Grade 6 Mathematics · Unit 6: Numerical and Algebraic ExpressionsStandard 6.AT.7

Lesson 6.6Identify Equivalent Algebraic Expressions

Start hereWords, worked example, and sentence starters

Name Date Period

Learning target I can show that two expressions are equivalent by simplifying and combining like terms.

Why This Matters At the music studio, two producers wrote formulas for the cost of renting equipment. Producer A wrote 3x + 2x + 10, and Producer B wrote 5x + 10. They look different, but are they really? Let's check: if x = 4, Producer A gets 3(4) + 2(4) + 10 = 30, and Producer B gets 5(4) + 10 = 30. Same result — they're equivalent expressions!

Start hereWords for this lesson

Read each word before you begin. Say it out loud.

WordWhat it meansExample
Equivalent ExpressionsSpanish: Expresiones equivalentes Expressions that look different but have the same value for every allowed variable value. 7c + 9 + 6c = 13c + 9
EquivalentSpanish: Equivalente Expressions that always have the same value. 2x + 4 and 2(x + 2): when x = 3, both = 10; when x = 7, both = 18 — always the same
SimplifySpanish: Simplificar To rewrite an expression in a shorter form that is still equal to the original. 7c + 9 + 6c → 13c + 9
Like TermsSpanish: Términos semejantes Terms with the same letter raised to the same power, like 2x and 5x. 7c and 6c
CombineSpanish: Combinar To add or subtract terms with the same letter into one. 7c + 6c = 13c
CoefficientSpanish: Coeficiente The number in front of a letter, like the 3 in 3x. the 13 in 13c

How it worksWorked example

These numbers are not on your problems. The steps are. Follow them with your own numbers.

Simplify 7c + 9 + 6c. Is it equivalent to 13c + 9?

  1. Like terms: 7c and 6c.
  2. Combine: 7c + 6c = 13c. The 9 stays alone.
  3. 7c + 9 + 6c = 13c + 9.

Answer: 13c + 9; yes, equivalent

Now you trySame steps, your turn

The steps are the same as the worked example. The numbers are yours. Do the work.

Let's check 2x + 3x and 5x. Try x = 2.

Answer:

Say it and write itSentence starters

Finish each sentence out loud with a partner. Then use them in your writing.

  • The like terms are and because .
  • I combined them to get .
  • The constant stays by itself because .
  • The two expressions equivalent because .

Word bank equivalentlike termscombinesimplifycoefficientconstanttermvariablesame valuegroupaddcompare

Watch out!A common mistake

A common mistake in Equivalent Expressions is adding the exponents instead of the coefficients when combining like terms

Grade 6 Mathematics · Unit 6: Numerical and Algebraic ExpressionsStandard 6.AT.7

Lesson 6.6Identify Equivalent Algebraic Expressions

Version ASupported practice

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can show that two expressions are equivalent by simplifying and combining like terms.

Why This Matters At the music studio, two producers wrote formulas for the cost of renting equipment. Producer A wrote 3x + 2x + 10, and Producer B wrote 5x + 10. They look different, but are they really? Let's check: if x = 4, Producer A gets 3(4) + 2(4) + 10 = 30, and Producer B gets 5(4) + 10 = 30. Same result — they're equivalent expressions!

Part 1Understand the ideamodels and meaning
  1. 1Circle the letter of the best answer. Show how you know.

    Which expression is equivalent to 4x + 3x?

    1. A7x
    2. B7x²
    3. C12x
    4. D43x

    Hint 4x and 3x share the exact same variable, x — that makes them like terms.

    Show your work
  2. 2Circle the letter of the best answer. Show how you know.

    Which expression is equivalent to 2(m + 5)?

    1. A2m + 10
    2. B2m + 5
    3. C7m
    4. Dm + 10

    Hint The 2 outside the parentheses must multiply both the m and the 5.

    Show your work
  3. 3Write the name of the correct group on each line.

    Sort each pair of expressions by whether they are equivalent or not.

    Groups: Equivalent Not Equivalent
    • 3x + 2x and 5x
    • 2(n + 4) and 2n + 8
    • 4a + 3 and 7a
    • x + x and 2x
    • 3y + 2 and 5y

    Hint Equivalent means always equal. Simplify the first expression in each pair and compare with the second.

Part 2Apply itreal-world problems
  1. 4Circle the letter of the best answer. Show how you know.

    Which property is shown: 3(x + 7) = 3x + 21?

    1. ACommutative
    2. BDistributive
    3. CAssociative
    4. DIdentity

    Hint Watch what happened to the 3: it multiplied the x AND the 7 across the parentheses.

    Show your work
  2. 5Circle the letter of the best answer. Show how you know.

    Simplify: 5n + 2n

    1. A7n
    2. B7n²
    3. C10n
    4. D52n

    Hint 5n and 2n both have the variable n — they're like terms, so they can combine.

    Show your work
Part 3Explain and correctreasoning
  1. 6Find the mistake. Explain it, then show the correct work.

    Spot the Distributive Mistake

    1. 1Expression3(x + 4)
    2. 2DistributeMultiply the 3 by the first term only: 3 · x + 4
    3. 3Result3(x + 4) = 3x + 4

    Which step has the mistake? Explain what went wrong, then show the correct work.

    Hint How many terms are inside the parentheses? The 3 multiplies each one.

    Correct work
    Correct answer:

Explain your thinkingHow did you decide which expressions belong in the same group? What strategies did you use?

Sentence starter I knew ___ and ___ are equivalent because when I ___ the terms, I got the same expression.

3-Point Exit Checklist: ☐ 1. Units & Labels Verified ☐ 2. Mathematical Reasoning Stated ☐ 3. Calculations Double-Checked
How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help
Grade 6 Mathematics · Unit 6: Numerical and Algebraic ExpressionsStandard 6.AT.7

Lesson 6.6Identify Equivalent Algebraic Expressions

Version BCore practice

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can show that two expressions are equivalent by simplifying and combining like terms.

Why This Matters At the music studio, two producers wrote formulas for the cost of renting equipment. Producer A wrote 3x + 2x + 10, and Producer B wrote 5x + 10. They look different, but are they really? Let's check: if x = 4, Producer A gets 3(4) + 2(4) + 10 = 30, and Producer B gets 5(4) + 10 = 30. Same result — they're equivalent expressions!

Part 1Understand the ideamodels and meaning
  1. 1Write the name of the correct group on each line.

    Match each expression with its simplified form.

    Groups: Correctly Matched
    • 8y + 2y → 10y
    • 3(a + 6) → 3a + 18
    • 5n + 4 + 2n → 7n + 4
    • 9p − 3p → 6p
    • 4(2x − 1) → 8x − 4
    • 6t + t → 7t
  2. 2Write the letter of the matching item on each line.

    Match each expression to its simplified equivalent.

    1. 6x + 2x
    2. 3(n + 5)
    3. 4a + 7 + a
    4. 2(3y − 4)
    5. 10m − 4m + 1
    6. 5(x + 2)
    • A8x
    • B3n + 15
    • C5a + 7
    • D6y − 8
    • E6m + 1
    • F5x + 10
  3. 3Complete the table. Show how you found each value.

    Simplify each expression. Then evaluate both the original and simplified form when x = 3 to verify they are equivalent.

    OriginalSimplifiedValue when x = 3
    2x + 5x + 1
    4(x + 2)
    3x + 4 + x − 1
    Show your work
  4. 4Decide whether each pair balances. Check Yes or No, then explain.

    Both equal 14 — they are equivalent because 3x + 2x = 5x.

    Left sideRight sideBalanced?
    Explain your choice
Part 2Explain and correctreasoning
  1. 5Find the mistake. Explain it, then show the correct work.

    Spot the Combining Mistake

    1. 1Expression7x + 3 + 2x
    2. 2CombineCombine every number you see: 7x + 2x + 3 = 9x + 3, then add the 9 and 3 too
    3. 3Result7x + 3 + 2x = 12x

    Which step has the mistake? Explain what went wrong, then show the correct work.

    Correct work
    Correct answer:

Explain your thinkingHow did you decide which expressions belong in the same group? What strategies did you use?

3-Point Exit Checklist: ☐ 1. Units & Labels Verified ☐ 2. Mathematical Reasoning Stated ☐ 3. Calculations Double-Checked
How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help
Grade 6 Mathematics · Unit 6: Numerical and Algebraic ExpressionsStandard 6.AT.7

Lesson 6.6Identify Equivalent Algebraic Expressions

ChallengeExtension

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can show that two expressions are equivalent by simplifying and combining like terms.

Why This Matters At the music studio, two producers wrote formulas for the cost of renting equipment. Producer A wrote 3x + 2x + 10, and Producer B wrote 5x + 10. They look different, but are they really? Let's check: if x = 4, Producer A gets 3(4) + 2(4) + 10 = 30, and Producer B gets 5(4) + 10 = 30. Same result — they're equivalent expressions!

Part 1Understand the ideamodels and meaning
  1. 1Complete the table. Show how you found each value.

    For each expression, find an equivalent expression using a DIFFERENT method (distributive property, combining like terms, or factoring).

    ExpressionEquivalent FormMethod Used
    4x + 8
    3a + 2a + 5
    6(n − 2)
    Show your work
Part 2Explain and correctreasoning
  1. 2Find the mistake. Explain it, then show the correct work.

    Find the Simplification Error

    1. 1Expression3x + 5 + 2x + 1
    2. 2Combine3x + 2x + 5 + 1
    3. 3Simplify5x² + 6

    Which step has the mistake? Explain what went wrong, then show the correct work.

    Correct work
    Correct answer:
  2. 3Answer in complete sentences.

    A student says 2x + 3 and 5x are equivalent because both equal 5 when x = 1. Is the student correct? Explain why testing one value is not enough to prove equivalence, and find a value of x where they give different results.

    Write your answer

Explain your thinkingHow did you decide which expressions belong in the same group? What strategies did you use?

3-Point Exit Checklist: ☐ 1. Units & Labels Verified ☐ 2. Mathematical Reasoning Stated ☐ 3. Calculations Double-Checked
How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help