6.AT.8 Lesson 8-3-group2 🟣 Group 2 · Challenge & Extension
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

8.3 Small Group · Group 2

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

Solve Multiplication and Division Equations · Inverse operation · Procedural Fluency · Reasoning & Critique

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — How do I solve a multiplication or division equation?
1Divide to undo multiplying, and multiply to undo dividing — always on both sides — 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. Solve 3x = 21. This means 3 times x equals 21.
  2. The inverse of multiplying by 3 is dividing by 3.
  3. I divide both sides by 3: 3x ÷ 3 = 21 ÷ 3.
  4. That leaves x = 7. Check: 3 × 7 = 21. True!
3Second Model — Try it together — then prove it
  1. Solve x ÷ 4 = 9. Is x multiplied or divided?
  2. It is divided by 4, so the inverse is multiplying both sides by 4.
  3. x = 9 × 4 = 36. Check: 36 ÷ 4 = 9. True?
  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
  • Solve Multiplication and Division Equations (Resolver ecuaciones de multiplicación y división) — Use inverse multiplication or division to isolate the variable.
  • Inverse operation (Operación inversa) — Two math actions that undo each other, like × and ÷.
  • Multiply (Multiplicar) — To add equal groups again and again.
  • Divide (Dividir) — To split into equal groups.
  • Isolate (Despejar) — To get the letter by itself on one side.
  • Coefficient (Coeficiente) — The number in front of a letter, like the 3 in 3x.
5Watch out
  • A common mistake in Solve Multiplication and Division Equations is applying the SAME operation again instead of the inverse — for example, seeing 6x = 42 and multiplying both sides by 6 to get x = 252, instead of dividing by 6 to get x = 7. Before you submit, ask: did I use the OPPOSITE operation shown in the equation, and did I apply it to both sides?
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 FILL TABLE

    Solve each equation, show your work, and check.

    EquationWorkSolutionCheck
    8m = 56
    y / 6 = 11
    12k = 96
    ✏️ Scratchpad / Reasoning
  2. 2 BALANCE SCALE
    Algebraic Balance Model
    5x = 40 8

    Solve 5x = 40 using the balance scale.

    ✏️ Workspace & Solution Steps
SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 3 MULTIPLE CHOICE

    Vault A is sealed with 4x = 32. Crack the lock: what is x?

    1. Ax = 8
    2. Bx = 128
    3. Cx = 28
    4. Dx = 36
    ✏️ Workspace & Solution Steps
  2. 4 MULTIPLE CHOICE

    Vault B is sealed with x / 3 = 6. Crack the lock: what is x?

    1. Ax = 18
    2. Bx = 2
    3. Cx = 9
    4. Dx = 3
    ✏️ Workspace & Solution Steps
  3. 5 MULTIPLE CHOICE

    Vault C is sealed with 5n = 45. Crack the lock: what is n?

    1. An = 9
    2. Bn = 225
    3. Cn = 40
    4. Dn = 50
    ✏️ Workspace & Solution Steps
SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 6 ERROR ANALYSIS

    Level 2 Extension - The Vault That Won't Open

    1. 1Equation:4x = 32
    2. 2Identify inverse:Inverse of multiplication is multiplication
    3. 3Apply to both sides:4x x 4 = 32 x 4
    4. 4Simplify:x = 128

    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