6.EE.A.4 6th Grade Expressions & Equations

Identifying Equivalent Expressions

Identify when two expressions are equivalent, meaning they name the same number regardless of which value is substituted into them.

How to explain it

At this standard, students identify when two expressions are equivalent using two strategies: (1) simplifying via combining like terms (CLT) or the distributive property (DP) until both expressions match, and (2) verifying by substituting a value. Students distinguish equivalent (same value for ALL substitutions) from simply equal for one specific value. Note: Problem 15 (D-1=V) verifies x=1 in both 2(x+3) and 2x+6; Problem 16 (R-D=ii) requires writing the expanded form and verifying with x=2.

The anchor students hold onto: Simplify both → same form? EQUIVALENT — Test values → all equal? EQUIVALENT

Equivalence underlies solutions: a value solves an equation when it makes both sides equal. Leads to 6.EE.B.5+B.6 (testing solutions) and 7.EE.A+B expression reasoning in 7th grade.

Worked examples

Example 1 Combining Like Terms

Are 4x + 3x and 7x equivalent?

Step 1Both terms have x — they are LIKE TERMS

Step 2Combine: 4x + 3x = 7x

Step 3Test x = 2: 8 + 6 = 14 and 7 × 2 = 14 OK

Answer4x + 3x = 7x — EQUIVALENT

Example 2 Distributive Property

2(x+5) vs. 2x+10: equivalent?

Step 1Distribute: 2(x + 5) = 2 × x + 2 × 5

Step 2Simplify: 2x + 10

Step 3Test x = 3: 2(8) = 16 and 6 + 10 = 16 OK

Answer2(x + 5) = 2x + 10 — EQUIVALENT

Common mistakes

What students write Thinking two expressions are equivalent because they give the same result for one value of x — equal for one value does not mean equivalent for all values.

The fix Equivalent means the same value for EVERY substitution. Test at least two values, or simplify both sides to the same form, to be certain.

Try this Student work: 4(x + 3) = 4x + 3 Conclusion: 4(x + 3) and 4x + 3 are equivalent. The student above made an error. Find and fix the error. Then test with x = 1 to confirm your correction.

What students write Only distributing to the first term: writing 4(x + 3) = 4x + 3 instead of 4x + 12.

The fix The factor multiplies EVERY term inside the parentheses: 4 × x AND 4 × 3 = 4x + 12. Distribute to all terms.

Teacher tip

Head off the two predictable errors before they happen. First: Equivalent means the same value for EVERY substitution. Test at least two values, or simplify both sides to the same form, to be certain. Second: The factor multiplies EVERY term inside the parentheses: 4 × x AND 4 × 3 = 4x + 12. Distribute to all terms.