6.NOS.1 Lesson 6-11-group2 🟣 Group 2 · Challenge & Extension
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6.11 Small Group · Group 2

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

Fraction Division Problem Solving · Model · Procedural Fluency · Real-World Applications

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — How do I solve a fraction division word problem?
1Total goes first (dividend), group size goes second (divisor): total ÷ group size. Then check your answer by multiplying it back — 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. Agent Torres has 2/3 gallon of dust and each test uses 1/6 gallon. I ask: how many tests fit?
  2. The total 2/3 is the dividend. The size of each test 1/6 is the divisor. My equation is 2/3 ÷ 1/6.
  3. Keep 2/3, change to ×, flip 1/6 to 6/1: 2/3 × 6/1 = 12/3 = 4.
  4. So she can run 4 tests. I check by multiplying back: 4 × 1/6 = 4/6 = 2/3. It matches the total, so the answer is reasonable.
3Second Model — Try it together — then prove it
  1. A ribbon is 3/4 yard and each bow uses 1/4 yard. The total 3/4 is the dividend; 1/4 is the divisor.
  2. Equation: 3/4 ÷ 1/4. Keep, change, flip: 3/4 × 4/1 = 12/4 = 3.
  3. So we can make 3 bows. Check: 3 × 1/4 = 3/4. It matches, so it is reasonable.
  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
  • Fraction Division Problem Solving (Resolución de problemas de división de fracciones) — Use fraction division to find an unknown number of groups or the size of each group.
  • Model (Modelo) — A picture or math way to show a problem so you can solve it.
  • Equation (Ecuación) — A math sentence with an equal sign showing both sides are the same.
  • Solution (Solución) — A number that makes the equation or inequality true.
  • Reasonableness (Razonabilidad) — Checking if your answer makes sense.
  • Inverse operations (Operaciones inversas) — Two math actions that undo each other, like × and ÷.
5Watch out
  • A common mistake in Fraction Division Problem Solving is reversing which fraction is the dividend and which is the divisor when setting up the equation from a word problem. For example: a pitcher holds 3/4 cup of punch, and each serving is 1/8 cup — how many servings? The correct setup is total ÷ group size = 3/4 ÷ 1/8 = 3/4 × 8/1 = 24/4 = 6 servings. A student who reverses the setup calculates 1/8 ÷ 3/4 = 1/8 × 4/3 = 4/24 = 1/6, an answer smaller than 1 that can't possibly be a serving count. Before you submit, check: does my answer make sense as a NUMBER OF GROUPS? If the setup gives a fraction less than 1 for a 'how many' question, you likely swapped the dividend and divisor.
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 FILL TABLE

    Solve each word problem by writing and solving an equation.

    ProblemEquationSolution
    A board is 7/8 foot long. Each shelf bracket needs 1/4 foot. How many brackets fit?
    A jug holds 5/6 liter. Each cup holds 1/3 liter. How many cups can be filled?
    A path is 3/4 mile long. A runner does laps of 3/8 mile. How many laps fit?
    ✏️ Scratchpad / Reasoning
  2. 2 DRAG SORT

    Sort: is the equation and solution CORRECT, or is there an error?

    Target Categories: Correct Equation and Solution Incorrect Equation or Solution
    SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
    1. 3 MULTIPLE CHOICE

      Room 1 — Evidence Locker: Agent Torres has 7/8 pound of casting clay to lift footprints. Each cast uses 1/8 pound. How many casts can she make to unlock the door?

      1. A7
      2. B6
      3. C8
      4. D1/7
      Rewrite each fraction
      Work it out
      Simplify
    2. 4 MULTIPLE CHOICE

      Room 2 — Surveillance Map: A suspect's trail is 5/6 of a mile. Hidden checkpoints are spaced every 1/6 of a mile. How many checkpoints are on the trail?

      1. A5
      2. B6
      3. C1/5
      4. D4
      Rewrite each fraction
      Work it out
      Simplify
    3. 5 MULTIPLE CHOICE

      Room 5 — Final Keypad: The vault key code equals the number of 3/10-liter vials Torres can fill from a 9/10-liter bottle of fingerprint solution. What is the code?

      1. A3
      2. B9
      3. C1/3
      4. D6
      Rewrite each fraction
      Work it out
      Simplify
    SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
    1. 6 ERROR ANALYSIS

      Level 2 Extension — Escape the Wrong Operation

      1. 1Problem:A tank holds 4/5 gallon of testing fluid. Each test uses 1/10 gallon. How many tests can be run to escape the lab?
      2. 2Student's equation:4/5 × 1/10
      3. 3Solve:4/5 × 1/10 = 4/50 = 2/25
      4. 4Answer:2/25 of a test

      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 mathematical model proves the solution because each step preserves quantitative equivalence.
    BUT An estimate provides a quick benchmark, but an exact proof is required for precision.
    SO The quantities follow the standard rule, so the final result is verified with certainty.
    📐 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