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 it means for two expressions to be equivalent — same output for every substitution, not just one
2. Use substitution of at least two values to test whether two expressions are equivalent, and recognize that one mismatch is enough to prove they are not
3. Confirm equivalence algebraically by simplifying both expressions using the distributive property and combining like terms

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 algebraically:

Both sides are identical — equivalent ✓

Pair B: Are These Equivalent?

Step 1 — Substitute :

Mismatch — stop. Not equivalent.

Note: , not . The constant matters.

Pair C: Are These Equivalent?

Step 1 — Substitute two values:

Step 2 — Distribute:

Equivalent ✓

Pair D: Are These Equivalent?

Step 1 — Substitute :

Mismatch — stop. Not equivalent.

These share the same symbols, but , 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.

Answers: Pair E and Pair F

Pair E: and — Not equivalent

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

Pair F: and — Equivalent ✓

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

What to Remember and Watch Out For

✓ Equivalence means same output for every value — a universal claim, not a lucky coincidence

✓ One mismatch is conclusive proof of non-equivalence — stop and record

✓ Algebra confirms what substitution can only suggest: simplify to the same form

Watch out: One matching test is not proof — it could be a coincidence (x + 6 and 2x both give 12 at x = 6)

Watch out: Distribute to every term — , not

Watch out: Shared symbols don't mean 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