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

3.6 Small Group · Group 2

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

Visual Models · Convert Measurements within the Same System · Conversion factor · Procedural Fluency

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — Converting inside one measurement system
1Write the conversion factor as a unit ratio (12 inches : 1 foot), then multiply to go to the smaller unit and divide to go to the larger one. The amount never changes — only the unit does — 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 check the ride height rule
  1. A ride requires riders to be at or above 44 inches tall. A child is 4 feet tall. The two measurements use different units, so I cannot compare them yet.
  2. Both units belong to the customary system, and there are 12 inches in one foot. That relationship is the conversion factor. As a ratio it is 12 : 1 — a unit ratio, because it compares an amount to exactly 1 foot.
  3. Inches are smaller than feet, so there will be MORE of them: I multiply. 4 × 12 = 48.
  4. The child is 48 inches tall. Now the comparison is fair: 48 inches is greater than 44 inches, so the child can go on the ride.
3Second Model — Try it together — then prove it
  1. Diamond wants to drink 4 liters of water. Her bottle is marked in milliliters.
  2. The conversion factor is 1,000 mL : 1 L. On a double number line, every 1 liter above matches 1,000 milliliters below.
  3. Milliliters are smaller, so multiply: 4 × 1,000 = 4,000. So 4 liters = 4,000 milliliters.
  4. Her bottle holds 500 milliliters. How many fills? Solve ? × 500 = 4,000. Since 8 × 500 = 4,000, she fills the bottle 8 times.
  5. Now prove it: say why that move had to work at all — not just that it did.
  6. Sentence frame — convince a skeptic: "My method works because ___ . It would stop working if ___ ."
4Mathematical Word Bank
  • Convert Measurements within the Same System (Convertir medidas dentro del mismo sistema) — Rewriting a measurement in a different unit of the SAME system, such as feet into inches or liters into milliliters.
  • Conversion factor (Factor de conversión) — The ratio that links two units in the same measurement system, such as 12 inches to 1 foot. Within one system it is exact.
  • Unit ratio (Razón unitaria) — A ratio that compares an amount to exactly 1 unit of another quantity, like 12 inches to 1 foot.
  • Convert (Convertir) — To change a measurement from one unit to another without changing how much it is.
  • Measurement system (Sistema de medidas) — A family of units that go together — customary units like inches and feet, or metric units like milliliters and liters.
  • Double number line (Recta numérica doble) — Two number lines lined up so matching tick marks show two units for the same amount.
5Watch out
  • A common mistake when converting within a measurement system is choosing the operation by habit instead of by unit size — multiplying every time. Converting 36 inches to feet by multiplying gives 36 × 12 = 432 feet, taller than a 40-story building. Ask first which unit is smaller: it takes MORE small units to make the same amount, so going to a smaller unit multiplies, and going to a larger unit divides.
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 FILL TABLE

    There are 1,000 milliliters in 1 liter. Complete the double number line as a table.

    LitersMilliliters
    1
    2
    4
    ✏️ Scratchpad / Reasoning
  2. 2 BAR MODEL
    TotalQuart 1Quart 2Quart 3

    A 3-quart stockpot is filled using a 1-cup scoop. There are 4 cups in 1 quart. Use the bar model to show how many scoops fill the pot.

    ✏️ Workspace & Solution Steps
SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 3 MULTIPLE CHOICE

    A tap fills 250 milliliters every 10 seconds. Will it fill a 3-liter jug in under 2 minutes?

    1. AYes — it delivers 3,000 mL in 120 seconds, exactly filling the jug at the 2-minute mark
    2. BNo — 250 mL in 10 seconds is far too slow for 3 liters
    3. CYes — 3 liters is only 300 mL, so it takes 12 seconds
    4. DThere is not enough information to tell
    ✏️ Workspace & Solution Steps
  2. 4 MULTIPLE CHOICE

    Which pair belongs to the SAME measurement system?

    1. ACups and quarts
    2. BInches and centimeters
    3. CPounds and grams
    4. DLiters and gallons
    ✏️ Workspace & Solution Steps
  3. 5 MULTIPLE CHOICE

    A shelf is 5 feet long. A row of books measures 58 inches. Which is longer, and by how much?

    1. AThe shelf, by 2 inches
    2. BThe books, by 53 inches
    3. CThe books, by 2 inches
    4. DThey are the same length
    ✏️ Workspace & Solution Steps
  4. 6 MULTIPLE CHOICE

    Claim: “Converting a measurement to a smaller unit always makes the number bigger.” Always, sometimes, or never true?

    1. AAlways — a smaller unit means more of them are needed to cover the same amount
    2. BSometimes — it depends whether you multiply or divide
    3. CNever — converting cannot change the number
    4. DSometimes — it is true for length but not for mass
    ✏️ 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
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