At a glance. One lesson of about 60 minutes. Students convert metric lengths, masses and capacities, use 2.54 cm per inch, and change a speed from kilometres per hour to metres per second in two steps.
Calculator: no calculator. The factors are powers of ten, or 2.54 with a whole number of inches that is a multiple of 10, so no calculator is needed. The reasoning is the skill.
What students will learn
Know
- There are 100 cm in a metre, 1000 g in a kilogram, 1000 m in a kilometre and 1000 mL in a litre.
- 1 inch is 2.54 cm.
- A smaller unit gives a bigger number for the same amount.
Understand
- A conversion factor is a rate; multiplying and dividing by it are inverse operations.
- The size of the unit tells you which operation to use, before any calculation.
- A rate with two units needs two conversions.
Do
- Convert between metric units of length, mass and capacity.
- Convert inches to centimetres and back.
- Convert kilometres per hour to metres per second and back.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: converting units correctly in both directions (questions 1 to 6).
- Criterion B. Investigating patterns: the first figure. Change the amount and the pair of units; ask what stays the same (the factor) and what changes (the number).
- Criterion C. Communicating: giving a reason for the direction of a conversion in words (question 9).
- Criterion D. Applying mathematics in real-life contexts: recipes, ribbon, leaks, cycling speeds (the hook and questions 4 to 8).
The one thing to do with the figure
Ask for a prediction before the buttons are pressed. “1.5 litres in millilitres: bigger or smaller than 1.5? Multiply or divide?” Take both directions from the room and ask which unit is smaller. The figure then shows the same amount on two scales.
Press each pair of units and ask which scale has the bigger numbers every time. Then go to the second figure and ask what 1 km and 1 h become, before the arrows are read.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook: 1.5 litres of stock, with predictions. The two-scale figure on the board. |
| 10 min | Key idea; the factor as a rate; multiply for smaller units, divide for bigger. |
| 10 min | The speed figure and both worked examples; students finish the faded example. |
| 5 min | Question 1, the check. If many choose A, ask how many metres in 1 km. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | 240 km; 35; 1 : 4. |
|---|---|
| Worked example 1 | 3.4 kg. |
| Worked example 2 | 7.5 m per second. |
| Faded example | 5.4 km per hour. |
| 1. Kilometres to metres | B, 4500 m. A uses 100, C uses 10000, D divides. |
| 2. Kilograms to grams | 3200. 3.2 × 1000 = 3200 g. |
| 3. Centimetres to metres | 6.5. 650 ÷ 100 = 6.5 m. |
| 4. Same unit first | 5. 1.25 × 1000 = 1250 mL, and 1250 ÷ 250 = 5. |
| 5. km per hour to m per second | 12.5. 45000 ÷ 3600 = 12.5 m per second. |
| 6. Inches to centimetres | 76.2. 30 × 2.54 = 76.2 cm. |
| 7. Time and capacity together | 1.8. 120 × 15 = 1800 mL = 1.8 L. |
| 8. Changing both units of a rate | 24000. 0.4 × 1000 = 400 mL per second, and 400 × 60 = 24000. |
| 9. Which way to convert | A. C ignores the time unit; B changes one unit, the wrong way; D divides when it should multiply. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1; 32 or 320 in question 2 | Used the wrong power of ten, or divided when the unit is smaller. | Write the factor as a sentence: there are 1000 metres in each kilometre. |
| B or D in question 9 | Changed only one unit, or divided by 3600. | Ask what 1 metre and 1 second each become, one unit at a time. |
| 0.0032 in question 2; C in question 9 | Divided to go to a smaller unit, or thinks a smaller unit means a smaller number. | Ask which unit is smaller, then whether there will be more or fewer of it. |
| 65000 in question 3 | Multiplied to go to a bigger unit. | Ask whether 650 cm can be 65000 m. |
| 65 in question 3 | Divided by 10 instead of 100. | Count the zeros in 100. |
| 0.005 or 312.5 in question 4 | Used litres and millilitres together. | Put both quantities in the same unit before dividing. |
| 45000 in question 5 | Stopped after kilometres to metres. | Check the unit: metres per hour is not metres per second. |
| 0.0125 or 162 in question 5 | Changed only the time, or multiplied by 3.6. | Ask whether metres per second should be a bigger number or a smaller one than km per hour. |
| 32.54 or 7.62 in question 6 | Added the factor, or dropped a zero. | Say it as 2.54 centimetres for each inch, thirty times. |
| 1800, 0.03 or 30 in question 7 | Stopped at millilitres, or treated minutes as seconds. | Change the time first, then the capacity. |
| 24 or 400 in question 8 | Changed only one of the two units. | Do the litres to millilitres, then the second to a minute, and name each unit. |
Questions worth asking
- Why does the number get bigger when the unit gets smaller? More small units are needed to make the same amount. Listen for a picture of a ruler.
- How do you know whether to multiply or divide? By asking which unit is smaller, not by remembering a rule. Listen for “more of them”.
- Why is 3.6 in the shortcut? 1000 ÷ 3600 = 1 ÷ 3.6. Listen for the two conversions combined.
- Is 100 cm per metre a rate or a ratio? A rate: the units differ. Link to Lesson 1.
Differentiation
A way in for anyone stuck
- Give a conversion table for the five factors on the page and ask for the sentence “There are…” each time.
- Draw both scales for the first few questions and mark the starting amount on each.
- Keep to km per hour that are multiples of 18 until the two-step method is secure.
Stretch
- Show that 1 m per second is 3.6 km per hour. (3600 ÷ 1000 = 3.6.)
- Which is faster, 20 m per second or 80 km per hour? By how much, in km per hour? (20 m per second is 72 km per hour, so 80 km per hour is faster by 8.)
Students learning in English as an additional language
The prefixes are the key to the metric words. Say “kilo” as a thousand, “centi” as a hundredth and “milli” as a thousandth, with an example each.
| Word | Plain meaning | Example |
|---|---|---|
| unit | a standard amount used for measuring | metre, gram, litre |
| conversion factor | the number of one unit in another | 100 cm in 1 m |
| kilo- | a thousand times | 1 kilometre is 1000 metres |
| centi- | one hundredth of | 1 centimetre is one hundredth of a metre |
| milli- | one thousandth of | 1 millilitre is one thousandth of a litre |
| convert | change from one unit to another without changing the amount | 2.6 kg to 2600 g |
Sentence starters for explaining aloud or in writing:
- There are … … in each …, so I multiply (divide) by …
- The number should get bigger (smaller) because …
- I changed the … first, then the …
Two things not to say
“Move the decimal point three places.” It works for powers of ten but hides which way and why, and it fails for 2.54.
“Multiply by 1000 to go from big to small.” Unit size is the idea; the rule follows from it.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- LengthChange 3.6 km to metres.
3600 m. - MassChange 450 g to kilograms.
0.45 kg.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- CookingA recipe needs 0.75 litres of coconut milk and a tin holds 250 mL. How many tins are needed?
3 tins: 0.75 L is 750 mL, and 750 ÷ 250 = 3. - SpeedA train travels at 72 km per hour. How many metres does it cover each second?
20 metres: 72 ÷ 3.6 = 20. - Earlier work (Lesson 2)A walker covers 9 km at 6 km per hour. How many minutes does that take?
90 minutes: 9 ÷ 6 = 1.5 hours, and 1.5 × 60 = 90.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA snail moves 2 mm every second. How far does it go in 1 hour, in metres?
2 × 3600 = 7200 mm, which is 7.2 m. - Choosing a unitWhich is faster, 25 metres per second or 100 kilometres per hour? Say how you know.
100 km per hour: 25 m per second is 25 × 3.6 = 90 km per hour.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The buttons and sliders work with the keyboard (tab to a control, then the arrow keys or Enter) and by touch. Under each figure, “Show me the steps” gives the same states in words.
No device? Use a metre rule marked in centimetres and a kitchen scale, and let students find 1000 g of rice by measuring.