6.AT.3 Lesson 3-7-group1 🟡 Group 1 · Support & Scaffolding
MASTERY CHECK
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3.7 Small Group · Group 1

Tier 2 Intervention · Concrete-Representational-Abstract (CRA) · Dual-Language Anchors

Convert Measurements Between Systems · Conversion factor · Procedural Fluency · Reasoning & Critique

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ CONCEPT SUMMARY & GUIDED NOTES: Let's build it together — 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 =.
  • Kilometers and miles both measure distance, but they come from different measurement systems, so the numbers cannot be compared directly. A conversion factor links the two systems, and a ratio table scales it up to the amount you need. We'll walk through a worked example, try one together, then you'll try a few with hints right there when you need them.
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.
3Mathematical 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.
4Watch 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 kilometer ≈ 0.6 mile. Complete the ratio table.

    KilometersMiles
    1
    10
    100
    ✏️ Scratchpad / Reasoning
  2. 2 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
SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 3 MULTIPLE CHOICE

    1 kilometer ≈ 0.6 mile. About how many miles is 5 kilometers?

    1. A3 miles
    2. B5.6 miles
    3. C8 miles
    4. D30 miles
    ✏️ Workspace & Solution Steps
  2. 4 MULTIPLE CHOICE

    1 pound ≈ 0.45 kilogram. About how many kilograms is 10 pounds?

    1. A4.5 kg
    2. B10.45 kg
    3. C22 kg
    4. D45 kg
    ✏️ Workspace & Solution Steps
  3. 5 MULTIPLE CHOICE

    Which pair of units comes from two DIFFERENT measurement systems?

    1. AMiles and kilometers
    2. BInches and feet
    3. CCups and quarts
    4. DGrams and kilograms
    ✏️ Workspace & Solution Steps
  4. 6 MULTIPLE CHOICE

    A sign in Mexico gives a distance of 80 kilometers. Using 1 km ≈ 0.6 mi, about how far is that in miles?

    1. AAbout 48 miles
    2. BAbout 80 miles
    3. CAbout 133 miles
    4. DAbout 8 miles
    ✏️ 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
Starter: My mathematical claim is that the solution is...
E Evidence
Starter: The evidence from the model/table demonstrates that...
R Reasoning
Starter: This proves my answer because the standard mathematical definition of...
Student Mastery Self-Assessment:
1 · Need More Support
2 · Getting Closer
3 · Got It / Solid
4 · Master / Can Teach It