MPP.4 Lesson 10-4-group2 🟣 Group 2 · Set B
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

10.4 Small Group · Group 2

Second Challenge Form · Non-Routine Extension · Same Standard, New Problems

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 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 1 MULTIPLE CHOICE

    The pedal gear turns 48 teeth per rotation. The largest rear gear needs 32 teeth for one full rotation. How many rotations does the rear wheel make for each pedal rotation?

    1. A1.5 rotations
    2. B16 rotations
    3. C0.5 rotations
    4. D2 rotations
    per 1
  2. 2 MULTIPLE CHOICE

    The smallest rear gear has 12 teeth. How many rotations will the rear wheel make for each rotation of the 48-tooth pedal gear?

    1. A4 rotations
    2. B36 rotations
    3. C3 rotations
    4. D0.25 rotations
    ✏️ Workspace & Solution Steps
  3. 3 MULTIPLE CHOICE

    You are about to ride up a steep hill. Which rear gear should the chain be on, and why?

    1. AThe 32-tooth gear, because 1.5 rotations per pedal makes pedaling easier at slow speeds
    2. BThe 12-tooth gear, because 4 rotations per pedal is always better
    3. CThe 32-tooth gear, because it makes the bike go faster
    4. DIt does not matter which gear you use on a hill
    ✏️ Workspace & Solution Steps
SECTION 2 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 4 OPEN RESPONSE

    How would the distance traveled differ with a smaller front wheel on the Penny Farthing? Use the 1 to 1 pedal ratio in your answer.

    ✏️ Mathematical Justification & Response
  2. 5 OPEN RESPONSE

    How did math help the bicycle's designers come up with ingenious solutions — first with the Penny Farthing, then with chains and gears?

    ✏️ Mathematical Justification & Response
  3. 6 ERROR ANALYSIS

    Fix our table's thinking

    1. 1The problem:You are about to ride up a steep hill. Which rear gear should the chain be on, and why?
    2. 2A classmate at our table answered:The 12-tooth gear, because 4 rotations per pedal is always better

    Which step contains the error? Explain the mathematical misconception and write the correct calculation below.

    ✏️ Corrected Mathematical Work & Explanation
📐 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.
Student Mastery Self-Assessment:
1 · Need More Support
2 · Getting Closer
3 · Got It / Solid
4 · Master / Can Teach It