Equivalent Expressions - Sixth Grade
Complete Notes & Formulas
1. What are Equivalent Expressions?
Definition
Equivalent expressions are expressions that
have the SAME VALUE for ALL values of the variable
They look different but mean the same thing!
Examples
• 2(x + 3) and 2x + 6 are equivalent
• 3x + 5x and 8x are equivalent
• 4 + x and x + 4 are equivalent
2. Properties of Addition
Commutative Property of Addition
Changing the ORDER doesn't change the sum
a + b = b + a
Example: 5 + 3 = 3 + 5
Associative Property of Addition
Changing the GROUPING doesn't change the sum
(a + b) + c = a + (b + c)
Example: (2 + 3) + 4 = 2 + (3 + 4)
Identity Property of Addition
Adding ZERO doesn't change the number
a + 0 = a
Example: 7 + 0 = 7
3. Properties of Multiplication
Commutative Property of Multiplication
Changing the ORDER doesn't change the product
a × b = b × a
Example: 4 × 5 = 5 × 4
Associative Property of Multiplication
Changing the GROUPING doesn't change the product
(a × b) × c = a × (b × c)
Example: (2 × 3) × 4 = 2 × (3 × 4)
Identity Property of Multiplication
Multiplying by ONE doesn't change the number
a × 1 = a
Example: 9 × 1 = 9
Zero Property of Multiplication
Multiplying by ZERO always equals zero
a × 0 = 0
Example: 100 × 0 = 0
4. Distributive Property
The Most Important Property!
Multiply the number OUTSIDE by EACH term inside
a(b + c) = ab + ac
a(b − c) = ab − ac
Example 1: Using Distributive Property
Simplify: 3(x + 5)
3(x + 5)
= 3 × x + 3 × 5
= 3x + 15
Answer: 3x + 15
Example 2: With Subtraction
Simplify: 4(2x − 3)
4(2x − 3)
= 4 × 2x − 4 × 3
= 8x − 12
Answer: 8x − 12
Visual: Area Model for Distributive Property
Example: 3(x + 4) using area model
Area = 3(x + 4) = 3x + 12
5. Factoring Using Distributive Property
What is Factoring?
Factoring is REVERSE of distributing
Take OUT the common factor
ab + ac = a(b + c)
Steps to Factor
Step 1: Find the Greatest Common Factor (GCF)
Step 2: Divide each term by the GCF
Step 3: Write as: GCF(remaining terms)
Example 1: Factor 6x + 9
Step 1: GCF of 6 and 9 is 3
Step 2: 6x ÷ 3 = 2x, and 9 ÷ 3 = 3
Step 3: 3(2x + 3)
Answer: 3(2x + 3)
Example 2: Factor 4x + 8y
Step 1: GCF of 4x and 8y is 4
Step 2: 4x ÷ 4 = x, and 8y ÷ 4 = 2y
Step 3: 4(x + 2y)
Answer: 4(x + 2y)
6. Combining Like Terms
What are Like Terms?
Like terms have the SAME variable
with the SAME exponent
Only coefficients can be different
Like Terms vs Unlike Terms
Like Terms ✓ | Unlike Terms ✗ |
---|---|
3x and 5x | 3x and 5y |
7y and -2y | 7x and 7x² |
4 and 9 (constants) | 4x and 9 |
How to Combine Like Terms
Add or subtract the COEFFICIENTS
Keep the VARIABLE the same
Example 1: Combine 4x + 7x
4x + 7x
= (4 + 7)x
= 11x
Answer: 11x
Example 2: Simplify 5x + 3 - 2x + 7
Step 1: Group like terms
(5x - 2x) + (3 + 7)
Step 2: Combine each group
3x + 10
Answer: 3x + 10
7. Writing Equivalent Expressions Using Properties
Methods to Create Equivalent Expressions
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)
Example: Multiple Equivalent Forms
Original: 2(3x + 4)
Equivalent Form 1: 6x + 8 (using distributive)
Equivalent Form 2: 2(4 + 3x) (using commutative)
Equivalent Form 3: 2 × 3x + 2 × 4 (expanded)
All equal 6x + 8!
8. How to Identify Equivalent Expressions
Method 1: Simplify Both Expressions
Simplify each expression completely
If they simplify to the SAME form, they're equivalent
Example: Are 3(x + 2) and 3x + 6 equivalent?
Expression 1: 3(x + 2)
= 3x + 6
Expression 2: 3x + 6
= 3x + 6
Yes! They are equivalent ✓
Method 2: Substitute a Value
Test: Are 2x + 4 and 2(x + 2) equivalent?
Let x = 3
Expression 1: 2(3) + 4 = 10
Expression 2: 2(3 + 2) = 10
Both equal 10, so they're equivalent! ✓
9. Using Strip Models for Equivalent Expressions
What is a Strip Model?
A visual bar divided into sections
Each section represents a term
Helps see if expressions are equivalent
Visual Example: x + x + x = 3x
Both strips show the same total length
Quick Reference: Properties Summary
Property | Formula | What It Does |
---|---|---|
Commutative (Addition) | a + b = b + a | Change order |
Associative (Addition) | (a + b) + c = a + (b + c) | Change grouping |
Distributive | a(b + c) = ab + ac | Multiply each term |
Identity (Addition) | a + 0 = a | Add zero |
Identity (Multiplication) | a × 1 = a | Multiply by one |
💡 Important Tips to Remember
✓ 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
🧠 Memory Tricks & Strategies
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!"
Master Equivalent Expressions! 🔢 ↔️ 🎯
Remember: Different looks, same value!