Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- (Lesson 2) A train travels at 80 km per hour for 3 hours. How far does it go?
Answer
240 km. Distance = speed × time = 80 × 3. - (Year 7, Unit 1, lesson 1) What is 0.35 × 100?
Answer
35. Each digit moves two places to the left. - (Year 8, Unit 1, lesson 2) Write 50 cm : 2 m as a ratio in its simplest form.
Answer
1 : 4. Put both in centimetres, 50 : 200, then divide by 50.
Words used in this lesson
- unit
- a standard amount used for measuring. For example: metre, gram, litre.
- conversion factor
- the number of one unit in another. For example: 100 cm in 1 m.
- kilo-
- a thousand times. For example: 1 kilometre is 1000 metres.
- centi-
- one hundredth of. For example: 1 centimetre is one hundredth of a metre.
- milli-
- one thousandth of. For example: 1 millilitre is one thousandth of a litre.
- convert
- change from one unit to another without changing the amount. For example: 2.6 kg to 2600 g.
Counting the same thing in a different unit
A recipe from a cook in Manila asks for 1.5 litres of stock. Your measuring jug only shows millilitres. Predict first: how many millilitres is 1.5 litres? Will the number be bigger or smaller than 1.5, and do you multiply or divide?
The figure draws the same amount on two scales, one above the other. Only the unit changes; the amount of stuff does not.
4 metres is 4 × 100 = 400 centimetres. There are 100 centimetres in each metre, so the smaller unit gives the bigger number.
Show me the steps
- 4 metres to centimetres: 4 × 100 = 400 centimetres.
- 3 kilograms to grams: 3 × 1000 = 3000 grams.
- 7 kilometres to metres: 7 × 1000 = 7000 metres.
- 5 litres to millilitres: 5 × 1000 = 5000 millilitres.
- 10 inches to centimetres: 10 × 2.54 = 25.4 centimetres. Going back, 25.4 ÷ 2.54 = 10 inches.
A conversion factor is a rate: 100 centimetres per metre, 1000 grams per kilogram, 2.54 centimetres per inch.
To change to a smaller unit, multiply by the factor: there are more of the smaller unit, so the number is bigger. To change to a bigger unit, divide by it: there are fewer, so the number is smaller.
A rate has two units, so change each one in turn. 36 km per hour is 36000 metres per hour (× 1000), and an hour is 3600 seconds (÷ 3600), so it is 10 metres per second.
Back to the stock: a litre holds 1000 millilitres, so 1.5 litres is 1.5 × 1000 = 1500 mL. The number got bigger because the millilitre is the smaller unit.
A common mistake: “A smaller unit means a smaller number.” A 6 m rope is 600 cm. The rope has not changed; there are just more of the smaller units in it. If the number goes the wrong way when you convert, you have multiplied when you should divide, or the other way round.
Two units in one rate
Speeds are rates, so they carry two units. Changing a speed from kilometres per hour to metres per second changes the distance unit and the time unit, one after the other.
36 km/h is 36 × 1000 = 36000 m/h, and 36000 ÷ 3600 = 10 m/s.
Show me the steps
- 18 km/h is 18000 m/h, and 18000 ÷ 3600 = 5 m/s.
- 36 km/h is 36000 m/h, and 36000 ÷ 3600 = 10 m/s.
- 54 km/h is 54000 m/h, and 54000 ÷ 3600 = 15 m/s.
- 90 km/h is 90000 m/h, and 90000 ÷ 3600 = 25 m/s.
- 108 km/h is 108000 m/h, and 108000 ÷ 3600 = 30 m/s. Dividing 108 by 3.6 gives the same 30 in one step.
Worked example: going to a bigger unit
A parcel has a mass of 3400 g. Give its mass in kilograms.
- A kilogram is the bigger unit, so there will be fewer kilograms than grams: divide.
- There are 1000 grams in a kilogram: 3400 ÷ 1000 = 3.4.
- 3.4 kg. Check by going back: 3.4 × 1000 = 3400 g.
Worked example: a speed
A cyclist rides at 27 km per hour. What is her speed in metres per second?
- Kilometres to metres: 27 × 1000 = 27000 metres per hour.
- Hours to seconds: there are 3600 seconds in an hour, so divide: 27000 ÷ 3600 = 7.5.
- 7.5 metres per second. Check with the shortcut: 27 ÷ 3.6 = 7.5.
Your go. The first step is given. A swimmer moves at 1.5 metres per second. Find her speed in kilometres per hour.
- An hour is 3600 seconds, so in an hour she swims 1.5 × 3600 = 5400 metres.
- Now change metres per hour to kilometres per hour, then check.
Check the answer
5400 ÷ 1000 = 5.4, so 5.4 km per hour. Check with the shortcut: 1.5 × 3.6 = 5.4.
Check before you go on
1. Which of these is the same distance as 4.5 km?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. How many grams are there in 3.2 kg?
3. A ribbon is 650 cm long. How many metres is that?
Level 2 · Building
4. A jug holds 1.25 litres. A glass holds 250 mL. How many glasses fill the jug?
5. A motorbike taxi in Kampala travels at 45 km per hour. How many metres per second is that?
6. A garden hose is 30 inches long. There are 2.54 cm in 1 inch. How many centimetres long is the hose?
Level 3 · Stretch
7. A leaking tap loses 15 mL every second. How many litres leak out in 2 minutes?
8. A machine fills 0.4 litres every second. How many millilitres does it fill every minute?
Something to think about
Is 1 inch = 2.54 cm a ratio or a rate? Decide, then say what you would lose by writing it as 1 : 2.54.
One way to think about it
It is a rate, because it compares two different units, inches and centimetres. Written as 1 : 2.54 the units are lost, and so is the way to tell which operation to use, because you could no longer see which number is in inches and which is in centimetres. The rate keeps its units, 2.54 centimetres per inch, and the units tell you whether to multiply or to divide.
Exit ticket
9. Divya changes 18 metres per second to kilometres per hour and gets 5 km per hour. What went wrong?