6.AT.7 Lesson 6-6-group2 🟣 Group 2 · Set B
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

6.6 Small Group · Group 2

Second Challenge Form · Non-Routine Extension · Same Standard, New Problems

Equivalent Expressions · Equivalent · Procedural Fluency · Reasoning & Critique

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — When are two expressions equivalent?
1Equivalent expressions give the same answer for every value of the variable — and you can say why it is true, and where it would stop being true.
  • You already can do this. So we go up a level, not up a number: find what is always true, show it a second way, and be ready to defend it with a reason instead of an answer.
2Worked Example — Watch me
  1. Producer A wrote 3x + 2x + 10 and Producer B wrote 5x + 10. Are they equivalent?
  2. I test x = 4. For A: 3(4) + 2(4) + 10 = 12 + 8 + 10 = 30.
  3. For B: 5(4) + 10 = 20 + 10 = 30.
  4. Both give 30, and 3x + 2x is just 5x, so they are equivalent.
3Second Model — Try it together — then prove it
  1. Let's check 2x + 3x and 5x. Try x = 2.
  2. 2(2) + 3(2) = 4 + 6 = 10, and 5(2) = 10. Same answer.
  3. So 2x + 3x and 5x are equivalent.
  4. Now prove it: say why that move had to work at all — not just that it did.
  5. Sentence frame — convince a skeptic: "My method works because ___ . It would stop working if ___ ."
4Mathematical Word Bank
  • Equivalent Expressions (Expresiones equivalentes) — Expressions that look different but have the same value for every allowed variable value.
  • Equivalent (Equivalente) — Expressions that always have the same value.
  • Simplify (Simplificar) — To rewrite an expression in a shorter form that is still equal to the original.
  • Like Terms (Términos semejantes) — Terms with the same letter raised to the same power, like 2x and 5x.
  • Combine (Combinar) — To add or subtract terms with the same letter into one.
  • Coefficient (Coeficiente) — The number in front of a letter, like the 3 in 3x.
5Watch out
  • A common mistake in Equivalent Expressions is adding the exponents instead of the coefficients when combining like terms — writing 3x + 2x as 5x² instead of 5x. Remember: 3x means 3 groups of x and 2x means 2 more groups of x, so you add the coefficients (3 + 2 = 5) and keep the variable's power the same (x¹, not x²).
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 MULTIPLE CHOICE

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

    1. A2m + 10
    2. B2m + 5
    3. Cm + 10
    4. D7m
    ✏️ Workspace & Solution Steps
SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 2 MULTIPLE CHOICE

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

    1. ADistributive
    2. BCommutative
    3. CAssociative
    4. DIdentity
    ✏️ Workspace & Solution Steps
SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 3 ERROR ANALYSIS

    Fix our table's thinking

    1. 1The problem:Which property is shown: 3(x + 7) = 3x + 21?
    2. 2A classmate at our table answered:Commutative

    Which step contains the error? Explain the mathematical misconception and write the correct calculation below.

    ✏️ Corrected Mathematical Work & Explanation
  2. 4 ERROR ANALYSIS

    Spot the Combining Mistake

    1. 1Expression:7x + 3 + 2x
    2. 2Combine:Combine every number you see: 7x + 2x + 3 = 9x + 3, then add the 9 and 3 too
    3. 3Result:7x + 3 + 2x = 12x

    Which step contains the error? Explain the mathematical misconception and write the correct calculation below.

    ✏️ Corrected Mathematical Work & Explanation
  3. 5 ERROR ANALYSIS

    Find the Simplification Error

    1. 1Expression:3x + 5 + 2x + 1
    2. 2Combine:3x + 2x + 5 + 1
    3. 3Simplify:5x² + 6

    Which step contains the error? Explain the mathematical misconception and write the correct calculation below.

    ✏️ Corrected Mathematical Work & Explanation
  4. 6 OPEN RESPONSE

    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.

    ✏️ Mathematical Justification & Response
📐 CER MATHEMATICAL PROOF MATRIX (CER 2.0) SMP.3 / Proof & Justification

Writing Task: Justify why your mathematical solution is accurate and complete.

C Claim
State your direct mathematical answer/claim with units.
E Evidence
Cite exact numbers, calculations, dimensions, or graph data.
R Reasoning
Explain the mathematical theorem, property, or definition connecting evidence to claim.
Student Mastery Self-Assessment:
1 · Need More Support
2 · Getting Closer
3 · Got It / Solid
4 · Master / Can Teach It