Properties of Operations Worksheet: 3rd Grade Review for Grade 4

Review of properties of operations from 3rd grade: commutative, associative, distributive, and identity properties.

Worksheet Review: 1 to 3
1 Commutative property review
Exercise 1
3rd Grade Review: Complete the equations using the commutative property: 5 + 8 = __ + 5 and 4 × 6 = __ × 4.
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
Addition
5 + 8 = 8 + 5
Multiplication
4 × 6 = 6 × 4
Results
Both equal 13 and 24
Step 1: Solve addition equation

5 + 8 = __ + 5

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

So 5 + 8 = 8 + 5

Step 2: Verify addition equation

5 + 8 = 13 and 8 + 5 = 13

Both sides are equal ✓

Step 3: Solve multiplication equation

4 × 6 = __ × 4

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

So 4 × 6 = 6 × 4

Step 4: Verify multiplication equation

4 × 6 = 24 and 6 × 4 = 24

Both sides are equal ✓

5 + 8 = 8 + 5 = 13
4 × 6 = 6 × 4 = 24
Final answer:

5 + 8 = 8 + 5 and 4 × 6 = 6 × 4. 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 review
Exercise 2
3rd Grade Review: Complete the equation using the associative property: (3 + 4) + 2 = 3 + (__ + 2) and (2 × 5) × 3 = 2 × (__ × 3).
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
(3 + 4) + 2 = 3 + (4 + 2)
Multiplication
(2 × 5) × 3 = 2 × (5 × 3)
Results
Both equal 9 and 30
Step 1: Solve addition equation

(3 + 4) + 2 = 3 + (__ + 2)

The associative property changes grouping

So (3 + 4) + 2 = 3 + (4 + 2)

The blank should contain 4

Step 2: Verify addition equation

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

3 + (4 + 2) = 3 + 6 = 9

Both sides are equal ✓

Step 3: Solve multiplication equation

(2 × 5) × 3 = 2 × (__ × 3)

The associative property changes grouping

So (2 × 5) × 3 = 2 × (5 × 3)

The blank should contain 5

Step 4: Verify multiplication equation

(2 × 5) × 3 = 10 × 3 = 30

2 × (5 × 3) = 2 × 15 = 30

Both sides are equal ✓

(3 + 4) + 2 = 3 + (4 + 2) = 9
(2 × 5) × 3 = 2 × (5 × 3) = 30
Final answer:

(3 + 4) + 2 = 3 + (4 + 2) and (2 × 5) × 3 = 2 × (5 × 3). 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 Identity properties review
Exercise 3
3rd Grade Review: Complete the equations using identity properties: 12 + 0 = __, 0 + 15 = __, 7 × 1 = __, 1 × 9 = __.
Definition:

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

Addition
12 + 0 = 12
Addition
0 + 15 = 15
Multiplication
7 × 1 = 7
Multiplication
1 × 9 = 9
Step 1: Apply additive identity property

12 + 0 = 12

Adding zero doesn't change the number

Step 2: Apply additive identity property (reverse)

0 + 15 = 15

Adding zero doesn't change the number

Step 3: Apply multiplicative identity property

7 × 1 = 7

Multiplying by one doesn't change the number

Step 4: Apply multiplicative identity property (reverse)

1 × 9 = 9

Multiplying by one doesn't change the number

12 + 0 = 12
0 + 15 = 15
7 × 1 = 7
1 × 9 = 9
Final answer:

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: Original number remains unchanged

3rd Grade Properties Review 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
Identity
a + 0 = a
Additive Identity
Identity
a × 1 = a
Multiplicative Identity
Worksheet Review: 4 to 5
4 Zero property review
Exercise 4
3rd Grade Review: Complete the equations using the zero property: 8 × 0 = __, 0 × 12 = __, 5 × 0 × 3 = __.
Definition:

Zero property of multiplication: Any number multiplied by zero equals zero. For any number a: a × 0 = 0 and 0 × a = 0. If zero appears anywhere in a multiplication, the result is zero.

First
8 × 0 = 0
Second
0 × 12 = 0
Third
5 × 0 × 3 = 0
Step 1: Apply zero property to first equation

8 × 0 = 0

Any number times zero equals zero

Step 2: Apply zero property to second equation

0 × 12 = 0

Zero times any number equals zero

Step 3: Apply zero property to third equation

5 × 0 × 3 = 0

When zero appears in multiplication, the whole result is zero

8 × 0 = 0
0 × 12 = 0
5 × 0 × 3 = 0
Final answer:

Any number multiplied by zero equals zero, regardless of other factors.

Applied rules:

• Zero dominance: Zero "takes over" in multiplication

• Universal: Applies to any number multiplied by zero

• Chain rule: If zero appears anywhere in a multiplication chain, result is zero

5 Mixed properties review
Exercise 5
3rd Grade Review: Identify which property is used in each equation: (6 + 2) + 3 = 6 + (2 + 3), 4 × 7 = 7 × 4, 9 × 1 = 9.
Definition:

Mixed properties application: Problems requiring identification of which property is being used in specific contexts.

Equation 1
(6 + 2) + 3 = 6 + (2 + 3)
Property
Associative
Equation 2
4 × 7 = 7 × 4
Property
Commutative
Equation 3
9 × 1 = 9
Property
Identity
Step 1: Identify property in first equation

(6 + 2) + 3 = 6 + (2 + 3)

The grouping of numbers changes but the result stays the same

This is the associative property of addition

Step 2: Identify property in second equation

4 × 7 = 7 × 4

The order of numbers changes but the result stays the same

This is the commutative property of multiplication

Step 3: Identify property in third equation

9 × 1 = 9

Multiplying by 1 doesn't change the number

This is the multiplicative identity property

(6 + 2) + 3 = 6 + (2 + 3) (Associative)
4 × 7 = 7 × 4 (Commutative)
9 × 1 = 9 (Identity)
Final answer:

First: Associative property, Second: Commutative property, Third: Identity property

Applied rules:

• Associative property: Grouping doesn't matter in addition

• Commutative property: Order doesn't matter in multiplication

• Identity property: Multiplying by 1 preserves the number

3rd Grade Properties Review 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

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 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: Zero is like a "black hole" in multiplication - anything times 0 is 0.
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 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)

• Identity: a + 0 = a and a × 1 = a

• Zero: a × 0 = 0

3rd Grade Properties Review: Progress Tracking
Exercise 6: Property Recognition Skills
Track your progress with different properties from 3rd grade:
Commutative, Associative, Identity, and Zero properties

Analysis: The chart shows your progress with different properties from 3rd grade.

  • Commutative property: Understanding of order changes
  • Associative property: Grasp of grouping concepts
  • Identity property: Recognition of neutral elements
  • Zero property: Understanding of zero's effect in multiplication
  • Overall review score: Comprehensive understanding

Questions & Answers

Question: Why do we need to review 3rd grade properties in 4th grade?

Answer: Reviewing 3rd grade properties is important because:

  • Foundation building: These properties form the basis for more complex math
  • Memory reinforcement: Regular review strengthens your understanding
  • Preparation: You'll use these properties in more advanced operations
  • Confidence: Solid foundation makes new concepts easier to learn

Strong knowledge of these fundamentals helps you excel in 4th grade math and beyond.

Question: I forgot the difference between commutative and associative properties. How can I remember?

Answer: Here's a helpful way to remember:

  • Commutative = COMMUTE: Numbers commute (travel) to different positions
  • Associative = ASSOCIATE: Numbers associate (group) differently with each other
  • Commutative: Changes order: a + b = b + a
  • Associative: Changes grouping: (a + b) + c = a + (b + c)

Think of commuting to school (changing location/order) versus forming different study groups (changing associations/groupings).

Question: How can I practice these properties effectively?

Answer: Here are effective practice strategies:

  • Start with simple numbers: Practice with small numbers first
  • Write examples: Create your own equations using each property
  • Use visual models: Draw arrays or number lines to understand concepts
  • Practice daily: Spend 10-15 minutes each day reviewing
  • Check your work: Always verify that both sides of equations are equal

Regular practice with increasing complexity builds strong mathematical understanding.