9.2 Small Group · Group 2

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

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

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

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 $135.
The number of tickets is the independent variable, so it goes first in each ordered pair: (1, 45), (2, 90), (3, 135).
I label one axis 'Number of tickets' and the other 'Total cost ($)', then plot each pair as a point.
The points rise to the right: as the number of tickets increases, the total cost increases.

3Second Model — Try it together — then prove it

The tram travels 1,200 feet every minute. Which quantity is independent — time or distance?
Time is independent, so our pairs look like (minutes, feet): (1, 1200), (2, 2400), (5, 6000).
The ascent takes 15 minutes, so the last point is (15, 18000) — because 15 times 1,200 is 18,000 feet.
Each point means: after THIS many minutes, the tram has traveled THIS many feet.
Now prove it: say why that move had to work at all — not just that it did.
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 MULTIPLE CHOICE
The ticket table and ticket graph are two representations. Which statement about them is true?
1. AThe graph shows a different relationship than the table
2. BOnly the table can show that cost increases with tickets
3. COnly the graph is a real mathematical representation
4. DBoth show the same relationship — the table with numerical values, the graph in a more visual way
✏️ Workspace & Solution Steps

SECTION 2 COMPUTATION & PROCEDURAL FLUENCY [Show Every Step]

2 COORDINATE GRID

Plot the ticket-cost relationship: one point for each number of tickets.
✏️ Workspace & Solution Steps

SECTION 3 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]

3 MULTIPLE CHOICE
After 10 minutes, the Sandia Peak tram has traveled 12,000 feet. The whole ascent is 18,000 feet. How much farther must it travel to reach the top?
1. A1,200 feet
2. B5,000 feet
3. C6,000 feet
4. D12,000 feet
✏️ Workspace & Solution Steps
4 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 starts higher up the vertical axis
2. BIts line is longer than the other one
3. CIts line rises more steeply — it gains more feet in the same minute
4. DYou cannot tell without plotting points for both
✏️ Workspace & Solution Steps
5 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. AAlways — growing together is what makes a line
2. BNever — real data never lines up
3. CSometimes — only when the quantity grows by the same amount each step
4. DSometimes — only when both quantities are measured in the same unit
✏️ Workspace & Solution Steps

SECTION 4 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]

6 OPEN RESPONSE

The ticket graph has “Number of tickets” across the bottom and “Total cost ($)” up the side. Rowan plots the first ticket as the point (45, 1). What did Rowan misunderstand — and what does his point actually claim about tickets and money?
✏️ Mathematical Justification & Response

✍️ 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