Properties of Operations Practice for Grade 4

Practice your understanding of properties of operations with these exercises on commutative, associative, distributive, and identity properties.

Practice Exercises: 1 to 3
1 Commutative property practice
Exercise 1
Practice: Fill in the blanks to make the equations true using the commutative property: 9 × 7 = __ × 9 and 15 + 8 = 8 + __.
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.

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
15 + 8 = 8 + 15
Results
Both equal 63 and 23
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

15 + 8 = 8 + __

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

So 15 + 8 = 8 + 15

Step 4: Verify second equation

15 + 8 = 23 and 8 + 15 = 23

Both sides are equal ✓

9 × 7 = 7 × 9 = 63
15 + 8 = 8 + 15 = 23
Final answer:

9 × 7 = 7 × 9 and 15 + 8 = 8 + 15. Both equations use the commutative property.

Applied rules:

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

• Verification: Always check that both sides are equal

• Pattern recognition: Identify when numbers switch positions

2 Associative property practice
Exercise 2
Practice: Complete the equation using the associative property: (6 + 4) + 3 = 6 + (__ + 3) and (2 × 5) × 4 = 2 × (__ × 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
(6 + 4) + 3 = 6 + (4 + 3)
Multiplication
(2 × 5) × 4 = 2 × (5 × 4)
Results
Both equal 13 and 40
Step 1: Solve addition equation

(6 + 4) + 3 = 6 + (__ + 3)

The associative property changes grouping

So (6 + 4) + 3 = 6 + (4 + 3)

The blank should contain 4

Step 2: Verify addition equation

(6 + 4) + 3 = 10 + 3 = 13

6 + (4 + 3) = 6 + 7 = 13

Both sides are equal ✓

Step 3: Solve multiplication equation

(2 × 5) × 4 = 2 × (__ × 4)

The associative property changes grouping

So (2 × 5) × 4 = 2 × (5 × 4)

The blank should contain 5

Step 4: Verify multiplication equation

(2 × 5) × 4 = 10 × 4 = 40

2 × (5 × 4) = 2 × 20 = 40

Both sides are equal ✓

(6 + 4) + 3 = 6 + (4 + 3) = 13
(2 × 5) × 4 = 2 × (5 × 4) = 40
Final answer:

(6 + 4) + 3 = 6 + (4 + 3) and (2 × 5) × 4 = 2 × (5 × 4). 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 practice
Exercise 3
Practice: Use the distributive property to solve: 4 × (7 + 3) and 6 × (9 - 2). 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
4 × (7 + 3)
Distributed
(4 × 7) + (4 × 3)
Result
40
Step 1: Apply distributive property to first expression

4 × (7 + 3) = (4 × 7) + (4 × 3)

Multiply 4 by each number inside the parentheses

Step 2: Calculate each multiplication

4 × 7 = 28 and 4 × 3 = 12

Step 3: Add the results

28 + 12 = 40

Step 4: Verify with direct calculation

4 × (7 + 3) = 4 × 10 = 40 ✓

Step 5: Apply distributive property to second expression

6 × (9 - 2) = (6 × 9) - (6 × 2)

Multiply 6 by each number inside the parentheses

Step 6: Calculate each multiplication

6 × 9 = 54 and 6 × 2 = 12

Step 7: Subtract the results

54 - 12 = 42

Step 8: Verify with direct calculation

6 × (9 - 2) = 6 × 7 = 42 ✓

4 × (7 + 3) = 40
6 × (9 - 2) = 42
Final answer:

4 × (7 + 3) = (4 × 7) + (4 × 3) = 28 + 12 = 40

6 × (9 - 2) = (6 × 9) - (6 × 2) = 54 - 12 = 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 Practice 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
Practice Exercises: 4 to 5
4 Identity and zero properties practice
Exercise 4
Practice: Simplify these expressions using identity and zero properties: 18 + 0, 0 + 25, 12 × 1, 1 × 30, 15 × 0, 0 × 8.
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
18 + 0 = 18
0 + 25 = 25
Multiplications
12 × 1 = 12
1 × 30 = 30
Zeros
15 × 0 = 0
0 × 8 = 0
Step 1: Apply additive identity property

18 + 0 = 18 (adding zero doesn't change the number)

0 + 25 = 25 (adding zero doesn't change the number)

Step 2: Apply multiplicative identity property

12 × 1 = 12 (multiplying by one doesn't change the number)

1 × 30 = 30 (multiplying by one doesn't change the number)

Step 3: Apply zero property

15 × 0 = 0 (any number times zero equals zero)

0 × 8 = 0 (any number times zero equals zero)

Step 4: Verify all results

All expressions follow the respective properties correctly

18 + 0 = 18
0 + 25 = 25
12 × 1 = 12
1 × 30 = 30
15 × 0 = 0
0 × 8 = 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 practice
Exercise 5
Practice: Identify which property is used in each step: 8 × (5 + 2) = 8 × (2 + 5) = (8 × 2) + (8 × 5).
Definition:

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

Step 1
8 × (5 + 2) = 8 × (2 + 5)
Step 2
8 × (2 + 5) = (8 × 2) + (8 × 5)
Results
Both equal 56
Step 1: Identify first property

8 × (5 + 2) → 8 × (2 + 5)

Inside the parentheses: 5 + 2 → 2 + 5

This is the commutative property of addition

Step 2: Identify second property

8 × (2 + 5) → (8 × 2) + (8 × 5)

This is the distributive property of multiplication over addition

Step 3: Verify the calculation

8 × (5 + 2) = 8 × 7 = 56

(8 × 2) + (8 × 5) = 16 + 40 = 56

Step 4: Confirm both properties

First step: Commutative property of addition

Second step: Distributive property

8 × (5 + 2) = (8 × 2) + (8 × 5) = 56
Final answer:

Step 1: Commutative property of addition (5 + 2 = 2 + 5)

Step 2: Distributive property (8 × (2 + 5) = (8 × 2) + (8 × 5))

Applied rules:

• Commutative property: Justifies 5 + 2 = 2 + 5

• Distributive property: Justifies 8 × (2 + 5) = (8 × 2) + (8 × 5)

• Verification: Both paths lead to the same result

Properties of Operations Practice 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 Practice Progress
Exercise 6: Property Mastery Levels
Track your progress with different properties through practice:
Commutative, Associative, Distributive, and Identity properties

Analysis: The chart shows your improvement with different properties through practice.

  • 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 often should I practice these properties to master them?

Answer: Consistent practice is key to mastering these properties:

  • Daily practice: Even 10-15 minutes daily is better than longer sessions once a week
  • Progressive difficulty: Start with simple numbers and advance to larger ones
  • Regular review: Revisit properties weekly to reinforce learning
  • Application: Use properties in real math problems to solidify understanding

Remember, practice makes permanent - consistent practice leads to lasting knowledge.

Question: What if I mix up the properties during practice?

Answer: Mixing up properties is completely normal during learning:

  • Slow down: Take time to identify what's happening in each step
  • Look for clues: Are numbers switching places? (Commutative)
  • Check grouping: Are parentheses moving? (Associative)
  • Seek help: Ask teachers or use visual aids when confused

Making mistakes is part of the learning process - analyze your errors to improve.

Question: How can I make my practice more effective?

Answer: Here are strategies for effective practice:

  • Variety: Mix different types of problems in each session
  • Self-testing: Quiz yourself regularly without looking at examples
  • Visual aids: Draw diagrams or use manipulatives to understand concepts
  • Real applications: Connect properties to real-world situations
  • Track progress: Keep a log of your practice and improvements

Active engagement with the material leads to better retention than passive reading.