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

9.4 Small Group · Group 2

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

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

    The solution to one problem was 41.7 cars. Would producing or washing a PARTIAL car make sense?

    1. ANo — cars come in whole numbers, so round up to the next whole car
    2. BYes — 0.7 of a car is a real amount you can wash
    3. CNo — so round down and accept a smaller total
    4. DYes — the equation is wrong if it gives a decimal
    ✏️ Workspace & Solution Steps
  2. 2 MULTIPLE CHOICE

    Would you recommend the student council charge $12 or $14.30 per car? Which reasoning is strongest?

    1. AEither can work: $12 needs at least 42 cars washed, while $14.30 needs only about 35 — so the choice depends on how many customers they expect
    2. B$12 is always better because a smaller number is always better
    3. C$14.30 is always better because more money per car is always better
    4. DNeither price can reach the $500 goal
    ✏️ Workspace & Solution Steps
  3. 3 MULTIPLE CHOICE

    The order requires 1,000 cars. Using 2.5h = 1,000, how many hours will the plant need?

    1. A400 hours
    2. B2,500 hours
    3. C997.5 hours
    4. D40 hours
    My work — one step per lineWhat I did to both sides
  4. 4 MULTIPLE CHOICE

    The plant needs 400 hours to fill the order. Running 24 hours a day, about how many days is that?

    1. AAbout 17 days
    2. B400 days
    3. C50 days
    4. D4 days
    ✏️ Workspace & Solution Steps
  5. 5 MULTIPLE CHOICE

    The goal is $500 at $12 per car. Using 12n = 500, at least how many cars must be washed?

    1. A42 cars
    2. B41 cars
    3. C40 cars
    4. D50 cars
    per 1
SECTION 2 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 6 ERROR ANALYSIS

    Fix our table's thinking

    1. 1The problem:The order requires 1,000 cars. Using 2.5h = 1,000, how many hours will the plant need?
    2. 2A classmate at our table answered:2,500 hours

    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