Solved Exercises on Properties of Operations in Math for Grade 4

Master properties of operations in math: commutative, associative, distributive, and identity properties through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Commutative property
Exercise 1
Demonstrate the commutative property of addition and multiplication with 9 + 4 and 6 × 3
Definition:

Commutative property: The order of numbers doesn't change the result in addition or multiplication. For addition: a + b = b + a. For multiplication: a × b = b × a.

Commutative property verification method:
  1. Take two numbers and perform the operation
  2. Switch the order of the numbers
  3. Perform the same operation again
  4. Compare the results
Addition
9 + 4 = 13
Switched
4 + 9 = 13
Result
Same!
Multiplication
6 × 3 = 18
Switched
3 × 6 = 18
Result
Same!
Step 1: Test addition

9 + 4 = 13

4 + 9 = 13

The order doesn't matter in addition!

Step 2: Test multiplication

6 × 3 = 18

3 × 6 = 18

The order doesn't matter in multiplication!

Step 3: Verify the property

Both operations give the same result regardless of order

9 + 4 = 4 + 9 = 13
6 × 3 = 3 × 6 = 18
Final answer:

The commutative property holds: changing the order of numbers doesn't change the result in addition and multiplication

Applied rules:

• Commutative property: Order doesn't matter in addition and multiplication

• Verification: Always check both orders to confirm

• Limitation: Does not apply to subtraction or division

2 Associative property
Exercise 2
Demonstrate the associative property with (5 + 3) + 7 and 5 + (3 + 7)
Definition:

Associative property: The grouping of numbers doesn't change the result in addition or multiplication. For addition: (a + b) + c = a + (b + c). For multiplication: (a × b) × c = a × (b × c).

Grouping 1
(5 + 3) + 7
Grouping 2
5 + (3 + 7)
Result
Both equal 15
Step 1: Solve first grouping

(5 + 3) + 7 = 8 + 7 = 15

First solve inside parentheses: 5 + 3 = 8

Then add: 8 + 7 = 15

Step 2: Solve second grouping

5 + (3 + 7) = 5 + 10 = 15

First solve inside parentheses: 3 + 7 = 10

Then add: 5 + 10 = 15

Step 3: Compare results

Both groupings equal 15, confirming the associative property

(5 + 3) + 7 = 5 + (3 + 7) = 15
Final answer:

The associative property holds: changing the grouping doesn't change the result in addition

Applied rules:

• Associative property: Grouping doesn't matter in addition and multiplication

• Parentheses: Always solve operations inside parentheses first

• Consistency: Result remains the same regardless of grouping

3 Distributive property
Exercise 3
Apply the distributive property to 4 × (6 + 3)
Definition:

Distributive property: Multiplying a number by a sum equals multiplying the number by each addend separately and then adding the results: a × (b + c) = (a × b) + (a × c).

Original
4 × (6 + 3)
Distributed
(4 × 6) + (4 × 3)
Result
36
Step 1: Solve inside parentheses first

4 × (6 + 3) = 4 × 9 = 36

First add: 6 + 3 = 9

Then multiply: 4 × 9 = 36

Step 2: Apply the distributive property

4 × (6 + 3) = (4 × 6) + (4 × 3)

Multiply 4 by each number inside the parentheses

Step 3: Calculate each multiplication

4 × 6 = 24 and 4 × 3 = 12

Step 4: Add the results

24 + 12 = 36

4 × (6 + 3) = (4 × 6) + (4 × 3) = 24 + 12 = 36
Final answer:

The distributive property holds: 4 × (6 + 3) = (4 × 6) + (4 × 3) = 36

Applied rules:

• Distributive property: Multiply the outside number by each term inside

• Order of operations: Parentheses first, then multiplication

• Verification: Both methods yield the same result

Properties of Operations Summary
\(a × (b + c) = (a × b) + (a × c)\)
Distributive Property
Commutative
a + b = b + a
Order doesn't matter
Commutative
a × b = b × a
Order doesn't matter
Associative
(a + b) + c = a + (b + c)
Grouping doesn't matter
Associative
(a × b) × c = a × (b × c)
Grouping doesn't matter
Distributive
a × (b + c) = (a × b) + (a × c)
Distribute multiplication over addition
Identity
a + 0 = a
Additive Identity
Identity
a × 1 = a
Multiplicative Identity
Zero
a × 0 = 0
Multiplication by Zero
Solution: Exercises 4 to 5
4 Identity properties
Exercise 4
Demonstrate identity properties with 15 + 0, 8 × 1, and 12 × 0
Definition:

Identity properties: Special numbers that don't change other numbers when combined with them. Additive identity: a + 0 = a. Multiplicative identity: a × 1 = a. Zero property: a × 0 = 0.

Addition
15 + 0 = 15
Multiplication
8 × 1 = 8
Zero
12 × 0 = 0
Step 1: Test additive identity

15 + 0 = 15

Adding zero doesn't change the number

Step 2: Test multiplicative identity

8 × 1 = 8

Multiplying by one doesn't change the number

Step 3: Test zero property

12 × 0 = 0

Any number multiplied by zero equals zero

Step 4: Verify the properties

Each identity property preserves or transforms the number as expected

15 + 0 = 15, 8 × 1 = 8, 12 × 0 = 0
Final answer:

The identity properties hold: adding 0, multiplying by 1, or multiplying by 0 behave predictably

Applied rules:

• Additive identity: Zero is the identity element for addition

• Multiplicative identity: One is the identity element for multiplication

• Zero property: Any number times zero equals zero

5 Mixed properties application
Exercise 5
Solve: 3 × (4 + 5) + 0 × 7 using multiple properties
Definition:

Mixed properties application: Problems that require using multiple properties together. We solve step by step, applying the correct property at each stage.

Original
3 × (4 + 5) + 0 × 7
After distributive
3 × 9 + 0
Final
27
Step 1: Solve inside parentheses

4 + 5 = 9

Step 2: Apply zero property

0 × 7 = 0 (any number times zero equals zero)

Step 3: Substitute values

3 × (4 + 5) + 0 × 7 = 3 × 9 + 0

Step 4: Complete the calculation

3 × 9 + 0 = 27 + 0 = 27

Step 5: Apply additive identity

27 + 0 = 27 (adding zero doesn't change the number)

3 × (4 + 5) + 0 × 7 = 27
Final answer:

Using multiple properties: 3 × (4 + 5) + 0 × 7 = 27

Applied rules:

• Order of operations: Solve parentheses first

• Zero property: 0 × 7 = 0

• Additive identity: 27 + 0 = 27

• Multiple properties: Different properties apply to different parts

Properties of Operations in Math Guide
\(a × (b + c) = (a × b) + (a × c)\)
Distributive Property
Key definitions:

Commutative property: The order of numbers doesn't change the result in addition or multiplication

Associative property: The grouping of numbers doesn't change the result in addition or multiplication

Distributive property: Multiplication can be distributed over addition or subtraction

Identity property: Special numbers that preserve the identity of other numbers when combined

Properties identification method:
  1. Look for order changes: If numbers switch places but result stays same → Commutative
  2. Look for grouping changes: If parentheses move but result stays same → Associative
  3. Look for distribution: If multiplication is spread across addition → Distributive
  4. Look for 0 or 1: If 0 or 1 appears → Identity or Zero property
Tip 1: Remember: "Commutative" sounds like "commute" - numbers commute (switch places).
Tip 2: "Associative" sounds like "associate" - numbers associate (group) differently.
Tip 3: The distributive property is like giving out items to a group.
Tip 4: Zero is like a "black hole" in multiplication - anything times 0 is 0.
Common error: Thinking subtraction and division are commutative (they're not!).
Memory aid: Think of properties as "rules" that make math easier to work with.
Properties to remember:

• Commutative: a + b = b + a and a × b = b × a

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

• Distributive: a × (b + c) = (a × b) + (a × c)

• Identity: a + 0 = a and a × 1 = a

• Zero: a × 0 = 0

Properties of Operations: Concept Map
Exercise 6: Property Relationships
Visualize how different properties relate to each other in mathematical operations:
Commutative, Associative, Distributive, and Identity properties

Analysis: The chart shows how different properties work together in mathematical operations.

  • Commutative property: Changes order but keeps result the same
  • Associative property: Changes grouping but keeps result the same
  • Distributive property: Breaks down complex operations into simpler ones
  • Identity property: Preserves the original number

Questions & Answers

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

Answer: These properties might seem obvious now, but they're actually very important because:

  • They help us solve problems faster and more efficiently
  • They allow us to rearrange numbers to make calculations easier
  • They form the foundation for more advanced math in higher grades
  • They help us understand why math works the way it does

For example, knowing the commutative property lets you rearrange 8 + 27 + 2 to (8 + 2) + 27 = 10 + 27 = 37, which is much easier to calculate!

Question: Are subtraction and division also commutative and associative like addition and multiplication?

Answer: No, subtraction and division are neither commutative nor associative. Let me show you why:

  • Subtraction is not commutative: 5 - 3 = 2, but 3 - 5 = -2. These are different!
  • Division is not commutative: 6 ÷ 2 = 3, but 2 ÷ 6 = 1/3. These are different!
  • Subtraction is not associative: (10 - 5) - 2 = 5 - 2 = 3, but 10 - (5 - 2) = 10 - 3 = 7. Different results!
  • Division is not associative: (12 ÷ 6) ÷ 2 = 2 ÷ 2 = 1, but 12 ÷ (6 ÷ 2) = 12 ÷ 3 = 4. Different results!

This is why we say these properties only apply to addition and multiplication.

Question: How can I remember the distributive property? It seems complicated.

Answer: Here are some helpful ways to remember the distributive property:

  • Visual analogy: Think of distributing candy to friends - you give the same amount to each person
  • Story method: "Multiply the outside number by each thing inside the parentheses"
  • Pattern: a(b + c) = ab + ac (the a multiplies both b and c)
  • Practice: Start with simple examples like 2(3 + 4) = 2×3 + 2×4 = 6 + 8 = 14

The distributive property is like multiplication reaching inside the parentheses to multiply each term!