Equivalent expressions are expressions that

have the SAME VALUE for ALL values of the variable

They look different but mean the same thing!

• 2(x + 3) and 2x + 6 are equivalent

• 3x + 5x and 8x are equivalent

• 4 + x and x + 4 are equivalent

Changing the ORDER doesn't change the sum

a + b = b + a

Example: 5 + 3 = 3 + 5

Changing the GROUPING doesn't change the sum

(a + b) + c = a + (b + c)

Example: (2 + 3) + 4 = 2 + (3 + 4)

Adding ZERO doesn't change the number

a + 0 = a

Example: 7 + 0 = 7

Changing the ORDER doesn't change the product

a × b = b × a

Example: 4 × 5 = 5 × 4

Changing the GROUPING doesn't change the product

(a × b) × c = a × (b × c)

Example: (2 × 3) × 4 = 2 × (3 × 4)

Multiplying by ONE doesn't change the number

a × 1 = a

Example: 9 × 1 = 9

Multiplying by ZERO always equals zero

a × 0 = 0

Example: 100 × 0 = 0

Multiply the number OUTSIDE by EACH term inside

a(b + c) = ab + ac

a(b − c) = ab − ac

Simplify: 3(x + 5)

Simplify: 4(2x − 3)

Example: 3(x + 4) using area model

Area = 3(x + 4) = 3x + 12

Factoring is REVERSE of distributing

Take OUT the common factor

ab + ac = a(b + c)

Step 1: Find the Greatest Common Factor (GCF)

Step 2: Divide each term by the GCF

Step 3: Write as: GCF(remaining terms)

Like terms have the SAME variable

with the SAME exponent

Only coefficients can be different

Add or subtract the COEFFICIENTS

Keep the VARIABLE the same

1. Use Distributive Property (expand)

2. Factor expressions (reverse distribute)

3. Combine like terms

4. Use commutative property (change order)

5. Use associative property (change grouping)

Original: 2(3x + 4)

Simplify each expression completely

If they simplify to the SAME form, they're equivalent

Test: Are 2x + 4 and 2(x + 2) equivalent?

A visual bar divided into sections

Each section represents a term

Helps see if expressions are equivalent

Both strips show the same total length

✓ Distributive Property: Multiply OUTSIDE by EACH term inside

✓ Factoring: Reverse of distributive (pull out GCF)

✓ Like Terms: Must have same variable AND exponent

✓ Combining: Add/subtract coefficients, keep variable

✓ Commutative: Change order (works for + and ×)

✓ Associative: Change grouping (works for + and ×)

✓ Equivalent: Same value for all variable values

✓ Check equivalence: Simplify both or substitute

✓ Area models help visualize distributive property

✓ Strip models help visualize equivalent expressions

Distributive Property:

"Distribute means hand out - multiply outside with each inside, no doubt!"

Commutative:

"Commute means travel around - order can change, result stays sound!"

Associative:

"Associate means hang out together - parentheses move, still same forever!"

Like Terms:

"Same variable, same power - that's when you can combine in the hour!"

Factoring:

"Pull out what's common, that's the way - GCF comes first, terms inside stay!"