Solved Exercises on Properties of Operations Lesson for Grade 4

Master properties of operations through this comprehensive lesson: commutative, associative, distributive, and identity properties through these 5 detailed exercises.

Lesson: Exercises 1 to 3
1 Commutative property lesson
Exercise 1
Lesson: Demonstrate the commutative property of addition and multiplication with 8 + 5 and 5 + 8, then 6 × 4 and 4 × 6
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
8 + 5 = 13
Switched
5 + 8 = 13
Result
Same!
Multiplication
6 × 4 = 24
Switched
4 × 6 = 24
Result
Same!
Step 1: Test addition

8 + 5 = 13

5 + 8 = 13

The order doesn't matter in addition!

Step 2: Test multiplication

6 × 4 = 24

4 × 6 = 24

The order doesn't matter in multiplication!

Step 3: Verify the property

Both operations give the same result regardless of order

8 + 5 = 5 + 8 = 13
6 × 4 = 4 × 6 = 24
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 lesson
Exercise 2
Lesson: Demonstrate the associative property with (3 + 4) + 5 and 3 + (4 + 5), then (2 × 3) × 4 and 2 × (3 × 4)
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).

Addition Grouping 1
(3 + 4) + 5
Addition Grouping 2
3 + (4 + 5)
Result
Both equal 12
Multiplication Grouping 1
(2 × 3) × 4
Multiplication Grouping 2
2 × (3 × 4)
Result
Both equal 24
Step 1: Solve addition first grouping

(3 + 4) + 5 = 7 + 5 = 12

First solve inside parentheses: 3 + 4 = 7

Then add: 7 + 5 = 12

Step 2: Solve addition second grouping

3 + (4 + 5) = 3 + 9 = 12

First solve inside parentheses: 4 + 5 = 9

Then add: 3 + 9 = 12

Step 3: Solve multiplication first grouping

(2 × 3) × 4 = 6 × 4 = 24

First solve inside parentheses: 2 × 3 = 6

Then multiply: 6 × 4 = 24

Step 4: Solve multiplication second grouping

2 × (3 × 4) = 2 × 12 = 24

First solve inside parentheses: 3 × 4 = 12

Then multiply: 2 × 12 = 24

(3 + 4) + 5 = 3 + (4 + 5) = 12
(2 × 3) × 4 = 2 × (3 × 4) = 24
Final answer:

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

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 lesson
Exercise 3
Lesson: Apply the distributive property to 3 × (5 + 2) and 4 × (6 + 1)
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).

Example 1
3 × (5 + 2)
Distributed
(3 × 5) + (3 × 2)
Result
21
Step 1: Solve inside parentheses first

3 × (5 + 2) = 3 × 7 = 21

First add: 5 + 2 = 7

Then multiply: 3 × 7 = 21

Step 2: Apply the distributive property

3 × (5 + 2) = (3 × 5) + (3 × 2)

Multiply 3 by each number inside the parentheses

Step 3: Calculate each multiplication

3 × 5 = 15 and 3 × 2 = 6

Step 4: Add the results

15 + 6 = 21

3 × (5 + 2) = (3 × 5) + (3 × 2) = 15 + 6 = 21
Final answer:

The distributive property holds: 3 × (5 + 2) = (3 × 5) + (3 × 2) = 21

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
Lesson: Exercises 4 to 5
4 Identity properties lesson
Exercise 4
Lesson: Demonstrate identity properties with 17 + 0, 0 + 23, 9 × 1, 1 × 14, 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.

Additive Identity 1
17 + 0 = 17
Additive Identity 2
0 + 23 = 23
Multiplicative Identity
9 × 1 = 9
Zero Property
12 × 0 = 0
Step 1: Test additive identity

17 + 0 = 17

Adding zero doesn't change the number

Step 2: Test additive identity with 0 first

0 + 23 = 23

Adding 23 to zero gives back 23

Step 3: Test multiplicative identity

9 × 1 = 9

Multiplying by one doesn't change the number

Step 4: Test zero property

12 × 0 = 0

Any number multiplied by zero equals zero

Step 5: Verify the properties

Each identity property preserves or transforms the number as expected

17 + 0 = 17, 0 + 23 = 23, 9 × 1 = 9, 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 lesson
Exercise 5
Lesson: Solve: 4 × (3 + 2) + 0 × 6 × 1 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
4 × (3 + 2) + 0 × 6 × 1
After distributive
4 × 5 + 0
Final
20
Step 1: Apply distributive property

4 × (3 + 2) = (4 × 3) + (4 × 2) = 12 + 8 = 20

OR solve inside parentheses first: 3 + 2 = 5, then 4 × 5 = 20

Step 2: Apply zero property

0 × 6 × 1 = 0 (any number times zero equals zero)

Step 3: Substitute values

4 × (3 + 2) + 0 × 6 × 1 = 20 + 0

Step 4: Apply additive identity

20 + 0 = 20 (adding zero doesn't change the number)

4 × (3 + 2) + 0 × 6 × 1 = 20
Final answer:

Using multiple properties: 4 × (3 + 2) + 0 × 6 × 1 = 20

Applied rules:

• Distributive property: 4 × (3 + 2) = (4 × 3) + (4 × 2)

• Zero property: 0 × 6 × 1 = 0

• Additive identity: 20 + 0 = 20

• Multiple properties: Different properties apply to different parts

Properties of Operations Comprehensive Lesson
\(a × (b + c) = (a × b) + (a × c)\)
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

Zero property: Multiplication by zero always results in zero

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: Lesson Overview
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
  • Zero property: Always results in zero when multiplying

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!