MPP.4 Lesson 10-4-group2 🟣 Group 2 · Challenge & Extension
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

10.4 Small Group · Group 2

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

ingenuity · rotation · Procedural Fluency · Real-World Applications

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — Math can power ingenious solutions
1With each pedal rotation, the bicycle travels the length of the perimeter of the wheel. The front wheel is large so the bicycle travels a greater distance with one pedal rotation — 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 use a gear ratio
  1. A later bicycle design uses a chain: the pedal gear has 48 teeth, and the largest rear gear has 32 teeth.
  2. For each rotation of the 48-tooth pedal gear, the wheel gear also turns 48 teeth.
  3. The 32-tooth gear needs to turn 32 teeth to make one full rotation, so I find the equivalent ratio: 48 ÷ 32 = 1.5.
  4. The rear wheel makes 1.5 rotations for each pedal rotation — this gear makes it easier to pedal at slower speeds or go uphill.
3Second Model — Try it together — then prove it
  1. The smallest rear gear has 12 teeth, and the pedal gear still turns 48 teeth per rotation.
  2. One full rotation of the 12-tooth gear takes 12 teeth.
  3. 48 ÷ 12 = 4, so the rear wheel makes 4 rotations for each pedal rotation.
  4. This gear makes it easier to pedal at faster speeds or go downhill.
  5. Now prove it: say why that move had to work at all — not just that it did.
  6. Sentence frame — convince a skeptic: "My method works because ___ . It would stop working if ___ ."
4Mathematical Word Bank
  • ingenuity (ingenio) — Coming up with different, and sometimes unique, solutions to problems.
  • rotation (rotación) — One full turn of a wheel, gear, or pedal all the way around.
  • gear (engranaje) — A toothed wheel that passes motion to another toothed wheel through a chain.
  • ratio (razón) — A comparison of two quantities, like pedal rotations to wheel rotations.
  • equivalent ratios (razones equivalentes) — Ratios that make the same comparison using different numbers.
5Watch out
  • Thinking a rear gear with MORE teeth makes the wheel spin more — it is the opposite: the 32-tooth gear gives only 1.5 rotations per pedal rotation, while the 12-tooth gear gives 4.
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 FILL TABLE

    The pedal gear has 48 teeth. Complete the table of wheel rotations per pedal turn.

    Rear gear (teeth)Ratio 48 ÷ teethWheel rotations per pedal turn
    12
    16
    24
    ✏️ Scratchpad / Reasoning
  2. 2 MATCHING GAME

    Distance = rotations × circumference. Match each ride to its distance.

    1. 4 wheel rotations, 7 ft circumference
    2. 3 wheel rotations, 6 ft circumference
    3. 2 wheel rotations, 7 ft circumference
    4. 48-tooth pedal gear with a 12-tooth rear gear
    • A28 feet
    • B18 feet
    • C14 feet
    • D4 rotations per pedal turn
SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 3 MULTIPLE CHOICE

    With the chain on the 12-tooth gear, the rider makes 10 pedal rotations. How many rotations does the rear wheel make?

    1. A40 rotations
    2. B14 rotations
    3. C22 rotations
    4. D120 rotations
    ✏️ Workspace & Solution Steps
  2. 4 MULTIPLE CHOICE

    The rider makes 3 full pedal rotations with the chain on the 32-tooth gear. How many rotations does the rear wheel make?

    1. A4.5 rotations
    2. B3 rotations
    3. C96 rotations
    4. D6 rotations
    ✏️ Workspace & Solution Steps
  3. 5 MULTIPLE CHOICE

    A designer adds a middle gear so the rear wheel makes exactly 2 rotations per rotation of the 48-tooth pedal gear. How many teeth must the new gear have?

    1. A24 teeth
    2. B22 teeth
    3. C96 teeth
    4. D16 teeth
    per 1
SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 6 OPEN RESPONSE

    When the chain is on the largest gear the rear wheel makes 1.5 rotations per pedal rotation; on the smallest gear it makes 4. What do you notice about the size of the wheel gears and the distance the bike travels?

    ✏️ Mathematical Justification & Response
✍️ 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