Ratio Reasoning: Convert Measurements Between Systems
I can use ratio reasoning to convert measurements between the customary and metric systems.
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🎯 Content Objective / Objetivo de contenido
I can use ratio reasoning to convert measurements between the customary and metric systems.
Today's Flow
Total pacing: ~45 min · Progress bar at top tracks your place
LAUNCH
⏱ ~10 min
⏱️ 3 MIN · THINK-PAIR-SHARE
A 5-kilometer race and a 5-mile race have the same number — 5. Before converting anything: which race is longer, and how do you know?
Check for Understanding #1
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
Border-Crossing Converter
The academy's kitchen orders from suppliers in two different countries. One ships in pounds and quarts, the other in kilograms and liters. Recipes, highway signs, and even medicine doses can be written in either system — and comparing them means converting between systems first, using a conversion factor that is almost always approximate.
Concept Launch
💡 Comparing across two measurement systems
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.
Converting Measurements Between Systems. 1. Identify the conversion ratio connecting systems (e.g. 1 in ≈ 2.54 cm, 1 kg ≈ 2.2 lbs). 2. Set up a ratio table or proportion aligning customary and metric units. 3. Multiply or divide by the conversion factor to compute the equivalent measurement
Check for Understanding #2
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
Your turn
VOCABULARY
⏱ ~8 min
| Term / Término | Meaning / Significado | Example / Ejemplo | Visual |
|---|---|---|---|
| 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. Reescribir una medida usual como métrica, o al revés, para poder comparar dos cantidades. |
100 kilometers ≈ 60 miles | |
| 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. La razón que relaciona una unidad de un sistema con una del otro, como 1 milla a unos 1.609 kilómetros. |
1 lb ≈ 0.45 kg · 1 in ≈ 2.54 cm | |
| Customary system Sistema usual |
The measurement system used in the United States: inches, feet, miles, ounces, pounds, cups, quarts, gallons. El sistema de medidas que se usa en Estados Unidos: pulgadas, pies, millas, onzas, libras, tazas, cuartos y galones. |
in · ft · mi · oz · lb · qt · gal | |
| Metric system Sistema métrico |
The measurement system used in most of the world, built on tens: centimeters, meters, kilometers, grams, kilograms, liters. El sistema de medidas que se usa en casi todo el mundo, basado en decenas: centímetros, metros, kilómetros, gramos, kilogramos y litros. |
cm · m · km · g · kg · mL · L | |
| Approximately Aproximadamente |
Close to, but not exactly equal. Conversions between systems are almost always approximate, so they use the ≈ sign. Cercano, pero no exactamente igual. Las conversiones entre sistemas casi siempre son aproximadas y usan el signo ≈. |
1 km ≈ 0.6 mi, not exactly 0.6 | |
| Ratio table Tabla de razones |
A table of equal ratios used to scale a conversion factor up to the amount you actually need. Una tabla de razones iguales que sirve para escalar un factor de conversión hasta la cantidad que necesitas. |
km: 1, 10, 100 · mi: 0.6, 6, 60 |
Which Word Fits?
Rewriting a customary measurement as a metric one gives an ___ answer, not an exact one.
Use It In a Sentence
Check for Understanding #3
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
Turn & Talk — Launch
A 5-kilometer race and a 5-mile race have the same number — 5. Before converting anything: which race is longer, and how do you know?
👂 Listen For
Students reason about UNIT SIZE before touching arithmetic — same number, different units, different lengths.
Extend: About how many kilometers long is the 5-mile race? Use 1 km ≈ 0.6 mi and defend your estimate.
EXPLORE & PRACTICE
⏱ ~18 min
Visual Modeling Workspace
Use the drawing tray below to annotate the visual model. Teacher: say "Click to reveal" on key steps.
Explore Activity
Sort each unit into the measurement system it belongs to. Converting between systems always crosses this line.
✍️ Explore Discourse
Why is a conversion between two systems written with ≈ instead of =, when a conversion inside one system (12 in = 1 ft) is exact?
Whiteboard Moment
Show your work clearly. Be ready to explain your thinking to a partner.
Turn & Talk — Explore
To convert between systems you used a conversion factor like 1 km ≈ 0.6 mi as a RATIO. How does writing it as the ratio 1 : 0.6 let a table do the converting for you?
👂 Listen For
Students connect the conversion factor to equivalent-ratio scaling — both parts multiplied by the same number.
Extend: A classmate converted 20 km by computing 20 ÷ 0.6. Critique that move: what question would 20 ÷ 0.6 actually answer?
Practice Check A
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?
✍️ Show Your Work
Explain why your answer is correct using today's vocabulary.
Practice Check B
1 kilometer ≈ 0.6 mile. About how many miles is 5 kilometers?
✍️ Show Your Work
Explain why your answer is correct using today's vocabulary.
Equivalent Ratio Sort
Complete the interactive activity using today's strategy.
✍️ Justify Your Thinking
Sort each label into the correct box.
A classmate turned in the work below. One step has a mistake. Read every step, find it, name it, and fix it.
Choose ONE option to show what you know — then do it in the workspace below.
Use evidence from today's lesson to complete each frame.
Today's key idea is: "Converting Measurements Between Systems. 1. Identify the conversion ratio connecting systems (e.g. 1 in ≈ 2.54 cm, 1 kg ≈ 2.2 lbs). 2. Set up a ratio table or proportion aligning customary and metric units. 3. Multiply or divide by the conversion factor to compute the equivalent measurement" — and it works because ___.
Because Convert Measurements Between Systems means ___, but a tricky part is ___, so I have to ___.
A common mistake with Convert Measurements Between Systems is ___. It happens because ___, and the fix is ___.
I can prove my answer is correct by ___, using Conversion factor to check my work.
✍️ TWR · WRITE 3 SENTENCES · 7 MIN
Converting Measurements Between Systems. 1. Identify the conversion ratio connecting systems (e.g. 1 in ≈ 2.54 cm, 1 kg ≈ 2.2 lbs). 2. Set up a ratio table or proportion aligning customary and metric units. 3. Multiply or divide by the conversion factor to compute the equivalent measurement because ___
Converting Measurements Between Systems. 1. Identify the conversion ratio connecting systems (e.g. 1 in ≈ 2.54 cm, 1 kg ≈ 2.2 lbs). 2. Set up a ratio table or proportion aligning customary and metric units. 3. Multiply or divide by the conversion factor to compute the equivalent measurement but ___
Converting Measurements Between Systems. 1. Identify the conversion ratio connecting systems (e.g. 1 in ≈ 2.54 cm, 1 kg ≈ 2.2 lbs). 2. Set up a ratio table or proportion aligning customary and metric units. 3. Multiply or divide by the conversion factor to compute the equivalent measurement so ___
🌱 TWR · GROW THE KERNEL · 6 MIN
Answer these to add detail
Sentence starters (tap to use)
Student Workspace
Fill in the table using today's strategy.
| Column A | Column B |
|---|---|
✏️ Sketch Your Strategy
Differentiation Paths
Step-by-step with a worked model and sentence frames.
1 kilometer ≈ 0.6 mile. About how many miles is 5 kilometers?
Core practice aligned to the standard.
Extension with error analysis or multi-step reasoning.
Partner Activity
Work with your partner on the practice problems at your differentiation path level. Explain each step using math vocabulary.
Check for Understanding #4
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
Real-World Connection
🌍 Math in the Wild
Evelyn is driving in Canada. The highway sign reads 100 kilometers per hour. She is used to speed limits in miles per hour, and 1 kilometer is about 0.6 mile.
✍️ Connection Reasoning
Is 100 kilometers per hour faster or slower than 100 miles per hour, and how do you know?
The conversion factor is 1 km ≈ ___ mi. Scaling that up, 100 kilometers ≈ ___ miles, so the sign means about ___ miles per hour. That is ___ than 100 miles per hour, because in one hour you would travel ___ distance.
Turn & Talk — Connect
Evelyn sees a 100 km/h speed sign in Canada, and at home her highway limit is 65 mi/h. One partner should convert with the table, the other with 100 × 0.6. Is 100 km/h faster or slower than her usual 65 mi/h?
👂 Listen For
Students explain the paradox — a larger number with a smaller unit can mean a slower speed — and both methods land on ≈60.
Extend: At what km/h reading would Evelyn actually be at her usual 65 mi/h? Work backwards and check your answer.
Summarize: Ratio Reasoning: Convert Measurements Between Systems
Converting Measurements Between Systems. 1. Identify the conversion ratio connecting systems (e.g. 1 in ≈ 2.54 cm, 1 kg ≈ 2.2 lbs). 2. Set up a ratio table or proportion aligning customary and metric units. 3. Multiply or divide by the conversion factor to compute the equivalent measurement
In your own words:
CLOSURE & REFLECT
⏱ ~8 min
Today I learned that ___ because ___.
One thing I am still not sure about is ___.
A highway sign in Canada reads 100 km/h. Using 1 km ≈ 0.6 mi, is that faster or slower than 100 mi/h?
Bonus Exit Check
1 pound ≈ 0.45 kilogram. About how many kilograms is 10 pounds?
✍️ Show Your Work
Explain why your answer is correct using today's vocabulary.
Reflection & Self-Assessment
Continue Learning
Launch the Full Interactive Activity
Students continue practice in the HTML lesson engine with auto-check, hints, and differentiation.
Family Connection
Share tonight's family homework and discuss one vocabulary word at home.
Open Family Homework ↗Teacher Notes
⏱️ Pacing Guide
- Launch & vocab: 12 min
- I Do / We Do / You Do: 15 min
- Explore & practice: 15 min
- Connect & closure: 8 min
Total: ~45 min
🎯 Listen For · Common Errors
• Students reason about UNIT SIZE before touching arithmetic — same number, different units, different lengths.
• Students connect the conversion factor to equivalent-ratio scaling — both parts multiplied by the same number.
• Students explain the paradox — a larger number with a smaller unit can mean a slower speed — and both methods land on ≈60.
• Students name the size-direction check, not just the multiplication or division they performed.
Common mistake: 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.
Answer Key (Teacher Appendix)
Hide this slide during presentation or move to the end of your copy.
✓ Practice 1: The 10–25 kg package — 31 pounds ≈ 0.45 × 31 ≈ 14 kilograms, which falls inside the 10–25 kilogram package.
✓ Practice 2: 3 miles — Scale the conversion factor by 5: 0.6 × 5 = 3 miles.
✓ Practice 3: 4.5 kg — Scale by 10: 0.45 × 10 = 4.5 kilograms.
✓ Practice 4: Miles and kilometers — Miles are customary and kilometers are metric, so comparing them requires a between-systems conversion. The other pairs stay inside one system.
✓ Exit ticket: Slower — 100 km/h is about 60 mi/h — Scale the conversion factor: 0.6 × 100 = 60, so 100 kilometers per hour ≈ 60 miles per hour — slower than 100 miles per hour.