MPP.4 Lesson 10-4-group1 🟡 Group 1 · Set B
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

10.4 Small Group · Group 1

Second Practice Form · Re-Teach, Homework or Retake · Same Standard, New Problems

ingenuity · rotation · Procedural Fluency · Real-World Applications

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ CONCEPT SUMMARY & GUIDED NOTES: Let's build it together — 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.
  • The Penny Farthing is powered by the rider: pushing the pedals rotates the front wheel directly, so each pedal rotation rotates the wheel one time — a 1 to 1 ratio. We'll walk through a worked example, try one together, then you'll try a few with hints right there when you need them.
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.
3Mathematical 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.
4Watch 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

    For each rotation of the 48-tooth pedal gear, how many teeth does the wheel gear turn?

    1. A48 teeth
    2. B32 teeth
    3. C12 teeth
    4. D1 tooth
    ✏️ Workspace & Solution Steps
  2. 2 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
    1. 1LabelWrite what each column is.
    2. 2Find per 1Divide to get one.
    3. 3ScaleMultiply up to what is asked.
    4. 4CheckDoes the size make sense?
  3. 3 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
  4. 4 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. 5 OPEN RESPONSE

    Name one ingenious solution — from this lesson or from your own life — where math helped solve an everyday problem.

    💬 Sentence Starter: One ingenious solution is ___ .<br>The math behind it is ___ .
    ✏️ Mathematical Justification & Response
  2. 6 OPEN RESPONSE

    Why do you think the larger gear makes it easier to pedal at slower speeds?

    💬 Sentence Starter: With the larger gear, each pedal rotation only turns the wheel ___ .<br>That makes each push ___ because ___ .
    ✏️ Mathematical Justification & Response
📐 CER MATHEMATICAL PROOF MATRIX (CER 2.0) SMP.3 / Proof & Justification

Writing Task: Justify why your mathematical solution is accurate and complete.

C Claim
Starter: My mathematical claim is that the solution is...
E Evidence
Starter: The evidence from the model/table demonstrates that...
R Reasoning
Starter: This proves my answer because the standard mathematical definition of...
Student Mastery Self-Assessment:
1 · Need More Support
2 · Getting Closer
3 · Got It / Solid
4 · Master / Can Teach It