6.AT.3 Lesson 3-9-group1 🟡 Group 1 · Support & Scaffolding
MASTERY CHECK
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3.9 Small Group · Group 1

Tier 2 Intervention · Concrete-Representational-Abstract (CRA) · Dual-Language Anchors

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

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ CONCEPT SUMMARY & GUIDED NOTES: Let's build it together — What are equivalent ratios?
1Multiply or divide both numbers by the same amount to make an equivalent ratio.
  • Equivalent ratios are ratios that mean the same thing. You make one by multiplying or dividing BOTH numbers by the same amount. We'll walk through a worked example, try one together, then you'll try a few with hints right there when you need them.
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.
3Mathematical 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.
4Watch 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 each pair of ratios: are they equivalent or not equivalent?

    Target Categories: Equivalent Ratios NOT Equivalent
    • 2 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
      SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
      1. 3 MULTIPLE CHOICE

        Which ratio is equivalent to 2:5?

        1. A4:10
        2. B3:6
        3. C2:10
        4. D5:2
        ✏️ Workspace & Solution Steps
      2. 4 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
      3. 5 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
      4. 6 MULTIPLE CHOICE

        What is the simplest form of the ratio 15:25?

        1. A3:5
        2. B5:3
        3. C1:5
        4. D15:25
        ✏️ Workspace & Solution Steps
      🗣️ 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
      Starter: My mathematical claim is that the solution is...
      E Evidence
      Starter: The evidence from the model/table demonstrates that...
      R Reasoning
      Starter: This proves my answer because the standard mathematical definition of...
      Student Mastery Self-Assessment:
      1 · Need More Support
      2 · Getting Closer
      3 · Got It / Solid
      4 · Master / Can Teach It