Direct Proportion and Proportional Graphs

1. 1 1 litre of petrol costs £1.60. Complete a table: 2 litres = £3.20, 3 litres = £4.80, 5 litres = £8.00. 'What do you notice?' (Always multiply litres by 1.6.) 'This is direct proportion — as one quantity increases, the other increases at the same rate.'
2. 2 In our example, cost = 1.6 × litres. Write as y = kx where k = 1.6 (the constant of proportionality). For any directly proportional relationship, y/x = k (constant ratio). Students verify with the table.
3. 3 Students plot (litres, cost) from the table. Join the points. 'Does the line pass through the origin?' (Yes.) 'This is the key feature of a directly proportional graph — it always passes through (0,0).'
4. 4 The gradient of the line = k = 1.6. 'Rise over run': pick two points, calculate (change in y) ÷ (change in x). 'The gradient of a proportional graph equals the constant of proportionality.'
5. 5 Show three tables of values. Students identify which show direct proportion (constant ratio) and which do not (e.g. y = x + 3 is linear but not proportional — it doesn't pass through the origin).
6. 6 Exchange rate: £1 = 1.18 euros. How many euros for £65? Graph it and read off. Recipe: 3 eggs make 12 biscuits — how many eggs for 20 biscuits? Students form and use y = kx.
7. 7 Connect to the unitary method: find the value of 1 unit first, then multiply. 'If 5 books cost £30, one book costs £6, so 8 books cost £48.' Show this is the same as using y = 6x.

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