Expressions and Properties

Properties are the rules that make math work - master them to simplify expressions!

THE BIG FOUR PROPERTIES:

1. COMMUTATIVE PROPERTY "You can change the ORDER, the result stays the same"

Addition: a + b = b + a Example: 5 + 3 = 3 + 5 = 8 Real-world: Getting $5, then $3 = Getting $3, then $5

Multiplication: a × b = b × a Example: 4 × 7 = 7 × 4 = 28 Real-world: 4 bags of 7 apples = 7 bags of 4 apples

⚠️ NOT for subtraction or division! 5 - 3 ≠ 3 - 5 12 ÷ 4 ≠ 4 ÷ 12

2. ASSOCIATIVE PROPERTY "You can change the GROUPING, the result stays the same"

Addition: (a + b) + c = a + (b + c)
Example: (2 + 3) + 4 = 2 + (3 + 4)
        5 + 4 = 2 + 7
        9 = 9 ✓

Multiplication: (a × b) × c = a × (b × c)
Example: (2 × 3) × 5 = 2 × (3 × 5)
        6 × 5 = 2 × 15
        30 = 30 ✓

⚠️ NOT for subtraction or division!

3. DISTRIBUTIVE PROPERTY "Multiply the outside number by each term inside parentheses"

Formula: a(b + c) = ab + ac

Examples: 3(x + 5) = 3x + 15 4(2x - 7) = 8x - 28 -2(3 + x) = -6 - 2x

Reverse (Factoring): 6x + 9 = 3(2x + 3) 10y - 15 = 5(2y - 3)

Real-world: You buy 3 packs of (4 pens + 2 pencils) = 3 packs × 4 pens + 3 packs × 2 pencils = 12 pens + 6 pencils

4. IDENTITY PROPERTIES "These special numbers don't change the value"

Additive Identity: a + 0 = a Example: 47 + 0 = 47 Zero doesn't change the value when adding!

Multiplicative Identity: a × 1 = a Example: 92 × 1 = 92 One doesn't change the value when multiplying!

BONUS: INVERSE PROPERTIES

Additive Inverse (Opposite): a + (-a) = 0 Example: 7 + (-7) = 0 Opposites sum to zero!

Multiplicative Inverse (Reciprocal): a × (1/a) = 1 (when a ≠ 0) Example: 5 × (1/5) = 1 Reciprocals multiply to one!

ZERO PROPERTY OF MULTIPLICATION: a × 0 = 0 Anything times zero equals zero! Example: 1,000,000 × 0 = 0

USING PROPERTIES TO SIMPLIFY:

Example 1: Mental Math Calculate: 25 × 13 × 4 Commutative: 25 × 4 × 13 = 100 × 13 = 1,300

Example 2: Combining Like Terms Simplify: 3x + 5 + 2x + 7 Commutative: 3x + 2x + 5 + 7 Combine: 5x + 12

Example 3: Distributive Property Expand: 5(x + 3) = 5x + 15

Factor: 6x + 12 = 6(x + 2)

WHY PROPERTIES MATTER: - Make mental math easier - Simplify algebraic expressions - Solve equations efficiently - Understand why math "works" - Foundation for all higher math!

Practice identifying and applying mathematical properties!

IDENTIFY THE PROPERTY: Name the property illustrated.

1. 7 + 9 = 9 + 7 2. 4 × (5 × 2) = (4 × 5) × 2 3. 3(x + 4) = 3x + 12 4. 18 + 0 = 18 5. 6 × 1 = 6 6. 8 + (-8) = 0 7. 12 × 5 = 5 × 12 8. 2(3x - 5) = 6x - 10

TRUE OR FALSE: Determine if the property applies.

9. Is 15 - 7 = 7 - 15? (Does commutative work for subtraction?) 10. Is 20 ÷ 5 = 5 ÷ 20? (Does commutative work for division?) 11. Does (8 - 4) - 2 = 8 - (4 - 2)? (Does associative work for subtraction?) 12. Is (x + 3) + 5 = x + (3 + 5)? (Associative for addition?)

USE DISTRIBUTIVE PROPERTY: Expand these expressions.

13. 4(x + 7) 14. 5(2y - 3) 15. -3(x + 8) 16. 6(4 - 2x) 17. -2(5x - 9)

FACTOR USING DISTRIBUTIVE PROPERTY: Write as a product (reverse of distributing).

18. 8x + 12 = ? 19. 15y - 10 = ? 20. 6x + 18 = ? 21. 20 - 12x = ?

SIMPLIFY USING PROPERTIES: Use commutative and associative properties.

22. 7 + 12 + 3 + 8 (Group to make friendly numbers) 23. 2 × 17 × 5 (Rearrange to use 2 × 5 = 10) 24. 3x + 5 + 2x + 9 (Combine like terms) 25. 4y + 7 - 2y + 3

WORD PROBLEMS: 26. Marcus multiplies 8 × 25 × 4 by rearranging it as (25 × 4) × 8 = 100 × 8 = 800. What property did he use?

27. Show how the distributive property explains why 6 × 19 = 6 × 20 - 6 × 1.

28. A store has 4 shelves with (5 books + 3 magazines) on each shelf. Write two equivalent expressions for the total items using the distributive property.

CHALLENGE:
29. Simplify: 2(3x + 4) + 5(x - 1)
    (Distribute first, then combine like terms)

30. Prove that 3(x + 2) - 2(x - 1) = x + 8 by simplifying the left side.