Solved Exercises on Properties of Multiplication in Grade 3

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

Solution: Exercises 1 to 3
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.

Method to Demonstrate:
  1. Calculate the first multiplication: a × b
  2. Calculate the second multiplication: b × a
  3. Compare the results: Both should be equal
  4. Visualize: Draw arrays to show the relationship
4 × 3 = 3 × 4
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Left Side
4 × 3 = 12
Right Side
3 × 4 = 12
Conclusion
12 = 12 ✓
Step 1: Calculate 4 × 3

4 groups of 3 items each: 3 + 3 + 3 + 3 = 12

Step 2: Calculate 3 × 4

3 groups of 4 items each: 4 + 4 + 4 = 12

Step 3: Compare results

Both equal 12, so 4 × 3 = 3 × 4

4 × 3 = 3 × 4 = 12
Final answer:

The commutative property is demonstrated: 4 × 3 = 3 × 4 = 12

Applied rules:

• Commutative Property: a × b = b × a for any numbers a and b

• Order Independence: Changing the order of factors doesn't change the product

• Visual Verification: Arrays show the same total regardless of orientation

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

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

(2 × 3) × 4 = 2 × (3 × 4)
Properties of Multiplication | Solved Exercises | Mathematics Grade 3

Solved Exercises on Properties of Multiplication in Grade 3

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

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

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

Method:
  1. Identify the factors: In 4 × 3, the factors are 4 and 3
  2. Reverse the order: Change to 3 × 4
  3. Calculate both sides: Compute 4 × 3 and 3 × 4
  4. Compare results: Verify that both products are equal
  5. State the property: Confirm the commutative property
4 × 3 = 3 × 4
Group 1
Group 2
Group 3
Group 4
4 groups of 3
↔
3 groups of 4
Group 1
Group 2
Group 3
Left side
4 × 3 = 12
Right side
3 × 4 = 12
Result
Equal!
Step 1: Calculate 4 × 3

4 groups of 3 items each = 3 + 3 + 3 + 3 = 12

Step 2: Calculate 3 × 4

3 groups of 4 items each = 4 + 4 + 4 = 12

Step 3: Compare results

12 = 12, so the order doesn't matter

4 × 3 = 3 × 4 = 12
Final answer:

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

Applied rules:

• Commutative Property: a × b = b × a for any numbers a and b

• Order Independence: Factors can be multiplied in any order

• Product Consistency: The result is always the same regardless of order

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

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

(2 × 3) × 4 = 2 × (3 × 4)
Left side
(2 × 3) × 4 = 6 × 4 = 24
Right side
2 × (3 × 4) = 2 × 12 = 24
Result
Equal!
Step 1: Calculate left side (2 × 3) × 4

First: 2 × 3 = 6, Then: 6 × 4 = 24

Step 2: Calculate right side 2 × (3 × 4)

First: 3 × 4 = 12, Then: 2 × 12 = 24

Step 3: Compare results

24 = 24, so grouping doesn't matter

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

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

Applied rules:

• Associative Property: (a × b) × c = a × (b × c) for any numbers a, b, and c

• Grouping Independence: Parentheses can be moved without changing the result

• Flexible Computation: Allows choosing the easiest order to multiply

3 Identity Property
Exercise 3
Show that 7 × 1 = 7 and 1 × 9 = 9 using the identity property of multiplication.
Definition:

Identity Property of Multiplication: Any number multiplied by 1 equals the original number. For any number a, a × 1 = a and 1 × a = a.

7 × 1 = 7 and 1 × 9 = 9
7 items
7 groups of 1
=
Just 7 items
9 items
First example
7 × 1 = 7
Second example
1 × 9 = 9
Result
Same number!
Step 1: Calculate 7 × 1

7 groups of 1 item each = 1 + 1 + 1 + 1 + 1 + 1 + 1 = 7

Step 2: Calculate 1 × 9

1 group of 9 items = 9

Step 3: Observe the pattern

Multiplying by 1 keeps the original number unchanged

7 × 1 = 7 and 1 × 9 = 9
Final answer:

7 × 1 = 7 and 1 × 9 = 9, demonstrating the identity property of multiplication

Applied rules:

• Identity Property: a × 1 = a and 1 × a = a for any number a

• Preservation: Multiplying by 1 preserves the original number

• Neutral Element: 1 is the multiplicative identity element

Multiplication Properties Overview
a × b = b × a
Commutative Property
Property 1
Commutative
a × b = b × a
Property 2
Associative
(a × b) × c = a × (b × c)
Property 3
Identity
a × 1 = a
Property 4
Zero
a × 0 = 0
Property Rule Example Commutative a × b = b × a 5 × 3 = 3 × 5 = 15 Associative (a × b) × c = a × (b × c) (2 × 3) × 4 = 2 × (3 × 4) = 24 Identity a × 1 = a 8 × 1 = 8 Zero a × 0 = 0 6 × 0 = 0
Key definitions:

Factor: A number that is multiplied by another number

Product: The result of multiplying two or more numbers

Property: A rule that is always true for a particular operation

How to Use Properties:
  1. Recognize the pattern: Identify which property applies to your problem
  2. Apply the rule: Use the property to simplify or rearrange
  3. Solve efficiently: Use properties to make calculations easier
  4. Verify results: Check that your answer follows the property
Commutative Property: Order doesn't matter in multiplication
Associative Property: Grouping doesn't matter in multiplication
Tip 1: Use the commutative property to put easier numbers together first.
Tip 2: The associative property lets you group numbers strategically.
Tip 3: Remember: anything times 0 equals 0, and anything times 1 stays the same.
Solution: Exercises 4 to 5
4 Zero Property
Exercise 4
Show that 5 × 0 = 0 and 0 × 8 = 0 using the zero property of multiplication.
Definition:

Zero Property of Multiplication: Any number multiplied by 0 equals 0. For any number a, a × 0 = 0 and 0 × a = 0.

5 × 0 = 0 and 0 × 8 = 0
5 groups of 0
No items in any group
=
Total = 0
0 groups of 8
First example
5 × 0 = 0
Second example
0 × 8 = 0
Result
Always 0!
Step 1: Calculate 5 × 0

5 groups with 0 items each = 0 + 0 + 0 + 0 + 0 = 0

Step 2: Calculate 0 × 8

0 groups of 8 items each = 0

Step 3: Observe the pattern

Multiplying by 0 always results in 0

5 × 0 = 0 and 0 × 8 = 0
Final answer:

5 × 0 = 0 and 0 × 8 = 0, demonstrating the zero property of multiplication

Applied rules:

• Zero Property: a × 0 = 0 and 0 × a = 0 for any number a

• Annihilation: Multiplying by 0 always yields 0

• Universal Rule: This property applies to all numbers

5 Mixed Properties Practice
Exercise 5
Use properties to simplify: (4 × 1) × 5 and 3 × (0 × 7)
Definition:

Mixed Properties: Using multiple properties together to simplify complex multiplication expressions.

(4 × 1) × 5 and 3 × (0 × 7)
First expression
(4 × 1) × 5 = 4 × 5 = 20
Second expression
3 × (0 × 7) = 3 × 0 = 0
Results
20 and 0
Step 1: Simplify (4 × 1) × 5

Using identity property: 4 × 1 = 4, so 4 × 5 = 20

Step 2: Simplify 3 × (0 × 7)

Using zero property: 0 × 7 = 0, so 3 × 0 = 0

Step 3: Apply properties strategically

Recognize which property simplifies the calculation fastest

(4 × 1) × 5 = 20 and 3 × (0 × 7) = 0
Final answer:

(4 × 1) × 5 = 20 and 3 × (0 × 7) = 0, using identity and zero properties respectively

Applied rules:

• Strategic Application: Use properties to simplify calculations efficiently

• Sequential Properties: Apply multiple properties in order

• Computational Advantage: Properties make complex problems easier

Advanced Properties Applications
(a × b) × c = a × (b × c)
Associative Property
Key definitions:

Factor: A number that divides another number evenly or is multiplied with another number

Product: The result obtained when two or more numbers are multiplied together

Property: A characteristic or rule that applies to mathematical operations

Problem-Solving with Properties:
  1. Identify the operation: Determine if you're working with multiplication
  2. Look for patterns: See if the expression fits a known property
  3. Apply the appropriate property: Use the matching rule
  4. Simplify the expression: Make the calculation easier
  5. Verify your result: Check that the property was applied correctly
Tip 1: The commutative property allows you to rearrange factors to make multiplication easier.
Tip 2: The associative property lets you group numbers in a way that simplifies the calculation.
Tip 3: Remember that 0 and 1 are special numbers in multiplication.
Tip 4: These properties help with mental math and checking your work.
Memory Aid: "Commutative" sounds like "commute" - factors can commute (switch places).
Memory Aid: "Associative" sounds like "associate" - factors can associate (group) differently.
Properties to remember:

• Commutative: a × b = b × a (order doesn't matter)

• Associative: (a × b) × c = a × (b × c) (grouping doesn't matter)

• Identity: a × 1 = a (multiplying by 1 doesn't change the number)

• Zero: a × 0 = 0 (multiplying by 0 always gives 0)

Commutative
a × b = b × a
Order doesn't matter
Associative
(a × b) × c = a × (b × c)
Grouping doesn't matter
Identity
a × 1 = a
Multiplying by 1
Zero
a × 0 = 0
Multiplying by 0

Questions & Answers

Question: Why does the commutative property work? Why does 4 × 3 equal 3 × 4?

Answer: Think of it this way:

  • 4 × 3 means 4 groups of 3 items each (3 + 3 + 3 + 3 = 12)
  • 3 × 4 means 3 groups of 4 items each (4 + 4 + 4 = 12)

Both give you 12 items total! You can also think of it as an array:

4 rows of 3: ⬜⬜⬜
                       ⬜⬜⬜
                       ⬜⬜⬜
                       ⬜⬜⬜

If you rotate this array, you get 3 rows of 4: ⬜⬜⬜⬜
                                                                                                                                                                                                                                                            ...... Properties of Multiplication | Solved Exercises | Mathematics Grade 3

Solved Exercises on Properties of Multiplication in Grade 3

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

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

Commutative Property of Multiplication: Changing the order of the factors does not change the product. In symbols: a × b = b × a

Method:
  1. Identify the two factors in the multiplication
  2. Multiply them in the original order
  3. Multiply them in the reverse order
  4. Compare the results to verify they are equal
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3 × 4 = 4 × 3
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Original
3 × 4
Reverse
4 × 3
Result
12
Step 1: Calculate the original multiplication

3 × 4 = 12 (3 groups of 4 = 12)

Step 2: Calculate the reversed multiplication

4 × 3 = 12 (4 groups of 3 = 12)

Step 3: Compare the results

Both multiplications equal 12, so 3 × 4 = 4 × 3

Step 4: State the property

This demonstrates the commutative property of multiplication

3 × 4 = 4 × 3 = 12
Final answer:

3 × 4 = 4 × 3 = 12, which proves the commutative property of multiplication

Applied rules:

• Commutative Property: a × b = b × a (order doesn't matter)

• Equality: Both sides of the equation must be equal

• Verification: Calculate both sides to confirm equality

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

Associative Property of Multiplication: Changing the grouping of the factors does not change the product. In symbols: (a × b) × c = a × (b × c)

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(2 × 3) × 4 = 2 × (3 × 4)
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Left side
(2 × 3) × 4
Right side
2 × (3 × 4)
Result
24
Step 1: Calculate the left side of the equation

(2 × 3) × 4 = 6 × 4 = 24

Step 2: Calculate the right side of the equation

2 × (3 × 4) = 2 × 12 = 24

Step 3: Compare the results

Both sides equal 24, so (2 × 3) × 4 = 2 × (3 × 4)

Step 4: State the property

This demonstrates the associative property of multiplication

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

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

Applied rules:

• Associative Property: (a × b) × c = a × (b × c) (grouping doesn't matter)

• Order of Operations: Parentheses are calculated first

• Verification: Calculate both sides to confirm equality

3 Identity Property
Exercise 3
Show that 7 × 1 = 7 and 1 × 9 = 9 using the identity property of multiplication.
Definition:

Identity Property of Multiplication: Any number multiplied by 1 equals the original number. In symbols: a × 1 = a and 1 × a = a

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7 × 1 = 7
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1 × 9 = 9
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First example
7 × 1
Second example
1 × 9
Results
7 and 9
Step 1: Calculate 7 × 1

7 × 1 = 7 (one group of 7 = 7)

Step 2: Calculate 1 × 9

1 × 9 = 9 (nine groups of 1 = 9)

Step 3: Analyze the results

7 × 1 = 7 and 1 × 9 = 9, showing the number stays the same

Step 4: State the property

This demonstrates the identity property of multiplication

7 × 1 = 7 and 1 × 9 = 9
Final answer:

7 × 1 = 7 and 1 × 9 = 9, which proves the identity property of multiplication

Applied rules:

• Identity Property: a × 1 = a and 1 × a = a (1 is the identity element)

• Preservation: The original number is preserved when multiplied by 1

• Commutative: The property works regardless of order

Multiplication Properties Overview
a × b = b × a
Commutative Property
🔄
Commutative
a × b = b × a
Order doesn't matter in multiplication. 3×4=4×3=12
🔗
Associative
(a×b)×c = a×(b×c)
Grouping doesn't matter. (2×3)×4=2×(3×4)=24
⭐
Identity
a × 1 = a
Any number times 1 equals itself. 7×1=7
❌
Zero
a × 0 = 0
Any number times 0 equals 0. 5×0=0
Key definitions:

Property: A rule that is always true for a mathematical operation

Factor: A number that is multiplied by another number

Product: The result of multiplying two or more numbers

Identity Element: A number that doesn't change other numbers when used in an operation

How to Identify Properties:
  1. Commutative: Look for switched order of numbers (a×b vs b×a)
  2. Associative: Look for changed grouping with parentheses
  3. Identity: Look for multiplication by 1
  4. Zero: Look for multiplication by 0
Common Uses: These properties help simplify calculations and solve problems more efficiently.
Memory Tip: "Commutative" sounds like "commute" - numbers commute or swap places.
Tip 1: Use commutative property to multiply in the order that's easiest for you.
Tip 2: Associative property helps when multiplying three or more numbers.
Tip 3: Remember: multiplying by 1 keeps the number, multiplying by 0 makes it zero.
Solution: Exercises 4 to 5
4 Zero Property
Exercise 4
Show that 5 × 0 = 0 and 0 × 8 = 0 using the zero property of multiplication.
Definition:

Zero Property of Multiplication: Any number multiplied by 0 equals 0. In symbols: a × 0 = 0 and 0 × a = 0

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5 × 0 = 0
0 × 8 = 0
First example
5 × 0
Second example
0 × 8
Results
0
Step 1: Calculate 5 × 0

5 × 0 = 0 (five groups of zero items = zero total)

Step 2: Calculate 0 × 8

0 × 8 = 0 (zero groups of eight items = zero total)

Step 3: Analyze the results

Both multiplications equal 0, showing that any number times 0 is 0

Step 4: State the property

This demonstrates the zero property of multiplication

5 × 0 = 0 and 0 × 8 = 0
Final answer:

5 × 0 = 0 and 0 × 8 = 0, which proves the zero property of multiplication

Applied rules:

• Zero Property: a × 0 = 0 and 0 × a = 0 (0 is the absorbing element)

• Universal Truth: This property holds for all numbers

• Commutative: The property works regardless of order

5 Mixed Properties Practice
Exercise 5
Identify which property is shown in each equation: (a) 6 × 2 = 2 × 6, (b) (3 × 4) × 5 = 3 × (4 × 5), (c) 9 × 1 = 9, (d) 7 × 0 = 0
Definition:

Property Identification: Recognizing which multiplication property is being demonstrated in a given equation by analyzing the structure and elements of the equation.

Equation (a)
6 × 2 = 2 × 6
Property
Commutative
Equation (b)
(3 × 4) × 5 = 3 × (4 × 5)
Property
Associative
Equation (c)
9 × 1 = 9
Property
Identity
Equation (d)
7 × 0 = 0
Property
Zero
Step 1: Analyze equation (a) 6 × 2 = 2 × 6

The factors have switched positions, so this is the commutative property

Step 2: Analyze equation (b) (3 × 4) × 5 = 3 × (4 × 5)

The grouping with parentheses has changed, so this is the associative property

Step 3: Analyze equation (c) 9 × 1 = 9

Any number multiplied by 1 equals itself, so this is the identity property

Step 4: Analyze equation (d) 7 × 0 = 0

Any number multiplied by 0 equals 0, so this is the zero property

(a) Commutative, (b) Associative, (c) Identity, (d) Zero
Final answer:

(a) Commutative property, (b) Associative property, (c) Identity property, (d) Zero property

Applied rules:

• Pattern Recognition: Look for specific structures in equations

• Commutative Pattern: Factors switch positions

• Associative Pattern: Parentheses group different factors

Advanced Properties Concepts
(a × b) × c = a × (b × c)
Associative Property
Key definitions:

Commutative Property: Order doesn't matter (a × b = b × a)

Associative Property: Grouping doesn't matter ((a × b) × c = a × (b × c))

Identity Property: Multiplying by 1 preserves the number (a × 1 = a)

Zero Property: Multiplying by 0 always results in 0 (a × 0 = 0)

Property Applications:
  1. Commutative: Makes multiplication easier by allowing you to multiply in any order
  2. Associative: Helps when multiplying three or more numbers by choosing convenient groupings
  3. Identity: Useful for understanding that 1 is neutral in multiplication
  4. Zero: Critical for understanding that 0 "absorbs" other numbers in multiplication
Tip 1: Use commutative property to multiply harder numbers first if it makes the problem easier.
Tip 2: Associative property is especially helpful with larger multiplication problems.
Tip 3: Remember that addition has similar properties, which helps with connections.
Tip 4: These properties don't apply to subtraction and division, which is important to remember.
Common Mistakes: Thinking these properties apply to subtraction or division, confusing with addition properties.
Memorization Aid: Remember that "commutative" means to "commute" or move around.
Properties comparison:

• Commutative: Changes order (a×b = b×a)

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

• Identity: Multiplies by 1 (a×1 = a)

• Zero: Multiplies by 0 (a×0 = 0)

Property
Formula
Example
Commutative
a × b = b × a
3 × 4 = 4 × 3
Associative
(a×b)×c = a×(b×c)
(2×3)×4 = 2×(3×4)
Identity
a × 1 = a
7 × 1 = 7
Zero
a × 0 = 0
5 × 0 = 0

Questions & Answers

Question: Why do we need to learn these properties? Don't we just multiply numbers?

Answer: Properties of multiplication are like "tricks" that make math easier and faster! Here's why they're important:

  • Commutative Property: If you don't know 7 × 3, but you know 3 × 7 = 21, then you know the answer!
  • Associative Property: For 2 × 3 × 5, you can do (2 × 3) × 5 = 6 × 5 = 30 OR 2 × (3 × 5) = 2 × 15 = 30. Both give the same answer!
  • Identity Property: Makes it easy to multiply by 1 without calculation
  • Zero Property: Makes it easy to multiply by 0 without calculation

These properties help you solve problems more quickly and understand how numbers work together!

Question: Do addition and multiplication have the same properties?

Answer: Addition and multiplication share some properties but have different ones too:

  • Both have: Commutative and Associative properties
  • Only multiplication has: Identity property (with 1) and Zero property
  • Only addition has: Identity property (with 0)

For example:

  • Addition: 3 + 4 = 4 + 3 (commutative) and (2 + 3) + 4 = 2 + (3 + 4) (associative)
  • Multiplication: 3 × 4 = 4 × 3 (commutative) and (2 × 3) × 4 = 2 × (3 × 4) (associative)

But multiplication has unique properties like 5 × 1 = 5 and 5 × 0 = 0.

Question: Do subtraction and division have these same properties?

Answer: No! Subtraction and division do NOT have commutative or associative properties:

  • Subtraction is not commutative: 5 - 3 ≠ 3 - 5 (because 2 ≠ -2)
  • Division is not commutative: 6 ÷ 2 ≠ 2 ÷ 6 (because 3 ≠ 1/3)
  • Subtraction is not associative: (8 - 4) - 2 ≠ 8 - (4 - 2) (because 2 ≠ 6)
  • Division is not associative: (8 ÷ 4) ÷ 2 ≠ 8 ÷ (4 ÷ 2) (because 1 ≠ 4)

However, subtraction and division do have identity elements:

  • Subtraction: a - 0 = a (subtracting 0 keeps the number)
  • Division: a ÷ 1 = a (dividing by 1 keeps the number)

This is why multiplication and addition are considered "nicer" operations than subtraction and division!