6.AT.7 Lesson 6-6-group2 🟣 Group 2 · Challenge & Extension
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

6.6 Small Group · Group 2

Tier 1 Extension · Non-Routine Synthesis · Misconception Traps · Rigorous CER Proofs

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 DRAG SORT

    Match each expression with its simplified form.

    Target Categories: Correctly Matched
    • 2 MATCHING GAME

      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 MULTIPLE CHOICE

      A track stacks 3x and then one more x. Which expression is equivalent to 3x + x?

      1. A4x
      2. B3x²
      3. C4
      4. D3x
      ✏️ Workspace & Solution Steps
    • 4 MULTIPLE CHOICE

      Producer B flipped a formula to 5 + 2x. Which expression is equivalent?

      1. A2x + 5
      2. B7x
      3. C10x
      4. D5x + 2
      ✏️ Workspace & Solution Steps
    • 5 MULTIPLE CHOICE

      Which expression is equivalent to 4x + 3x?

      1. A7x
      2. B12x
      3. C7x²
      4. D43x
      ✏️ Workspace & Solution Steps
    SECTION 2 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
    1. 6 ERROR ANALYSIS

      Level 2 Extension — The False Remix

      1. 1Expression:2x + 6
      2. 2Claim:2x and 6 are like terms, so combine them
      3. 3Result:2x + 6 = 8x

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

      ✏️ Corrected Mathematical Work & Explanation
    🗣️ MATHEMATICAL DISCOURSE & TALK MOVES SMP.3 / Construct Viable Arguments

    Group Discussion Prompt: How does your visual model justify your mathematical solution?

    🗣️ Partner A:
    Partner A: "I modeled this by identifying the relationship and..."
    👂 Partner B:
    Partner B: "I agree with your step because the standard mathematical rule states..."
    ✍️ The Writing Revolution (TWR) · Sentence Expansion Because · But · So

    Complete each sentence stem to demonstrate precise mathematical reasoning:

    BECAUSE The ratio remains equivalent because both quantities are scaled by the exact same multiplicative factor.
    BUT Two ratios may look similar, but inverting the order of terms fundamentally changes the comparison.
    SO The recipe requires 3 parts flour to 2 parts water, so the unit rate is 1.5 cups of flour per cup of water.
    📐 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.
    ✍️ AUTHOR YOUR OWN EXTENSION CHALLENGE

    Create an original multi-step word problem aligned to this standard. Include constraints, and write the complete step-by-step mathematical proof below.

    ✏️ Author Workspace & Complete Solution Key
    Student Mastery Self-Assessment:
    1 · Need More Support
    2 · Getting Closer
    3 · Got It / Solid
    4 · Master / Can Teach It