lesson
A correct calculation can answer the wrong model
1 · Learn
Mathematical modelling selects relationships and assumptions to represent a situation. A correct solution of those equations is useful only if the assumptions fit the question. Evaluate units, plausible ranges and omitted constraints.
A linear cost model with a fixed fee should not be treated as direct proportion through the origin. Checking one value is weaker than checking the model's structure against the scenario.
2 · Worked example
A service costs 5 units plus 2 per item: C=5+2n. Doubling n from 3 to 6 changes cost from 11 to 17, not 22, because the fee is paid once.
3 · Your turn
Using C=5+2n, find n when C=21 and explain what n represents.
Check your answer
n=8, the number of items, assuming the stated model and permitted whole-item quantities.
4 · Apply your learning
Compare a model prediction with a supplied observation and identify whether a discrepancy suggests calculation error, measurement uncertainty or a poor assumption.
Adult guidance and safety
This sequence includes extension mathematics and assumes secure earlier work. Check prerequisite understanding and the learner's readiness before assigning extension tasks. Require derivations, accurate diagrams and varied practice; a correct short example is not full assessment evidence. Exam-board and tier-specific pathways are outside this edition.
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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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