Investigations and mathematical modelling
Finding patterns, generalising, justifying, and building models.
Review these first
Learning objectives
What you need to be able to do
Teacher-mapped phrasing — check against the official Cambridge syllabus for exact wording.
- 3.1.1Investigate patterns, form and test generalisations, and apply the modelling cycle.Supplement
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Investigations and mathematical modelling
0607 includes an investigation paper, which is unlike anything in 0580. It rewards a systematic approach far more than clever guessing.
The investigation method
- Try small cases first. Work out the simplest examples completely — n = 1, 2, 3 — and record them clearly in a table.
- Look for a pattern in the results, checking differences between terms.
- Form a generalisation — a formula in terms of n.
- Test it on a case you have not yet used. This is where marks are won or lost.
- Justify or explain why the rule works, if you can. An explanation earns more credit than a formula alone.
- Extend the problem: what if the condition changed?
Presentation is assessed
Show your working in an organised table, state results in clear sentences, and define what your variables mean. An examiner who cannot follow your reasoning cannot award method marks, however correct the final formula.
The modelling cycle
For applied questions the process is: Assumptions → build the model → solve → interpret → validate against reality → refine. Stating your assumptions explicitly is a marked step. A model that assumes constant speed, no air resistance, or a fixed growth rate should say so.
Judging a model
Every model simplifies. A good answer says both what the model predicts and where it becomes unreliable — for example, an exponential population model that eventually predicts impossible numbers because it ignores limited food and space. Recognising the limitation is not a weakness in your answer; it is the mark.
Think of it like this
An investigation is detective work: gather evidence from small cases, form a theory, then deliberately test it against a case you have not seen. A theory never tested against fresh evidence is a guess.
Worked examples
Method, step by step
Investigate the number of handshakes when everyone in a group of n people shakes hands once with everyone else.
- 1Try small cases: n = 2 gives 1 handshake; n = 3 gives 3; n = 4 gives 6; n = 5 gives 10.
- 2Tabulate and look for a pattern — the differences are 2, 3, 4, suggesting a quadratic rule.
- 3The values 1, 3, 6, 10 are the triangular numbers, giving the generalisation n(n − 1)/2.
- 4Test on an unused case: n = 6 gives 6 × 5 / 2 = 15, and counting confirms 15.
- 5Justify: each of the n people shakes n − 1 hands, but each handshake is counted twice, hence dividing by 2.
The number of handshakes is n(n − 1)/2, justified because each of n people shakes n − 1 hands and every handshake is counted twice.
Common misconceptions
- Jumping to a formula from two data points. Two points fit infinitely many rules, so more cases are needed.
- Never testing the generalisation. Verifying it on an unused case is a distinct and frequently awarded mark.
- Presenting a model as exact. Stating its assumptions and limitations is part of a full answer.
In the exam
- Always build a table of results for small n before hunting for a formula. It makes patterns visible and shows the examiner your method.
- After stating a general rule, write one line testing it on a further case — "checking n = 6: formula gives 21, and counting gives 21 ✓".