Properties of Multiplication Worksheets for 3rd Grade Students

Master properties of multiplication: understand the rules, practice with exercises, and learn visual strategies through these 10 detailed problems.

Properties of Multiplication: Exercises 1 to 5
1 Commutative Property
Exercise 1
Show that 4 × 3 = 3 × 4 using the commutative property.
Definition:

Commutative Property of Multiplication: The order of factors does not change the product. For any numbers a and b: a × b = b × a.

Property method:
  1. Calculate the left side of the equation (4 × 3)
  2. Calculate the right side of the equation (3 × 4)
  3. Compare the results
Left Side
4 × 3 = 12
Right Side
3 × 4 = 12
Result
Both equal 12
Step 1: Calculate the left side

4 × 3 = 12

Step 2: Calculate the right side

3 × 4 = 12

Step 3: Compare the results

Both sides equal 12, confirming the commutative property

4 × 3 = 3 × 4 = 12
Final answer:

4 × 3 = 3 × 4 = 12, proving the commutative property of multiplication

Applied rules:

• Commutative property: Changing the order of factors doesn't change the product

• Order independence: a × b = b × a for any numbers a and b

• Product preservation: The result remains the same regardless of order

2 Associative Property
Exercise 2
Show that (2 × 3) × 4 = 2 × (3 × 4) using the associative property.
Definition:

Associative Property of Multiplication: The grouping of factors does not change the product. For any numbers a, b, and c: (a × b) × c = a × (b × c).

Left Side
(2 × 3) × 4 = 24
Right Side
2 × (3 × 4) = 24
Result
Both equal 24
Step 1: Calculate the left side

(2 × 3) × 4 = 6 × 4 = 24

Step 2: Calculate the right side

2 × (3 × 4) = 2 × 12 = 24

Step 3: Compare the results

Both sides equal 24, confirming the associative property

(2 × 3) × 4 = 2 × (3 × 4) = 24
Final answer:

(2 × 3) × 4 = 2 × (3 × 4) = 24, proving the associative property of multiplication

Applied rules:

• Associative property: Changing the grouping of factors doesn't change the product

• Grouping independence: (a × b) × c = a × (b × c) for any numbers a, b, and c

• Product preservation: The result remains the same regardless of grouping

3 Identity Property
Exercise 3
Show that 7 × 1 = 7 using the identity property.
Definition:

Identity Property of Multiplication: Multiplying any number by 1 gives the same number. For any number a: a × 1 = a.

Left Side
7 × 1 = 7
Right Side
7
Result
Both equal 7
Step 1: Calculate the left side

7 × 1 = 7

Step 2: Compare with the right side

The right side is simply 7

Step 3: Verify the property

Both sides equal 7, confirming the identity property

7 × 1 = 7
Final answer:

7 × 1 = 7, proving the identity property of multiplication

Applied rules:

• Identity property: Multiplying by 1 doesn't change the number

• One as identity: One is the identity element for multiplication

• Number preservation: a × 1 = a for any number a

Properties of Multiplication: Exercises 6 to 10
4 Zero Property
Exercise 4
Show that 9 × 0 = 0 using the zero property.
Definition:

Zero Property of Multiplication: Multiplying any number by 0 gives 0. For any number a: a × 0 = 0.

Left Side
9 × 0 = 0
Right Side
0
Result
Both equal 0
Step 1: Calculate the left side

9 × 0 = 0

Step 2: Compare with the right side

The right side is simply 0

Step 3: Verify the property

Both sides equal 0, confirming the zero property

9 × 0 = 0
Final answer:

9 × 0 = 0, proving the zero property of multiplication

Applied rules:

• Zero property: Multiplying by zero always results in zero

• Zero multiplication: Any number times zero equals zero

• Product of zero: a × 0 = 0 for any number a

5 Commutative Property Practice
Exercise 5
Complete: 6 × 5 = __ × 6
Definition:

Commutative Property of Multiplication: The order of factors does not change the product. We can use this property to find missing numbers.

Given
6 × 5 = __ × 6
Rule
Swap the factors
Result
5 × 6
Step 1: Identify the known factors

We have 6 × 5 on the left side

Step 2: Apply the commutative property

The missing number must be 5 to maintain the same product

Step 3: Verify the answer

6 × 5 = 30 and 5 × 6 = 30

6 × 5 = 5 × 6
Final answer:

6 × 5 = 5 × 6, so the missing number is 5

Applied rules:

• Commutative property: a × b = b × a

• Factor swapping: The factors can be switched

• Product equality: The result remains the same

6 Associative Property Practice
Exercise 6
Complete: (4 × 2) × 3 = 4 × (__ × 3)
Definition:

Associative Property of Multiplication: The grouping of factors does not change the product. We can use this to find missing numbers.

Given
(4 × 2) × 3 = 4 × (__ × 3)
Rule
Regroup the factors
Result
4 × (2 × 3)
Step 1: Identify the pattern

We have (4 × 2) × 3 on the left side

Step 2: Apply the associative property

We need to group 2 and 3 together on the right side

Step 3: Identify the missing number

The missing number is 2

Step 4: Verify the answer

(4 × 2) × 3 = 8 × 3 = 24 and 4 × (2 × 3) = 4 × 6 = 24

(4 × 2) × 3 = 4 × (2 × 3)
Final answer:

(4 × 2) × 3 = 4 × (2 × 3), so the missing number is 2

Applied rules:

• Associative property: (a × b) × c = a × (b × c)

• Factor grouping: Factors can be grouped differently

• Product preservation: The result remains the same

7 Identity Property Practice
Exercise 7
Complete: __ × 1 = 12
Definition:

Identity Property of Multiplication: Multiplying any number by 1 gives the same number. We can use this to find missing numbers.

Given
__ × 1 = 12
Rule
Identity property
Result
12
Step 1: Identify the known information

We know the result is 12 and one factor is 1

Step 2: Apply the identity property

Any number multiplied by 1 equals itself

Step 3: Identify the missing number

The missing number must be 12

Step 4: Verify the answer

12 × 1 = 12

12 × 1 = 12
Final answer:

12 × 1 = 12, so the missing number is 12

Applied rules:

• Identity property: a × 1 = a

• One as identity: One preserves the number

• Number preservation: The number stays the same

8 Zero Property Practice
Exercise 8
Complete: 15 × __ = 0
Definition:

Zero Property of Multiplication: Multiplying any number by 0 gives 0. We can use this to find missing numbers.

Given
15 × __ = 0
Rule
Zero property
Result
0
Step 1: Identify the known information

We know the result is 0 and one factor is 15

Step 2: Apply the zero property

Only when multiplying by 0 can we get 0 as a result

Step 3: Identify the missing number

The missing number must be 0

Step 4: Verify the answer

15 × 0 = 0

15 × 0 = 0
Final answer:

15 × 0 = 0, so the missing number is 0

Applied rules:

• Zero property: a × 0 = 0

• Zero multiplication: Only 0 gives 0 as product

• Product of zero: Any number times 0 equals 0

9 Commutative Property Challenge
Exercise 9
Which expression is equivalent to 8 × 7?
Definition:

Commutative Property of Multiplication: The order of factors does not change the product. This allows us to rearrange factors.

Given
8 × 7
Rule
Swap the factors
Result
7 × 8
Step 1: Identify the original expression

We have 8 × 7

Step 2: Apply the commutative property

Switch the order of the factors

Step 3: Write the equivalent expression

8 × 7 = 7 × 8

Step 4: Verify the equivalence

8 × 7 = 56 and 7 × 8 = 56

8 × 7 = 7 × 8
Final answer:

7 × 8 is equivalent to 8 × 7

Applied rules:

• Commutative property: a × b = b × a

• Factor swapping: The order of factors can be changed

• Product equality: Both expressions yield the same result

10 Associative Property Challenge
Exercise 10
Which expression is equivalent to (5 × 2) × 4?
Definition:

Associative Property of Multiplication: The grouping of factors does not change the product. This allows us to regroup factors.

Given
(5 × 2) × 4
Rule
Regroup the factors
Result
5 × (2 × 4)
Step 1: Identify the original expression

We have (5 × 2) × 4

Step 2: Apply the associative property

Regroup the factors differently

Step 3: Write the equivalent expression

(5 × 2) × 4 = 5 × (2 × 4)

Step 4: Verify the equivalence

(5 × 2) × 4 = 10 × 4 = 40 and 5 × (2 × 4) = 5 × 8 = 40

(5 × 2) × 4 = 5 × (2 × 4)
Final answer:

5 × (2 × 4) is equivalent to (5 × 2) × 4

Applied rules:

• Associative property: (a × b) × c = a × (b × c)

• Factor regrouping: The grouping of factors can be changed

• Product preservation: The result remains the same regardless of grouping

Complete Summary: Properties of Multiplication
Key Concepts & Definitions
Multiplication:
Multiplication is a mathematical operation that represents repeated addition. For example, 3 × 4 means adding 3 four times: 3 + 3 + 3 + 3 = 12.
Factors:
The numbers being multiplied together in a multiplication problem. In 5 × 3 = 15, the factors are 5 and 3.
Product:
The result of multiplying two or more numbers together. In 5 × 3 = 15, the product is 15.
Properties of Multiplication:
Special rules that govern how multiplication behaves with different numbers and arrangements.
Core Rules & Principles
Commutative Property:
  • Rule: a × b = b × a
  • Meaning: The order of factors doesn't change the product
  • Example: 4 × 5 = 5 × 4 = 20
Associative Property:
  • Rule: (a × b) × c = a × (b × c)
  • Meaning: The grouping of factors doesn't change the product
  • Example: (2 × 3) × 4 = 2 × (3 × 4) = 24
Identity Property:
  • Rule: a × 1 = a
  • Meaning: Multiplying by 1 doesn't change the number
  • Example: 7 × 1 = 7
Zero Property:
  • Rule: a × 0 = 0
  • Meaning: Multiplying by 0 always results in 0
  • Example: 9 × 0 = 0
Step-by-Step Methods
Method for Identifying Properties:
  1. Look at the equation carefully
  2. Identify what has changed between the two sides
  3. Determine if factors have been reordered (commutative)
  4. Determine if factors have been regrouped (associative)
  5. Look for multiplication by 1 (identity) or 0 (zero)
Examples & Applications
Commutative Property Examples:
  • 3 × 6 = 6 × 3 = 18
  • 8 × 2 = 2 × 8 = 16
  • 10 × 4 = 4 × 10 = 40
  • 1 × 9 = 9 × 1 = 9
Associative Property Examples:
  • (2 × 3) × 5 = 2 × (3 × 5) = 30
  • (4 × 2) × 3 = 4 × (2 × 3) = 24
  • (1 × 6) × 4 = 1 × (6 × 4) = 24
  • (5 × 2) × 2 = 5 × (2 × 2) = 20
Identity Property Examples:
  • 5 × 1 = 5
  • 12 × 1 = 12
  • 1 × 8 = 8
  • 25 × 1 = 25
Zero Property Examples:
  • 7 × 0 = 0
  • 0 × 15 = 0
  • 100 × 0 = 0
  • 0 × 3 = 0
Tips, Tricks & Common Pitfalls
Helpful Tips:
  • Remember: "Commutative = Commute (move around)" - order can change
  • Remember: "Associative = Associate (group together)" - grouping can change
  • Think of 1 as the "identity" - it keeps the number the same
  • Remember that 0 "zeros out" any number it multiplies
Common Mistakes to Avoid:
  • Confusing commutative and associative properties
  • Thinking that subtraction and division have the same properties
  • Forgetting that the zero property always results in 0
  • Mixing up the identity elements (1 for multiplication, 0 for addition)
Memory Aids & Takeaways
Key Points to Remember:
  • "Order doesn't matter in multiplication" (Commutative)
  • "Grouping doesn't matter in multiplication" (Associative)
  • "Multiplying by 1 doesn't change the number" (Identity)
  • "Multiplying by 0 always gives 0" (Zero)
a × b = b × a
Commutative Property
(a × b) × c = a × (b × c)
Associative Property
a × 1 = a
Identity Property
a × 0 = 0
Zero Property
Tip 1: Remember that the commutative property allows you to switch the order of numbers in multiplication.
Tip 2: The associative property lets you regroup numbers when multiplying three or more numbers.
Tip 3: Multiplying by 1 doesn't change the original number.
Tip 4: Multiplying by 0 always results in 0, no matter what the other number is.
Visual Guide: Properties of Multiplication
Multiplication Properties
🔢
Commutative Property:

Order of factors can be changed without affecting the product

Associative Property:

Grouping of factors can be changed without affecting the product

Identity Property:

Multiplying by 1 doesn't change the number

Zero Property:

Multiplying by 0 always results in 0

4
×
3
=
3
×
4
=
12
(2×3)
×
4
=
2
×
(3×4)
=
24
Commutative

• a × b = b × a

• Order can change

• Example: 5×6 = 6×5

Associative

• (a×b)×c = a×(b×c)

• Grouping can change

• Example: (2×3)×4 = 2×(3×4)

Identity

• a × 1 = a

• Multiplied by 1

• Example: 7×1 = 7

Zero

• a × 0 = 0

• Multiplied by 0

• Example: 9×0 = 0

Commutative
4×5 = 5×4
Associative
(3×2)×4 = 3×(2×4)
Identity
8×1 = 8
Zero
6×0 = 0
Property Formula Example
Commutative a × b = b × a 3 × 4 = 4 × 3 = 12
Associative (a × b) × c = a × (b × c) (2 × 3) × 4 = 2 × (3 × 4) = 24
Identity a × 1 = a 7 × 1 = 7
Zero a × 0 = 0 9 × 0 = 0
Challenge 1
5 × 3 = 3 × ?
5
Challenge 2
(4 × 2) × 3 = 4 × (? × 3)
2
Challenge 3
? × 1 = 15
15
Challenge 4
12 × ? = 0
0

Questions & Answers

Question: Why do we need to learn these multiplication properties? They seem obvious.

Answer: These properties might seem obvious now, but they're actually very powerful tools! They help you:

  • Make mental math easier (like knowing that 6 × 7 is the same as 7 × 6)
  • Solve problems more quickly by choosing the easiest order
  • Check your work by trying a different arrangement
  • Understand why math works the way it does

For example, if you know 7 × 6 = 42, then you automatically know 6 × 7 = 42 too!

Question: My child is confusing the commutative and associative properties. How can I help them remember the difference?

Answer: Use these memory aids:

  • Commutative: Think "commute" - like going back and forth. The numbers move positions (a × b = b × a).
  • Associative: Think "associate" - like grouping friends together. The parentheses move but the numbers stay in the same order.

Commutative changes the order of numbers, associative changes the grouping of numbers.

Question: Do these properties work for addition too?

Answer: Yes! Addition has the same properties:

  • Commutative: a + b = b + a (like 3 + 5 = 5 + 3)
  • Associative: (a + b) + c = a + (b + c) (like (2 + 3) + 4 = 2 + (3 + 4))
  • Identity: a + 0 = a (adding zero doesn't change the number)

However, there's no zero property for addition since adding zero doesn't change the number.