### Grade 3 Mathematics Module 1, Topic C, Lesson 7

```Lesson 7 3•1
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 7
Objective: Demonstrate the commutativity of multiplication, and practice
related facts by skip-counting objects in array models.
Suggested Lesson Structure
Fluency Practice

Application Problem

Concept Development

Student Debrief

Total Time
(13 minutes)
(5 minutes)
(32 minutes)
(10 minutes)
(60 minutes)
Fluency Practice (13 minutes)
 Group Counting 3.OA.1
(3 minutes)
 Divide Equal Groups 3.OA.2
(5 minutes)
 Multiply with Twos 3.OA.7
(5 minutes)
Group Counting (3 minutes)
Note: Group counting reviews interpreting multiplication as repeated addition. Counting by twos and threes
in this activity anticipates work with those factors in Topic C.
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Let’s count by twos. (Direct students to count forward and backward to 20, emphasizing the 8 to 10,
10 to 12, and 18 to 20 transitions.)
Let’s count by threes. (Direct students to count forward and backward to 30, periodically changing
directions. Emphasize the 9 to 12, 18 to 21, and 27 to 30 transitions.)
Divide Equal Groups (5 minutes)
Materials: (S) Personal white board
Note: Students directly relate repeated addition to division. They interpret the unknown in division. This
activity reviews Lesson 6.
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(Project an array with 2 groups of 4.) Say the total as a repeated addition sentence.
4 + 4 = 8.
Write a division sentence for 8 divided into 2 equal groups.
(Write 8 ÷ 2 = 4.)
Lesson 7:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Demonstrate the commutativity of multiplication, and practice
related facts by skip-counting objects in array models.
10/21/14
1.C.3
Lesson 7 3•1
NYS COMMON CORE MATHEMATICS CURRICULUM
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Below that division sentence write a division sentence dividing 8 into 4 equal groups.
(Write 8 ÷ 4 = 2.)
Continue with possible sequence: 5 groups of 3, 3 groups of 4, and 6 groups of 2.
Multiply with Twos (5 minutes)
Materials: (S) Personal white board, twos array (Fluency Template), blank paper
Note: Students unit count objects in an array and write
multiplication sentences that match the count-by in
anticipation of this lesson’s objective.
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Twos array fluency template
paper to cover all but the first row of dots.
How many twos show?
1 two.
Say the multiplication sentence to represent the array
that’s shown and solve.
1 × 2 = 2.
Uncover another row.
Continue this sequence having students uncover twos for 2 × 2, 3 × 2, 10 × 2, 4 × 2, 5 × 2, 6 × 2, 7 × 2, 9 × 2,
and 8 × 2.
Application Problem (5 minutes)
Anna picks 24 flowers. She makes equal bundles of flowers and
gives 1 bundle to each of her 7 friends. She keeps a bundle for
herself too. How many flowers does Anna put in each bundle?
Note: This problem reviews equal groups division from Lesson 5
where the unknown represents the size of the group. The
problem’s complexity is in understanding that the flowers are
divided equally into 8 bundles, not 7, in order to include a
bundle for Anna. Students might choose to solve by drawing a
division array learned in Lesson 6 or a number bond learned in
Lesson 3.
Lesson 7:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Demonstrate the commutativity of multiplication, and practice
related facts by skip-counting objects in array models.
10/21/14
1.C.4
Lesson 7 3•1
NYS COMMON CORE MATHEMATICS CURRICULUM
Concept Development (32 minutes)
Materials: (S) Personal white board
Problem 1: Rotate arrays 90 degrees.
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Turn your personal white board so it’s horizontal.
Draw a line down the middle to make two sides. On
the left, skip-count by twos 4 times and write each
number.
(Write 2, 4, 6, 8.)
On the right side, skip-count by fours 2 times and write
each number.
(Write 4, 8.)
How are the count-bys related?
The first one is 4 twos and the second one is 2 fours.
Under each count-by, draw an array to match it.
(Draw arrays shown below.)
What do you notice about the arrays? Do they both
have 4 groups of 2?
(Discuss.)
Do they both have 2 groups of 4?
Yes. The one on the right has 2 rows of 4. If you turn it
sideways, then the one on the left does too. Or, you
can just see that it has 2 vertical rows of 4.
It’s the same array turned different ways. We have a
special name for rows when they are vertical. We call
them columns.
Prompt students to write and solve equations to show the total
objects in each array. Continue with the following possible
examples:


Final student personal white board
2, 4, 6, 8
4, 8
4×2=
8
2×4=
8
NOTES ON
MULTIPLE MEANS
OF REPRESENTATION:
If necessary, students may keep the
twos array template from the fluency
activity Multiply with Two alongside
their workspace to help them draw
arrays from skip-counting.
NOTE ON
VOCABULARY:
The word column was originally
introduced in Grade 2, Module 6 but is
treated as new vocabulary in this
lesson.
2 × 5 array
7 × 2 array
As you circulate, guide students to notice that factors switch places, and help them to relate the change to
the rotated array. For example, write 4 × 2 = 2 × 4 and ask students to discuss how they know it is true.
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Depending on how we look at an array, columns or rows can be the number of groups. Discuss with
your partner how you know that’s true.
Lesson 7:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Demonstrate the commutativity of multiplication, and practice
related facts by skip-counting objects in array models.
10/21/14
1.C.5
Lesson 7 3•1
NYS COMMON CORE MATHEMATICS CURRICULUM
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(Discuss.)
Problem 2: Interpreting rows and columns in rotated arrays.
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MP.7
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Turn your board so it’s vertical. Draw an array that shows 8 equal groups of 2. How many rows of 2
did you draw?
8 rows.
How many columns of 8 did you draw?
2 columns.
Write an equation to match the array. Don’t solve it yet.
(Write 8 × 2 = ____.)
Rotate your board so that it’s horizontal. How many rows of 8 do you have now?
2 rows.
How many columns of 2?
8 columns.
Write an equation to match the array. Don’t solve it yet.
(Write 2 × 8 = ____.)
Explain to your partner using the words columns and rows why your equation changed.
When the array turned, the columns and rows switched.  Columns became rows and rows
became columns. They both represent equal groups. It depends on how you look at the array.
Will 8 × 2 and 2 × 8 have the same total?
Yes!
How do you know?
They have the same array.  Two groups of 8 and 8 groups of 2 are the same.
Prompt students to skip-count to find the totals of the array in both positions.
Work through the following examples to build vocabulary and understanding of commutativity: 6 × 2 and
2 × 9.
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When we multiply, changing the order of the factors
doesn’t change the total. We say the factors are
commutative. That means they can switch around.
Tell your partner what commutative means.
It means numbers can switch around.  The factors
change places in a multiplication equation, but the
total doesn’t change.  Addition works the same way.
What we’ve explored today is called the commutative
property.
Lesson 7:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
NOTES ON
MULTIPLE MEANS
OF REPRESENTATION:
Students need not master the words
commutative or commutative property
(3.OA.5). However, they will need to
be familiar with the vocabulary moving
forward in this module.
Demonstrate the commutativity of multiplication, and practice
related facts by skip-counting objects in array models.
10/21/14
1.C.6
Lesson 7 3•1
NYS COMMON CORE MATHEMATICS CURRICULUM
Problem Set (10 minutes)
Students should do their personal best to complete the
Problem Set within the allotted 10 minutes. For some
classes, it may be appropriate to modify the assignment
by specifying which problems they work on first. Some
problems do not specify a method for solving. Students
should solve these problems using the RDW approach
used for Application Problems.
Student Debrief (10 minutes)
Lesson Objective: Demonstrate the commutativity of
multiplication, and practice related facts by skip-counting
objects in array models.
Extension:
The Student Debrief is intended to invite reflection and
active processing of the total lesson experience. Invite
students to review their solutions for the Problem Set.
They should check work by comparing answers with a
partner before going over answers as a class. Look for
misconceptions or misunderstandings that can be
addressed in the Debrief. Guide students in a conversation
to debrief the Problem Set and process the lesson.
You may choose to use any combination of the questions

Discuss the usefulness of unit counting to solve
multiplication problems.

Build fluency by having students unit count to find
the answer to the following expressions without
the help of an array. They can keep track of the
count using fingers.
 3 twos, 2 threes
 4 twos, 2 fours
 2 eights, 8 twos
 2 tens, 10 twos
Discuss the commutativity of multiplication and
how it relates to equal groups, columns, rows, and
rotating arrays.
Relate the commutative property of multiplication


Lesson 7:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Demonstrate the commutativity of multiplication, and practice
related facts by skip-counting objects in array models.
10/21/14
1.C.7
Lesson 7 3•1
NYS COMMON CORE MATHEMATICS CURRICULUM
to the commutative property of addition to help students connect it to prior learning.
Exit Ticket (3 minutes)
After the Student Debrief, instruct students to complete the Exit Ticket. A review of their work will help you
assess the students’ understanding of the concepts that were presented in the lesson today and plan more
effectively for future lessons. You may read the questions aloud to the students.
Lesson 7:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Demonstrate the commutativity of multiplication, and practice
related facts by skip-counting objects in array models.
10/21/14
1.C.8
Lesson 7 Problem Set 3•1
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
Date
1. a. Count by twos 6 times.
2. a. Count by sixes 2 times.
_____, _____, _____, _____, _____, _____
_______, _______
b. Draw an array to match your count-by.
b. Draw an array to match your count-by.
c. Write an equation to represent the total number
c. Write an equation to represent the total number
________ × ________ = ________
________ × ________ = ________
3. a. Turn your paper to look at the arrays in Problems 1 and 2 in different ways. What is the same and
b. Why are the factors in your equations in a different order?
4. Skip-count to find the total for each expression. Write an equation to match your count-by. The first one
is done for you.
a. 6 twos: 6 × 2 = 12 ___
d. 2 sevens:
_____
Extension:
b. 2 sixes:
e. 9 twos:
_____
g. 11 twos:
___
f. 2 nines:
____________
h. 2 twelves:
___
_____
c. 7 twos: _______
Lesson 7:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Demonstrate the commutativity of multiplication, and practice
related facts by skip-counting objects in array models.
10/21/14
1.C.9
Lesson 7 Problem Set 3•1
NYS COMMON CORE MATHEMATICS CURRICULUM
5. Write and solve a different equation to describe each array.
6. Ms. Nenadal writes 2 × 7 = 7 × 2 on the board. Do you agree or disagree? Draw arrays to help explain
7. Find the missing factor to make each equation true.
5 × 2 = 2 × _____
_____ × 8 = 8 × 2
2 × 10 = _____× 2
2 × _____ = 9 × 2
8. Jada gets 2 new packs of erasers. Each pack has 6 erasers in it.
a. Draw an array to show how many erasers Jada has altogether.
b. Write and solve an equation to describe the array.
c. Use the commutative property to write and solve a different equation for the array.
Lesson 7:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Demonstrate the commutativity of multiplication, and practice
related facts by skip-counting objects in array models.
10/21/14
1.C.10
Lesson 7 Exit Ticket 3•1
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
Date
2×5=5×2
Do you agree or disagree with the statement in the box? Draw arrays and use skip-counting to explain your
thinking.
Lesson 7:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Demonstrate the commutativity of multiplication, and practice
related facts by skip-counting objects in array models.
10/21/14
1.C.11
Lesson 7 Homework 3•1
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
Date
1. a. Count by twos 7 times.
2. a. Count by sevens 2 times.
_____, _____, _____, _____, _____ ,_____, _____
_______, _______
b. Draw an array to match your count-by.
b. Draw an array to match your count-by.
c. Write an equation to represent the total number
c. Write an equation to represent the total number
________ × ________ = ________
________ × ________ = ________
3. a. Turn your paper to look at the arrays in Problems 1 and 2 in different ways. What is the same and
b. Why are the factors in your equations in a different order?
4. Skip-count to find the total for each expression. Write an equation to match your count-by. The first one
is done for you.
a. 2 twos: 2 × 2 = 4
d. 2 fours: _______________
g. 2 fives: ______________
b. 3 twos: _____________
e. 4 twos: ________________
h. 6 twos: ______________
c. 2 threes: ____________
f. 5 twos: ________________
i. 2 sixes: ______________
Lesson 7:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Demonstrate the commutativity of multiplication, and practice
related facts by skip-counting objects in array models.
10/21/14
1.C.12
Lesson 7 Homework 3•1
NYS COMMON CORE MATHEMATICS CURRICULUM
5. Write and solve a different equation to describe each array.
6. Angel writes 2 × 8 = 8 × 2 in his notebook. Do you agree or disagree? Draw arrays to help explain your
thinking.
7. Find the missing factor to make each equation true.
2×6=6×
___________
_____ × 2 = 2 × 7
9 × 2 = _____ × 9
2 × ____ = 10 × 2
8. Tamia buys 2 bags of candy. Each bag has 7 pieces of candy in it.
a. Draw an array to show how many pieces of candy Tamia has altogether.
b. Write and solve an equation to describe the array.
c. Use the commutative property to write and solve a different equation for the array.
Lesson 7:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Demonstrate the commutativity of multiplication, and practice
related facts by skip-counting objects in array models.
10/21/14
1.C.13
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 7 Fluency Template 3•1
twos array
Lesson 7:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Demonstrate the commutativity of multiplication, and practice
related facts by skip-counting objects in array models.
10/21/14