6.AT.2 Lesson 3-8-group2 🟣 Group 2 · Challenge & Extension
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

3.8 Small Group · Group 2

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

Unit Rate Problem Solving · Unit Rate · Procedural Fluency · Reasoning & Critique

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — How do I use unit rates to find the better buy?
1Divide the price by the number of items to get the cost per unit. The lower cost per unit is the better buy — 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. Fun Supply Co. sells 500 tokens for $60. I want the cost for 1 token.
  2. I divide the price by the tokens: $60 ÷ 500 = $0.12.
  3. So this deal is $0.12 per token. That is the unit rate.
  4. Now I can compare it fairly to a deal with a different number of tokens.
3Second Model — Try it together — then prove it
  1. Arcade Depot sells 350 tokens for $38.50. What do we divide?
  2. We divide price by tokens: $38.50 ÷ 350 = ?
  3. What is the cost per token, and is it lower than $0.12? ($0.11 — yes, the better buy.)
  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
  • Unit Rate Problem Solving (Resolución de problemas con tasas unitarias) — Find an amount per 1 unit and use it to compare choices or solve a problem.
  • Unit Rate (Tasa unitaria) — A rate for just 1 of something, like cost for 1 item.
  • Better Buy (Mejor compra) — The choice that costs less for each item.
  • Comparison (Comparación) — Looking at two or more amounts to see how they are related.
  • Per Unit (Por unidad) — For each single item.
  • Rate (Tasa) — A ratio comparing two amounts with different units, like miles per hour.
5Watch out
  • A common mistake in Solve Problems with Unit Rates is comparing the total prices instead of the cost per item — the deal with the smaller total can still cost more per unit. Always reduce each option to its unit rate before you decide, then pick the lower one.
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 FILL TABLE

    Find the unit rate for each deal and identify the better buy.

    OptionDealUnit Rate
    Store A Batteries
    Store B Batteries
    Store A Candy
    Store B Candy
    ✏️ Scratchpad / Reasoning
  2. 2 DRAG SORT

    Three friends each walked different distances in different times. Sort them from fastest to slowest.

    Target Categories: Fastest Middle Slowest
    SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
    1. 3 MULTIPLE CHOICE

      The snack bar compares two popcorn deals: 12 oz for $3.00 or 20 oz for $4.00. Which is the better buy?

      1. A20 oz for $4.00 — $0.20 per oz
      2. B12 oz for $3.00 — $0.25 per oz
      3. CThey cost the same per ounce
      4. DNot enough information
      ✏️ Workspace & Solution Steps
    2. 4 MULTIPLE CHOICE

      The delivery van brings arcade supplies 150 miles in 3 hours. What is the unit rate in miles per hour?

      1. A50 miles per hour
      2. B45 miles per hour
      3. C153 miles per hour
      4. D30 miles per hour
      per 1
    3. 5 MULTIPLE CHOICE

      The ticket counter sells redemption tickets two ways: 8 tickets for $2.00 or 15 tickets for $3.00. Which costs less per ticket?

      1. A15 tickets for $3.00 — $0.20 each
      2. B8 tickets for $2.00 — $0.25 each
      3. CThey cost the same per ticket
      4. DCannot determine
      ✏️ Workspace & Solution Steps
    SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
    1. 6 OPEN RESPONSE

      Two arcade supply companies offer token deals. Company X: 300 tokens for $36. Company Y: 500 tokens for $55. The arcade needs 1,000 tokens. Find the unit rate for each company, determine which is cheaper per token, and calculate the total cost of buying 1,000 tokens from the better deal.

      ✏️ Mathematical Justification & Response
    🗣️ 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