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

9.3 Small Group · Group 2

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

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 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 FILL TABLE

    A canoe rents for $15 per hour: c = 15h. Complete the table.

    Hours (h)SubstituteCost in dollars (c)
    2
    5
    8
    ✏️ Scratchpad / Reasoning
  2. 2 MATCHING GAME

    Match each scenario to the equation that models it.

    1. Canoe rental: $15 per hour
    2. Gym membership: $24.95 per month
    3. Football tickets: $45 each
    4. Tram climbing 1,200 feet per minute
    • Ac = 15h
    • Bc = 24.95m
    • Cc = 45t
    • Dd = 1200m
SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 3 MULTIPLE CHOICE

    The friends have $90 to spend. Using c = 15h, for how many hours can they rent the canoe?

    1. A6 hours
    2. B5 hours
    3. C75 hours
    4. D1,350 hours
    My work — one step per lineWhat I did to both sides
  2. 4 MULTIPLE CHOICE

    How are the canoe graph and the equation c = 15h related?

    1. AEvery point on the graph is a pair (h, c) that makes the equation true
    2. BThe graph and the equation show different relationships
    3. CThe equation only works for the points shown on the graph
    4. DThe graph replaces the equation, so only one can be used
    ✏️ Workspace & Solution Steps
  3. 5 MULTIPLE CHOICE

    A different gym charges $19.95 per month. How does Miguel's equation change for this gym?

    1. AThe rate changes: c = 19.95m instead of c = 24.95m
    2. BThe variable changes: c = 24.95x instead of c = 24.95m
    3. CThe operation changes: c = 19.95 + m
    4. DNothing changes — all gyms have the same equation
    ✏️ Workspace & Solution Steps
  4. 6 MULTIPLE CHOICE

    What is the benefit of an equation over a table for the membership costs?

    1. AThe equation gives the total cost for ANY number of months, not just the rows listed
    2. BThe equation lists more rows than the table
    3. CThe equation is the only representation that shows the cost increasing
    4. DThe table is always wrong and the equation is always right
    ✏️ 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