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Direct proportion and scaling
Adjusting recipes, comparing unit prices, and the unitary method.
If 1 cake costs 50p, then 2 cakes cost £1 and 3 cakes cost £1.50. When one amount goes up, the other goes up in step with it. That is called direct proportion. In this lesson you will learn how to:
- spot when two amounts are in direct proportion, and when they are not
- scale a recipe up or down for more or fewer people
- use the unitary method: find the value of one first
- draw a double number line to see it
- find the best buy in a shop by comparing prices
1. Going up together
Example: a box holds 6 eggs
Each box adds another 6 eggs. 4 boxes is double 2 boxes, so 4 boxes hold double the eggs: 12 × 2 = 24. A table like this is called a proportion table.
Fill the basket
- Tap the pack on the shelf to put one in your basket. Tap a pack in your basket to put it back.
- Watch the table grow, one row for each pack. Then press Check.
2. In proportion or not?
In proportion: apples at 20p each
Double the apples and the cost doubles. Every cost is 20p × the number of apples.
Not in proportion: a taxi
The taxi costs £3 to get in, then £2 for each mile. 2 miles is double 1 mile, but £7 is not double £5.
Proportion detective
- Read the case and look at the table. Does doubling one amount double the other?
- Tap your verdict. The detective will explain.
3. Scaling a recipe
Up: pancakes for 12 people instead of 4
- 200 g flour
- 2 eggs
- 300 ml milk
- 600 g flour
- 6 eggs
- 900 ml milk
12 ÷ 4 = 3, so 12 people is 3 times as many. Multiply every amount by 3.
Down: pancakes for 2 people
- 200 g flour
- 2 eggs
- 300 ml milk
- 100 g flour
- 1 egg
- 150 ml milk
2 people is half of 4 people, so divide every amount by 2.
A halfway step: pancakes for 6 people
- 200 g flour
- 2 eggs
- 300 ml milk
- 100 g flour
- 1 egg
- 150 ml milk
- 300 g flour
- 3 eggs
- 450 ml milk
No whole number times 4 makes 6. So go through 2 people first: divide by 2, then multiply by 3.
Recipe kitchen
- Step 1: pick what to do to every amount, to go from the old number of people to the new one. Step 2: work out each amount.
- Two with no slips moves you up a level. Level 1 multiplies, level 2 divides, level 3 needs a halfway step, level 4 goes through 1 person, and level 5 works backwards.
4. The unitary method: find one first
Example: 5 pens cost £3.50. How much do 8 pens cost?
- Work in pence:£3.50 = 350p
- Find one pen (divide by 5):350 ÷ 5 = 70, so 1 pen costs 70p
- Find 8 pens (multiply by 8):70 × 8 = 560
See it on a double number line: a tap fills 12 litres in 3 minutes. How much in 7 minutes?
Two number lines, side by side. Whatever you do to the top line, do to the bottom line. ÷ 3 takes you from 3 minutes to 1 minute, and from 12 litres to 4 litres. Then × 7 gives 28 litres in 7 minutes.
Jump along the double number line
- Tap the tick for 1 on the top line, then pick its partner on the bottom line. Then tap the tick you need, and pick its partner.
- Two with no slips moves you up a level. Level 2 brings bigger jumps; level 3 brings money.
5. Best buys
Price of one can
240 ÷ 4 = 60, so Pack A is 60p a can. 330 ÷ 6 = 55, so Pack B is 55p a can. Pack B is better value.
Price of 100 g
300 g is 3 lots of 100 g: 120 ÷ 3 = 40p. 500 g is 5 lots: 175 ÷ 5 = 35p. The big box is better value.
Best buy shop
- Tap each ? to work out the price of one (or of 100 g). Then tap the best buy.
- A shelf with no slips moves you up a level. Level 2 sells by weight; level 3 has three packs.
6. Proportion workshop
Fill in the proportion table step by step. Level 1 finds one; level 2 finds one, then many; level 3 works in pence; level 4 brings bigger numbers; level 5 works backwards to find how many.
7. Practice: pick your level
Start with Foundation if you are in Year 4 or just starting. Move up when you are getting most of them right.
The mistakes that cost marks
- Adding instead of multiplying. Going from 4 people to 6 people by adding 2 to every amount. find the number to multiply by: halve for 2 people, then × 3 for 6 people.
- Changing only one amount. Tripling the flour but forgetting the eggs and the milk. every amount in the recipe changes by the same number.
- Dividing the wrong way round. For 5 pens costing 350p, working out 5 ÷ 350. the price of one is the cost divided by how many: 350 ÷ 5 = 70p.
- Mixing pounds and pence. Writing 8 pens at 70p as £560. work in pence, then change back: 560p = £5.60.
- Picking the cheapest pack. Saying the pack with the smallest price is the best buy. compare the price of one, or the price of 100 g.
- Thinking everything is in proportion. Saying a 2-mile taxi ride costs double a 1-mile ride. check first: does double one amount give double the other?
Quick checklist
Tick each one when you can do it without help.
All ticked. Brilliant work! You have finished Lesson 4.1.
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