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

3.2 Small Group · Group 2

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

Rates and Unit Rates · Rate · Procedural Fluency · Reasoning & Critique

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — What is a unit rate, and how does it help me find the better buy?
1To find a unit rate, divide the total cost by the number of items. A unit rate that gives a cost is called a unit price, and the lower unit price 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. Booth A charges $3 for 5 games. I want the cost for just 1 game.
  2. I divide the cost by the games: $3 ÷ 5 = $0.60.
  3. So Booth A costs $0.60 per game. That is the unit rate.
  4. Because this unit rate is money for one item, it has its own name: it is the unit price.
  5. Now I can compare this booth's unit price fairly with any other booth's.
3Second Model — Try it together — then prove it
  1. A booth charges $4 for 8 games. What do we divide?
  2. We divide cost by games: $4 ÷ 8 = ?
  3. What is the unit price, the cost for just 1 game? ($0.50 per game.)
  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
  • Rates and Unit Rates (Tasas y tasas unitarias) — A rate compares different units; a unit rate compares a quantity to 1 unit.
  • Rate (Tasa) — A ratio comparing two amounts with different units, like miles per hour.
  • Unit Rate (Tasa unitaria) — A rate for just 1 of something, like cost for 1 item.
  • Per (Por) — For each one. Example: 5 dollars per book.
  • Ratio (Razón) — A way to compare two amounts, like 3 to 2.
  • Unit price (Precio unitario) — A unit rate that gives the cost of exactly 1 item, like $0.60 per game.
5Watch out
  • A common mistake in Rates and Unit Rates is comparing total costs instead of unit rates: a student sees Pack X at $4.00 for 10 tokens and Pack Y at $6.00 for 16 tokens and picks Pack X just because $4.00 is the smaller number, without dividing to find the cost per token ($0.40 for Pack X vs. $0.375 for Pack Y — Pack Y is actually the better buy). A second common mistake is dividing backwards — finding tokens ÷ cost instead of cost ÷ tokens — which turns 'dollars per token' into 'tokens per dollar' and gives an answer that looks reasonable but answers the wrong question. Always divide the total cost by the number of items, and check that your unit rate answers 'the cost for ONE item,' not the reverse.
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 FILL TABLE

    Complete the table to find unit rates and determine the best deal.

    DealQuantityCostUnit Rate
    Tokens Pack A
    Tokens Pack B
    Tokens Pack C
    ✏️ Scratchpad / Reasoning
  2. 2 DRAG SORT

    Sort each situation by whether the unit rate is 'per item' or 'per time'.

    Target Categories: Rate Per Item Rate Per Time
    SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
    1. 3 MULTIPLE CHOICE

      The Star Blaster cabinet scored 90 points in 3 games. What is the unit rate in points per game?

      1. A30 points per game
      2. B270 points per game
      3. C3 points per game
      4. D93 points per game
      per 1
    2. 4 MULTIPLE CHOICE

      Nova: 78 points in 6 min. Pulse: 75 points in 5 min. Which scores faster (more points per minute)?

      1. APulse, at 15 points per minute
      2. BNova, at 13 points per minute
      3. CThey score at the same rate
      4. DThere is not enough information
      ✏️ Workspace & Solution Steps
    3. 5 MULTIPLE CHOICE

      Machine A: 150 coins for 5 plays. Machine B: 168 coins for 8 plays. Which is the better value (fewer coins per play)?

      1. AMachine B, at 21 coins per play
      2. BMachine A, at 30 coins per play
      3. CThey are the same value
      4. DMachine A, because it costs fewer coins total
      ✏️ Workspace & Solution Steps
    SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
    1. 6 OPEN RESPONSE

      An arcade sells tokens in three packs: 25 tokens for $5, 60 tokens for $10.80, and 100 tokens for $19. A student says the 100-token pack is always the best deal because you get the most tokens. Is the student correct? Find all three unit rates and explain.

      ✏️ 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