Solved Exercises on Properties of Operations in Multiplication for Grade 4

Master multiplication properties: commutative, associative, distributive, and identity properties through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Commutative property of multiplication
Exercise 1
Demonstrate the commutative property of multiplication with 7 × 3 and 3 × 7
Definition:

Commutative property of multiplication: The order of numbers doesn't change the product. For any numbers a and b: a × b = b × a.

Commutative property verification method:
  1. Take two numbers and multiply them in one order
  2. Multiply the same numbers in reverse order
  3. Compare the results
  4. Confirm they are equal
First Order
7 × 3 = 21
Second Order
3 × 7 = 21
Result
Same!
Step 1: Multiply in first order

7 × 3 = 21

Think of 7 groups of 3: 3 + 3 + 3 + 3 + 3 + 3 + 3 = 21

Step 2: Multiply in reverse order

3 × 7 = 21

Think of 3 groups of 7: 7 + 7 + 7 = 21

Step 3: Compare results

Both operations give the same result: 21

Step 4: Verify the property

7 × 3 = 3 × 7 = 21

7 × 3 = 3 × 7 = 21
Final answer:

The commutative property of multiplication holds: 7 × 3 = 3 × 7 = 21

Applied rules:

• Commutative property: Order doesn't matter in multiplication

• Verification: Always check both orders to confirm

• Generalization: This works for any numbers: a × b = b × a

2 Associative property of multiplication
Exercise 2
Demonstrate the associative property with (4 × 2) × 3 and 4 × (2 × 3)
Definition:

Associative property of multiplication: The grouping of numbers doesn't change the product. For any numbers a, b, and c: (a × b) × c = a × (b × c).

Grouping 1
(4 × 2) × 3
Grouping 2
4 × (2 × 3)
Result
Both equal 24
Step 1: Solve first grouping

(4 × 2) × 3 = 8 × 3 = 24

First solve inside parentheses: 4 × 2 = 8

Then multiply: 8 × 3 = 24

Step 2: Solve second grouping

4 × (2 × 3) = 4 × 6 = 24

First solve inside parentheses: 2 × 3 = 6

Then multiply: 4 × 6 = 24

Step 3: Compare results

Both groupings equal 24, confirming the associative property

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

The associative property of multiplication holds: (4 × 2) × 3 = 4 × (2 × 3) = 24

Applied rules:

• Associative property: Grouping doesn't matter in multiplication

• Parentheses: Always solve operations inside parentheses first

• Consistency: Result remains the same regardless of grouping

3 Distributive property over addition
Exercise 3
Apply the distributive property to 5 × (3 + 2)
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
5 × (3 + 2)
Distributed
(5 × 3) + (5 × 2)
Result
25
Step 1: Solve inside parentheses first

5 × (3 + 2) = 5 × 5 = 25

First add: 3 + 2 = 5

Then multiply: 5 × 5 = 25

Step 2: Apply the distributive property

5 × (3 + 2) = (5 × 3) + (5 × 2)

Multiply 5 by each number inside the parentheses

Step 3: Calculate each multiplication

5 × 3 = 15 and 5 × 2 = 10

Step 4: Add the results

15 + 10 = 25

5 × (3 + 2) = (5 × 3) + (5 × 2) = 15 + 10 = 25
Final answer:

The distributive property holds: 5 × (3 + 2) = (5 × 3) + (5 × 2) = 25

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

Multiplication Properties Summary
\(a × (b + c) = (a × b) + (a × c)\)
Distributive Property
Commutative
a × b = b × a
Order 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 × 1 = a
Multiplicative Identity
Zero
a × 0 = 0
Multiplication by Zero
Solution: Exercises 4 to 5
4 Identity and zero properties
Exercise 4
Demonstrate identity and zero properties with 9 × 1, 1 × 12, and 8 × 0
Definition:

Identity property of multiplication: Multiplying any number by one leaves it unchanged: a × 1 = a and 1 × a = a. Zero property of multiplication: Any number multiplied by zero equals zero: a × 0 = 0 and 0 × a = 0.

Identity 1
9 × 1 = 9
Identity 2
1 × 12 = 12
Zero
8 × 0 = 0
Step 1: Test multiplicative identity

9 × 1 = 9

Multiplying by one doesn't change the number

Step 2: Test identity with 1 first

1 × 12 = 12

Multiplying one by any number gives that number

Step 3: Test zero property

8 × 0 = 0

Any number multiplied by zero equals zero

Step 4: Verify the properties

Identity property preserves the number, zero property results in zero

9 × 1 = 9, 1 × 12 = 12, 8 × 0 = 0
Final answer:

The identity and zero properties hold: multiplying by 1 preserves the number, multiplying by 0 results in zero

Applied rules:

• Multiplicative identity: One is the identity element for multiplication

• Zero property: Any number times zero equals zero

• Commutativity: Both properties work in either order

5 Mixed properties application
Exercise 5
Solve: 2 × (3 + 4) × 1 + 0 × 5 using multiple properties
Definition:

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

Original
2 × (3 + 4) × 1 + 0 × 5
After distribution
(2 × 3 + 2 × 4) × 1 + 0
Final
14
Step 1: Apply distributive property

2 × (3 + 4) = (2 × 3) + (2 × 4) = 6 + 8 = 14

Step 2: Apply zero property

0 × 5 = 0 (any number times zero equals zero)

Step 3: Substitute values

2 × (3 + 4) × 1 + 0 × 5 = 14 × 1 + 0

Step 4: Apply identity property

14 × 1 = 14 (multiplying by one doesn't change the number)

Step 5: Complete the calculation

14 × 1 + 0 = 14 + 0 = 14

2 × (3 + 4) × 1 + 0 × 5 = 14
Final answer:

Using multiple properties: 2 × (3 + 4) × 1 + 0 × 5 = 14

Applied rules:

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

• Zero property: 0 × 5 = 0

• Identity property: 14 × 1 = 14

• Multiple properties: Different properties apply to different parts

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

Commutative property: The order of numbers doesn't change the product in multiplication

Associative property: The grouping of numbers doesn't change the product in multiplication

Distributive property: Multiplication can be distributed over addition or subtraction

Identity property: Multiplying by one preserves the identity of the other number

Zero property: Multiplying 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 1 or 0: If 1 or 0 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: One is the "do nothing" number in multiplication.
Common error: Thinking subtraction and division have these properties (they don't!).
Memory aid: Think of properties as "rules" that make multiplication easier to work with.
Properties to remember:

• Commutative: a × b = b × a

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

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

• Identity: a × 1 = a

• Zero: a × 0 = 0

Multiplication Properties: Efficiency Comparison
Exercise 6: Property Applications
Compare the efficiency of calculations with and without using multiplication properties:
Regular calculation vs. property-based calculation

Analysis: The chart shows how using multiplication properties makes calculations more efficient.

  • Commutative property: Allows choosing easier number combinations
  • Associative property: Enables strategic grouping for simpler calculations
  • Distributive property: Breaks down complex multiplications into simpler ones
  • Identity property: Recognizing multiplication by 1 saves computation
  • Zero property: Immediate recognition of zero result

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 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 multiplication works the way it does

For example, knowing the commutative property lets you choose the easier calculation: 7 × 8 or 8 × 7. Knowing the distributive property helps with mental math like 6 × 17 = 6 × (10 + 7) = 60 + 42 = 102.

Question: Does the zero property work if zero is in the middle of a multiplication, like 5 × 0 × 7?

Answer: Yes, absolutely! The zero property works no matter where zero appears in a multiplication. Here's why:

  • 5 × 0 × 7 = (5 × 0) × 7 = 0 × 7 = 0
  • Or you could think of it as: 5 × 0 × 7 = 5 × (0 × 7) = 5 × 0 = 0
  • As soon as you multiply by zero anywhere in the sequence, the entire result becomes zero

Zero is like a "contagion" in multiplication - if it touches any part of the multiplication, everything becomes zero!

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!