6B Properties of Operations Study Guide

Key Vocabulary

Commutative Property — you can swap the order and get the same answer (a + b = b + a). (propiedad conmutativa)

Associative Property — you can regroup the numbers and get the same answer ((a + b) + c = a + (b + c)). (propiedad asociativa)

Distributive Property — multiply a number by each term inside the parentheses: a(b + c) = ab + ac. (propiedad distributiva)

Identity Property — adding 0 or multiplying by 1 does not change the number. (propiedad de identidad)

Visual Examples

Commutative (swap order)
3 + 7 = 7 + 3 = 10
4 × 5 = 5 × 4 = 20
Think: "commute" = move around

Associative (regroup)
(2 + 3) + 4 = 2 + (3 + 4) = 9
(2 × 3) × 4 = 2 × (3 × 4) = 24
Think: "associate" = group with friends

Distributive (distribute = share)
5(x + 3) = 5 · x + 5 · 3 = 5x + 15
3(2x − 4) = 3 · 2x − 3 · 4 = 6x − 12
Think: pass out the multiplier to EACH term inside

Identity (stay the same)
7 + 0 = 7 (additive identity)
7 × 1 = 7 (multiplicative identity)

Step-by-Step: Using the Distributive Property

1 Look for a number outside the parentheses.
2 Multiply that number by EACH term inside the parentheses.
3 Write the new expression without parentheses.

Example: 4(x + 6)
= 4 · x + 4 · 6
= 4x + 24

Formula Box

Distributive Property: a(b + c) = ab + ac

Commutative: a + b = b + a | a × b = b × a

Associative: (a + b) + c = a + (b + c)

Watch Out!

Commutative does NOT work for subtraction or division. 5 − 3 ≠ 3 − 5. Only addition and multiplication can swap.
Distribute to EVERY term inside. 3(x + 4) = 3x + 12, NOT 3x + 4.
Watch the signs! 2(x − 5) = 2x − 10. The minus stays minus.

Practice Problems

1. Use the distributive property to expand: 3(x + 7). Write your answer without parentheses.

2. Which property does this show? 9 + 4 = 4 + 9. Type: commutative, associative, or distributive.

3. Use the distributive property to expand: 5(2x − 3). Write your answer without parentheses.