×_
3 =_
15
_
4 ×_
5 =_
20
_
4 ×_
2 =_
8
_
3 ×_
5 =_
15
_
5 ×_
4 =_
20
_
198
© Houghton Mifflin Harcourt Publishing Company
3.
2.
Math Talk
Mathematical Processes
Use Math Talk to focus on students’ understanding
of how an array can be used to show a product.
Use the checked exercises for Quick Check. Students
should show their answers for the Quick Check on the
MathBoard.
3
2
Quick Check
1
Enrich
Logical / Mathematical
Small Group
• Write the following statement on the board, and ask students to find
the missing factor.
If 3 × 4 = 12, then 4 × ? = 12. 3
• Have students create their own If, then statements using the
Commutative Property of Multiplication.
• Tell students to leave question marks in their statements in place of
one factor or the product.
• Ask students to write their statements on the board and then have the
class solve them.
Go to Go to thinkcentral.com for additional enrichment
activities in the Enrich Activity Guide.
IF
THEN
a student misses the checked exercises
Differentiate Instruction with
RtI Tier 1 Lesson 30
Lesson 6.5
198
Name
Problem
Problem Solving
Solving
Problem Solving
5.
Problems
Problems 6–11 require students to apply what
they learned about the Commutative Property
of Multiplication to find unknown factors. Guide
students to see that Problems 6 and 11 can each be
solved simply by changing the order of the factors,
using the Commutative Property of Multiplication.
Both properties state that you can do the operation in any order and get the
same answer.
Algebra Write the unknown factor.
Problem 13 requires students to use more than one
step to solve the problem. Students may draw arrays
to help them solve the problem.
6.
7 ×3
3×7=_
7.
9.
6 =4×9
6×_
10.
12.
Ask students why they might need to use a related
fact to solve a multiplication problem.
• If you know that 9 × 4 = 36, what other fact do
you know? Explain. 4 × 9 = 36. Possible answer: the
This array shows 3 × 6 = 18.
Springboard to Learning Have students circle the
first factor and count the rows to check that the
number of rows is the same as the circled factor.
Have students draw a rectangle around the second
factor and a rectangle around the top row. Then
have them count the number in the row and check
that it is the same as the second factor.
13.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (tr) ©Houghton Mifflin Harcourt
Example
5
5×8=8×_
59 bird stickers
14.
Write Math
Write two different word problems about
12 birds to show 2 × 6 and 6 × 2. Solve each problem.
Possible problems: Tim drew 2 bird cages with 6 birds in
each cage. How many birds did he draw? 12 birds; Lisa drew
6 bird cages with 2 birds in each cage. How many birds did
she draw? 12 birds.
Module 6 • Lesson 5 199
RtI Tier 1 Lesson 30
1
M
Math
on the Spot
Video Tutor
V
199 Module 6
11.
Multi-Step There are 4 rows of
6 bird stickers in Don’s sticker album. There
are 7 rows of 5 bird stickers in Lindsey’s album.
How many bird stickers do they have in all?
2
Enrich 30
Name
Name
LESSON
30
Enrich 30
1
Algebra • Commutative Property
of Multiplication
3.4.E, 3.4.K
Multiplication by Arrangement
OBJECTIVE Model the Commutative Property of Multiplication and use it to find products.
Solve.
The Commutative Property of Multiplication states that when you
change the order of the factors, the product stays the same.
There are 4 rows of 5 tiles.
There are 5 rows of 4 tiles.
Think: 4 equal groups of 5
Think: 5 equal groups of 4
5 + 5 + 5 + 5 = 20
4 + 4 + 4 + 4 + 4 = 20
Multiply. 4 × 5 = 20
Multiply. 5 × 4 = 20
1.
Sara made an array with 10 tiles.
The array had 2 rows. How many
tiles were in each row?
3.
Leslie wants to display 12 seashells
in equal rows. She starts to draw
this array.
2.
Kelly put 15 jars of spices in
the cabinet. There were 5 jars
in each row. How many rows
did Kelly make?
4.
This array shows how José displays
the 18 rocks in his collection.
5 tiles
The factors are 4 and 5. The product is 20.
Write a multiplication sentence for the array.
Possible sentences are shown.
1.
2.
3 × 7 = 21
© Houghton Mifflin Harcourt Publishing Company
Math on the Spot videos are in the
Interactive Student Edition and at
thinkcentral.com.
2 ×9
3×6=_
Jenna used pinecones to make 18 peanut
butter bird feeders. She hung the same
number of feeders in each of 6 trees. Draw
an array to show how many feeders she
put in each tree. Check students’ drawings.
3
Through the Math on the Spot Video Tutor,
students will be guided through an interactive
solving of this type of H.O.T. problem. Use this
video to also help students solve the H.O.T.
problem in the Interactive Student Edition. With
these videos and the H.O.T. problems, students
will build skills needed in the TEXAS assessment.
3 ×8=4×6
_
8.
3 bird feeders in each tree.
She put _
order of the factors does not change the product.
COMMON ERRORS
C
2
4 × 5 = 10 × _
Problem
Problem Solving
Solving
Go Deeper
E
Error
Students write an incorrect multiplication fact
for an array.
Write Math
How is the Commutative Property of Multiplication
like the Commutative Property of Addition?
3.
2 × 5 = 10
1×8=8
Write a multiplication sentence for the model. Then use the
Commutative Property of Multiplication to write a related
multiplication sentence.
What are two ways Leslie can
complete the array?
What are three other ways José can
display his rocks in equal rows?
Draw 2 more circles in
Possible answer: he can
each row to show 2 rows
make 6 rows of 3 rocks,
of 6 or draw another
2 rows of 9 rocks, or
row to show 3 rows of 4
9 rows of 2 rocks.
circles.
5.
5.
4.
3 rows
6.
Mark arranges cans in 8 rows with 3 cans
in each row. Using the same total number of cans, how
many different ways can Mark make equal rows of cans?
List the ways.
8 ways: 1 row of 24 cans, 2 rows of 12 cans, 3 rows of
4
2
×
×
2
4
=
=
Number and Operations
8
8
4
3
×
×
3
4
12
= 12
=
4
6
×
×
6
4
24
= 24
8 cans, 4 rows of 6 cans, 6 rows of 4 cans, 8 rows of
=
3 cans, 12 rows of 2 cans, or 24 rows of 1 can.
59
Enrich
© Houghton Mifflin Harcourt Publishing Company
E30
Mathematical Processes
Model ¥ Reason ¥ Communicate
Daily
Daily Assessment
Assessment Task
Task
3
Fill in the bubble for the correct answer choice.
You can use objects or models to solve.
15.
Daily Assessment Task
add 4 + 5
C
multiply 5 × 4
B
add 5 + 5
D
multiply 6 × 5
1
Can students use the Commutative Property of
Multiplication to find products?
Sebastian has 4 equal rows of action figures. There are
5 action figures in each row. One way to find the number
of action figures is to multiply 4 × 5. What is another way
to find the number of action figures?
A
2
THEN
IF
NO
•
Soar to Success Math
Warm-Up 12.32
16.
17.
Use Tools Rachel has 24 stamps. She arranges them in
3 rows with 8 stamps in each row. What is another way
she could arrange the 24 stamps?
A
2 rows of 8 stamps
B
8 rows of 3 stamps
C
8 rows of 4 stamps
D
6 rows of 3 stamps
YES
•
Enrich 30
Homework and Practice
Lesson 6.5
TEXAS Test Prep Coach
Test Prep Coach helps teachers to identify common
errors that students can make.
Multi-Step Mr. Diaz sets out 6 rows of glasses with
3 glasses in each row. Mrs. Diaz sets out 3 rows
of glasses with 6 glasses in each row. How many
glasses do Mr. and Mrs. Diaz set out in all?
In the Test Prep exercise, if students selected:
A
36
C
18
A They did not choose related facts.
B
54
D
24
B They did not recognize that the order of the
factors was the same.
D They did not recognize that the operations
were different.
Communicate Which is an example of the
Commutative Property of Multiplication?
A
2×6=3×4
C
2×4=4×2
B
4×1=4×1
D
2+3=3×2
200
© Houghton Mifflin Harcourt Publishing Company
TEXAS Test Prep
18.
•
?
Essential Question
Write
Math
How can you use the Commutative Property of
Multiplication to find products?
Possible answer: when I know one fact, I can use the
Commutative Property to find the product of the related fact.
For example, if I know 2 × 7 = 14, then I know that
7 × 2 = 14.
Differentiated Centers Kit
Games
Games
Multiplication
Bingo
Students practice
multiplication
facts through 10.
Literature
Here’s What I Do
Students read about using
multiplication tables to
win a computer game
about multiplication.
Activities
Diamond Derby
Students complete
purple Activity Card
15 by practicing
multiplication facts
through 10 by 10.
Lesson 6.5
200
5
6.5
ALGEBRA
Number and Operations—3.4.E, 3.4.K
Algebraic Reasoning—3.5.B Also 3.4.D
MATHEMATICAL PROCESSES 3.1.C, 3.1.E
Name
7.
Commutative Property of Multiplication
Write a multiplication sentence for each model. Then use the
Commutative Property of Multiplication to write a related
multiplication sentence.
1.
3.
2.
3 ×_
4 =_
12
_
2 ×_
9 =_
18
_
4 ×_
3 =_
12
_
9 ×_
2 =_
18
_
Josh made 14 potholders. He put
an equal number of potholders in
each of 2 boxes. Write two related
multiplication sentences to show
how many potholders Josh made.
4.
9.
Lu Chen has 21 rocks in a collection.
He puts the same number of rocks
in each of 3 boxes. Draw an array
to show how many rocks are in
each box.
2 ×_
7 =_
14
_
TEXAS Test Prep
Lesson
Lesson Check
Check
There are 7 vases with 5 flowers in
each vase. Gina multiplies 7 × 5 to
find the number of flowers. What is
another way to find the number of
flowers?
8.
A
add 7 + 5
A
add 6 + 8
B
add 5 + 7
B
multiply 6 × 8
C
multiply 5 × 7
C
add 8 + 6
D
multiply 7 × 7
D
multiply 6 × 6
Demi has 36 toy cars. She puts them
in 4 rows with 9 in each row. What is
another way Demi can arrange the
toy cars to show the Commutative
Property of Multiplication?
10.
B
3 rows of 12
C
9 rows of 4
A
3×2=2×3
D
2 rows of 18
B
2×2=3×3
C
3×2=1×6
D
3×2=1×6
6 rows of 6
7 ×_
2 =_
14
_
11.
© Houghton Mifflin Harcourt Publishing Company
5.
Serena makes 5 belts. Each belt has
4 beads. Serena writes 5 × 4 = 20
to show the number of beads. Write
another multiplication sentence to
show the number of beads.
6.
Dikembe has 3 packs of 10 baseball
cards. He writes 3 × 10 = 30 to show
the number of cards. Write another
multiplication sentence to show the
number of baseball cards.
4 × 5 = 20
10 × 3 = 30
Module 6 • Lesson 5
Homework and Practice
Use the Homework and Practice pages to provide
students with more practice on the concepts and
skills of this lesson.
201-202
Module 6
201
202
Risa makes this model.
Which of the following can Risa
write to show the Commutative
Property of Multiplication?
A
Possible drawing:
Problem
Problem Solving
Solving
A photo album has 8 pages with
6 photos on each page. Cory
multiplies 8 × 6 to find the number
of photos. What is another way to
find the number of photos?
Multi-Step Juan puts 3 rows of
5 stickers in a book. Then he puts
5 rows of 3 stickers on the next page.
How many stickers does Juan put in
the book?
12.
Multi-Step One classroom has
4 rows of 5 chairs. Another
classroom has 5 rows of 4 chairs.
How many chairs are there in both
classrooms?
A
15
A
20
B
16
B
19
C
30
C
25
D
25
D
40
© Houghton Mifflin Harcourt Publishing Company
Ho mewo rk
and Practice