3.7 Small Group · Group 2

Second Challenge Form · Non-Routine Extension · Same Standard, New Problems

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

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

Evelyn sees a Canadian highway sign reading 100 kilometers per hour and wonders whether that is faster or slower than 100 miles per hour.
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.
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.
So 100 kilometers ≈ 60 miles, which means 100 kilometers per hour ≈ 60 miles per hour.
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

Terrance needs flea medicine for a dog weighing 31 pounds, but the packages are labeled in kilograms.
The conversion factor is 1 pound ≈ 0.45 kilogram, so the ratio is 1 : 0.45.
Scale it with a ratio table: 10 lb ≈ 4.5 kg, 20 lb ≈ 9 kg, 30 lb ≈ 13.5 kg, 40 lb ≈ 18 kg.
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.
Now prove it: say why that move had to work at all — not just that it did.
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 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]

1 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
2 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
3 MULTIPLE CHOICE
A dog weighs 31 pounds. Flea medicine comes in packages for 4–10 kg, 10–25 kg, and 25–40 kg. Which package is correct?
1. AThe 10–25 kg package
2. BThe 4–10 kg package
3. CThe 25–40 kg package
4. DNone — 31 is above every range
✏️ Workspace & Solution Steps

SECTION 2 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]

4 ERROR ANALYSIS
Fix our table's thinking
1. 1The problem:1 kilometer ≈ 0.6 mile. About how many miles is 5 kilometers?
2. 2A classmate at our table answered:8 miles
Which step contains the error? Explain the mathematical misconception and write the correct calculation below.

✏️ Corrected Mathematical Work & Explanation
5 ERROR ANALYSIS
Find the error
1. 1Setup:A recipe calls for 2 pounds of flour. The scale is metric. (1 lb ≈ 0.45 kg)
2. 2Step 1:The conversion factor is 1 : 0.45.
3. 3Step 2:0.45 × 2 = 0.9, so 2 pounds ≈ 0.9 kilogram.
4. 4Step 3:2 pounds = 0.9 kilogram exactly.
Which step contains the error? Explain the mathematical misconception and write the correct calculation below.

✏️ Corrected Mathematical Work & Explanation
6 OPEN RESPONSE

The Golden Gate Bridge can hold about 35,924,000 pounds. An average midsize car weighs about 1,524 kilograms. Using 1 kilogram ≈ 2.2 pounds, about how many midsize cars could be on the bridge at one time? Explain each step, and say why your answer is an estimate.
✏️ Mathematical Justification & Response

📐 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.

Student Mastery Self-Assessment:

1 · Need More Support

2 · Getting Closer

3 · Got It / Solid

4 · Master / Can Teach It