MPP.4 Lesson 10-4-group1 🟡 Group 1 · Support & Scaffolding
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

10.4 Small Group · Group 1

Tier 2 Intervention · Concrete-Representational-Abstract (CRA) · Dual-Language Anchors

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 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 DRAG SORT

    Sort each rear gear by how the wheel responds to one pedal turn.

    Target Categories: More than one wheel rotation One rotation or less
    • 12-tooth rear gear
    • 16-tooth rear gear
    • 24-tooth rear gear
    • 48-tooth rear gear
    • 60-tooth rear gear
    • 96-tooth rear gear
  2. 2 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
SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 3 MULTIPLE CHOICE

    On the Penny Farthing, what is the ratio of pedal rotations to front-wheel rotations?

    1. A1 to 1
    2. B1 to 2
    3. C2 to 1
    4. D1 to 4
    ✏️ Workspace & Solution Steps
  2. 4 MULTIPLE CHOICE

    How far does the Penny Farthing travel with each pedal rotation?

    1. AThe length of the perimeter of the front wheel
    2. BThe height of the rider
    3. CThe length of the pedal
    4. DIt does not move a fixed distance
    ✏️ Workspace & Solution Steps
  3. 5 MULTIPLE CHOICE

    Why was the front wheel of the Penny Farthing so large?

    1. ASo the bicycle travels a greater distance with one pedal rotation
    2. BSo the bicycle is easier to stop
    3. CSo the rider sits closer to the ground
    4. DSo the rear wheel spins faster
    ✏️ Workspace & Solution Steps
SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 6 OPEN RESPONSE

    How does the size of the front wheel affect the distance the bicycle travels?

    💬 Sentence Starter: A bigger wheel travels ___ with each pedal rotation because ___ .<br>A smaller wheel would travel ___ .
    ✏️ 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
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