6.AT.3 Lesson 3-5-group2 🟣 Group 2 · Challenge & Extension
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

3.5 Small Group · Group 2

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

Compare Ratios · Unit rate · Procedural Fluency · Real-World Applications

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — How do we compare two ratios?
1Compare ratios fairly by making one part equal or by finding the unit rate — 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 Reyes uses 3 tbsp cocoa for 5 oz milk (3:5). Chef Tran uses 4 tbsp for 7 oz milk (4:7).
  2. I cannot compare yet because the milk amounts are different.
  3. I scale both to 35 oz of milk. For Reyes: 5 x 7 = 35, so 3 x 7 = 21 tbsp, giving 21:35.
  4. For Tran: 7 x 5 = 35, so 4 x 5 = 20 tbsp, giving 20:35.
  5. Now the milk is equal, so I compare cocoa: 21 > 20, so Chef Reyes's cocoa is more chocolatey.
3Second Model — Try it together — then prove it
  1. Store A sells 2 pens for $6. Store B sells 3 pens for $12. Which is the better deal?
  2. Find each unit rate, the price for 1 pen. Store A: 6 divided by 2 = $3 per pen.
  3. Store B: 12 divided by 3 = $4 per pen. Store A is cheaper at $3 per pen.
  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
  • Compare Ratios (Comparar razones) — Decide which ratio is greater, less, or equal by using equivalent ratios or unit rates.
  • Unit rate (Tasa unitaria) — A rate for just 1 of something, like cost for 1 item.
  • Equivalent ratio (Razón equivalente) — Two ratios that mean the same thing.
  • Compare (Comparar) — To look at ratios and see which is bigger, smaller, or 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.
  • Common denominator (Denominador común) — A bottom number that two fractions can share so you can compare them.
5Watch out
  • A common mistake in Compare Ratios is thinking the ratio with the bigger raw number automatically wins — like assuming Chef Tran's 8 tablespoons of cocoa must be more chocolatey than Chef Reyes's 6 tablespoons, without checking that the milk amounts (14 oz vs. 10 oz) are different. Never compare the raw numbers alone — always scale both ratios to the same amount or find each unit rate first.
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 DRAG SORT

    Match each scenario to the correct answer: Which ratio is greater?

    Target Categories: Correct Comparison Incorrect Comparison
    • 2 FILL TABLE

      Two frosting recipes both mix drops of food coloring to make green frosting for a cake. Recipe A: 3 drops blue to 5 drops yellow. Recipe B: 4 drops blue to 7 drops yellow. Scale both to a common amount of yellow to compare which recipe is more blue.

      RecipeBlue DropsYellow Drops
      Recipe A
      Recipe B
      Recipe A (scaled to 35)
      Recipe B (scaled to 35)
      ✏️ Scratchpad / Reasoning
    SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
    1. 3 MULTIPLE CHOICE

      Chef Academy's snack bar sells 3 mini quiches for $6 or 5 mini quiches for $8. Which is the better deal per quiche?

      1. A5 quiches for $8
      2. B3 quiches for $6
      3. CThey cost the same
      4. DCannot determine
      ✏️ Workspace & Solution Steps
    2. 4 MULTIPLE CHOICE

      Two soup recipes use salt. Recipe A uses 4 tsp salt for 10 cups of broth. Recipe B uses 3 tsp salt for 8 cups of broth. Which soup is saltier per cup?

      1. ARecipe B
      2. BRecipe A
      3. CThey are equally salty
      4. DCannot determine
      ✏️ Workspace & Solution Steps
    3. 5 MULTIPLE CHOICE

      Chef Mara's lemon glaze uses 6 oz of lemon juice for 9 cupcakes. Chef Devi's uses 4 oz for 7 cupcakes. Whose glaze has more lemon per cupcake?

      1. AChef Mara
      2. BChef Devi
      3. CSame amount
      4. DCannot determine
      ✏️ Workspace & Solution Steps
    SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
    1. 6 ERROR ANALYSIS

      Level 2 Extension — Chef Omar's Glaze Comparison

      1. 1Read the problem:Chef Omar compares two glazes: 3 oz syrup for 8 oz water and 5 oz syrup for 16 oz water. Which is sweeter per ounce of water?
      2. 2Look at the syrup:Glaze 1 has 3 oz syrup. Glaze 2 has 5 oz syrup.
      3. 3Compare the syrup amounts:Since 5 > 3, Glaze 2 must have more syrup per ounce of water.
      4. 4Conclusion:Glaze 2 is sweeter per ounce of 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