Properties of Addition and Multiplication for 3rd Grade Students

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

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

Commutative Property of Addition: The order of numbers in addition does not change the sum. For any numbers a and b: a + b = b + a.

Property method:
  1. Calculate the left side of the equation (3 + 5)
  2. Calculate the right side of the equation (5 + 3)
  3. Compare the results
Left Side
3 + 5 = 8
Right Side
5 + 3 = 8
Result
Both equal 8
Step 1: Calculate the left side

3 + 5 = 8

Step 2: Calculate the right side

5 + 3 = 8

Step 3: Compare the results

Both sides equal 8, confirming the commutative property

3 + 5 = 5 + 3 = 8
Final answer:

3 + 5 = 5 + 3 = 8, proving the commutative property of addition

Applied rules:

• Commutative property: Changing the order of addends doesn't change the sum

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

• Sum preservation: The total remains the same regardless of order

2 Commutative Property of Multiplication
Exercise 2
Show that 4 × 6 = 6 × 4 using the commutative property.
Definition:

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

Left Side
4 × 6 = 24
Right Side
6 × 4 = 24
Result
Both equal 24
Step 1: Calculate the left side

4 × 6 = 24

Step 2: Calculate the right side

6 × 4 = 24

Step 3: Compare the results

Both sides equal 24, confirming the commutative property

4 × 6 = 6 × 4 = 24
Final answer:

4 × 6 = 6 × 4 = 24, 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

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

Associative Property of Addition: The grouping of numbers in addition does not change the sum. For any numbers a, b, and c: (a + b) + c = a + (b + c).

Left Side
(2 + 3) + 4 = 9
Right Side
2 + (3 + 4) = 9
Result
Both equal 9
Step 1: Calculate the left side

(2 + 3) + 4 = 5 + 4 = 9

Step 2: Calculate the right side

2 + (3 + 4) = 2 + 7 = 9

Step 3: Compare the results

Both sides equal 9, confirming the associative property

(2 + 3) + 4 = 2 + (3 + 4) = 9
Final answer:

(2 + 3) + 4 = 2 + (3 + 4) = 9, proving the associative property of addition

Applied rules:

• Associative property: Changing the grouping of addends doesn't change the sum

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

• Sum preservation: The total remains the same regardless of grouping

Properties of Addition and Multiplication: Exercises 6 to 10
4 Associative Property of Multiplication
Exercise 4
Show that (2 × 3) × 4 = 2 × (3 × 4) using the associative property.
Definition:

Associative Property of Multiplication: The grouping of numbers in multiplication 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

5 Identity Property of Addition
Exercise 5
Show that 7 + 0 = 7 using the identity property of addition.
Definition:

Identity Property of Addition: Adding zero to any number results in the same number. For any number a: a + 0 = a.

Left Side
7 + 0 = 7
Right Side
7
Result
Both equal 7
Step 1: Calculate the left side

7 + 0 = 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 + 0 = 7
Final answer:

7 + 0 = 7, proving the identity property of addition

Applied rules:

• Identity property: Adding zero doesn't change the number

• Zero as identity: Zero is the identity element for addition

• Number preservation: a + 0 = a for any number a

6 Identity Property of Multiplication
Exercise 6
Show that 9 × 1 = 9 using the identity property of multiplication.
Definition:

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

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

9 × 1 = 9

Step 2: Compare with the right side

The right side is simply 9

Step 3: Verify the property

Both sides equal 9, confirming the identity property

9 × 1 = 9
Final answer:

9 × 1 = 9, proving the identity property of multiplication

Applied rules:

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

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

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

7 Zero Property of Multiplication
Exercise 7
Show that 5 × 0 = 0 using the zero property of multiplication.
Definition:

Zero Property of Multiplication: Multiplying any number by zero results in zero. For any number a: a × 0 = 0.

Left Side
5 × 0 = 0
Right Side
0
Result
Both equal 0
Step 1: Calculate the left side

5 × 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

5 × 0 = 0
Final answer:

5 × 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

8 Commutative Property of Addition
Exercise 8
Show that 12 + 8 = 8 + 12 using the commutative property.
Definition:

Commutative Property of Addition: The order of numbers in addition does not change the sum. For any numbers a and b: a + b = b + a.

Left Side
12 + 8 = 20
Right Side
8 + 12 = 20
Result
Both equal 20
Step 1: Calculate the left side

12 + 8 = 20

Step 2: Calculate the right side

8 + 12 = 20

Step 3: Compare the results

Both sides equal 20, confirming the commutative property

12 + 8 = 8 + 12 = 20
Final answer:

12 + 8 = 8 + 12 = 20, proving the commutative property of addition

Applied rules:

• Commutative property: Changing the order of addends doesn't change the sum

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

• Sum preservation: The total remains the same regardless of order

9 Commutative Property of Multiplication
Exercise 9
Show that 7 × 3 = 3 × 7 using the commutative property.
Definition:

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

Left Side
7 × 3 = 21
Right Side
3 × 7 = 21
Result
Both equal 21
Step 1: Calculate the left side

7 × 3 = 21

Step 2: Calculate the right side

3 × 7 = 21

Step 3: Compare the results

Both sides equal 21, confirming the commutative property

7 × 3 = 3 × 7 = 21
Final answer:

7 × 3 = 3 × 7 = 21, 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

10 Associative Property of Addition
Exercise 10
Show that (1 + 2) + 3 = 1 + (2 + 3) using the associative property.
Definition:

Associative Property of Addition: The grouping of numbers in addition does not change the sum. For any numbers a, b, and c: (a + b) + c = a + (b + c).

Left Side
(1 + 2) + 3 = 6
Right Side
1 + (2 + 3) = 6
Result
Both equal 6
Step 1: Calculate the left side

(1 + 2) + 3 = 3 + 3 = 6

Step 2: Calculate the right side

1 + (2 + 3) = 1 + 5 = 6

Step 3: Compare the results

Both sides equal 6, confirming the associative property

(1 + 2) + 3 = 1 + (2 + 3) = 6
Final answer:

(1 + 2) + 3 = 1 + (2 + 3) = 6, proving the associative property of addition

Applied rules:

• Associative property: Changing the grouping of addends doesn't change the sum

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

• Sum preservation: The total remains the same regardless of grouping

Complete Summary: Properties of Addition and Multiplication
Key Concepts & Definitions
Commutative Property:
The commutative property states that changing the order of numbers in addition or multiplication does not change the result. For addition: a + b = b + a. For multiplication: a × b = b × a.
Associative Property:
The associative property states that changing the grouping of numbers in addition or multiplication does not change the result. For addition: (a + b) + c = a + (b + c). For multiplication: (a × b) × c = a × (b × c).
Identity Property:
The identity property states that adding zero to any number or multiplying any number by one leaves the number unchanged. For addition: a + 0 = a. For multiplication: a × 1 = a.
Core Rules & Principles
Commutative Property:
  • a + b = b + a (addition)
  • a × b = b × a (multiplication)
  • Order can be changed without affecting the result
Associative Property:
  • (a + b) + c = a + (b + c) (addition)
  • (a × b) × c = a × (b × c) (multiplication)
  • Grouping can be changed without affecting the result
Identity Property:
  • a + 0 = a (zero is the additive identity)
  • a × 1 = a (one is the multiplicative identity)
  • The number remains unchanged
Zero Property:
  • a × 0 = 0
  • Any number multiplied by zero equals zero
Step-by-Step Methods
Method for Verifying Properties:
  1. Identify the property being tested
  2. Calculate the left side of the equation
  3. Calculate the right side of the equation
  4. Compare the results
  5. Confirm they are equal
Examples & Applications
Commutative Property Examples:
  • 3 + 5 = 5 + 3 = 8
  • 4 × 6 = 6 × 4 = 24
  • 12 + 7 = 7 + 12 = 19
  • 9 × 2 = 2 × 9 = 18
Associative Property Examples:
  • (2 + 3) + 4 = 2 + (3 + 4) = 9
  • (5 × 2) × 3 = 5 × (2 × 3) = 30
  • (1 + 4) + 6 = 1 + (4 + 6) = 11
  • (3 × 4) × 2 = 3 × (4 × 2) = 24
Identity Property Examples:
  • 7 + 0 = 7
  • 9 × 1 = 9
  • 15 + 0 = 15
  • 12 × 1 = 12
Tips, Tricks & Common Pitfalls
Helpful Tips:
  • Remember: "commute" means to move or change places
  • Remember: "associate" means to group together
  • Zero is the identity for addition, one is the identity for multiplication
  • These properties only apply to addition and multiplication, not subtraction or division
Common Mistakes to Avoid:
  • Applying commutative property to subtraction (a - b ≠ b - a)
  • Applying commutative property to division (a ÷ b ≠ b ÷ a)
  • Confusing the identity elements (zero for addition, one for multiplication)
  • Thinking that zero is the identity for multiplication (it's not!)
Memory Aids & Takeaways
Key Points to Remember:
  • "Commutative = Commute (move around)" - order can change
  • "Associative = Associate (group together)" - grouping can change
  • "Identity = Stays the same" - adding zero or multiplying by one
  • "Zero times anything is zero" - always true
  • These properties help make calculations easier!
a + b = b + a
Commutative Property of Addition
a × b = b × a
Commutative Property of Multiplication
(a + b) + c = a + (b + c)
Associative Property of Addition
(a × b) × c = a × (b × c)
Associative Property of Multiplication
a + 0 = a
Identity Property of Addition
a × 1 = a
Identity Property of Multiplication
a × 0 = 0
Zero Property of Multiplication
Tip 1: Remember that commutative property allows you to switch the order of numbers in addition and multiplication.
Tip 2: The associative property lets you regroup numbers when adding or multiplying three or more numbers.
Tip 3: Adding zero or multiplying by one doesn't change the original number.
Tip 4: These properties don't work for subtraction or division!
Visual Guide: Properties of Addition and Multiplication
Math Properties
🧮
Commutative Property:

Order can be changed without affecting the result

Associative Property:

Grouping can be changed without affecting the result

Identity Property:

Adding zero or multiplying by one leaves the number unchanged

3
+
5
=
5
+
3
=
8
4
×
6
=
6
×
4
=
24
(2+3)
+
4
=
2
+
(3+4)
=
9
Addition Properties

• Commutative: a + b = b + a

• Associative: (a + b) + c = a + (b + c)

• Identity: a + 0 = a

Multiplication Properties

• Commutative: a × b = b × a

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

• Identity: a × 1 = a

• Zero: a × 0 = 0

Commutative
a + b = b + a
3 + 5 = 5 + 3
Order can change
Commutative
a × b = b × a
4 × 6 = 6 × 4
Order can change
Associative
(a + b) + c = a + (b + c)
(2 + 3) + 4 = 2 + (3 + 4)
Grouping can change
Associative
(a × b) × c = a × (b × c)
(2 × 3) × 4 = 2 × (3 × 4)
Grouping can change
a + 0 = a
Identity of Addition
a × 1 = a
Identity of Multiplication
a × 0 = 0
Zero Property
Property Formula Example
Commutative (Addition) a + b = b + a 3 + 5 = 5 + 3 = 8
Commutative (Multiplication) a × b = b × a 4 × 6 = 6 × 4 = 24
Associative (Addition) (a + b) + c = a + (b + c) (2 + 3) + 4 = 2 + (3 + 4) = 9
Associative (Multiplication) (a × b) × c = a × (b × c) (2 × 3) × 4 = 2 × (3 × 4) = 24
Identity (Addition) a + 0 = a 7 + 0 = 7
Identity (Multiplication) a × 1 = a 9 × 1 = 9
Zero Property a × 0 = 0 5 × 0 = 0
Challenge 1
(3 + 4) + 5
3 + (4 + 5) = 12
Challenge 2
8 × 1
8
Challenge 3
6 + 0
6
Challenge 4
7 × 0
0

Questions & Answers

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

Answer: These properties might seem obvious, but they're very important! They help you:

  • Make mental math easier (like switching 2 + 99 to 99 + 2)
  • Understand why math works the way it does
  • Solve more complex problems efficiently
  • Check your work and catch mistakes

For example, if you know that 3 + 5 = 5 + 3, you can solve whichever version is easier for you!

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

Answer: Use these memory aids:

  • Commutative: "Commuter" - like commuting to work, numbers commute or move around (a + b = b + a)
  • Associative: "Associate" - like associating with friends, numbers group together differently ((a + b) + c = a + (b + c))

Commutative changes the order, associative changes the grouping.

Question: Do these properties work for subtraction and division too?

Answer: No, these properties do NOT work for subtraction and division! Let's see why:

  • Subtraction: 5 - 3 = 2, but 3 - 5 = -2 (not the same!)
  • Division: 6 ÷ 2 = 3, but 2 ÷ 6 = 1/3 (not the same!)

The commutative, associative, and identity properties only work for addition and multiplication!