THE NEWARK PUBLIC SCHOOLS
THE OFFICE OF MATHEMATICS
Grade 3
Operations and Algebraic Thinking
OA.5 & OA.6
2012 COMMON CORE STATE STANDARDS ALIGNED MODULES
Office of Mathematics
THE NEWARK PUBLIC SCHOOLS
MATHTASKS
Operations and Algebraic Thinking - 3.OA.5. & 3.OA.6
Understand properties of multiplication and the relationship
between multiplication and division.
Goal:
Students will apply properties of operations as strategies to multiply and divide. Examples:
If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of
multiplication). 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10,
then 3 × 10 = 30. (Associative property of multiplication). Knowing that 8 × 5 = 40 and 8
× 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive
property). In addition, students will understand division as an unknown-factor problem. For
example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.
Prerequisites:
Essential Questions:
What is multiplication?
What are the properties of multiplication?
How can multiplication be used when
dividing?
Whole Numbers
Simple Counting
Addition
Subtraction
Factors
Embedded Mathematical Practices
MP.1 Make sense of problems and persevere in solving them
MP.2 Reason abstractly and quantitatively
MP.3 Construct viable arguments and critique the reasoning
of others
MP.4 Model with mathematics
Lesson 5
MP.5 Use appropriate tools strategically
3. OA.5 & 3.OA.6
Golden Problem
MP.6 Attend to precision
MP.7 Look for and make use of structure
MP.8 Look for and express regularity in
Lesson 4
repeated reasoning
3. OA.6 Using multiplication to divide
Lesson 3
3. OA.5 Properties of Multiplication
Lesson 2
3. OA.5 Commutative Property and
Multiplicative Identity Property of One
Lesson 1
3. OA.5Understanding Multiplication
Lesson Structure:
Introductory Task
Prerequisite Skills
Focus Questions
Guided Practice
Homework
Journal Question
Page 2 of 37
Multiplication Concepts
Multiplication can be defined in terms of repeated addition. For example, 3 × 6 can be viewed as 6 + 6 + 6. More generally, for any
positive integer n, n × b can be represented as n × b = b + b + … + b, where the sum on the right consists of n addends.
A rectangular array provides a visual model for multiplication. For example, the product 3 × 6 can be represented as
By displaying 18 dots as 3 rows with 6 dots in each row, this array provides a visual representation of 3 × 6 as 6 + 6 + 6. An
equivalent area model can be made in which the dots of the array are replaced by unit squares.
Besides representing 3 × 6 as an array of 18 unit squares, this model also shows that the area of a rectangle with a height of 3 units and
a base of 6 units is 3 × 6 square units, or 18 square units.
Multiplication is a binary operation that operates on a pair of numbers to produce another number. Given a pair of numbers a and b
called factors, multiplication assigns them a value a × b = c, called their product.
Multiplication has certain fundamental properties that are of great importance in arithmetic. The Commutative Property of
Multiplication states that changing the order in which two numbers are multiplied does not change the product. That is, for all
numbers a and b, a × b = b × a.
The array model can be used to make this plausible. For example, because 3 × 6 = 6 × 3, an array with 3 rows and 6 dots in each row
has the same number of dots as an array with 6 rows and 3 dots in each row.
Another important property of multiplication is the Identity Property of Multiplication. It states that the product of any number and 1
is that number. That is, for all numbers a, a × 1 = 1 × a = a.
The Zero Property of Multiplication states that when a number is multiplied by zero, the product is zero. That is, for all numbers a,
a × 0 = 0 × a = 0.
Page 3 of 37
Teaching Tips
Digit Name vs. Digit Value
Teaching Tip 1
Stress place value in multiplication by distinguishing between the name of
the digit and the value it stands for. The 2 in 24 stands for 2 × 10 = 20, not
2. Base-10 blocks and area model diagrams emphasize the value that each
digit stands for because they use expanded notation to build the answer.
Drawing Rectangles for an Area Model
Teaching Tip 2
The area model is an alternative and efficient way to multiply. Encourage
students to draw rectangles, even though the rectangles may not be drawn
to scale. If students need to use base-10 blocks as a transitional step,
change the numbers in the problems to match the quantity of blocks that
are available.
Using an Area Model to Record Multiplication
Teaching Tip 3
Is it okay to permit students to use the area model as a recording
method for multiplication? Yes. An area model not only helps to explain
why the standard algorithm commonly taught in the United States for
multiplication works, it is an efficient recording alternative. Some students
(especially visual learners and those who have difficulty keeping numbers
lined up in multiplication problems) may prefer it. Furthermore, this
method has certain benefits. It illuminates important mathematical
concepts (such as the distributive property), allows for computational
flexibility (expanded notations allow students to use derived facts), and
reinforces the concept of area. Finally, when students take algebra, they
are likely to see the area model when they learn to multiply and factor
polynomials.
Page 4 of 37
Multiple Representations to Multiplication
Distributive
Property
In the identity 3(4 + 5)
= 3(4) + 3(5), the 3 is
“distributed” over the
4 and the 5.
a(b + c ) = ab + ac
and
(b + c )a = ba + ca
Commutative
Properties of
Multiplication
a•b=b•a
(3 •4) •5 =12 • 5 = 60
or
3 • (4•5) =3 •20 = 60
3•4=4•3
Associative
Properties of
Multiplication
(a
c=a
(b
)
Area Model
4
2 ·2
·3
12
4
Array Model
Interpret products of
whole numbers
5 × 7 as the total
number of objects
in 5 groups of 7
objects each
Page 5 of 37
3.OA.5: LESSON 13.OA.5: LESSON 1
Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4 × 6
= 24 is also known. (Commutative property of multiplication). 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 ×
2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication). Knowing that 8 × 5 = 40
and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property).
Introductory Task
Guided Practice
Collaborative
Homework
Assessment
Below is an array that shows 6 x 3 = 18. Look at the array below. Design your own array that would depict the
problem 8 x 4. What does 8 x 4 equal? Would the answer change if the problem was 4 x 8? Use your array to
help you answer these questions.
Example
6 x 3=18
Focus Questions
Question 1: What is multiplication and what does it tell us?
Question 2: How is multiplication related to addition?
Journal Question
Why is understanding how to
multiply useful? Describe one
way you could use multiplication
in your daily life.
Page 6 of 37
3.OA.5: LESSON 1
Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4 × 6
= 24 is also known. (Commutative property of multiplication). 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 ×
2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication). Knowing that 8 × 5 = 40
and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property).
Introductory Task
Guided Practice
Collaborative
Homework
Assessment
Use the arrays below to help you identify the multiplication problems that are being described.
1.
2.
_____X_____ or _____ X _____
3.
_____X_____ or _____ X _____
4.
_____X_____ or _____ X _____
5.
_____X_____ or _____ X _____
6.
_____X_____ or _____ X _____
_____X_____ or _____ X _____
Page 7 of 37
3.OA.5: LESSON 1
Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4 × 6
= 24 is also known. (Commutative property of multiplication). 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 ×
2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication). Knowing that 8 × 5 = 40
and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property). MP:
Make sense of problems and persevere in solving them, Reason abstractly and quantitatively, Model with
Introductory Task
Guided Practice
Collaborative
Homework
Assessment
Create your own array below that identifies the multiplication problems that are listed.
7.
8.
6 x 5 or 5 x 6
9.
3 x 7 or 7 x 3
10.
1 x 9 or 9 x 1
11.
10 x 3 or 3 x 10
12..
4 x 5 or 5 x 4
8 x 3 or 3 x 8
Page 8 of 37
3.OA.5: LESSON 1
Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4 × 6
= 24 is also known. (Commutative property of multiplication). 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 ×
2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication). Knowing that 8 × 5 = 40
and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property). MP:
Make sense of problems and persevere in solving them, Reason abstractly and quantitatively, Model with
mathematics, Use appropriate tools strategically, and attend to precision.
Introductory Task
Guided Practice
Collaborative
Name____________________________
Homework
Assessment
Date_____________________
Below you will need to either identify the multiplication problem that is described by an array, or create
your own array to describe a multiplication problem that is given. Use the arrays below to help you identify
the multiplication problems that are being described.
1.
2.
_____X_____ or _____ X _____
3.
_____X_____ or _____ X _____
4.
_____X_____ or _____ X _____
_____X_____ or _____ X _____
Page 9 of 37
5.
6.
_____X_____ or _____ X _____
_____X_____ or _____ X _____
Create your own array below to that identifies the multiplication problems that are listed.
7.
8.
6 x 5 or 5 x 6
9.
3 x 10 or 10 x 3
10.
1 x 9 or 9 x 1
4 x 3 or 3 x 4
11.
12..
4 x 5 or 5 x 4
8 x 3 or 3 x 8
Page 10 of 37
3.OA.5: LESSON 2
Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then
4 × 6 = 24 is also known. (Commutative property of multiplication). 3 × 5 × 2 can be found by 3 × 5 = 15,
then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication). Knowing that
8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive
property). MP: Make sense of problems and persevere in solving them, Reason abstractly and
quantitatively, Model with mathematics, Use appropriate tools strategically, and attend to precision.
Introductory Task
Guided Practice
Collaborative
Homework
Assessment
Do the diagrams below show the same quantity of stars? Describe how you can determine the answer using
multiplication.
Diagram A
Focus Questions
Question 1: What strategies can be used to multiply?
Question 2: How do the numbers 1 and 0 effect
multiplication?
Diagram B
Journal Question
In your own words, describe what
happens when you multiply two
numbers together. Explain what is
special about multiplying by 1.
Page 11 of 37
3.OA.5: LESSON 2
Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then
4 × 6 = 24 is also known. (Commutative property of multiplication). 3 × 5 × 2 can be found by 3 × 5 = 15,
then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication). Knowing that
8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive
property). MP: Make sense of problems and persevere in solving them, Reason abstractly and
quantitatively, Model with mathematics, Use appropriate tools strategically, and attend to precision.
Introductory Task
Guided Practice
Collaborative
Homework
Assessment
Identify the multiplication problems that are pictured below.
1.
______ X _____ = ______
or
_____ X _____ = _______
______ X _____ = ______
or
_____ X _____ = _______
______ X _____ = ______
or
_____ X _____ = _______
2.
3.
Page 12 of 37
3.OA.5: LESSON 2
Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then
4 × 6 = 24 is also known. (Commutative property of multiplication). 3 × 5 × 2 can be found by 3 × 5 = 15,
then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication). Knowing that
8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive
property). MP: Make sense of problems and persevere in solving them, Reason abstractly and
quantitatively, Model with mathematics, Use appropriate tools strategically, and attend to precision.
Introductory Task
Guided Practice
Collaborative
Homework
Assessment
4.
______ X _____ = ______
or
_____ X _____ = _______
______ X _____ = ______
or
_____ X _____ = _______
______ X _____ = ______
or
_____ X _____ = _______
5.
6.
7.
______ X _____ = ______
or
_____ X _____ = _______
Page 13 of 37
3.OA.5: LESSON 2
Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then
4 × 6 = 24 is also known. (Commutative property of multiplication). 3 × 5 × 2 can be found by 3 × 5 = 15,
then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication). Knowing that
8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive
property). MP: Make sense of problems and persevere in solving them, Reason abstractly and
quantitatively, Model with mathematics, Use appropriate tools strategically, and attend to precision.
Introductory Task
Guided Practice
Collaborative
Homework
Assessment
Create pictorial representations of the multiplication problems listed below
8.
1
or
__7__ X _ ___ = _______
4
or
__6__ X _ ___ = _______
9
or
__2__ X _ ___ = _______
__ __ X __7__ = ______
1
9.
__ __ X __6__ = ______
4
10.
__ __ X __2__ = ______
9
Page 14 of 37
3.OA.5: LESSON 2
Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then
4 × 6 = 24 is also known. (Commutative property of multiplication). 3 × 5 × 2 can be found by 3 × 5 = 15,
then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication). Knowing that
8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive
property). MP: Make sense of problems and persevere in solving them, Reason abstractly and
quantitatively, Model with mathematics, Use appropriate tools strategically, and attend to precision.
Introductory Task
Guided Practice
Collaborative
Name: ____________________________
Homework
Assessment
Date: _______________________
Identify the multiplication problems that are pictured below.
1.
______ X _____ = ______
or
_____ X _____ = _______
______ X _____ = ______
or
_____ X _____ = _______
______ X _____ = ______
or
_____ X _____ = _______
2.
3.
Page 15 of 37
3.OA.5: LESSON 2
Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then
4 × 6 = 24 is also known. (Commutative property of multiplication). 3 × 5 × 2 can be found by 3 × 5 = 15,
then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication). Knowing that
8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive
property). MP: Make sense of problems and persevere in solving them, Reason abstractly and
quantitatively, Model with mathematics, Use appropriate tools strategically, and attend to precision.
Introductory Task
Guided Practice
Collaborative
Homework
Assessment
4.
______ X _____ = ______
or
_____ X _____ = _______
______ X _____ = ______
or
_____ X _____ = _______
______ X _____ = ______
or
_____ X _____ = _______
______ X _____ = ______
or
_____ X _____ = _______
5.
6.
7.
Page 16 of 37
3.OA.5: LESSON 2
Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then
4 × 6 = 24 is also known. (Commutative property of multiplication). 3 × 5 × 2 can be found by 3 × 5 = 15,
then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication). Knowing that
8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive
property). MP: Make sense of problems and persevere in solving them, Reason abstractly and
quantitatively, Model with mathematics, Use appropriate tools strategically, and attend to precision.
Introductory Task
Guided Practice
Collaborative
Homework
Assessment
Create pictorial representations of the multiplication problems listed below
8.
1
or
__9__ X _ ___ = _______
4
or
__5__ X _ ___ = _______
7
or
__2__ X _ ___ = _______
__ __ X __9__ = ______
1
9.
__ __ X __5__ = ______
4
10.
__ __ X __2__ = ______
7
Page 17 of 37
3.OA.5: LESSON 3
Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4
× 6 = 24 is also known. (Commutative property of multiplication). 3 × 5 × 2 can be found by 3 × 5 = 15,
then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication). Knowing that
8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive
property). MP: Make sense of problems and persevere in solving them, Reason abstractly and
quantitatively, Model with mathematics, Use appropriate tools strategically, and attend to precision.
Introductory Task
Guided Practice
Collaborative
Homework
Assessment
Zack wants to hang his 8 model airplanes from his bedroom ceiling with wire. Each model airplane needs 6
inches of wire. How many inches of wire will Zack need to hang all 8 of his model airplanes? If Zack bought
two more airplanes, how much more wire would he need?
Focus Questions
Question 1: Does it matter what order the numbers are in
when you multiply?
Question 2: Is there more than one way to multiply and get the
same result?
Journal Question
If 7x9=63, and 2+5=7, does
(2x9)+(5x9)=63? Explain your
answer in words or pictures.
Page 18 of 37
3.OA.5: LESSON 3
Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4
× 6 = 24 is also known. (Commutative property of multiplication). 3 × 5 × 2 can be found by 3 × 5 = 15,
then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication). Knowing that
8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive
property). MP: Make sense of problems and persevere in solving them, Reason abstractly and
quantitatively, Model with mathematics, Use appropriate tools strategically, and attend to precision.
Introductory Task
Guided Practice
Collaborative
Homework
Assessment
Look at the example below. Notice how larger groups can be broken into smaller pieces to make multiplying
easier. In this case 7x8=56. However, 7 can be broken down into 2+5. So, 2x8=16 and 5x8=40, and
40+16=56.
7x 8 = 56
5 x 8 = 40
2 x 8 = 16
56
Look at the arrays below. Try to determine what multiplication problem the original array represents. Then
try to break it into small pieces like the example.
1.
____X____=____
____X____=____
______
____X____=____
+ ______
______
Page 19 of 37
3.OA.5: LESSON 3
Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4
× 6 = 24 is also known. (Commutative property of multiplication). 3 × 5 × 2 can be found by 3 × 5 = 15,
then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication). Knowing that
8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive
property). MP: Make sense of problems and persevere in solving them, Reason abstractly and
quantitatively, Model with mathematics, Use appropriate tools strategically, and attend to precision.
Introductory Task
Guided Practice
Collaborative
Homework
Assessment
2.
____X____=____
____X____=____
______
____X____=____
+ ______
______
3.
____X____=____
____X____=____
______
____X____=____
+ ______
______
Page 20 of 37
3.OA.5: LESSON 3
Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4
× 6 = 24 is also known. (Commutative property of multiplication). 3 × 5 × 2 can be found by 3 × 5 = 15,
then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication). Knowing that
8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive
property). MP: Make sense of problems and persevere in solving them, Reason abstractly and
quantitatively, Model with mathematics, Use appropriate tools strategically, and attend to precision.
Introductory Task
4.
Guided Practice
Collaborative
Homework
Assessment
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
xxxxxxxxxxxx
____X____=____
____X____=____
______
____X____=____
+ ______
______
5.
OOOOOOOOO
OOOOOOOOO
OOOOOOOOO
OOOOOOOOO
OOOOOOOOO
OOOOOOOOO
OOOOOOOOO
OOOOOOOOO
OOOOOOOOO
OOOOOOOOO
OOOOOOOOO
OOOOOOOOO
____X____=____
____X____=____
______
____X____=____
+ ______
______
Page 21 of 37
3.OA.5: LESSON 3
Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4
× 6 = 24 is also known. (Commutative property of multiplication). 3 × 5 × 2 can be found by 3 × 5 = 15,
then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication). Knowing that
8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive
property). MP: Make sense of problems and persevere in solving them, Reason abstractly and
quantitatively, Model with mathematics, Use appropriate tools strategically, and attend to precision.
Introductory Task
6.
Guided Practice
Collaborative
Homework
Assessment
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____X____=____
____X____=____
______
____X____=____
+ ______
______
7.
____X____=____
____X____=____
______
____X____=____
+ ______
______
Page 22 of 37
3.OA.5: LESSON 3
Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4
× 6 = 24 is also known. (Commutative property of multiplication). 3 × 5 × 2 can be found by 3 × 5 = 15,
then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication). Knowing that
8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive
property). MP: Make sense of problems and persevere in solving them, Reason abstractly and
quantitatively, Model with mathematics, Use appropriate tools strategically, and attend to precision.
Introductory Task
Guided Practice
Collaborative
Homework
Assessment
8.
____X____=____
____X____=____
______
____X____=____
+ ______
______
9.
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


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








____X____=____
____X____=____
______
____X____=____
+ ______
______
10.
____X____=____
____X____=____
______
____X____=____
+ ______
______
Page 23 of 37
3.OA.5: LESSON 3
Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4
× 6 = 24 is also known. (Commutative property of multiplication). 3 × 5 × 2 can be found by 3 × 5 = 15,
then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication). Knowing that
8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive
property). MP: Make sense of problems and persevere in solving them, Reason abstractly and
quantitatively, Model with mathematics, Use appropriate tools strategically, and attend to precision.
Introductory Task
Guided Practice
Name _______________________
Collaborative
Homework
Assessment
Date __________________
Look at the arrays that follow. Try to determine what multiplication problem the original array represents.
Then try to break it into small pieces like the example.
1.
____X____=____
____X____=____
______
____X____=____
+ ______
______
2.





____X____=____
____X____=____





____X____=____
______
+ ______
______
Page 24 of 37
3.OA.5: LESSON 3
Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4
× 6 = 24 is also known. (Commutative property of multiplication). 3 × 5 × 2 can be found by 3 × 5 = 15,
then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication). Knowing that
8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive
property). MP: Make sense of problems and persevere in solving them, Reason abstractly and
quantitatively, Model with mathematics, Use appropriate tools strategically, and attend to precision.
Introductory Task
Guided Practice
Collaborative
Homework
Assessment
3.
____X____=____
____X____=____
______
____X____=____
+ ______
______
4.
xxxxxxxxxxxxxxx
xxxxxxxxxxxxxxx
xxxxxxxxxxxxxxx
xxxxxxxxxxxxxxx
____X____=____
xxxxxxxxxxxxxxx
xxxxxxxxxxxxxxx
xxxxxxxxxxxxxxx
xxxxxxxxxxxxxxx
____X____=____
______
____X____=____
+ ______
______
5.
OOOOOOOOOOO
OOOOOOOOOOO
OOOOOOOOOOO
OOOOOOOOOOO
OOOOOOOOOOO
OOOOOOOOOOO
____X____=____
OOOOOOOOOOO
OOOOOOOOOOO
OOOOOOOOOOO
OOOOOOOOOOO
OOOOOOOOOOO
OOOOOOOOOOO
____X____=____
______
____X____=____
+ ______
______
Page 25 of 37
3.OA.5: LESSON 3
Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4
× 6 = 24 is also known. (Commutative property of multiplication). 3 × 5 × 2 can be found by 3 × 5 = 15,
then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication). Knowing that
8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive
property). MP: Make sense of problems and persevere in solving them, Reason abstractly and
quantitatively, Model with mathematics, Use appropriate tools strategically, and attend to precision.
Introductory Task
6.
Guided Practice
Collaborative
Homework
Assessment












____X____=____
____X____=____
______
____X____=____
+ ______
______
7.
____X____=____
____X____=____
______
____X____=____
+ ______
______
Page 26 of 37
3.OA.5: LESSON 3
Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4
× 6 = 24 is also known. (Commutative property of multiplication). 3 × 5 × 2 can be found by 3 × 5 = 15,
then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication). Knowing that
8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive
property). MP: Make sense of problems and persevere in solving them, Reason abstractly and
quantitatively, Model with mathematics, Use appropriate tools strategically, and attend to precision.
Introductory Task
8.
Guided Practice



____X____=____
Collaborative
Homework
Assessment



____X____=____
______
____X____=____
+ ______
______
9.





____X____=____





____X____=____
______
____X____=____
+ ______
______
10.







____X____=____







____X____=____
______
____X____=____
+ ______
______
Page 27 of 37
3.OA.6: LESSON 4
Understand division as an unknown-factor problem. For example, find32 ÷ 8 by finding the number that
makes 32 when multiplied by 8.
Introductory Task
Guided Practice
Collaborative
Homework
Assessment
Farmer Fahy was rounding up his sheep when a thunderstorm came rolling in. He was not sure if he had all of
the sheep herded into the barn. He knows that he has 12 sheep. When he looked under the stall in the barn, he
counted 48 feet. Tell Farmer Fahy if he has all of his sheep, or if he needs to go back outside in the
thunderstorm to look for any that are missing.
Focus Questions
Question 1: How can multiplication help with division?
Question 2: What does division really mean?
Journal Question
What are we doing with number
when we divide? Write a journal
entry as if you were trying to
explain division to another
student. Use examples.
Page 28 of 37
3.OA.6: LESSON 4
Understand division as an unknown-factor problem. For example, find32 ÷ 8 by finding the number that
makes 32 when multiplied by 8.
Introductory Task
Guided Practice
Collaborative
Homework
Assessment
Determine the missing number in these division problems by using multiplication.
1.
2.
__
18
24
x,÷
x,÷
3
__
2
Page 29 of 37
3.OA.6: LESSON 4
Understand division as an unknown-factor problem. For example, find32 ÷ 8 by finding the number that
makes 32 when multiplied by 8.
Introductory Task
Guided Practice
3.
Collaborative
Homework
Assessment
4.
36
12
x,÷
x,÷
__
4
5.
__
2
6.
__
40
21
x,÷
x,÷
4
__
7
Page 30 of 37
3.OA.6: LESSON 4
Understand division as an unknown-factor problem. For example, find32 ÷ 8 by finding the number that
makes 32 when multiplied by 8.
Introductory Task
Guided Practice
7.
Collaborative
Homework
Assessment
8.
15
35
x,÷
x,÷
5
__
9.
5
__
10.
__
32
12
x,÷
x,÷
4
4
__
Page 31 of 37
3.OA.6: LESSON 4
Understand division as an unknown-factor problem. For example, find32 ÷ 8 by finding the number that
makes 32 when multiplied by 8.
Introductory Task
Guided Practice
Collaborative
Name _______________________
Homework
Assessment
Date __________________
Determine the missing number in these division problems by using multiplication.
1.
2.
42
10
x,÷
x,÷
__
__
3.
10
4.
6
60
12
x,÷
x,÷
__
6
__
Page 32 of 37
3.OA.6: LESSON 4
Understand division as an unknown-factor problem. For example, find32 ÷ 8 by finding the number that
makes 32 when multiplied by 8.
Introductory Task
Guided Practice
5.
Collaborative
Homework
Assessment
6.
32
36
x,÷
x,÷
__
8
7.
6
__
8.
12
48
18
x,÷
x,÷
__
3
__
Page 33 of 37
3.OA.6: LESSON 4
Understand division as an unknown-factor problem. For example, find32 ÷ 8 by finding the number that
makes 32 when multiplied by 8.
Introductory Task
Guided Practice
9.
Collaborative
Homework
Assessment
10.
18
60
x,÷
x,÷
__
2
11.
5
__
12.
__
56
81
x,÷
x,÷
__
Page 34 of 37
3.OA.5-6: LESSON 5
Understand division as an unknown-factor problem. For example, find32 ÷ 8 by finding the number that
makes 32 when multiplied by 8.
Introductory Task
Guided Practice
Collaborative
Homework
Assessment
Juan is having his birthday party at the amusement park. He and his friends have broken up into two equal
groups of four, so that their parents can chaperone them easily. His mom has bought a total of 72 ride tickets
for Juan and each of his friends. How many tickets will each group get? Use pictures, mathematical operations,
and words to explain your answer.
Focus Questions
Question 1: What strategies can be used to find answer to a
multiplication problem?
Question 2: Can you determine the answer to a division
problem by multiplying? If so, how?
Journal Question
If you had to explain how to
multiply to a 2nd grader, how would
you do it? Provide examples so that
they can understand.
Page 35 of 37
LESSON 5 RUBRIC
GOLDEN PROBLEM
Score
Description
3
Student has an understanding of multiplication and division. Student
correctly determines the amount of children (including Juan) to be 8. In
addition, the student correctly identifies the total number of tickets
needed to be is 72. The student correctly determines the amount of
tickets each group gets is 36 (72 total tickets divided by 2 groups). The
student then identifies that the amount of tickets per group (36) must be
divided by the amount of people in each group (36/4). The students
identifies that each person will get 9 tickets in each group (9x4 = 36). All
of the information and explains his/her conclusion through the use of
mathematical language, pictures and diagrams, and/or mathematical
processes.
Student has an understanding of multiplication and division, however the
student does not identify each the amount of tickets each student is to
receive. Student has an understanding of dividing the amount of tickets
(72) by 2 for the 2 groups, however does not identify what each student
should get. The student shows his/her work, however, has limited
explanation through the use of language, pictures, diagrams, and/or
mathematical processes.
Student may determine how many children are at the party, but fails to
figure out the total number of tickets that are needed. The student does
not show work and has flaws in their approach to answer the problem.
Does not address task, unresponsive, unrelated or inappropriate.
2
1
0
Page 36 of 37
Third Grade CCSSM Fluencies
Skills
Multiply/divide within 100 (By end of year, know from memory all
products of two one‐digit numbers)
Add/subtract within 1000
Skill builders for the above fluencies.
1. Addition Worksheet Two Plus Two Digit Addition Version 3
Answer Key
2. Addition Worksheet Three Plus Two Digit Addition Version 3
Answer Key
3. Multiplication M + N Two Minute Test Version 1
Answer Key
4. Multiplication M + N Two Minute Test Version 2
Answer Key
Page 37 of 37