6.NOS.1 Lesson 6-10-group2 🟣 Group 2 · Challenge & Extension
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

6.10 Small Group · Group 2

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

Divide Mixed Numbers · Mixed Number · Procedural Fluency · Real-World Applications

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — How do I divide with mixed numbers?
1Always convert a mixed number to an improper fraction first, then divide using Keep, Change, Flip — 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 Chen has a 3 1/2-mile route cut into 1/4-mile segments. I want 3 1/2 ÷ 1/4.
  2. First I convert 3 1/2. Multiply the whole by the bottom: 3 × 2 = 6. Add the top: 6 + 1 = 7. So 3 1/2 = 7/2.
  3. Now Keep, Change, Flip: keep 7/2, change ÷ to ×, flip 1/4 to 4/1.
  4. Multiply: 7/2 × 4/1 = 28/2.
  5. Simplify: 28/2 = 14. So 3 1/2 ÷ 1/4 = 14 segments.
3Second Model — Try it together — then prove it
  1. Let's try 2 1/2 ÷ 1/2. First convert 2 1/2: 2 × 2 = 4, plus 1 = 5, so 2 1/2 = 5/2.
  2. Keep 5/2, change to ×, flip 1/2 to 2/1.
  3. Multiply: 5/2 × 2/1 = 10/2 = 5. So 2 1/2 ÷ 1/2 = 5.
  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 Mixed Numbers (Dividir números mixtos) — Change mixed numbers to improper fractions, then multiply by the reciprocal.
  • Mixed Number (Número mixto) — A whole number plus a fraction, like 2 1/3.
  • Improper Fraction (Fracción impropia) — A fraction where the top is bigger than or equal to the bottom, like 7/4.
  • Convert (Convertir) — To change a number to a new form but keep the same value.
  • Simplify (Simplificar) — To write a fraction with smaller numbers that names the same amount, like 2/4 = 1/2.
  • Reciprocal (Recíproco) — A fraction turned upside down. You use it in keep-change-flip.
5Watch out
  • A common mistake in Divide Mixed Numbers is dividing the whole-number parts and fraction parts separately instead of converting first — for example, treating 4 1/2 ÷ 1 1/2 as (4 ÷ 1) + (1/2 ÷ 1/2) = 5. Always convert each mixed number to one improper fraction first, then divide using Keep, Change, Flip: 4 1/2 ÷ 1 1/2 = 9/2 ÷ 3/2 = 9/2 × 2/3 = 3.
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 FILL TABLE

    Convert each mixed number to an improper fraction, then divide.

    ProblemConvert to ImproperKeep-Change-FlipQuotient
    3 1/2 ÷ 1/4
    1 3/4 ÷ 1/2
    2 1/3 ÷ 7/6
    4 1/5 ÷ 3/5
    ✏️ Scratchpad / Reasoning
  2. 2 MATCHING GAME

    Match each mixed number division to its quotient.

    1. 2 1/4 ÷ 3/4
    2. 4 1/2 ÷ 9/8
    3. 1 2/3 ÷ 5/6
    4. 3 1/3 ÷ 2/3
    5. 5 1/4 ÷ 3/4
    6. 1 1/2 ÷ 1/4
    • A3
    • B4
    • C2
    • D5
    • E7
    • F6
SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 3 MULTIPLE CHOICE

    Mission 1: The stakeout shift is 4 1/2 hours long. Each agent covers a 1 1/2-hour watch. How many agent watches fill one shift?

    1. A3
    2. B6
    3. C2 1/2
    4. D4 1/2
    ✏️ Workspace & Solution Steps
  2. 4 MULTIPLE CHOICE

    Mission 2: A surveillance route is 3 miles long, split into 1 1/2-mile patrol sections. How many sections are on the route?

    1. A2
    2. B1 1/2
    3. C4 1/2
    4. D3
    ✏️ Workspace & Solution Steps
  3. 5 MULTIPLE CHOICE

    Mission 3: There are 7 1/2 feet of caution tape to seal a crime scene. Each doorway needs 2 1/2 feet. How many doorways can be sealed?

    1. A3
    2. B5
    3. C2 1/2
    4. D10
    ✏️ Workspace & Solution Steps
SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 6 ERROR ANALYSIS

    Find Liam's Mistake

    1. 1Problem:3 1/2 ÷ 1 1/4
    2. 2Convert first number:3 1/2 = 7/2
    3. 3Convert second number:1 1/4 = 5/4
    4. 4Divide:7/2 ÷ 5/4 = 7/2 × 5/4 = 35/8 = 4 3/8

    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