6.AT.11 Lesson 9-3-group2 🟣 Group 2 · Set B
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

9.3 Small Group · Group 2

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

equation · variable · Procedural Fluency · Real-World Applications

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — From a pattern in the table to an equation
1When the total is the rate times the amount, write it as an equation: dependent variable = rate × independent variable — 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 write Miguel's equation
  1. The quantities are the number of months of membership and the total cost. The total cost depends on the number of months.
  2. In the table, each month adds $24.95: 1 month is $24.95, 2 months are $49.90, 3 months are $74.85.
  3. The pattern in words: the total cost is $24.95 times the number of months.
  4. Define the variables — let m = number of months and c = total cost in dollars — and write the equation: c = 24.95m.
  5. Now ANY number of months works: for 10 months, c = 24.95 × 10 = 249.50.
3Second Model — Try it together — then prove it
  1. A canoe rents for $15 per hour. Which quantity is independent — hours or total cost?
  2. The hours of rental is independent; the total cost depends on it.
  3. Say the pattern in words: the total cost equals $15 times the number of hours.
  4. Let h = hours rented and c = total cost. What equation matches? c = 15h.
  5. Check it: a 3-hour rental is 15 × 3 = 45 dollars, and a 5-hour rental is 15 × 5 = 75 dollars.
  6. Now prove it: say why that move had to work at all — not just that it did.
  7. Sentence frame — convince a skeptic: "My method works because ___ . It would stop working if ___ ."
4Mathematical Word Bank
  • equation (ecuación) — A mathematical sentence with an equals sign that shows two amounts are the same.
  • variable (variable) — A letter that represents a quantity that can change, or vary.
  • rate (tasa) — A fixed amount of one quantity for every one unit of the other quantity.
  • represent (representar) — To show a relationship using a tool such as a table, a graph, or an equation.
  • define the variables (definir las variables) — To state exactly what quantity each letter in an equation stands for.
5Watch out
  • Writing the equation with addition — c = 24.95 + m — instead of multiplication. The fee is charged EVERY month, so the total is the rate times the number of months; always test the equation against a row of the table.
SECTION 1 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 1 MULTIPLE CHOICE

    The friends spent exactly $120 on the canoe. Solve 15h = 120. How long was the rental?

    1. A8 hours
    2. B105 hours
    3. C9 hours
    4. D1,800 hours
    My work — one step per lineWhat I did to both sides
  2. 2 MULTIPLE CHOICE

    Using c = 24.95m, what is the total cost of 10 months of membership?

    1. A$249.50
    2. B$34.95
    3. C$224.55
    4. D$2,495.00
    My work — one step per lineWhat I did to both sides
  3. 3 MULTIPLE CHOICE

    Using c = 15h, what is the total cost of a 5-hour canoe rental?

    1. A$75
    2. B$20
    3. C$60
    4. D$80
    My work — one step per lineWhat I did to both sides
SECTION 2 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 4 OPEN RESPONSE

    The canoe shop's equation is c = 15h. A rival shop advertises c = 12h + 6 and claims “we are ALWAYS cheaper.” Test the claim with at least two values of h, then tell the truth about when each shop is the better deal.

    ✏️ Mathematical Justification & Response
  2. 5 ERROR ANALYSIS

    Fix our table's thinking

    1. 1The problem:What is the benefit of an equation over a table for the membership costs?
    2. 2A classmate at our table answered:The table is always wrong and the equation is always right

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

    ✏️ Corrected Mathematical Work & Explanation
  3. 6 OPEN RESPONSE

    The two equations in this lesson, c = 24.95m and c = 15h, look alike. Say what they have in common that makes both PROPORTIONAL. Then write an equation for a gym charging a $30 joining fee plus $19.95 a month, and give two different tests that show it is not proportional.

    ✏️ 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
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