6.AT.3 Lesson 3-7-group2 🟣 Group 2 · Challenge & Extension
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

3.7 Small Group · Group 2

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

Convert Measurements Between Systems · Conversion factor · Procedural Fluency · Real-World Applications

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — Comparing across two measurement systems
1Write the conversion factor as a ratio (1 km ≈ 0.6 mi), then scale it with a ratio table until it reaches your amount. Between systems the result is approximate, so write ≈, not = — 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 compare two speed limits
  1. Evelyn sees a Canadian highway sign reading 100 kilometers per hour and wonders whether that is faster or slower than 100 miles per hour.
  2. Kilometers are metric and miles are customary, so 100 and 100 cannot be compared as they stand. Both measure distance, so a conversion factor links them: 1 kilometer ≈ 0.6 mile, or the ratio 1 : 0.6.
  3. I build a ratio table from that unit ratio. Kilometers 1, 10, 100 pair with miles 0.6, 6, 60 — each column multiplies both numbers by the same factor.
  4. So 100 kilometers ≈ 60 miles, which means 100 kilometers per hour ≈ 60 miles per hour.
  5. Now the comparison is fair: 100 miles per hour is faster, because in one hour you would travel 100 miles instead of only about 60.
3Second Model — Try it together — then prove it
  1. Terrance needs flea medicine for a dog weighing 31 pounds, but the packages are labeled in kilograms.
  2. The conversion factor is 1 pound ≈ 0.45 kilogram, so the ratio is 1 : 0.45.
  3. Scale it with a ratio table: 10 lb ≈ 4.5 kg, 20 lb ≈ 9 kg, 30 lb ≈ 13.5 kg, 40 lb ≈ 18 kg.
  4. 31 pounds is just past 30, so the dog is a little more than 13.5 kilograms. That falls inside the package for dogs between 10 and 25 kilograms.
  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 Between Systems (Convertir medidas entre sistemas) — Rewriting a customary measurement as a metric one, or the reverse, so two amounts can be compared.
  • Conversion factor (Factor de conversión) — The ratio that links a unit in one system to a unit in the other, such as 1 mile to about 1.609 kilometers.
  • Customary system (Sistema usual) — The measurement system used in the United States: inches, feet, miles, ounces, pounds, cups, quarts, gallons.
  • Metric system (Sistema métrico) — The measurement system used in most of the world, built on tens: centimeters, meters, kilometers, grams, kilograms, liters.
  • Approximately (Aproximadamente) — Close to, but not exactly equal. Conversions between systems are almost always approximate, so they use the ≈ sign.
  • Ratio table (Tabla de razones) — A table of equal ratios used to scale a conversion factor up to the amount you actually need.
5Watch out
  • A common mistake converting between systems is comparing the bare numbers without converting at all — deciding that 5 kilometers and 5 miles are the same distance because both say 5. They are not: 5 kilometers ≈ 3 miles. A second mistake is writing = instead of ≈. Conversion factors between systems are rounded (1 mile is about 1.609344 kilometers), so the result is approximate no matter how carefully you multiply.
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 FILL TABLE

    1 pound ≈ 0.45 kilogram. Complete the ratio table, then use it to place a 31-pound dog.

    PoundsKilograms
    1
    10
    30
    40
    ✏️ Scratchpad / Reasoning
  2. 2 DRAG SORT

    Converting between systems: sort each conversion by what happens to the number.

    Target Categories: The number gets smaller The number gets bigger
    • Kilometers → miles
    • Miles → kilometers
    • Pounds → kilograms
    • Kilograms → pounds
    • Inches → centimeters
    • Centimeters → inches
SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 3 MULTIPLE CHOICE

    Rosa runs a 5-kilometer race. Her cousin runs 3.5 miles. Using 1 km ≈ 0.6 mi, who ran farther?

    1. AThe cousin — 5 km is about 3 miles, and 3.5 miles is farther than that
    2. BRosa — 5 is greater than 3.5
    3. CThey tie exactly
    4. DYou cannot compare kilometers with miles
    ✏️ Workspace & Solution Steps
  2. 4 MULTIPLE CHOICE

    Why do conversions between systems use the ≈ symbol?

    1. AThe conversion factors are rounded, so results are approximate
    2. BBecause the two amounts are actually different
    3. CBecause metric units are always larger
    4. DBecause you are allowed to guess
    ✏️ Workspace & Solution Steps
  3. 5 MULTIPLE CHOICE

    A sign reads 80 km. Dev multiplies by 0.6 to get 48 miles. Mia divides by 1.6 and gets 50 miles. Why do two correct-looking methods disagree?

    1. ABoth are right — 0.6 and 1/1.6 are near but not equal, so the two roundings drift apart
    2. BDev is wrong — you must always divide when converting
    3. CMia is wrong — you must always multiply when converting
    4. DOne of them made an arithmetic mistake
    ✏️ Workspace & Solution Steps
  4. 6 MULTIPLE CHOICE

    Claim: “Converting pounds to kilograms always makes the number smaller.” Always, sometimes, or never true?

    1. AAlways — a kilogram is heavier than a pound, so fewer of them cover the same mass
    2. BSometimes — it depends how heavy the object is
    3. CNever — converting cannot change the number
    4. DSometimes — it is true above 1 pound and false below it
    ✏️ 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