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

3.5 Small Group · Group 2

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

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 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 1 MULTIPLE CHOICE

    Chef A uses 2 cups of cheese for every 8 crackers. Chef B uses 3 cups of cheese for every 9 crackers. Who uses more cheese per cracker?

    1. AChef B
    2. BChef A
    3. CThey use the same amount
    4. DCannot determine
    ✏️ Workspace & Solution Steps
  2. 2 MULTIPLE CHOICE

    Which ratio is greater: 3:4 or 5:8?

    1. A3:4
    2. B5:8
    3. CThey are equal
    4. DCannot determine
    ✏️ Workspace & Solution Steps
  3. 3 MULTIPLE CHOICE

    Store A sells 3 mangoes for $6. Store B sells 5 mangoes for $9. Which store offers a better price per mango?

    1. AStore B
    2. BStore A
    3. CSame price
    4. DCannot determine
    ✏️ Workspace & Solution Steps
SECTION 2 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 4 ERROR ANALYSIS

    Fix our table's thinking

    1. 1The problem:Chef A uses 2 cups of cheese for every 8 crackers. Chef B uses 3 cups of cheese for every 9 crackers. Who uses more cheese per cracker?
    2. 2A classmate at our table answered:They use the same amount

    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

    1. 1Read the problem:Compare Chef Reyes's dressing (4 tbsp oil : 6 tbsp vinegar) to Chef Tran's dressing (5 tbsp oil : 8 tbsp vinegar). Which dressing is oilier?
    2. 2Find a common denominator:LCD of 6 and 8 is 24.
    3. 3Scale each ratio:Chef Reyes: 6 × 4 = 24, so oil = 4 × 4 = 16, giving 16:24. Chef Tran: to reach 24 he also multiplies by 4: oil = 5 × 4 = 20, giving 20:24.
    4. 4Conclusion:Since 20 > 16, Chef Tran's dressing is oilier.

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

    ✏️ Corrected Mathematical Work & Explanation
  3. 6 ERROR ANALYSIS

    Find Kai's Mistake

    1. 1Read the problem:Compare 3:5 and 4:7. Which ratio is greater?
    2. 2Find common denominator:LCD of 5 and 7 is 35
    3. 3Scale the ratios:3:5 = 21:35 (3 × 7 = 21). For 4:7, Kai multiplies by 7 again: 4 × 7 = 28, giving 28:35.
    4. 4Conclusion:Since 28 > 21, the ratio 4:7 is greater.

    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