3.3 Small Group · Group 2

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

Ratio Tables · Ratio table · Procedural Fluency · Reasoning & Critique

Name: Date: Period: Learning Goal:

■ ADVANCED CONCEPT ANCHOR: Push further — How do ratio tables work?

1To keep a ratio the same, multiply both numbers by the same scale factor — 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 pancake recipe is 2 cups of mix for every 3 cups of milk, so the ratio is 2:3.
To make 2 batches, my scale factor is 2.
I multiply both numbers by 2: 2 x 2 = 4 cups of mix, and 3 x 2 = 6 cups of milk.
Now my row is 4:6, which is the same ratio as 2:3.
I do NOT add 2 to each number; I multiply both by the same factor.

3Second Model — Try it together — then prove it

A drink uses 1 cup of juice for every 4 cups of water, so the ratio is 1:4.
To double it, what scale factor do we use? We use 2.
Multiply both by 2: 1 x 2 = 2 cups of juice and 4 x 2 = 8 cups of water, giving 2:8.
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

Ratio Tables (Tablas de razones) — Tables that organize pairs of equivalent ratios.
Ratio table (Tabla de razones) — A table of equal ratios that shows a pattern.
Equivalent ratio (Razón equivalente) — Two ratios that mean the same thing, like 1:2 and 2:4.
Scale factor (Factor de escala) — The number you multiply both parts of a ratio by to get an equal ratio.
Pattern (Patrón) — A rule that numbers follow again and again.
Additive pattern (Patrón aditivo) — Building new rows of a ratio table by adding each column's own constant amount, so the two columns grow together.

5Watch out

A common mistake in Ratio Tables is scaling only ONE part of the ratio, or adding the scale factor to a part instead of multiplying by it — for example, turning 2:3 into 6:3 (multiplying only the first number by 3 and forgetting the second) or into 4:5 (adding the scale factor 2 to the second number instead of multiplying, since 3+2=5 but 3×2=6). Before you submit, multiply BOTH numbers by the same scale factor and check that your new ratio simplifies back to the original.

SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]

1 FILL TABLE

Complete the ratio table for lemon juice to water in Chef Reyes's lemonade recipe (1 cup lemon juice for every 4 cups water).

Pitchers Lemon Juice (cups) Water (cups)
1
2
5
10

✏️ Scratchpad / Reasoning
2 DRAG SORT
Which rows belong in a ratio table for 3:4? Sort them into correct and incorrect.

Target Categories: Belongs in 3:4 Table Does NOT Belong

SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]

3 MULTIPLE CHOICE
The food truck sells tacos at a steady rate: 2 tacos cost $5. At that rate, how much do 6 tacos cost?
1. A$15
2. B$11
3. C$12
4. D$30
✏️ Workspace & Solution Steps
4 MULTIPLE CHOICE
A smoothie recipe at the truck uses 4 cups of mango for every 3 cups of pineapple. How many cups of mango are needed for 9 cups of pineapple?
1. A12
2. B10
3. C13
4. D9
✏️ Workspace & Solution Steps

SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]

5 OPEN RESPONSE

A recipe uses 4 cups of flour for every 3 cups of sugar. A student filled in a ratio table: 4:3, 8:6, 12:9, 14:12. Find and fix the error in the last row. Explain what went wrong.
✏️ Mathematical Justification & Response
6 ERROR ANALYSIS
Level 2 Extension — Catch the Food Truck Error
1. 1Read the order:The hot sauce mix uses a ratio of 2 spoons of pepper to 5 spoons of tomato (2:5). Find an equivalent row for a bigger batch.
2. 2Pick a scale factor:To make the batch bigger, Maya picks a scale factor of 3. A scale factor should multiply BOTH parts.
3. 3Scale the pepper:2 + 1 = 3 spoons of pepper
4. 4Scale the tomato:5 + 1 = 6 spoons of tomato
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

Pitchers	Lemon Juice (cups)	Water (cups)
1
2
5
10