Problem Solving and Investigation
Problem Solving and Investigation assesses how you model unfamiliar situations, form equations, prove statements, generalise patterns, and communicate reasoning. The mathematics may come from any strand, so the focus is on strategy and justification rather than a single procedure.
Core rules
- Define variables and units before forming equations. Draw a diagram and mark given information.
- Identify the required result and work backwards if necessary. Check that your answer satisfies the original constraints.
- Proof strategies: direct proof, proof by contradiction, and induction. For induction, prove the base case, assume the statement for (n=k), then prove it for (n=k+1).
- Generalise from patterns only after testing several cases; a pattern is not a proof.
- Interpret the final answer in context and state a clear conclusion.
Worked example
Prove that the sum of two consecutive odd integers is divisible by 4. Let the integers be (2n+1) and (2n+3). Their sum is (4n+4=4(n+1)), which is a multiple of 4. Since (n) is an integer, (n+1) is an integer, so the sum is divisible by 4.
Traps
Not defining variables; assuming the statement to be proved; ignoring domain restrictions or units; accepting an extraneous solution without checking; giving examples instead of a proof; finishing without a conclusion that answers the question.