6.AT.3 Lesson 3-9-group2 🟣 Group 2 · Set B
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

3.9 Small Group · Group 2

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

Equivalent Ratios · Ratio · Procedural Fluency · Real-World Applications

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — What are equivalent ratios?
1Multiply or divide both numbers by the same amount to make an equivalent ratio — 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
  1. Chef Montoya's soup serves 8 people, but she needs to serve 32.
  2. I find the scale factor: 32 divided by 8 = 4.
  3. The recipe uses 3 cups of tomatoes to 2 cups of broth, a 3:2 ratio.
  4. I multiply both by 4: 3 x 4 = 12 cups of tomatoes, and 2 x 4 = 8 cups of broth.
  5. So 3:2 and 12:8 are equivalent ratios, and the flavor stays the same.
3Second Model — Try it together — then prove it
  1. A ratio is 2:5. We want to make it bigger using a scale factor of 3.
  2. Multiply both by 3: 2 x 3 = 6 and 5 x 3 = 15.
  3. So 2:5 and 6:15 are equivalent ratios.
  4. Now prove it: say why that move had to work at all — not just that it did.
  5. Sentence frame — convince a skeptic: "My method works because ___ . It would stop working if ___ ."
4Mathematical Word Bank
  • Equivalent Ratios (Razones equivalentes) — Ratios that compare quantities in the same relationship.
  • Ratio (Razón) — A way to compare two amounts, like 3 to 2.
  • Rate (Tasa) — A ratio comparing two amounts with different units, like miles per hour.
  • Proportion (Proporción) — A math sentence saying two ratios are equal.
  • Simplify (Simplificar) — To write a ratio with smaller numbers that still compares the same way. 6:8 and 3:4 are the same ratio.
  • Ratio table (Tabla de razones) — A table of equal ratios that shows a pattern.
5Watch out
  • A common mistake in Equivalent Ratios is ADDING the scale factor to one part instead of MULTIPLYING both parts by it. For example, scaling 2:5 by a factor of 4 should give 8:20 (2×4 and 5×4) — not 6:9 (2+4 and 5+4). Adding changes the taste of the recipe (and the value of the ratio); only multiplying or dividing both numbers by the same amount keeps it equivalent.
SECTION 1 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 1 MULTIPLE CHOICE

    A recipe uses 4 eggs for every 6 cups of flour. Which ratio is equivalent?

    1. A2:3
    2. B8:10
    3. C4:3
    4. D6:4
    ✏️ Workspace & Solution Steps
  2. 2 MULTIPLE CHOICE

    Which ratio is NOT equivalent to 6:9?

    1. A3:5
    2. B2:3
    3. C12:18
    4. D18:27
    ✏️ Workspace & Solution Steps
  3. 3 MULTIPLE CHOICE

    Marcus earns $45 for every 3 hours of work. How much does he earn in 8 hours?

    1. A$120
    2. B$90
    3. C$135
    4. D$105
    ✏️ Workspace & Solution Steps
SECTION 2 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 4 ERROR ANALYSIS

    Fix our table's thinking

    1. 1The problem:The lemonade station mixes 5 cups of lemon juice to 2 cups of sugar (5:2). Which batch keeps the SAME taste?
    2. 2A classmate at our table answered:7:4

    Which step contains the error? Explain the mathematical misconception and write the correct calculation below.

    ✏️ Corrected Mathematical Work & Explanation
  2. 5 ERROR ANALYSIS

    Spot the Common Mistake — Simplifying a Ratio

    1. 1Set up the ratio:A student wants to write 12:18 in simplest form.
    2. 2Find the common factor:Both 12 and 18 can be divided by 6.
    3. 3Divide each part:12 ÷ 6 = 2 and 18 ÷ 3 = 6, so they wrote 2:6.
    4. 4Answer:12:18 simplifies to 2:6

    Which step contains the error? Explain the mathematical misconception and write the correct calculation below.

    ✏️ Corrected Mathematical Work & Explanation
  3. 6 ERROR ANALYSIS

    Find Marcus's Mistake

    1. 1Set up the ratio:3 cups : 5 cups = ? : 20 cups
    2. 2Find the multiplier:20 ÷ 5 = 4
    3. 3Apply to both parts:3 + 4 = 7
    4. 4Answer:7 cups : 20 cups

    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