lesson
Identify what stays constant
1 · Learn
In direct proportion y = kx, the ratio y/x stays constant. In inverse proportion y = k/x, the product xy stays constant. The second model can describe a fixed amount shared across different numbers of equal groups.
A real situation needs assumptions. Doubling workers does not always halve completion time if workers interfere or some tasks cannot be shared. State the idealised conditions before using the model.
2 · Worked example
At a constant £3 per item with no fixed fee, cost = 3 × quantity is direct proportion. Sharing 24 counters equally among n groups gives 24/n per group, an inverse relationship.
3 · Your turn
In y = 24/x, find y for x = 3 and x = 6.
Check your answer
8 and 4. Doubling x halves y in this stated model.
4 · Apply your learning
Compare tables for direct and inverse models and test which ratio or product remains constant.
Adult guidance and safety
Ask for the reasoning and a check, not only a numerical result. Supply accurate diagrams, coordinate grids and data displays where required. Use varied examples and revisit prerequisite gaps; these short teaching models are not a completed assessment programme.
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Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
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