Properties of Operations Worksheet for Grade 4

Comprehensive worksheet exercises on properties of operations: commutative, associative, distributive, and identity properties.

Worksheet Exercises: 1 to 3
1 Commutative property exercise
Exercise 1
Worksheet: Complete the equations using the commutative property of multiplication: 9 × 7 = __ × 9 and 12 × 5 = __ × 12.
Definition:

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

Solution method:
  1. Identify the property: Is the order of numbers changing?
  2. Apply the rule: Switch the numbers in the correct position
  3. Verify: Both sides should be equal
Equation 1
9 × 7 = 7 × 9
Equation 2
12 × 5 = 5 × 12
Results
Both equal 63 and 60
Step 1: Solve first equation

9 × 7 = __ × 9

According to the commutative property, we need to switch the order

So 9 × 7 = 7 × 9

Step 2: Verify first equation

9 × 7 = 63 and 7 × 9 = 63

Both sides are equal ✓

Step 3: Solve second equation

12 × 5 = __ × 12

According to the commutative property, we need to switch the order

So 12 × 5 = 5 × 12

Step 4: Verify second equation

12 × 5 = 60 and 5 × 12 = 60

Both sides are equal ✓

9 × 7 = 7 × 9 = 63
12 × 5 = 5 × 12 = 60
Final answer:

9 × 7 = 7 × 9 and 12 × 5 = 5 × 12. Both equations use the commutative property.

Applied rules:

• Commutative property: Order doesn't matter in multiplication

• Verification: Always check that both sides are equal

• Pattern recognition: Identify when numbers switch positions

2 Associative property exercise
Exercise 2
Worksheet: Complete the equation using the associative property: (8 + 6) + 4 = 8 + (__ + 4) and (3 × 5) × 2 = 3 × (__ × 2).
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
(8 + 6) + 4 = 8 + (6 + 4)
Multiplication
(3 × 5) × 2 = 3 × (5 × 2)
Results
Both equal 18 and 30
Step 1: Solve addition equation

(8 + 6) + 4 = 8 + (__ + 4)

The associative property changes grouping

So (8 + 6) + 4 = 8 + (6 + 4)

The blank should contain 6

Step 2: Verify addition equation

(8 + 6) + 4 = 14 + 4 = 18

8 + (6 + 4) = 8 + 10 = 18

Both sides are equal ✓

Step 3: Solve multiplication equation

(3 × 5) × 2 = 3 × (__ × 2)

The associative property changes grouping

So (3 × 5) × 2 = 3 × (5 × 2)

The blank should contain 5

Step 4: Verify multiplication equation

(3 × 5) × 2 = 15 × 2 = 30

3 × (5 × 2) = 3 × 10 = 30

Both sides are equal ✓

(8 + 6) + 4 = 8 + (6 + 4) = 18
(3 × 5) × 2 = 3 × (5 × 2) = 30
Final answer:

(8 + 6) + 4 = 8 + (6 + 4) and (3 × 5) × 2 = 3 × (5 × 2). Both equations use the associative property.

Applied rules:

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

• Parentheses: Show the grouping of numbers

• Consistency: Result remains the same regardless of grouping

3 Distributive property exercise
Exercise 3
Worksheet: Use the distributive property to solve: 6 × (8 + 4) and 7 × (9 - 3). Show your work.
Definition:

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

Expression 1
6 × (8 + 4)
Distributed
(6 × 8) + (6 × 4)
Result
72
Step 1: Apply distributive property to first expression

6 × (8 + 4) = (6 × 8) + (6 × 4)

Multiply 6 by each number inside the parentheses

Step 2: Calculate each multiplication

6 × 8 = 48 and 6 × 4 = 24

Step 3: Add the results

48 + 24 = 72

Step 4: Verify with direct calculation

6 × (8 + 4) = 6 × 12 = 72 ✓

Step 5: Apply distributive property to second expression

7 × (9 - 3) = (7 × 9) - (7 × 3)

Multiply 7 by each number inside the parentheses

Step 6: Calculate each multiplication

7 × 9 = 63 and 7 × 3 = 21

Step 7: Subtract the results

63 - 21 = 42

Step 8: Verify with direct calculation

7 × (9 - 3) = 7 × 6 = 42 ✓

6 × (8 + 4) = 72
7 × (9 - 3) = 42
Final answer:

6 × (8 + 4) = (6 × 8) + (6 × 4) = 48 + 24 = 72

7 × (9 - 3) = (7 × 9) - (7 × 3) = 63 - 21 = 42

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 Worksheet Guide
\(a × (b + c) = (a × b) + (a × c)\)
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
Worksheet Exercises: 4 to 5
4 Identity and zero properties exercise
Exercise 4
Worksheet: Simplify these expressions using identity and zero properties: 23 + 0, 0 + 37, 18 × 1, 1 × 45, 29 × 0, 0 × 14.
Definition:

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

Additions
23 + 0 = 23
0 + 37 = 37
Multiplications
18 × 1 = 18
1 × 45 = 45
Zeros
29 × 0 = 0
0 × 14 = 0
Step 1: Apply additive identity property

23 + 0 = 23 (adding zero doesn't change the number)

0 + 37 = 37 (adding zero doesn't change the number)

Step 2: Apply multiplicative identity property

18 × 1 = 18 (multiplying by one doesn't change the number)

1 × 45 = 45 (multiplying by one doesn't change the number)

Step 3: Apply zero property

29 × 0 = 0 (any number times zero equals zero)

0 × 14 = 0 (any number times zero equals zero)

Step 4: Verify all results

All expressions follow the respective properties correctly

23 + 0 = 23
0 + 37 = 37
18 × 1 = 18
1 × 45 = 45
29 × 0 = 0
0 × 14 = 0
Final answer:

All expressions follow the identity and zero properties: adding 0 or multiplying by 1 preserves the number, multiplying by 0 results in 0.

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 exercise
Exercise 5
Worksheet: Identify which property is used in each step: 5 × (7 + 2) = 5 × (2 + 7) = (5 × 2) + (5 × 7).
Definition:

Mixed properties application: Problems requiring multiple properties in sequence. Identify each property used at each step.

Step 1
5 × (7 + 2) = 5 × (2 + 7)
Step 2
5 × (2 + 7) = (5 × 2) + (5 × 7)
Results
Both equal 45
Step 1: Identify first property

5 × (7 + 2) → 5 × (2 + 7)

Inside the parentheses: 7 + 2 → 2 + 7

This is the commutative property of addition

Step 2: Identify second property

5 × (2 + 7) → (5 × 2) + (5 × 7)

This is the distributive property of multiplication over addition

Step 3: Verify the calculation

5 × (7 + 2) = 5 × 9 = 45

(5 × 2) + (5 × 7) = 10 + 35 = 45

Step 4: Confirm both properties

First step: Commutative property of addition

Second step: Distributive property

5 × (7 + 2) = (5 × 2) + (5 × 7) = 45
Final answer:

Step 1: Commutative property of addition (7 + 2 = 2 + 7)

Step 2: Distributive property (5 × (2 + 7) = (5 × 2) + (5 × 7))

Applied rules:

• Commutative property: Justifies 7 + 2 = 2 + 7

• Distributive property: Justifies 5 × (2 + 7) = (5 × 2) + (5 × 7)

• Verification: Both paths lead to the same result

Properties of Operations Worksheet Summary
\(a × (b + c) = (a × b) + (a × c)\)
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 Worksheet Performance
Exercise 6: Property Mastery Tracking
Track your progress with different properties through worksheet exercises:
Commutative, Associative, Distributive, and Identity properties

Analysis: The chart shows your progress with different properties through worksheet exercises.

  • Commutative property: Practice switching number order
  • Associative property: Work on regrouping numbers
  • Distributive property: Focus on multiplication distribution
  • Identity property: Recognize neutral elements
  • Zero property: Understand zero's effect in multiplication

Questions & Answers

Question: How should I approach completing these worksheet exercises?

Answer: Here's a systematic approach to worksheet completion:

  • Read carefully: Understand what each question is asking
  • Identify the property: Determine which property is being tested
  • Apply the rule: Use the appropriate property correctly
  • Show your work: Write down each step clearly
  • Verify your answer: Check that both sides of equations are equal

Work through problems systematically rather than rushing through them.

Question: What if I make mistakes on the worksheet?

Answer: Making mistakes is a natural part of learning:

  • Don't panic: Mistakes help you learn better
  • Review your work: Go back and check each step
  • Understand the error: Figure out why you made the mistake
  • Learn from it: Apply the correction to future problems
  • Ask for help: Consult your teacher if you're stuck

The most important thing is to understand why the mistake happened.

Question: How can I check my answers on the worksheet?

Answer: Here are effective checking strategies:

  • Calculate both sides: Work out the value of both sides of equations
  • Verify equality: Ensure both sides equal the same number
  • Apply the property: Make sure you used the correct property
  • Use alternative methods: Solve the same problem differently to confirm
  • Look for patterns: Check that the transformation follows property rules

Always double-check your work to catch any errors before submitting.