Identifying Equivalent Expressions

In this lesson:

• Explain what it means for two expressions to be equivalent
• Use substitution to test expression pairs
• Confirm equivalence with algebraic simplification

What You Will Be Able to Do

By the end of this lesson, you should be able to:

1. State what "equivalent" means: same output for every substitution, not just one
2. Test two expressions with substitution; one mismatch proves they are not equivalent
3. Confirm equivalence by simplifying both expressions algebraically

How Would You Convince a Skeptic?

In 6.EE.A.3 you combined like terms — became .

How would you convince a skeptic that these two expressions are always equal, for any value of ?

Equivalence: Same Output, Every Input

"The two expressions name the same number regardless of which value is substituted."
— CCSS 6.EE.A.4

Two expressions are equivalent when they produce the same output for every possible input — not just one or two values we happen to test.

Testing y + y + y and 3y

Every row matches — the expressions produce the same output for each value of we tried.

Does One Match Prove Equivalence?

Look at these two expressions: and

Test :

Both give 12. Are they equivalent?

One Matching Test Is Not Enough

The coincidence trap: and agreed at only — not equivalent.

Does One Match Prove It?

True or false:

If two expressions give the same result when , they are equivalent.

Think before advancing — what would you need to show to prove equivalence?

Substitution Tests vs. Algebraic Proof

• Disprove — one mismatch is conclusive; stop there
• Suggest — matches support equivalence but can't prove it for every value
• Algebra proves it — simplify both expressions to the same form; the match holds for all values

Testing and Confirming Equivalence: Two Steps

Step 1 — Substitute two or more values

• Any mismatch → not equivalent, stop
• All rows match → go to Step 2

Step 2 — Verify algebraically

• Simplify both expressions; if they reach the same form → equivalent

Pair A: Are These Equivalent?

Step 1 — substitute :

Step 2 — simplify: — identical, equivalent ✓

Pair B: Is , Not

Step 1 — substitute :

is a counterexample (a value where they disagree) — not equivalent.

Pair C: Are These Equivalent?

Step 1 — substitute two values:

Step 2 — distribute: — equivalent ✓

Pair D: Are These Equivalent?

Step 1 — substitute :

Mismatch — not equivalent. , not .

Your Turn: Classify These Pairs

Apply the two-step method to each pair.

Pair E: and

Pair F: and

Substitute two values first. Mismatch → done. All match → Step 2 algebra. Work both before advancing.

Answer: Pair E Is Not Equivalent

and

• Substitute : vs. ✗
• Mismatch at Step 1 — , not

Answer: Pair F Is Equivalent

and

• Substitute : vs. ✓; : vs. ✓
• Step 2: ✓

What to Remember and Watch Out For

✓ Equivalence = same output for every value, not just one

✓ One mismatch → not equivalent; stop

✓ Algebra proves it; substitution suggests

One match isn't proof — might be coincidence

Distribute every term: , not

Shared symbols ≠ equivalent:

Coming Up Next: Solving Equations

6.EE.B.5 asks the opposite question:

Which specific value makes two non-equivalent expressions equal?

That's solving an equation. The key contrast:

• Equivalence = interchangeable for all values
• Equation solution = equal at one specific value