3.4 Small Group · Group 2

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

Graph Ratio Tables · Coordinate plane · Procedural Fluency · Real-World Applications

Name: Date: Period: Learning Goal:

■ ADVANCED CONCEPT ANCHOR: Push further — How do we graph a ratio table?

1The points from equal ratios form a straight line that passes through the origin (0, 0) — 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

The recipe uses 2 ounces of sauce for every 1 sundae.
I put sundaes on the x-axis and ounces of sauce on the y-axis.
For 1 sundae I need 2 ounces, so I plot the ordered pair (1, 2): right 1, up 2.
For 2 sundaes I need 4 ounces, so I plot (2, 4): right 2, up 4.
When I connect (1, 2), (2, 4), (3, 6), they form a straight line through the origin (0, 0).

3Second Model — Try it together — then prove it

A recipe uses 1 cup of sugar for every 3 cups of flour.
For 1 cup of sugar, the ordered pair is (1, 3): right 1, up 3.
For 2 cups of sugar, what is the pair? It is (2, 6): right 2, up 6.
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 Ratio Tables (Representar tablas de razones en una gráfica) — Plot the paired values from a ratio table as ordered pairs.
Coordinate plane (Plano cartesiano) — A grid with a line going across and a line going up to plot points.
Ordered pair (Par ordenado) — Two numbers (x, y) that tell where a point is on a grid.
Linear pattern (Patrón lineal) — Points that make a straight line on a grid.
Proportional (Proporcional) — Two amounts that grow together at the same rate.
Origin (Origen) — The point (0, 0) where the two grid lines cross.

5Watch out

A common mistake in Graph Ratio Tables is writing the ordered pair in the wrong order — putting the second table value first instead of matching (x, y) to the table's column order. For example, if a recipe table shows 2 cups of sugar for every 4 cups of flour, a student might plot (4, 2) instead of (2, 4), landing the point in a completely different spot on the grid. Before you plot, check: which quantity goes on the x-axis, and does your ordered pair start with that number?

SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]

1 FILL TABLE

Use the graph pattern y = 5x to complete the ratio table, then identify what the ordered pairs would be.

x y Ordered Pair
1
2
4
6

✏️ Scratchpad / Reasoning

SECTION 2 COMPUTATION & PROCEDURAL FLUENCY [Show Every Step]

2 COORDINATE GRID

Chef Reyes uses 3 cups of flour for every 1 cup of sugar. Plot the ordered pairs from the ratio table: (1,3), (2,6), (3,9), (4,12).
✏️ Workspace & Solution Steps

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

3 MULTIPLE CHOICE
Recipe: 1 cup berries to 2 scoops yogurt. With cups on x and scoops on y, which pair shows 3 cups?
1. A(3, 6)
2. B(6, 3)
3. C(3, 5)
4. D(3, 3)
✏️ Workspace & Solution Steps
4 MULTIPLE CHOICE
On the smoothie graph, every point follows the rule scoops = 2 × cups. Which equation describes the pattern between x (cups) and y (scoops)?
1. Ay = 2x
2. By = x + 2
3. Cy = x ÷ 2
4. Dx = 2y
✏️ Workspace & Solution Steps

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

5 OPEN RESPONSE

A student plots the points (1, 3), (2, 6), (3, 9), and (4, 15) from a ratio table. She says they form a straight line because they go up. Is she correct? Explain how to check.
✏️ Mathematical Justification & Response
6 ERROR ANALYSIS
Level 2 Extension — The Swapped Smoothie Point
1. 1Read the table:Cups of blueberries: 2 | Scoops of yogurt: 4 (rule: 2 scoops per cup)
2. 2Name the axes:Cups go on the x-axis, scoops go on the y-axis
3. 3Write the ordered pair:(4, 2)
4. 4Plot the point:Go right 4, up 2 on the grid
Which step contains the error? Explain the mathematical misconception and write the correct calculation below.

✏️ Corrected Mathematical Work & Explanation

🗣️ MATHEMATICAL DISCOURSE & TALK MOVES SMP.3 / Construct Viable Arguments

Group Discussion Prompt: How does your visual model justify your mathematical solution?

🗣️ Partner A:
Partner A: "I modeled this by identifying the relationship and..."

👂 Partner B:
Partner B: "I agree with your step because the standard mathematical rule states..."

✍️ 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

x	y	Ordered Pair
1
2
4
6