6.NOS.1 Lesson 6-9-group2 🟣 Group 2 · Challenge & Extension
MASTERY CHECK
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6.9 Small Group · Group 2

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

Divide Whole Numbers by Fractions · Whole Number · Procedural Fluency · Real-World Applications

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — How do I divide a whole number by a fraction?
1Dividing by a fraction is the same as multiplying by its reciprocal (the fraction flipped over) — 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 Park has 4 pounds of powder and each bag holds 1/2 pound. I want 4 ÷ 1/2.
  2. Keep the 4. Change ÷ to ×. Flip 1/2 to make 2/1, which is just 2.
  3. Now I multiply: 4 × 2 = 8.
  4. So 4 ÷ 1/2 = 8. He can fill 8 bags. That makes sense: each pound fills 2 bags, and 4 pounds × 2 = 8.
3Second Model — Try it together — then prove it
  1. Let's try 3 ÷ 1/4 together. What do we keep? The 3.
  2. Change ÷ to ×, then flip 1/4 to 4/1, which is 4.
  3. Multiply: 3 × 4 = 12. So 3 ÷ 1/4 = 12.
  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
  • Divide Whole Numbers by Fractions (Dividir números enteros entre fracciones) — Find how many fractional groups fit in a whole number by multiplying by the reciprocal.
  • Whole Number (Número entero) — A counting number with no fraction or decimal, like 0, 1, 2, 3.
  • Fraction (Fracción) — A number that shows part of a whole, like 3/4.
  • Reciprocal (Recíproco) — A fraction turned upside down.
  • Keep, Change, Flip (Mantener, cambiar, invertir) — A way to divide fractions: keep the first, change ÷ to ×, flip the second.
  • Quotient (Cociente) — The answer when you divide.
5Watch out
  • A common mistake in Divide Whole Numbers by Fractions is doing 'Keep, Change' but skipping the 'Flip' — students multiply by the divisor as it is instead of by its reciprocal. For example, for 6 ÷ 3/4, a student might write 6 × 3/4 = 4 1/2 and stop there, but the divisor 3/4 must be flipped to 4/3 first: 6 × 4/3 = 24/3 = 8. Before you submit, check: did I flip the second fraction (the divisor) upside down before multiplying?
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 FILL TABLE

    Complete the table: Whole Number ÷ Fraction. Use Keep, Change, Flip.

    ExpressionRewrite as MultiplicationQuotient
    5 ÷ 1/4
    8 ÷ 1/2
    3 ÷ 1/6
    7 ÷ 1/3
    ✏️ Scratchpad / Reasoning
  2. 2 MATCHING GAME

    Match each division expression to its quotient.

    1. 4 ÷ 1/3
    2. 7 ÷ 1/2
    3. 5 ÷ 1/6
    4. 2 ÷ 1/8
    5. 9 ÷ 1/4
    6. 6 ÷ 1/5
    • A12
    • B14
    • C30
    • D16
    • E36
    • F30
SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 3 MULTIPLE CHOICE

    Detective Park has 4 cups of flour. Each batch of evidence-cookies needs 1/2 cup. How many batches can he make?

    1. A8
    2. B2
    3. C4
    4. D6
    Rewrite each fraction
    Work it out
    Simplify
  2. 4 MULTIPLE CHOICE

    The crew has 8 quarts of sports drink. Each detective's water bottle holds 4/5 of a quart. How many bottles can be filled?

    1. A10
    2. B5
    3. C32/5
    4. D12
    Rewrite each fraction
    Work it out
    Simplify
  3. 5 MULTIPLE CHOICE

    There are 3 bags of trail mix. Each snack pouch needs 2/3 of a bag. How many full pouches can be filled, and is there a leftover?

    1. A4 1/2 pouches (4 full, half a pouch left)
    2. B4 pouches exactly
    3. C2 pouches
    4. D6 pouches
    ✏️ Workspace & Solution Steps
  4. 6 MULTIPLE CHOICE

    Detective Park bakes with 9 cups of batter. Each muffin pan holds 3/4 of a cup. How many pans can he fill?

    1. A12
    2. B6
    3. C27/4
    4. D9
    Rewrite each fraction
    Work it out
    Simplify
🗣️ 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