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

9.2 Small Group · Group 2

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

graph · table of values · Procedural Fluency · Real-World Applications

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — Two pictures of the same relationship
1Each row of the table becomes one ordered pair — (independent variable, dependent variable) — plotted as one point on the graph — 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 build the ticket graph
  1. Each ticket costs $45, so the total cost is 45 times the number of tickets: 1 ticket is $45, 2 tickets are $90, 3 tickets are $135.
  2. The number of tickets is the independent variable, so it goes first in each ordered pair: (1, 45), (2, 90), (3, 135).
  3. I label one axis 'Number of tickets' and the other 'Total cost ($)', then plot each pair as a point.
  4. The points rise to the right: as the number of tickets increases, the total cost increases.
3Second Model — Try it together — then prove it
  1. The tram travels 1,200 feet every minute. Which quantity is independent — time or distance?
  2. Time is independent, so our pairs look like (minutes, feet): (1, 1200), (2, 2400), (5, 6000).
  3. The ascent takes 15 minutes, so the last point is (15, 18000) — because 15 times 1,200 is 18,000 feet.
  4. Each point means: after THIS many minutes, the tram has traveled THIS many feet.
  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
  • graph (gráfica) — A picture on a coordinate plane that shows how two quantities are related.
  • table of values (tabla de valores) — A list of matched pairs of values that shows a relationship using numbers.
  • ordered pair (par ordenado) — Two numbers written as (x, y) that name one point on a graph. The independent variable comes first.
  • axis (eje) — One of the two number lines that frame a graph. Each axis is labeled with a quantity from the problem.
  • coordinates (coordenadas) — The numbers in an ordered pair that tell how far to move along each axis to reach a point.
5Watch out
  • Reversing the coordinates of a point — writing (cost, tickets) instead of (tickets, cost). The independent variable always comes first in the ordered pair.
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 MULTIPLE CHOICE

    A different event's ticket graph passes through the point (4, 100). What does one ticket cost, and where would the point for 7 tickets sit?

    1. A$25 each, and the point would be (7, 175)
    2. B$25 each, and the point would be (175, 7)
    3. C$400 each, and the point would be (7, 2800)
    4. DThere is not enough information to find the cost of one ticket
    ✏️ Workspace & Solution Steps
SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 2 MULTIPLE CHOICE

    Two trams both start at the bottom at 0 minutes. One climbs 1,200 feet each minute, the other 900. Both are graphed on the same axes with minutes across. Without working out a single point, how can you tell which line belongs to the faster tram?

    1. AIts line rises more steeply — it gains more feet in the same minute
    2. BIts line starts higher up the vertical axis
    3. CIts line is longer than the other one
    4. DYou cannot tell without plotting points for both
    ✏️ Workspace & Solution Steps
  2. 3 MULTIPLE CHOICE

    Claim: “If two quantities grow together, the points on their graph always lie on a straight line.” Always, sometimes, or never true?

    1. ASometimes — only when the quantity grows by the same amount each step
    2. BAlways — growing together is what makes a line
    3. CNever — real data never lines up
    4. DSometimes — only when both quantities are measured in the same unit
    ✏️ Workspace & Solution Steps
SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 4 OPEN RESPONSE

    Every graph in this lesson rises to the right. Invent a situation where the dependent quantity goes DOWN as the independent quantity goes up. Give three ordered pairs, say what each quantity is, and describe what your graph looks like.

    ✏️ Mathematical Justification & Response
  2. 5 OPEN RESPONSE

    On the tram graph — minutes across, feet climbed up the side — a classmate marks the point (0, 0) and says it is meaningless. Do you agree? Say what (0, 0) claims about the tram, and why the line passes through it.

    ✏️ Mathematical Justification & Response
  3. 6 ERROR ANALYSIS

    Fix our table's thinking

    1. 1The problem:Why is the line on the football ticket graph dashed instead of solid?
    2. 2A classmate at our table answered:The graph is unfinished

    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