Solved Exercises on Properties and Operations on Numbers for Grade 4

Master number properties and operations: commutative, associative, distributive properties, and operational rules through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Commutative property
Exercise 1
Demonstrate the commutative property of addition and multiplication with 5 + 3 and 4 × 2
Definition:

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

Commutative property method:
  1. Take any two numbers
  2. Add/multiply them in one order
  3. Add/multiply them in reverse order
  4. Compare the results
Addition
5 + 3 = 8
Reversed
3 + 5 = 8
Result
Same!
Multiplication
4 × 2 = 8
Reversed
2 × 4 = 8
Result
Same!
Step 1: Test addition

5 + 3 = 8 and 3 + 5 = 8

The order doesn't matter in addition!

Step 2: Test multiplication

4 × 2 = 8 and 2 × 4 = 8

The order doesn't matter in multiplication!

Step 3: Verify the property

Both operations show the same result regardless of order

Addition: 5 + 3 = 3 + 5 = 8
Multiplication: 4 × 2 = 2 × 4 = 8
Final answer:

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

Applied rules:

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

• Not applicable: Subtraction and division are not commutative

• Verification: Always check both orders to confirm

2 Associative property
Exercise 2
Demonstrate the associative property with (2 + 3) + 4 and 2 + (3 + 4)
Definition:

Associative property: Grouping numbers differently doesn't change the result. For addition: (a + b) + c = a + (b + c). For multiplication: (a × b) × c = a × (b × c).

Grouping 1
(2 + 3) + 4
Grouping 2
2 + (3 + 4)
Result
Both equal 9
Step 1: Solve first grouping

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

First solve inside parentheses: 2 + 3 = 5

Then add: 5 + 4 = 9

Step 2: Solve second grouping

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

First solve inside parentheses: 3 + 4 = 7

Then add: 2 + 7 = 9

Step 3: Compare results

Both groupings equal 9, confirming the associative property

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

The associative property holds: changing the grouping doesn't change the result for 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 3 × (4 + 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
3 × (4 + 2)
Distributed
(3 × 4) + (3 × 2)
Result
18
Step 1: Solve inside parentheses first

3 × (4 + 2) = 3 × 6 = 18

First add: 4 + 2 = 6

Then multiply: 3 × 6 = 18

Step 2: Apply the distributive property

3 × (4 + 2) = (3 × 4) + (3 × 2)

Multiply 3 by each number inside the parentheses

Step 3: Calculate each multiplication

3 × 4 = 12 and 3 × 2 = 6

Step 4: Add the results

12 + 6 = 18

3 × (4 + 2) = (3 × 4) + (3 × 2) = 12 + 6 = 18
Final answer:

The distributive property holds: 3 × (4 + 2) = (3 × 4) + (3 × 2) = 18

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 and Operations Summary
\(a + b = b + a\)
Commutative 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 additive and multiplicative identity properties with 7 + 0 and 9 × 1
Definition:

Identity properties: Adding 0 or multiplying by 1 leaves a number unchanged. Additive identity: a + 0 = a. Multiplicative identity: a × 1 = a.

Addition
7 + 0 = 7
Multiplication
9 × 1 = 9
Result
No change!
Step 1: Test additive identity

7 + 0 = 7

Adding zero doesn't change the number

Step 2: Test multiplicative identity

9 × 1 = 9

Multiplying by one doesn't change the number

Step 3: Verify the properties

Both operations leave the original number unchanged

7 + 0 = 7 and 9 × 1 = 9
Final answer:

The identity properties hold: adding 0 or multiplying by 1 preserves the original number

Applied rules:

• Additive identity: Zero is the identity element for addition

• Multiplicative identity: One is the identity element for multiplication

• Preservation: These operations don't alter the original value

5 Zero property and combining properties
Exercise 5
Combine properties to solve: 5 × (3 + 0) × 1 and demonstrate the zero property
Definition:

Zero property: Any number multiplied by 0 equals 0: a × 0 = 0. Also, multiplying by 0 anywhere in a chain of multiplications results in 0.

Expression
5 × (3 + 0) × 1
Simplified
5 × 3 × 1
Result
15
Step 1: Apply additive identity

3 + 0 = 3 (adding zero doesn't change the number)

Step 2: Simplify the expression

5 × (3 + 0) × 1 = 5 × 3 × 1

Step 3: Apply multiplicative identity

Multiplying by 1 doesn't change the result

5 × 3 × 1 = 5 × 3 = 15

Step 4: Demonstrate zero property

If we had 5 × 0 × 3, the result would be 0

Any number multiplied by 0 equals 0

5 × (3 + 0) × 1 = 15
Final answer:

Combining properties: 5 × (3 + 0) × 1 = 15. Zero property: any number × 0 = 0

Applied rules:

• Multiple properties: Several properties can be used together

• Zero dominance: If 0 appears in multiplication, result is always 0

• Order of operations: Solve parentheses first

Detailed Properties and Operations Guide
\(a × (b + c) = (a × b) + (a × c)\)
Distributive Property
Key definitions:

Commutative property: Order of numbers doesn't affect the result in addition or multiplication

Associative property: Grouping of numbers doesn't affect the result in addition or multiplication

Distributive property: Multiplication distributes over addition

Identity property: Adding 0 or multiplying by 1 doesn't change a number

Zero property: Any number multiplied by 0 equals 0

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

Property Applications: Visual Comparison
Exercise 6: Property Effects
Compare the effects of different properties on calculations:
Commutative vs Associative vs Distributive properties

Analysis: The chart shows how different properties affect 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

Questions & Answers

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

Answer: Great question! While these properties might seem obvious now, 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!