7.4
Distributive Property
ALGEBRA
?
Essential Question How can you use the Distributive Property
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 arrays, area models
How can you use the
Distributive Proper t y 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, properties of operations
Algebraic Reasoning—3.5.B Represent and solve
one- and two-step multiplication problems within 100 using
arrays, and equations
Also 3.4.D, 3.5.D
MATHEMATICAL PROCESSES
3.1.C Select tools, technology, and techniques
3.1.D Communicate mathematical ideas and reasoning
3.1.E Create and use representations
Are You Ready?
Lesson Opener
Making Connections
Invite students to tell you what they know about multiplying by 1, 5, and 6.
What happens when you multiply any number by 1? (The answer is the same number.)
What are different ways to multiply by 5? (Count by 5, add 5s) What are different ways
to multiply by 6? (Count by 6, add 6s)
Access Prior Knowledge
Use the Are You Ready? 7.4 in the
Assessment Guide to assess students’
understanding of the prerequisite skills
for this lesson.
Vocabulary
Distributive Property
Using the Digital Lesson
Go to Multimedia eGlossary at
thinkcentral.com
You may wish to ask the students if they find it easier to multiply by 1 and 6 than to
multiply by 7.
Materials
Learning Task
What is the problem the students are trying to solve? Connect the story to the problem.
• How far does Doc’s neighbor drive each day for work? (7 miles)
• How many days per week does Doc’s neighbor drive to and from work? (5 days)
• What operation could you use to find the total number of miles Doc’s neighbor
drives for work per week? (Multiplication or addition)
• What multiplication expression could you use to find the total number of miles
Doc’s neighbor drives for work per week? (5 × 7)
• What are different ways to solve 5 × 7? (Count by 7s, add 7 + 7 + 7 + 7 + 7,
multiply 5 × 6 and add another 5)
Literacy and Mathematics
• Have students discuss the meaning of distribute in a larger context. Ask one
student to distribute pieces of paper to each student to demonstrate the
meaning of the word.
• Have students write a short story about why Doc wants to know how many miles his
neighbor drives.
square tiles
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 7.4 229A
Name
7.4
Unlock the Problem
?
Students should understand that they need to
multiply the number of fish by the cost of each fish
to find the total amount spent.
ALGEBRA
Number and
Operations—3.4.E, 3.4.K
Algebraic Reasoning—
3.5.B Also 3.4.D, 3.5.D
MATHEMATICAL PROCESSES
3.1.C, 3.1.D, 3.1.E
Distributive Property
Essential Question
How can you use the Distributive Property to find products?
Unlock
Unlock the
the Problem
Problem
Mark bought 6 new fish for his
aquarium. He paid $7 for each fish.
How much did he spend in all?
Have a volunteer read the definition of the
Distributive Property.
6 groups of $7
• Circle the numbers you will use to
solve the problem.
Find 6 × $7.
Activity
• Describe the groups in this problem.
You can use the Distributive
Property to solve the problem.
Work with students to make the array and
break it apart.
The Distributive Property states that
multiplying a sum by a number is the
same as multiplying each addend by the
number and then adding the products.
• How can you use the smaller arrays to find the
total? Find the total for each smaller array and add them
together.
Activity
• Does breaking apart a larger array into 2 smaller
arrays make the problem easier to solve? Explain.
Materials ■ square tiles
Make an array with tiles to show
6 rows of 7.
Yes; Possible explanation: I can use basic facts I already
know to find the total for each smaller array and then add
the totals to find the answer.
Remember
sum—the answer to an addition problem
addends—the numbers being added
Hands
On
Break apart the array to make two smaller
arrays for facts you know.
• Why do you think this is called the Distributive
Property? Possible answer: because I am breaking apart
the array so 6 multiplies both 2 and 5.
6×7=
Mathematical Processes
Use Math Talk to focus on students’ understanding
of using a model to draw conclusions and explain
reasons for their conclusions. Have several students
draw their arrays on the board and write the
multiplication problems they used to find 6 × 7.
English Language Learners
© Houghton Mifflin Harcourt Publishing Company
Math Talk
■
6×5
Possible answers: 6 × (6 + 1); 6 × (4 + 3)
6×7=■
6 × 7 = 6 × (5 + 2)
Think: 7 = 5 + 2
Math Talk
6 × 7 = (6 × 5) + (6 × 2) Multiply each addend by 6.
Mathematical Processes
30 + _
12
6×7=_
Add the products.
42
6×7=_
ELL Language Support
ELPS
Beginning: Activity 36
2.E.3, 3.D.2, 3.F.2
Intermediate: Activity 22
1.A.1, 3.G.1, 4.C.1
Strategy: Identify Relationships
Advanced: Activity 57
2.C.4, 3.D.2, 3.E
Materials: square tiles
Advanced High: Activity 30
1.F, 2.I.3, 3.H.3
229 Module 7
What other ways could
you break apart the 6 × 7
array?
42 for his new fish.
So, Mark spent $ _
Leveled Activities
thinkcentral.com for the ELL Activity
Guide containing these leveled activities.
6×2
Module 7 229
Kinesthetic / Auditory
Small Group
ELPS 2.C.4, 2.E.3, 3.D.2
• Students can understand the word distributive when they connect to
the word distribute.
• Put students in groups of 3. Give 1 student 7 tiles.
• Distribute the 7 tiles among the other students. How did you distribute
the tiles? I gave 4 to Sam and 3 to Nina.
• Distribute sounds like distributive. What do you think the Distributive
Property involves? distributing multiplication
• Show students how the Distributive Property “distributes”
multiplication by showing that 6 × (2 + 5) = (6 × 2) + (6 × 5).
Share
Share and
and Show
Show
Draw a line to show how you could break apart
this 6 × 8 array into two smaller arrays for
facts you know. Possible answers given.
1.
Share and Show
The first problem connects to the learning model.
Have students use the MathBoard to explain their
thinking.
6 and _
4
• What numbers do you multiply? _
6 and _
4
_
•
24 + _
24
What numbers do you add? _
Use the checked exercises for Quick Check. Students
should show their answers for the Quick Check on the
MathBoard.
Math Talk
4 +_
4 )
6 × 8 = 6 × (_
Mathematical Processes
6 ×_
4 ) + (_
6 ×_
4 )
6 × 8 = (_
Why do you have to
add to find the total
product when you use the
Distributive Property?
24 + _
24
6×8=_
48
6×8=_
3
2
Possible answer: I broke the
larger array into two smaller
arrays, so I have to add their
products to find the total product.
Write one way to break apart the array.
Then find the product. Possible answers given.
2.
Quick Check
1
a student misses the checked exercises
IF
3.
THEN
5 × (4 + 4) or (2 + 3) × 8
4 × (4 + 3) or (2 + 2) × 7
or (2 × 8) + (3 × 8); 40
or (2 × 7) + (2 × 7); 28
Problem
Problem Solving
Solving
COMMON ERRORS
C
E
Error
When using the Distributive Property,
students may break a number into its factors
instead of its addends.
Check students’ arrays and answers.
Shade tiles to make an array that shows
a fact with 7, 8, or 9 as a factor. Write the
fact. Explain how you found the product.
Possible explanation: I shaded 9 rows of 7 tiles.
I can think 7 = 4 + 3. I can write
9 × 7 = 9 × (4 + 3); 9 × 7 = (9 × 4) + (9 × 3);
9 × 7 = 36 + 27 = 63
230
© Houghton Mifflin Harcourt Publishing Company
4.
Differentiate Instruction with
RtI Tier 1 Lesson 34
Example 6 × 9 = (6 × 3) + (6 × 3) The student
broke apart 9 into two factors, 3 and 3, instead of
two addends, such as 3 and 6.
Springboard to Learning Encourage students to
write out their addition sentences before writing
their multiplication sentences. For example: 6 ×
9 = 6 × (6 + 3) = (6 × 6) + (6 × 3). Also have
students use square tiles to reinforce the concept
of breaking apart a larger array into two smaller
arrays.
If students complete the checked exercises
correctly, they may continue with Problem 4.
Enrich
Logical / Mathematical
Individual
• Write the following problem on the board.
9×6=?
• Challenge students to apply the Distributive Property, breaking apart
one factor into 3 addends. Have them show each step of the process.
Possible answers: (2 × 6) + (3 × 6) + (4 × 6) = 12 + 18 + 24 = 54
(3 × 6) + (3 × 6) + (3 × 6) = 18 + 18 + 18 = 54
• Then have students write another fact and repeat the process.
Go to Go to thinkcentral.com for additional enrichment
activities in the Enrich Activity Guide.
Lesson 7.4
230
Name
Problem
Problem Solving
Solving
Problem Solving
5.
Problems
Problem 5 requires students to analyze a statement to
find an error and justify their reasoning.
No; possible answer: Robin can find 8 × 7 by multiplying 4 × 7
and doubling it. If she finds 3 × 7 and doubles it, she is finding 6 × 7.
Go Deeper
6.
To extend their thinking, in Problem 5 students must
first recognize that there is an error through their
understanding of the Distributive Property. Then they
should explain why Robin’s way isn’t correct. Their
explanation should include the correct answer.
8 × $9 = ■
8 × 9 = (4 × 9) + (5 × 9)
8 × 9 = 36 + 45
8 × 9 = 81
Possible explanation: Brandon broke apart the
If students have trouble finding the error that was
made by Brandon, have students solve the problem
and then compare their answer to his to find the
error.
factor 9 and multiplied by 9. He should have
broken apart the factor 8 or multiplied by 8.
Some students might realize that Brandon could
have used the Distributive Property to solve
(4 × 8) + (5 × 8) to get the correct answer of 72.
Have students share any possible strategies they
found to solve the problem.
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.
Multi-Step Brandon needs 8 boxes of
spinners for his fishing club. The cost of each box
is $9. How much will Brandon pay?
Look at how Brandon solved the problem.
Find and describe his error.
For Problem 6, have students read the problem and
discuss what they need to find. This problem requires
higher order thinking skills and multiple steps to find
and describe an error. Then they need to find the
correct answer, showing each step.
Use the array to help solve the problem and correct Brandon’s error.
8 × 9 = (4 + 4) × 9
4 ×_
9 ) + (_
4 ×_
9 )
8 × 9 = (_
© Houghton Mifflin Harcourt Publishing Company
M
Math
on the Spot
Video Tutor
V
Sense or Nonsense? Robin says, “I can
find 8 × 7 by multiplying 3 × 7 and doubling it.“ Does
her statement make sense? Justify your answer.
36 + _
36
8×9=_
72
8×9=_
72 for the spinners.
So, Brandon will pay $ _
3
RtI Tier 1 Lesson 34
2
1
Enrich 35
Name
Name
Math on the Spot videos are in the
Interactive Student Edition and at
thinkcentral.com.
Module 7 • Lesson 4 231
LESSON
34
3.4.E, 3.4.K
United Arrays
Draw an array for each clue. Then use the arrays
to solve each problem.
A garden has 4 rows of 7 corn stalks. How many corn stalks
in all are in the garden?
You can use the Distributive Property to break an array
into smaller arrays to help you find the answer.
1.
Craig spent $27 to buy 3
calendars. The next day, he spent
another $18 for more calendars.
Each calendar cost the same
amount. How many calendars did
Craig buy?
2.
On Monday, Mrs. Jones spent
$32 on 4 books. On Tuesday, she
spent $16 on more books. Each
book cost the same amount. How
many books did Mrs. Jones buy?
3.
Hailey spent $12 to buy 2 fish.
Her cousin spent double the
amount on fish. Each fish cost
the same amount. How many fish
do Hailey and her cousin have
altogether?
Find 4 × 7.
Step 1 Make an array to show 4 rows of 7.
© Houghton Mifflin Harcourt Publishing Company
4
4
4
4
231 Module 7
×
×
×
×
7
7
7
7
4×4
16
+ 4×3
+ 12
= 28
6 books
= 4 × (4 + 3)
= (4 × 4) + (4 × 3)
= 16 + 12
=
28
So, there are 28 corn stalks in all in the garden.
Write one way to break apart
the array. Then find the product.
Possible answers are given.
1.
6 fish
2.
4.
(3 × 4) + (3 × 5) or
(2 × 5) + (1 × 5) or
(1 × 9) + (2 × 9); 27
(3 × 3) + (3 × 2);
Number and Operations
Check students’
arrays.
5 calendars
4 rows of 7, or 4 × 7
Step 2 Break apart the array to make two
smaller arrays for facts you know.
Step 3 Write the multiplication for the new
arrays. Multiply and then add the
products to find the answer.
Enrich 35
1
Algebra • Distributive Property
OBJECTIVE Use the Distributive Property to find products by breaking apart arrays.
Explain how you used the arrays to solve each problem.
Possible answer: I drew two arrays for each
problem then added the number of rows to
find the number of items.
67
Enrich
© Houghton Mifflin Harcourt Publishing Company
E35
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 may use models or strategies to solve.
7.
8.
9.
36
C
48
B
42
D
64
(5 + 4) × 6
B
(3 + 3) × 6
C
(5 + 5) × 6
D
(5 + 3) × 6
THEN
IF
NO
•
Soar to Success Math
Warm-Up 12.22, 12.23
At a grocery store, the cans of nuts are arranged in
9 rows. There are 6 cans in each row. Which shows
one way to find the total number of cans of nuts?
A
1
Can students use the Distributive Property to
find products?
Use Diagrams Maurice planted 8 rows of popcorn
plants. There are 6 plants in each row. The array
shows Maurice’s garden. How many popcorn plants
does Maurice have?
A
2
YES
•
•
Enrich 35
Homework and Practice
Lesson 7.4
TEXAS Test Prep Coach
Multi-Step Jillian bought 6 books for $7 each. She paid
with $50. How much change should Jillian receive?
A
$37
C
$43
B
$18
D
$8
Test Prep Coach helps teachers to identify common
errors that students can make.
In the Test Prep exercise, if students selected:
A They broke the product into addends instead of
the factors
TEXAS Test Prep
B They chose an example of the Associative Property
Representation Which number sentence below is an
example of the Distributive Property?
A
6 × 8 = 40 + 8
B
(8 × 2) × 3 = 8 × (2 × 3)
C
6×8=8×6
D
6 × 8 = (6 × 2) + (6 × 6)
232
C The chose an example of the Commutative
Property.
© Houghton Mifflin Harcourt Publishing Company
10.
?
Essential Question
Write
Math
How can you use the Distributive Property to find
products? Possible answer: I can break apart one of the
factors and multiply each addend by the other factor. Then I
can add the products to find the answer.
Differentiated Centers Kit
Literature
Collections
Times Four
Students read the book
and determine how to use
multiplication to find the
total number of objects in
each collection.
Activities
Factor Spin
Students complete
purple Activity
Card 7 by using
randomly generated
factors to practice
multiplication facts
for 1, 2, 4, and 5.
Activities
Hurray for Arrays!
Students complete
blue Activity Card
15 by using arrays to
model multiplication
facts.
Lesson 7.4
232
5
7.4
ALGEBRA
Number and Operations—3.4.E, 3.4.K
Algebraic Reasoning—3.5.B Also 3.4.D, 3.5.D
MATHEMATICAL PROCESSES 3.1.C, 3.1.D, 3.1.E
Name
Fill in the bubble completely to show your answer.
5.
Distributive Property
Write one way to break apart the array. Then find
the product. Possible answers are given for 1–2.
1.
3.
TEXAS Test Prep
Lesson
Lesson Check
Check
2.
4 × (4 + 4) or (1 + 3) × 8 or
5 × (4 + 2) or (2 + 3) × 6 or
(2 × 8) + (2 × 8); 32
(2 × 6) + (3 × 6); 30
7.
Shade tiles to make an array that
shows a fact with 5, 6, or 7 as a factor.
Explain how you found the product.
Possible explanation: I shaded 7 rows
There are 8 tables in the art class.
Five students sit at each table. How
many students are in the art class?
13
A
67
B
40
B
40
C
45
C
42
D
85
D
13
Jonas buys 6 packages of corn. Each
package has 8 ears of corn. Which
way can Jonas use to show how
many ears of corn he buys?
8.
(4 + 2) × 8
B
(3 + 2) × 8
A
(4 × 9) + (5 × 9)
(6 + 6) × 8
B
(3 × 5) + (6 × 5)
(4 + 4) × 6
C
(4 × 5) + (5 × 9)
D
(3 × 5) + (9 × 5)
D
Check students’ arrays and answers.
9.
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
4.
Shawn makes a tile design by using 7 rows of 8 tiles.
Explain how Shawn can break apart his design to find
how many tiles he used.
Possible explanation: Shawn can break apart the 7 to show 3 + 4. Then he can multiply
and add: (8 × 3) + (8 × 4) = 24 + 32 = 56
Module 7 • Lesson 4
Homework and Practice
Use the Homework and Practice pages to provide
students with more practice on the concepts and
skills of this lesson.
233-234
Module 7
A store sells bags of 9 apples.
Mr. Hyon buys 5 bags of apples to
make pies. Which way can Mr. Hyon
use to show how many apples
he buys?
A
7 × 5 = 7 × (2 + 3); 7 × 5 = (7 × 2)
+ (7 × 3); 7 × 5 = 14 + 21 or 35.
Seven large trucks are in a parking
lot. Each truck has 6 wheels. How
many wheels are there in all?
A
C
of 5 tiles. I can think 5 = 2 + 3. I can write
6.
233
234
Multi-Step Minka buys 8 CDs
for $6 each. She pays with three
$20 bills. How much change should
Minka receive?
10.
Multi-Step Kent buys 5 bags of
potatoes that cost $7 each. He gives
the clerk $40. How much change
should Kent receive?
A
$48
A
$35
B
$12
B
$12
C
$34
C
$5
D
$68
D
$52
© Houghton Mifflin Harcourt Publishing Company
Ho mewo rk
and Practice
Module 7 Assessment
Formative Assessment
Vocabulary
Vocabulary
Vocabulary
of Multiplication
Distributive Property
multiple
1. A ___
of 4 is any product that has 4 as
one of its factors.
2.
Use the Module Assessment to assess students’
learning and progress. The formative assessment
provides the opportunity to adjust teaching
methods for individual or whole class instruction.
Commutative Property
Choose the best term from the box to complete
the sentence.
multiple
(p. 217)
Distributive
This is an example of the ___
Property.
3 × 8 = (3 × 6) + (3 × 2)
This property states that multiplying a sum by a
number is the same as multiplying each addend
by the number and then adding the products. (p. 229)
Concepts
Concepts and
and Skills
Skills
Write one way to break apart the array.
Then find the product. q TEKS 3.4.E, 3.4.K Possible answers are given.
3.
4.
5 × (3 + 4) or (2 + 3) × 7
4 × (5 + 4) or (2 + 2) × 9
or (2 × 7) + (3 × 7); 35
or (2 × 9) + (2 × 9); 36
© Houghton Mifflin Harcourt Publishing Company
Find the product.
5.
3
3×1=_
9.
2
×
1
_
q TEKS 3.4.E, 3.4.K
6.
10.
2
30
5×6=_
6
×
_6
36
11.
7.
6
×
7
_
42
28
4×7=_
12.
6
×
3
_
18
8.
20
2 × 10 = _
13.
3
×
8
_
24
Module 7
235
Data-Driven Decision Making
3
2
1
Based on the results of the Module 7 Assessment, use the following
i
resources to strengthen
individual or whole class instruction.
Item
Lesson
TEKS*
Common Error
Intervene With
RtI* Tier 1 Lessons
Soar to
Success Math
3, 4
7.4
3.4.E, 3.4.K
May not understand how to find a product by breaking
apart an array.
34
12.22, 12.23
5–13
7.1–7.2
3.4.E, 3.4.K
May find an incorrect product when multiplying
31, 32
12.21, 12.22
12.23, 12.25
*TEKS—Texas Essential Knowledge and Skills; RtI—Response to Intervention
Module 7 Assessment 235
Module 7 Assessment
Name
Module 7 Assessment
Fill in the bubble for the correct answer choice.
You may use models to solve.
14.
TEXAS Test Prep
Lori saw 6 lightning bugs. They each had 6 legs.
How many legs did the lightning bugs have in all?
q TEKS 3.4.E, 3.4.K
15.
A
36
B
6
C
12
D
24
Zach gave his dog 1 dog biscuit each day, for 5 days.
Then he gave his dog 2 dog biscuits a day for the next
5 days. How many biscuits did Zach give to his dog?
q TEKS 3.4.E, 3.4.K
17.
Items
1
3–13
2
14–17
B
30
C
15
D
7
Annette bought 4 boxes of pencils. If there are 8 pencils in
each box, how many pencils did she buy? q TEKS 3.4.E, 3.4.K
A
32
B
12
C
16
D
24
Shelly can paint 4 pictures in a day. How many
pictures can she paint in 7 days? q TEKS 3.4.E, 3.4.K
Record your answer and fill in the bubbles on
the grid. Be sure to use the correct place value.
Depth of Knowledge
DOK Level
5
2
8
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
236
Data-Driven Decision Making
© Houghton Mifflin Harcourt Publishing Company
16.
A
3
2
1
Item
Lesson
TEKS*
Common Error
Intervene With
RtI* Tier 1 Lessons
Soar to
Success Math
14
7.3
3.4.E, 3.4.K
May find an incorrect product when multiplying with 6
33
12.26, 12.28
15
7.2
3.4.E, 3.4.K
May find an incorrect product when multiplying with 5
in a two-step problem
32
12.22, 12.23
16
7.1
3.4.E, 3.4.K
May find an incorrect product when multiplying with 4
31
12.21, 12.25
17
7.1
3.4.E, 3.4.K
May mark a gridded response item incorrectly when
multiplying with 4
31
12.21, 12.25
*TEKS—Texas Essential Knowledge and Skills; RtI—Response to Intervention
236
Module 7 Assessment