At a glance. One lesson of about 60 minutes. It assumes metric length conversions from Year 8 and scientific notation from Year 9. Students convert with prefixes from nano to tera, use squared and cubed factors for areas and volumes, and convert speeds and densities.
Calculator: GDC optional. Every conversion is a power of ten, so the lesson works by hand. The optional section shows GDC users the ×10ˣ key for powers of ten and reminds them that the GDC cannot choose the factor.
What students will learn
Know
- The prefixes T, G, M, k, c, m, μ and n as powers of ten.
- 1 m² = 10 000 cm², 1 m³ = 1 000 000 cm³ and 1 m³ = 1000 litres.
- 1 cm³ = 1 mL.
Understand
- For an area the length factor is used twice, and for a volume three times, because the units are squares and cubes.
- Changing to a smaller unit gives a larger number, and to a bigger unit a smaller number.
- A compound unit such as km/h or kg/m³ is converted one part at a time.
Do
- Convert lengths, areas and volumes between metric units.
- Convert between prefixes using powers of ten.
- Convert speeds and densities between units.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: prefixes, area and volume factors and compound units (questions 1 to 9).
- Criterion B. Investigating patterns: the first figure. Students predict the factor for each of the nine settings and state the general rule (the length factor to the power 1, 2 or 3).
- Criterion C. Communicating: writing the factor before the calculation and stating whether the number should grow or shrink (worked examples, faded example).
- Criterion D. Applying mathematics in real-life contexts: turf, solar power, a rainwater tank, a cheetah, a memory drive, gold and a syringe; the map-scale what-if.
The one thing to do with the figure
Make the factor visible before it is computed. In the first figure, press Area and ask “How many small squares, and how big is each?” before reading the text. The answer 100 squares of 100 cm² settles 1 m² = 10 000 cm² without a rule.
In the second figure, ask students to predict the power of ten for a pair of rungs, then check.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the turf. Collect predictions for 1 m² in cm²; most say 100. |
| 12 min | First figure with predictions; key idea; settle the hook (14 m²). |
| 8 min | Second figure; the warning card; both worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, go back to the Area setting. |
| 20 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 9. |
The answers
| Warm-up | 320 000; 450 cm; 10⁷. |
|---|---|
| Hook | 1 m² = 10 000 cm²; 350 cm × 400 cm = 140 000 cm² = 14 m² (or 3.5 m × 4 m). |
| Worked examples | 3500 cm²; 0.0075 mm; 72 km/h. |
| Faded example | 4200 cm³ = 4.2 litres. |
| 1. An area conversion | B. A uses the length factor; C uses 1000; D squares the 4 too. |
| 2. Nano to micro | 0.12. 120 ÷ 1000. |
| 3. Giga to mega | 1400. 1.4 × 1000. |
| 4. Square metres to square centimetres | 18 000. 1.8 × 10 000. |
| 5. Cubic metres to litres | 750. 1 m³ = 1000 litres. |
| 6. Metres per second to kilometres per hour | 108. 30 × 3600 ÷ 1000 (that is, × 3.6). |
| 7. Two prefixes in one problem | 500 000. 2 × 10⁶ ÷ 4. |
| 8. Converting a compound unit | 19 300. 19.3 × 10⁶ ÷ 1000. |
| 9. Millilitres to cubic millimetres | C. A uses the length factor; B the area factor; D divides. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Used the length factor, used 1000, or squared the 4. | Draw 1 m² as 100 by 100 and count the rows. |
| 120 000 or 1.2 in question 2 | Multiplied going to a bigger unit, or used 100 for a prefix step. | Ask: is a micrometre bigger or smaller than a nanometre? So more or fewer of them? |
| 0.0014 or 1 400 000 in question 3 | Divided going to a smaller unit, or went two steps. | Point to giga and mega on the ladder and count the steps. |
| 180 or 1800 in question 4 | Used the length factor or 1000 for an area. | Ask for the factor first, written as 100 × 100. |
| 750 000, 0.75 or 7.5 in question 5 | Stopped at cm³, treated 1 m³ as 1 litre, or used the area factor. | Ask how many litres fit in 1 m³ (1000), then scale. |
| 1800 or 0.03 in question 6 | Converted only to minutes, or only the distance. | Convert one part at a time: metres in an hour, then kilometres. |
| 500 or 0.5 in question 7 | Counted tera to mega as one step, or ignored the prefixes. | Write both as powers of ten and divide them. |
| 19 300 000 or 0.0193 in question 8 | Converted only the volume or only the mass. | Ask for the mass of a whole cubic metre, in grams, then in kilograms. |
| A, B or D in question 9 | Used the length or area factor, or divided. | Ask how many 1 mm cubes fit along, across and up a 1 cm cube. |
Questions worth asking
- Why is 1 m² not 100 cm²? Listen for “a square metre is 100 cm along both sides, so it is 100 rows of 100 squares”.
- When you change to a smaller unit, should the number get bigger or smaller? Why? Bigger: smaller units fit more times into the same amount.
- How would you convert km/h to m/s? Multiply by 1000 for metres, divide by 3600 for seconds: overall divide by 3.6.
- Why might a country keep its own units? Habit, existing infrastructure and cost of change; a discussion of measurement systems as human agreements.
Differentiation
A way in for anyone stuck
- Give a prefix ladder card (as in the second figure) and have students count steps before calculating.
- Draw every area conversion as a square and every volume conversion as a cube, labelled with the side in both units.
- Ask “bigger or smaller number?” before every calculation and write the answer down first.
Stretch
- 1 inch = 2.54 cm exactly. How many cm² are there in 1 square inch, and how many cm³ in 1 cubic inch?
- A hectare is a square 100 m by 100 m. How many hectares are there in 1 km²? Explain with a picture.
Students learning in English as an additional language
The prefixes come from Greek and Latin words (kilo, a thousand; milli, a thousandth; micro, small; nano, dwarf), which helps some students remember them. “Per” means “for each”: 30 metres per second is 30 metres for each second.
| Word | Plain meaning | Example |
|---|---|---|
| prefix | a part at the start of a unit that multiplies it by a power of ten | kilo means × 1000 |
| square centimetre | the area of a square 1 cm by 1 cm, written cm² | a postage stamp is about 6 cm² |
| cubic metre | the volume of a cube 1 m along each edge, written m³ | a large packing crate is about 1 m³ |
| litre | a unit of volume equal to 1000 cm³ | a 1.5 litre bottle |
| density | mass for each unit of volume | gold: 19.3 g/cm³ |
| speed | distance for each unit of time | 30 m/s |
| conversion factor | the number you multiply or divide by to change units | 1000 for km to m |
Sentence starters for explaining aloud or in writing:
- The factor is … because it is a length / an area / a volume.
- The new unit is smaller, so the number gets …
- … to … is … steps on the ladder, so I multiply by …
Two things not to say
“Just move the decimal point.” Without the factor, students move it the wrong number of places for areas and volumes; say what the factor is first.
“Multiply when you go down, divide when you go up” as a rule on its own. Give the reason (smaller units, more of them) so it survives a new unit.
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.
- PrefixesWrite 3.6 Mm in km.
3600 km. - AreaConvert 52 000 cm² to m².
5.2 m².
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA high-speed train in Japan travels at 320 km/h. What is that in m/s, to 3 significant figures?
88.9 m/s. - Earlier work (Year 9)Work out (4 × 10⁵) × (3 × 10⁻²) in scientific notation.
1.2 × 10⁴.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA cube has volume 8 cm³. What is its volume in mm³, and what is its edge in mm?
8000 mm³; the edge is 2 cm = 20 mm. - CompoundWater has density 1 g/cm³. What is the mass of 1 m³ of water in kg?
1 000 000 g = 1000 kg.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider or button, then the arrow keys or Enter) and by touch. “Show me the steps” gives the same states in words.
No device? A metre stick and a square of card 10 cm by 10 cm show 1 m² = 100 such squares of 100 cm² each. The With a GDC section is optional.