Study guide · Practice
A better way to practise maths problems
Progress comes from understanding why a solution works, not just collecting answers. Use this process whenever a question feels unfamiliar.
Read, represent, plan
First identify what is given and what the question asks for. Note restrictions, such as an integer condition or a denominator that cannot be zero. Draw a diagram for geometry, define variables, or rewrite the expression in a familiar form. Then choose a possible approach and ask which theorem or definition justifies it.
A worked example: a quadratic equation
Question: Find all real values of x satisfying x² − 5x + 6 = 0.
The product of the constant terms must be 6 and their sum must be −5. Factoring gives (x − 2)(x − 3) = 0. Therefore x = 2 or x = 3. Substituting each value into the original equation confirms both solutions.
The important habit is not merely recognising the factorisation: it is checking that the result answers the original question. In problems with square roots, denominators or logarithms, a domain check is especially important because an algebraic manipulation can introduce invalid answers.
When you get stuck
- Spend a reasonable amount of time trying a diagram, a simpler case or a different representation.
- Review the relevant definition or a similar example, rather than immediately reading the entire solution.
- If you need the solution, close it and reproduce the reasoning independently.
- Try a related question later to test whether the idea has transferred.
Build a useful practice set
Begin with direct questions to check the concept. Add questions that combine ideas, and later mix topics so that you must choose the method yourself. Include previous exam questions relevant to the current syllabus, but do not assume that a past pattern guarantees the format of a future paper.
Review the process, not just the score
For each error, ask whether the cause was a missing concept, a flawed strategy, an arithmetic slip or a missed condition. Write the correction in your own words. Revisit the question after a gap and see if you can solve it without the original solution.