This module is about investigating a mathematical situation: trying cases in an organised way, spotting a pattern, stating it as a rule, testing the rule and explaining why it works. It is most closely tied to the International Mathematics route (0607), where investigative work is part of the course. Check the current Cambridge syllabus page for how investigation tasks are assessed in your exam year, and confirm your route with your exam centre.
The skills are useful well beyond one paper. Anyone who can tabulate cases, write a rule with algebra and say where it stops working is stronger at sequences and pattern rules and at proof-style questions in every topic. If you are unsure whether you are on 0580 or 0607, read the route comparison first.
What should you know before starting?
You need three things. First, you should be able to find a term of a simple sequence and describe the gap between terms. Second, you should be comfortable substituting into an expression such as 3n + 1.
Third, you should be able to expand and factorise simple algebra.
If any of those feel shaky, revise them in sequences and pattern rules and algebraic structure first. The non-calculator working trainer is a quick way to keep your arithmetic checks honest.
One orienting example
Four friends meet and each shakes hands once with every other friend. How many handshakes are there?
List them systematically: A with B, C, D gives 3. B with C, D gives 2 more. C with D gives 1 more.
Total 3 + 2 + 1 = 6. For five friends the total is 4 + 3 + 2 + 1 = 10.
Already a pattern is forming: each new person adds one more handshake than the last person added. The rest of this module shows how to turn that observation into a rule, test it, prove it and state where it applies.
In what order should you study the lessons?
- Generate systematic cases for a pattern: a pattern you cannot see comes from cases you have not organised.
- Move from observed cases to a conjecture: turn a table into a precise statement you can test.
- Test a counterexample to a conjecture: one clear failure beats ten agreeing cases.
- Explain a general rule with algebra: algebra covers every case at once.
- State limits of a pattern-based conclusion: say how far your evidence goes and no further.
Then try the investigations practice set with the lessons closed.
What traps catch students in investigations?
The first trap is stopping at two or three cases. A pattern that holds for 1, 2 and 3 can still fail at 4.
The second trap is writing a rule and calling it proved because a few numbers fit. Checking examples is testing, not explaining. The third is forgetting to say which values the rule applies to, such as “for n ≥ 2”.
A fourth trap is tidy arithmetic with no words. Investigation marks usually reward the reasoning you write down, so say what you did and why.
How to use the practice set
Work through the questions in order and write every step, as if someone else will read your reasoning. Open each answer only after you have a complete attempt. The end of the set routes each kind of error back to a lesson.
Students who can follow a worked investigation but freeze on a fresh one often need a teacher to ask the right next question. That is something we do in online one-to-one Mathematics tuition.