6.AT.11 Lesson 9-4-group2 🟣 Group 2 · Challenge & Extension
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

9.4 Small Group · Group 2

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

solution · substitute · Procedural Fluency · Real-World Applications

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — Use the equation to answer a real question
1Write the relationship as an equation, substitute the known value, solve for the unknown, then interpret the solution in context — 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 test the car order
  1. The number of cars produced depends on the number of hours of operation. Every hour, the plant produces 2.5 cars.
  2. Let h = hours and c = cars. The equation is c = 2.5h. Check with the table: 8 hours gives 2.5 × 8 = 20 cars per shift.
  3. The order needs 1,000 cars, so substitute: 2.5h = 1,000.
  4. Divide both sides by 2.5: h = 1,000 ÷ 2.5 = 400 hours.
  5. Interpret it: 400 hours running 24 hours a day is about 17 days — about 2.5 weeks. That fits inside 3 weeks only if the plant runs 24 hours a day, 7 days a week.
3Second Model — Try it together — then prove it
  1. Last year the council washed 34 cars and raised $408. What did they charge per car? Divide: 408 ÷ 34 = $12.
  2. At $12 per car, let n = cars washed and t = total raised: t = 12n.
  3. The goal is $500, so 12n = 500 means n = 500 ÷ 12, which is about 41.7 — so they must wash at least 42 cars, because 41 cars raise only 41 × 12 = $492.
  4. Or fix the cars instead: if they predict 35 cars, then 35p = 500 gives p = 500 ÷ 35, about $14.29 — so they must charge at least $14.30 per car.
  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
  • solution (solución) — A value of the variable that makes an equation true.
  • substitute (sustituir) — To replace a variable with a number so you can compute.
  • at least (al menos) — That amount or more — the smallest amount that still works.
  • predict (predecir) — To make a reasonable guess about a future amount using information you already have.
  • justify (justificar) — To explain WHY an answer or recommendation makes sense, using the mathematics.
5Watch out
  • Adding the rate instead of multiplying — writing 2.5 + 8 = 10.5 cars for an 8-hour shift instead of 2.5 × 8 = 20. The rate applies to EVERY hour, so the total is always rate × amount.
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 FILL TABLE

    The assembly line produces 2.5 cars per hour. Complete the production table.

    Hours (h)Substitute c = 2.5hCars produced (c)
    4
    8 (one shift)
    100
    ✏️ Scratchpad / Reasoning
  2. 2 MATCHING GAME

    Connect the four representations of the assembly line: scenario, equation, table, and graph.

    1. Scenario: 2.5 cars every hour
    2. Table row: h = 8, c = 20
    3. Graph point (400, 1000)
    4. Equation c = 20s
    • AEquation: c = 2.5h
    • BGraph point (8, 20)
    • CThe 1,000-car order takes 400 hours
    • DCars counted by 8-hour shifts
SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 3 MULTIPLE CHOICE

    The council predicts 35 cars this year. Using 35p = 500, at least how much must they charge per car to reach $500?

    1. AAbout $14.30
    2. B$12.00
    3. C$14.00
    4. D$465.00
    ✏️ Workspace & Solution Steps
  2. 4 MULTIPLE CHOICE

    How could the council CHECK that charging $14.30 for 35 cars meets the goal?

    1. AMultiply 35 × 14.30 = 500.50 and confirm it is at least $500
    2. BDivide 35 by 14.30 and confirm it is at least $500
    3. CAdd 35 + 14.30 and confirm it is at least $500
    4. DNo check is possible after solving
    ✏️ Workspace & Solution Steps
  3. 5 MULTIPLE CHOICE

    Suppose the variable s counts 8-hour SHIFTS instead of hours. Which equation gives the cars produced?

    1. Ac = 20s
    2. Bc = 2.5s
    3. Cc = 8s
    4. Dc = 10.5s
    ✏️ Workspace & Solution Steps
  4. 6 MULTIPLE CHOICE

    If the plant runs 24 hours a day, 7 days a week for the full 3 weeks, how many hours is that — and can it fill the 1,000-car order?

    1. A504 hours — yes, because 504 hours is more than the 400 hours needed
    2. B168 hours — no, that is less than 400 hours
    3. C72 hours — no, that is less than 400 hours
    4. D504 hours — no, because 504 is less than 1,000
    ✏️ Workspace & Solution Steps
✍️ The Writing Revolution (TWR) · Sentence Expansion Because · But · So

Complete each sentence stem to demonstrate precise mathematical reasoning:

BECAUSE The ratio remains equivalent because both quantities are scaled by the exact same multiplicative factor.
BUT Two ratios may look similar, but inverting the order of terms fundamentally changes the comparison.
SO The recipe requires 3 parts flour to 2 parts water, so the unit rate is 1.5 cups of flour per cup of water.
📐 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