6.AT.3 Lesson 3-9-group2 🟣 Group 2 · Challenge & Extension
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3.9 Small Group · Group 2

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

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 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 DRAG SORT

    Sort these ratios — which are equivalent to 3:4 and which are not?

    Target Categories: Equivalent to 3:4 NOT Equivalent to 3:4
    • 2 FILL TABLE

      Find the missing value that makes each pair of ratios equivalent.

      Ratio 1Ratio 2Missing Value
      3:7
      5:8
      4:6
      ✏️ Scratchpad / Reasoning
    SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
    1. 3 MULTIPLE CHOICE

      Chef Montoya's dressing uses 3 spoons of oil to 4 spoons of vinegar (3:4). Which recipe card shows an EQUIVALENT ratio?

      1. A9:12
      2. B4:3
      3. C3:8
      4. D6:7
      ✏️ Workspace & Solution Steps
    2. 4 MULTIPLE CHOICE

      The lemonade station mixes 5 cups of lemon juice to 2 cups of sugar (5:2). Which batch keeps the SAME taste?

      1. A15:6
      2. B10:6
      3. C5:4
      4. D7:4
      ✏️ Workspace & Solution Steps
    3. 5 MULTIPLE CHOICE

      Which ratio is equivalent to 2:5?

      1. A4:10
      2. B3:6
      3. C2:10
      4. D5:2
      ✏️ Workspace & Solution Steps
    SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
    1. 6 ERROR ANALYSIS

      Level 2 Extension — The Scaled-Up Pancake Batter

      1. 1Set up the ratio:3 cups flour : 8 cups water = ? : 24 cups water
      2. 2Find the multiplier:24 ÷ 8 = 3
      3. 3Apply to the flour:3 + 3 = 6
      4. 4Answer:6 cups flour : 24 cups water

      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