📐 6th Grade Math · Expressions Anchor Charts
6.EE.3
Generating Equivalent Expressions
Apply properties of operations to write equivalent expressions
Equivalent expressions have the same value for every value of the variable.
We use properties of operations to rewrite them in different — but equal — forms.
Key Properties
📦 Distributive Property
a(b + c) = ab + ac
e.g. 3(x + 4) = 3x + 12
🔄 Commutative Property
a + b = b + a
e.g. 2x + 5 = 5 + 2x
🧲 Associative Property
(a+b)+c = a+(b+c)
e.g. (3x + 2) + 4 = 3x + (2+4)
🗂 Combining Like Terms
ax + bx = (a+b)x
e.g. 5x + 3x = 8x
How To Simplify — Step by Step
- 1 Use the Distributive Property to expand any parentheses if needed.
- 2 Identify like terms — same variable AND same exponent.
- 3 Combine like terms by adding or subtracting their coefficients.
- 4 Write the simplified equivalent expression.
Worked Example
✏️ Simplify: 3(x + 4) + 2x − 5
3(x + 4) + 2x − 5
→
3x + 12 + 2x − 5
(Distribute the 3)
→
3x + 2x + 12 − 5
(Group like terms)
→
5x + 7
(Combine like terms)
Like terms MUST have the same variable and same exponent. You can add 3x + 2x, but you CANNOT combine 3x + 2 because x and a number are NOT like terms!
🧠 Check Your Understanding
🪜 These questions are scaffolded — they start simple and get harder. Work through them in order!
LEVEL 1 · Identify Like Terms
Which pair of terms are like terms?
✅ Yes! 5x and 2x are like terms because they have the SAME variable (x) and the same exponent (1). You can combine them: 5x + 2x = 7x.
❌ Like terms must have the SAME variable AND the same exponent. 5x and 2x both have an x — that's what makes them like terms! A number alone (like 3) and a variable term (like 3x) are NOT like terms.
LEVEL 2 · Combine Like Terms
Simplify:
4x + 7 + 3x + 1✅ Great work! Group x-terms: 4x + 3x = 7x. Group constants: 7 + 1 = 8. Final answer: 7x + 8.
❌ Group like terms separately! x-terms: 4x + 3x = 7x. Constants: 7 + 1 = 8. Final answer: 7x + 8. Remember — you can only combine terms that are alike!
LEVEL 3 · Distributive Property
Which expression is equivalent to
3(x + 5)?✅ Correct! Distribute 3 to BOTH terms inside: 3 · x = 3x and 3 · 5 = 15. So 3(x + 5) = 3x + 15.
❌ Don't forget to multiply 3 by EACH term inside the parentheses: 3 · x = 3x AND 3 · 5 = 15. The answer is 3x + 15, not 3x + 5 (you must distribute to the 5 too!).
LEVEL 4 · Distribute + Combine
Simplify:
2(x + 4) + 3x✅ Excellent two-step work! Step 1 — Distribute: 2(x+4) = 2x + 8. Step 2 — Rewrite: 2x + 8 + 3x. Step 3 — Combine: 2x + 3x = 5x. Answer: 5x + 8.
❌ Two steps needed! First distribute: 2(x+4) = 2x + 8. Then combine like terms: 2x + 8 + 3x → (2x + 3x) + 8 = 5x + 8. Don't forget to distribute before combining!
LEVEL 5 · Multi-Step Challenge ⭐
Which expression is equivalent to
2(2x + 1) + 3x + 3?✅ You nailed it! Step 1 — Distribute: 2(2x+1) = 4x + 2. Step 2 — Rewrite all terms: 4x + 2 + 3x + 3. Step 3 — Combine x-terms: 4x + 3x = 7x. Step 4 — Combine constants: 2 + 3 = 5. Answer: 7x + 5.
❌ Let's work step by step: ① Distribute first: 2(2x+1) = 4x + 2. ② Rewrite: 4x + 2 + 3x + 3. ③ Combine x-terms: 4x + 3x = 7x. ④ Combine constants: 2 + 3 = 5. Answer: 7x + 5.
6.EE.4
Identifying Equivalent Expressions
Recognize when two expressions are equivalent — and prove it!
Two expressions are equivalent if they produce the same value no matter what number you substitute for the variable. We can verify this by simplifying both sides or by substituting a value.
Two Ways to Check Equivalence
🔬 Method 1: Simplify Both Sides
Fully simplify each expression using properties. If they reduce to the same expression, they are equivalent.
🔢 Method 2: Substitute a Value
Plug in a number for the variable in both expressions. If they give the same answer, they may be equivalent — test a second value to be sure!
Worked Example — Simplify Method
✏️ Are 2(3x + 1) and 6x + 2 equivalent?
2(3x + 1)
→
6x + 2
(Distribute the 2)
✅ YES — both sides simplify to 6x + 2. They are equivalent!
Worked Example — Substitution Method
✏️ Are 3x + 3 and 3(x + 1) equivalent? (Let x = 5)
3(5) + 3 = 18
and
3(5 + 1) = 18
→
Both = 18 ✓
Try x = 2: 3(2)+3=9 and 3(3)=9 ✓
✅ YES — They are equivalent expressions!
NOT Equivalent — Common Mistake
⚠️ Is 2x + 4 equivalent to 2(x + 4)?
2(x + 4)
→
2x + 8
(Distribute: 2·4 = 8)
❌ NO — 2x + 4 ≠ 2x + 8. They are NOT equivalent!
One substitution is NOT enough proof — you must fully simplify OR test multiple values. One matching answer could be a coincidence!
🧠 Check Your Understanding
🪜 These questions are scaffolded — they start simple and get harder. Work through them in order!
LEVEL 1 · True or False
True or False:
x + x + x is equivalent to 3x.✅ TRUE! x + x + x means adding three of the same term. Since they are like terms: x + x + x = 3x. This works for ANY value of x, not just x = 3!
❌ It is TRUE! x + x + x is adding x three times, which equals 3x — just like 1 apple + 1 apple + 1 apple = 3 apples! Note: x³ means x multiplied by itself (x · x · x), which is different.
LEVEL 2 · Distribute & Compare
Is
2(x + 3) equivalent to 2x + 6?✅ Yes, they are equivalent! Distribute: 2 · x = 2x and 2 · 3 = 6. So 2(x+3) = 2x + 6. Both expressions will always give the same result for any value of x.
❌ Distribute carefully: 2(x+3) means multiply 2 by EACH term. 2 · x = 2x AND 2 · 3 = 6. So 2(x+3) = 2x + 6 — they ARE equivalent!
LEVEL 3 · Spot the Mistake
A student says
4(x + 2) is equivalent to 4x + 2. What mistake did they make?✅ You caught it! The student forgot to distribute 4 to the 2. The correct answer is 4 · x = 4x AND 4 · 2 = 8, so 4(x+2) = 4x + 8, not 4x + 2.
❌ The student's error was only distributing to the x. You must multiply 4 by EVERY term inside: 4 · x = 4x and 4 · 2 = 8. So 4(x+2) = 4x + 8. The student stopped too early!
LEVEL 4 · Select Two ⭐
Which expressions are equivalent to
Select two answer choices.
12a + 6b?Select two answer choices.
✅ Both C and D equal 12a + 6b!
• C: 6(2a + b) = 6·2a + 6·b = 12a + 6b ✓
• D: 2(6a + 3b) = 2·6a + 2·3b = 12a + 6b ✓
• B: 3(4a+2) = 12a + 6 ✗ (missing the b!)
• C: 6(2a + b) = 6·2a + 6·b = 12a + 6b ✓
• D: 2(6a + 3b) = 2·6a + 2·3b = 12a + 6b ✓
• B: 3(4a+2) = 12a + 6 ✗ (missing the b!)
❌ Distribute each option and compare to 12a + 6b:
• C: 6(2a+b) = 12a+6b ✓
• D: 2(6a+3b) = 12a+6b ✓
• B: 3(4a+2) = 12a+6 ✗ (6, not 6b!)
• A: 18ab multiplies the variables ✗
• C: 6(2a+b) = 12a+6b ✓
• D: 2(6a+3b) = 12a+6b ✓
• B: 3(4a+2) = 12a+6 ✗ (6, not 6b!)
• A: 18ab multiplies the variables ✗
LEVEL 5 · GCF & Two Variables 🏆
Which expression is equivalent to
15x + 33y?✅ You found the GCF! The greatest common factor of 15 and 33 is 3. Divide each term: 15x ÷ 3 = 5x and 33y ÷ 3 = 11y. So 15x + 33y = 3(5x + 11y). Check: 3·5x + 3·11y = 15x + 33y ✓
❌ Find the GCF of 15 and 33 — it's 3! Factor it out: 15x ÷ 3 = 5x and 33y ÷ 3 = 11y. Answer: 3(5x + 11y). Watch out for D: 3(11x+5y) = 33x+15y — the coefficients are swapped! Always check by distributing back.