Key Stage 3 maths · Ratio, proportion and rates of change

KS3 converting units, scale drawings and maps: a parent’s guide

This topic is about moving confidently between sizes. In Key Stage 3 children change freely between related units of length, mass, time, capacity, area and volume, use scale factors with diagrams and maps, and write one quantity as a fraction of another, including fractions bigger than 1.

Units, scale and fractions of quantities, explained

Children start converting between units in primary school, including with decimals by Year 6. At Key Stage 3 the programme of study expects them to “change freely” between related units in either direction, including area and volume. The metric relationships come from powers of 10: 1 cm is 10 mm, 1 m is 100 cm, 1 km is 1,000 m, 1 kg is 1,000 g, 1 tonne is 1,000 kg and 1 litre is 1,000 ml. Time is different, with 60 seconds in a minute, 60 minutes in an hour and 24 hours in a day. A good habit is the sense check: a smaller unit means more of them, so converting 2.5 km to metres should give a bigger number, 2,500, not a smaller one.

Area and volume need more care. One metre is 100 cm, but one square metre is a 100 cm by 100 cm square, which holds 100 × 100 = 10,000 square centimetres. In the same way, one cubic metre is 100 × 100 × 100 = 1,000,000 cubic centimetres. Capacity links to volume too, since 1 ml is the same as 1 cm³, so a litre is 1,000 cm³. These conversions come up in practical questions about carpet, paint and containers.

Scale factors describe how much bigger or smaller a shape is made. When a shape is enlarged by scale factor 3, every length is multiplied by 3. Scale diagrams and maps work in the same way, using a ratio such as 1 : 50 on a plan or 1 : 25,000 on a walking map. The ratio says that 1 cm on the drawing stands for 50 cm or 25,000 cm in real life. Children measure on the diagram, multiply by the scale to get the real length, and then convert into a sensible unit. To draw a plan they work the other way, dividing real lengths by the scale.

The final idea is expressing one quantity as a fraction of another. If a child’s walk takes 45 minutes out of a 2-hour afternoon, they first put both quantities into the same unit, 45 minutes and 120 minutes, and then write 45/120, which simplifies to 3/8. The fraction can also be bigger than 1: 90 cm compared with 60 cm is 90/60 = 3/2, or 1½, which tells us the first is one and a half times the second. This links directly to ratio and to percentage change, which children also study in Key Stage 3.

Words your child will hear

Metric units: Units based on powers of 10, such as millimetres, metres, grams and litres.
Convert: Change a measurement from one unit to another without changing its size.
Scale factor: The number by which every length is multiplied in an enlargement or reduction.
Scale: The ratio of a length on a drawing or map to the real length, such as 1 : 25,000.
Scale drawing: A drawing in which every length is in the same proportion to the real object.
Capacity: How much a container holds, measured in litres or millilitres.
Equivalent fractions: Fractions with the same value, such as 45/120 and 3/8.

What the National Curriculum says

In Key Stage 3, pupils should be taught to:

Change freely between related standard units [for example time, length, area, volume/capacity, mass]
Use scale factors, scale diagrams and maps
Express 1 quantity as a fraction of another, where the fraction is less than 1 and greater than 1

Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.

A worked example: reading a map

On a walking map with scale 1 : 25,000, two villages are 6.4 cm apart and joined by a straight path. How far apart are they in real life, in kilometres? The path is part of a 4 km circular walk: what fraction of the walk is it?

The scale 1 : 25,000 means 1 cm on the map is 25,000 cm in real life. Multiply the measured length by 25,000.
```
6.4 × 25,000 = 160,000 cm
```
Convert centimetres to metres by dividing by 100.
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160,000 ÷ 100 = 1,600 m
```
Convert metres to kilometres by dividing by 1,000.
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1,600 ÷ 1,000 = 1.6 km
```
Sense check: the real distance is far bigger than the map distance, as it should be. A map of this scale gives 4 cm for each kilometre, and 6.4 cm is a little over 1.5 km.
Write the distance as a fraction of the whole walk. Both quantities are in kilometres, so 1.6 out of 4.
```
1.6/4 = 16/40 = 2/5
```

The villages are 1.6 km apart, which is 2/5 of the 4 km walk.

Common mistakes and how to help

Using the length conversion for area, so 1 m² = 100 cm²

Children know 1 m = 100 cm and apply the same factor to square metres. But a square metre has 100 rows with 100 square centimetres in each.

What helps: Draw a 1 m square, mark a 10 cm grid and count. Then think of the single row of 100 squares and 100 such rows. Volume works the same way, with the factor cubed.

Multiplying when they should divide (or the reverse) with scales

With a 1 : 50 plan, some children divide a measured 8 cm by 50 and get 0.16 cm, forgetting that real life is bigger than the drawing.

What helps: Ask whether the answer should be bigger or smaller before calculating. Real rooms are larger than plans, so measured length × scale; to draw a plan, real length ÷ scale.

Writing one quantity as a fraction of another without matching the units

A child asked for 45 minutes as a fraction of 2 hours writes 45/2 because they only see the numbers 45 and 2.

What helps: Always convert to the same unit first. 45 minutes and 120 minutes gives 45/120 = 3/8. A quick check is that 45 minutes is less than 2 hours, so the fraction must be below 1.

Believing a fraction cannot be larger than 1

Primary school fractions are usually parts of a whole, so 90/60 looks like a mistake and children turn it round to 60/90.

What helps: Ask which quantity is bigger. If the first is larger, the fraction is greater than 1. 90 cm is 1½ times 60 cm. Changing the fraction to a mixed number makes sense of it.

How to help at home

Use a real map. On a family walk or road trip, measure the distance between two places on a map with a ruler and use the scale to work out the real distance. Compare it with a signpost or the route distance, and talk about why a straight-line distance on the map is shorter than a winding road or path.
Draw a plan of a room. Measure a bedroom, choose a scale such as 1 cm for every 50 cm, and draw it on squared paper with the furniture. Check that the bed fits on the plan before moving it for real.
Convert in the kitchen. Recipes mix grams, kilograms, millilitres and litres. Ask your child to convert a quantity, such as 0.75 kg of flour to grams or 350 ml of milk to litres, before measuring it out.
Compare quantities as fractions. How much of the evening was spent on homework: 40 minutes out of 3 hours? Put both in minutes, write 40/180, and simplify it to 2/9. Then ask whether that is more or less than a quarter of the evening.

Builds on

Year 6: Ratio, percentages and scale factors

Leads on to

Key Stage 4: Compound units, proportion and growth
Key Stage 4: Proportion equations and rates of change