8.2
ALGEBRA
Associative Property
of Multiplication
?
Essential Question How can you use the Associative Property
of Multiplication to find products?
Texas Essential
Knowledge and Skills
Number and Operations—3.4.F
Recall facts to multiply up to 10 by 10 with automaticity and
recall the corresponding division facts;
3.4.K Solve one-step and two-step problems involving
multiplication and division within 100
How can you use the
Associative Proper t y of
Multiplication to f ind
products?
Algebraic Reasoning—3.5.B Represent and solve oneand two-step multiplication and division problems within 100
Also 3.4.D, 3.4.G, 3.5.D
MATHEMATICAL PROCESSES
3.1.E Create and use representations
3.1.G Display, explain, and justify mathematical ideas
and arguments
Are You Ready?
Access Prior Knowledge
Lesson Opener
Making Connections
Invite students to tell you what they know about different modes of transportation.
Explain to students that the Mole Mountain Express described in the problem is like a
taxi, shuttle, or bus.
Have you ever taken a taxi, a shuttle, or a bus? What was it like? Have you ever taken
bags with you on a trip? How many?
Use the Are You Ready? 8.2 in the
Assessment Guide to assess students’
understanding of the prerequisite skills
for this lesson.
Vocabulary
Associative Property of Multiplication
Go to Multimedia eGlossary at
thinkcentral.com
Using the Digital Lesson
You may wish to model the stated problem with students using pictures or
manipulatives, such as counters.
Learning Task
What is the problem the students are trying to solve? Connect the story to the problem.
Resources
• How many passengers go each time? (4 passengers each trip)
• How many bags does each passenger take? (2 bags per passenger)
• How could you find the number of bags delivered on one trip? (Multiply the
number of passengers, 4, by the number of bags per passenger, 2. Or, add the
number of bags per passenger, 2, 4 times.)
• What expression could you use to find the number of bags delivered on one trip?
(4 × 2 or 2 + 2 + 2 + 2)
Literacy and Mathematics
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
• Have students look for clue words in the problem that indicate that multiplication
might be used to solve it.
iTools Virtual
Manipulatives
• Have students write a short story about one of the travelers. Encourage them to
include details such as what the traveler has in his/her bag and what he/she is doing
at Mole Mountain.
Soar to Success Math
Online Intervention
Lesson 8.2 243A
Name
8.2
ALGEBRA
Connect
?
Students relate the Associative Property of Addition
to the Associative Property of Multiplication.
Have students draw a quick picture of the roller
coaster train to check their understanding of the
problem.
Use the Math Idea to remind students to complete
the operation inside the parentheses first.
Unlock
Unlock the
the Problem
Problem
Each car on the roller coaster
has 2 rows of seats. Each row has
2 seats. There are 3 cars in each
train. How many seats are on
each roller coaster train?
After students complete the problem with the new
grouping, ask the following questions:
• Does it matter which factors you multiply first?
No; whichever way I multiply the factors, the product will
still be the same.
explanation: yes; multiplying by 2 is easier for me,
so (3 × 2) × 2 was easier because I multiplied by
2 both times.
You can change the grouping with
parentheses and the product is the same.
3 × (2 × 2) = ■
(3 × 2) × 2 = ■
6
12
_
×2=_
© Houghton Mifflin Harcourt Publishing Company
4
12
3×_
=_
Math Talk
So, there are 3 cars with 4 seats in
each car.
Mathematical Processes
Explain why the products
3 × (2 × 2) and (3 × 2) × 2
are the same.
12 seats on each roller
There are _
coaster train.
Possible explanation: it does not matter how the factors
are grouped. The products will be the same.
Module 8 243
Kinesthetic / Verbal
Small Group
ELL Language Support
ELPS
Beginning: Activity 8
2.I.4, 3.G.1, 3.H.3
Intermediate: Activity 3
2.D.2, 2.E.3, 3.F.2
Strategy: Creative Grouping
Advanced: Activity 41
4.F.3, 4.F.8
Materials: square tiles
Advanced High: Activity 43
4.F.8, 4.G.2, 4.G.4
ELPS 1.A.2, 2.D.2, 2.I.4
• Partner advanced English learners with beginning English learners.
• Have beginning English learners use tiles to show (2 + 3) + 4; then
show 2 + (3 + 4).
• Have the advanced English learner help explain each model and how
each shows the Associative Property of Addition.
(2
243 Module 8
3 cars with 2 rows of 2 seats
You can use an array
to show 3 × (2 × 2).
Leveled Activities
Go to thinkcentral.com for the ELL Activity
Guide containing these leveled activities.
• Underline what you need to find.
• Describe the grouping of the seats.
Use an array.
• Was one grouping easier than the other for
finding the final answer? Explain. Possible
English Language Learners
Math Idea
Always multiply the numbers
inside the parentheses first.
The Associative Property of Multiplication states
that when the grouping of the factors is changed,
the product is the same. It is also called the
Grouping Property of Multiplication.
• What does each separate array show? Possible
answer: 2 rows of 2 seats each; 2 × 2
Use Math Talk to focus on students’ understanding
that changing the grouping of the factors does not
change the product.
How can you use the Associative Property of
Multiplication to find products?
(2 + 3) + 4 = 2 + (3 + 4)
The example uses arrays to represent the problem.
Mathematical Processes
Essential Question
Connect You have learned the Associative
Property of Addition. When the grouping of the
addends is changed, the sum stays the same.
Unlock the Problem
Math Talk
Number and
Operations—3.4.F, 3.4.K
Algebraic Reasoning—
3.5.B Also 3.4.D, 3.5.D
MATHEMATICAL PROCESSES
3.1.E
Associative Property
of Multiplication
1
3)
1
4
• Beginning English learners should ask for clarification if they do not
understand a word or phrase.
Example
Use the Commutative and Associative Properties.
You can also change the order of the factors.
The product is the same.
Example
(4 × 3) × 2 = ■
This example shows that you can change the order of
the factors and the product will be the same.
4 × (3 × 2) = ■
4 × (3 × 2) = ■
Associative Property
6 =_
24
4×_
4 × (2 × 3) = ■
Commutative Property
(4 × 2) × 3 = ■
Associative Property
• How could the Commutative and Associative
Properties be used to make (3 × 8) × 2 easier
to solve? Possible answer: change the order and the
grouping to 8 × (3 × 2) so that you can use a basic fact,
8 × 6, instead of 24 × 2 or 16 × 3.
8 ×3=_
24
_
Share
Share and
and Show
Show
1.
• How can multiplication properties help you find
products? Possible answer: you can change the grouping
Find the product of 5, 2, and 3. Write another way to
group the factors. Is the product the same? Why?
or the order of the factors to find products of facts you
know.
(5 × 2) × 3 = 30; Possible answer: I can regroup the factors as 5 × (2 × 3) = 30;
the product is the same because the grouping does not change the product.
Share and Show
Write another way to group the factors. Then find
the product. Possible groupings are given for 2–5.
4.
(2 × 1) × 7
3.
3 × (3 × 4)
2 × (1 × 7)
(3 × 3) × 4
14
36
5 × (2 × 5)
5.
3 × (2 × 6)
(5 × 2) × 5
(3 × 2) × 6
50
36
Algebra Find the unknown factor.
6.
9.
4 ) = 56
7 × (2 × _
3 )
42 = 7 × (2 × _
7.
10.
Use the checked exercises for Quick Check. Students
should show their answers for the Quick Check on the
MathBoard.
5 )×6
60 = (2 × _
8.
1 ) = 40
8 × (5 × _
11.
8 × (2 × 2) = 32
_
0 × (25 × 1)
0=_
244
3
2
Quick Check
1
© Houghton Mifflin Harcourt Publishing Company
2.
The first problem connects to the learning model.
Have students use the MathBoard to explain their
thinking.
IF
THEN
a student misses the checked exercises
Differentiate Instruction with
RtI Tier 1 Lesson 37
Problems
Exercises 6–11 require students to extend their
thinking to find an unknown factor when 2 factors
and a product are given. Remind students to apply
the Associative Property to help solve these problems.
Enrich
Visual
Individual
• Challenge students to fill in the unknown numbers so that the product
of the numbers in each row and each column equal 24.
2
6
2
3
4
2
4
1
6
• Have students create their own grids in which the numbers in each
row and each column have the same product.
Go to Go to thinkcentral.com for additional enrichment
activities in the Enrich Activity Guide.
Lesson 8.2
244
Name
Problem
Problem Solving
Solving
Problem Solving
Use the graph for 12–13.
12.
36 seats; Check students’ drawings.
13.
Problem 13 requires students to interpret a graph to
solve this multistep problem involving subtraction,
multiplication, and addition.
In Problem 14, students use the Commutative and
Associative Properties and find that changing the order
or grouping of factors does not change the product.
14.
Have students compare and contrast the
Commutative and Associative Properties of
Multiplication. They should describe how they are
the same and how they are different.
Example 3 × (2 × 6) = 3 × 12
Roller Coaster
Write Math
Sense or Nonsense? Each week,
Kelly works 2 days for 4 hours each day and
earns $5 an hour. Len works 5 days for 2 hours
each day and earns $4 an hour. Kelly says
they both earn the same amount. Does this
statement make sense? Explain.
Show Your Work
Yes; they both earn $40. (2 × 4) × $5 = $40 and
(5 × 2) × $4 = $40
15.
© Houghton Mifflin Harcourt Publishing Company
COMMON ERRORS
C
Multi-Step A Kingda Ka
train has 4 seats per car, but the last car
has only 2 seats. How many seats are
on one Kingda Ka train?
18 seats
Go Deeper
E
Error
Students may not change the grouping so
they can work with a fact they know.
7
6
5
4
3
2
1
0
Ki
ng
da
Ka
St
ee
lF
or
ce
M
r.
Fr
ee
ze
Problems
Students should be able to verbalize the difference:
with the Commutative Property, the order of the
factors changes; with the Associative Property, the
grouping changes while the order of the factors
stays the same. They should also recognize that the
product stays the same when applying either the
Commutative or Associative Property.
Roller Coasters
Multi-Step Each car on the Steel
Force train has 3 rows with 2 seats
in each row. How many seats are
on the train? Draw a quick picture.
Cars per Train
Have students read problem 12 and discuss what they
need to find. Students have to read a graph and draw
a picture to help solve a multiplication problem with
three factors.
Representations What number sentence
does this array represent? Write another way to
group the factors.
4 × (3 × 2) = 24; Possible answer: (4 × 3) × 2 = 24
Module 8 • Lesson 2 245
Springboard to Learning Have students look
for factors that have a 1-digit product. These
combinations will allow students to find the
product of another fact.
3
RtI Tier 1 Lesson 37
2
1
Enrich 37
Name
Name
M
Math
on the Spot
Video Tutor
V
Math on the Spot videos are in the
Interactive Student Edition and at
thinkcentral.com.
245 Module 8
Enrich 37
1
Algebra • Associative Property of
Multiplication
3.4.F, 3.4.K
Matching Factors and Products
OBJECTIVE Use the Associative Property of Multiplication to multiply with three factors.
You can use the Associative Property of Multiplication
to multiply with 3 factors. If you change the grouping of factors,
the product remains the same.
Complete the number sentence in Column A. Then circle
the correct product in Column B. In Column C, use 3
factors to write a number sentence for the product in
Column B that is not circled.
Find 4 × (3 × 1).
Find (4 × 3) × 1.
Possible answers for Column C are given.
Step 1 Start inside the parentheses.
Make 3 groups of 1 counter.
Step 1 Start inside the parentheses.
Make 4 groups of 3 counters.
(3 × 1)
(4 × 3)
Step 2 Multiply by 4, the number
outside the parentheses. Make
4 groups of the counters in Step 1.
Step 2 Multiply by 1, the number
outside the parentheses. Make
1 group of the counters in Step 1.
4 × (3 × 1)
(4 × 3) × 1
Step 3 Count the total number of
counters. 12 counters
© Houghton Mifflin Harcourt Publishing Company
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.
LESSON
37
Step 3 Count the total number of
counters. 12 counters
So, 4 × (3 × 1) = 12 and (4 × 3) × 1 = 12.
Column A
1. (2 × 3) × 2
2. 2 × (4 × 2)
18
18
14
2.
7 3 (2 3 3) 5
42
42
16
(2 3 4) 3 2 5 16
3.
8 3 (2 3 2) 5
32
48
32
8 3 (2 3 3) 5 48
4.
(1 3 2) 3 7 5
14
14
36
(2 3 3) 3 6 5 36
5.
1 3 (3 3 2) 5
6
6
5
(1 3 1) 3 5 5 5
6.
3 3 (8 3 1) 5
24
12
24
(3 3 2) 3 2 5 12
7.
(3 3 2) 3 6 5
36
36
40
(2 3 5) 3 4 5 40
8.
(3 3 3) 3 5 5
45
54
45
(3 3 3) 3 6 5 54
9.
9 3 (3 3 1) 5
27
27
35
(7 3 1) 3 5 5 35
10.
(7 3 1) 3 4 5
28
60
28
6 3 (5 3 2) 5 60
3. 2 × (3 × 1)
2 × (3 × 2); 12 (2 × 4) × 2; 16 (2 × 3) × 1; 6
4. 5 × (7 × 1)
5. 8 × (4 × 1)
(2 3 1) 3 7 5 14
6. 2 × (2 × 6)
(5 × 7) × 1; 35 (8 × 4) × 1; 32 (2 × 2) × 6; 24
Number and Operations
Column C
(9 3 1) 3 2 5
Possible groupings are given.
Write another way to group the factors. Then find the product.
Column B
1.
73
Explain how you decided which factors to group in Column C.
11.
Possible answer: I grouped factors together
to use multiplication facts I know.
Enrich
© Houghton Mifflin Harcourt Publishing Company
E37
Mathematical Processes
Model ¥ Reason ¥ Communicate
Daily
Daily Assessment
Assessment Task
Task
3
Daily Assessment Task
Fill in the bubble for the correct answer choice. You can
use an array, area model, or a strip diagram to represent
and solve.
16.
B
17.
C
36
12
D
9
B
$64
C
$48
$40
D
$32
NO
7
C
10
B
1
D
5
•
Soar to Success Math
Warm-Up 12.50
YES
•
•
Enrich 37
Homework and Practice
Lesson 8.2
Multi-Step Mrs. Lee has 25 baskets of books. She wants
to put them on the 2 bookcases in her classroom. Each
bookcase has 4 shelves. Each shelf can hold 3 baskets.
How many baskets will be left over?
A
THEN
IF
Emma helps at a farm 2 days each week. She works
4 hours each day and earns $8 for each hour. How
much money does Emma earn each week?
A
18.
18
1
Can students use the Associative Property of
Multiplication to find products?
Michael plays a fishing game at the school carnival. He
plays 4 times and catches 3 fish each time. For each fish
Michael catches, he wins a set of 3 toy planes. How many
toy planes does Michael win?
A
2
TEXAS Test Prep Coach
Test Prep Coach helps teachers to identify common
errors that students can make.
In the Test Prep exercise, if students selected:
A They multiplied 3 × 3, then added 2.
B They only multiplied 3 × 3.
TEXAS Test Prep
C They added 2 + 3 + 3.
Mark has 2 rows of 3 car models on each of his 3 shelves.
How many car models does he have?
A
11
B
9
C
8
D
18
© Houghton Mifflin Harcourt Publishing Company
19.
246
?
Essential Question
Write
Math
How can you use the Associative Property of
Multiplication to find products? Possible answer:
I can change the way the factors are grouped to work
with numbers I know, and still get the same answer.
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 8.2
246
5
8.2
ALGEBRA
Number and Operations—3.4.F, 3.4.K
Algebraic Reasoning—3.5.B Also 3.4.D, 3.5.D
MATHEMATICAL PROCESSES 3.1.E
Name
8.
Associative Property of Multiplication
Write another way to group the factors. Then find the product.
1.
3.
(3 × 2) × 5
2.
Possible groupings
are given for 1–4.
6 × (2 × 2)
10
B
30
C
20
30
24
D
25
4.
2 × (3 × 6)
40
36
Mario’s sticker books have 4 rows of
3 stickers in each row. How many
stickers are in Mario’s books?
Draw a picture.
24 stickers;
Check students’ drawings.
10.
Sticker Books
Name
Number of Books
Mario
Zeff, Maggie, and Jani each make
2 kites. Each kite has 4 tails. How
many tails are there on all of the
kites?
A
8
B
24
C
12
D
18
A
$42
B
$21
C
$12
D
$35
11.
There are 6 boxes of paints. Each box
has 3 rows of 2 paint cans. Which
shows how many paint cans in all?
A
6 + (3 × 2) = 12
B
(6 × 3) + 2 = 20
C
(6 × 3) × 2 = 18
D
6 × (3 × 2) = 36
Pang
Key: Each
= 1 sticker book
12.
Multi-Step Gwen has 30 action
figures. She puts them in 2 boxes.
Each box holds 3 rows of 4 action
figures. How many action figures
will be left over?
Pang’s sticker books have 3 rows
of 3 stickers. He uses 5 stickers.
31 stickers
How many stickers are still in the sticker books? ______
7.
Tamisha’s sticker books have 3 rows of 3 stickers in
each row. If Mario’s sticker books have 4 rows of 3
stickers in each row, who has more stickers? Explain.
A
24
B
6
Tamisha has 3 × (3 × 3) = 27; Mario has 2 × (4 × 3) = 24; 27 > 24,
C
8
D
21
so Tamisha has more stickers.
Module 8 • Lesson 2
Homework and Practice
Use the Homework and Practice pages to provide
students with more practice on the concepts and
skills of this lesson.
Module 8
Ana works at the library 3 days each
week. She works for 2 hours each day
and earns $7 each hour. How much
money does Ana earn each week?
Tamisha
6.
247-248
9.
(2 × 3) × 6
(5 × 2) × 4
Use the pictograph for 5–7.
© Houghton Mifflin Harcourt Publishing Company
A
(6 × 2) × 2
Problem
Problem Solving
Solving
5.
Malcolm put stamps on 5 pages
of an album. He puts 3 rows of
2 stamps on each page. How many
stamps does Malcolm have in his
album?
3 × (2 × 5)
5 × (2 × 4)
TEXAS Test Prep
Lesson
Lesson Check
Check
247
248
13.
Multi-Step Reggie has 65 quilt
patches. He makes 2 quilts with
5 rows of 6 patches in each quilt. How
many quilt patches will be left over?
A
43
B
8
C
5
D
52
© Houghton Mifflin Harcourt Publishing Company
Ho mewo rk
and Practice