Checking Your Own Working: Catching Maths Errors Before the Marker Does
Markers see a particular tragedy every exam season: solutions that are one sign-slip away from full marks, wrong in a way the student could have caught in fifteen seconds — because the answer contradicts the question, or common sense, or the student's own estimate made two lines earlier. Checking is a learnable skill with specific moves, and it is worth more marks per minute than any other activity in a maths paper.
This article teaches the working checker: estimation before calculation, substitution after it, the sanity questions for each topic, and how to budget checking time inside a real exam without sacrificing unfinished questions.
Key takeaways
- Estimate before you calculate: a rough expected size turns the final answer into something checkable rather than merely obtained.
- Substitute solutions back into the original equation — not your rearrangement of it — because the original is where the truth lives.
- Every topic has signature checks: discriminants predict root counts, percentages have direction, probabilities live between 0 and 1.
- Budget checking deliberately: quick per-question checks as you go, one reserved block at the end for the expensive ones.
Estimate first: give future-you a target
The estimate is a prediction made while you are still thinking clearly about the problem's meaning, before the machinery starts. 398 × 21 is about 400 × 20 = 8,000. A jacket reduced by 30% from 85 should cost a bit under 60. The square root of 50 is a little over 7. Each estimate takes seconds and converts your final answer from an assertion into a claim that can be verified: 8,358 passes; 83,580 fails on sight, exposing the slipped decimal while the trail is warm. Students resist estimating as extra work; it is the opposite — it is the cheapest error-detection system in mathematics.
For equations, substitution is the master check, and its target matters: substitute into the original question, not into any line of your working. Solving x² − 5x + 6 = 0 and getting x = 3 and x = −2: test in the original — 9 − 15 + 6 = 0 ✓; 4 + 10 + 6 = 20 ✗. The false root is caught in seconds, and the error hunt now has a location: it lives in the working, not the question. A subtle refinement: substitute with the arithmetic done fresh, not by re-reading the same line — a check that retraces the original steps merely re-commits the original slip, which is why students 'check' wrong work and pass it. Fresh computation, different order if possible.
Each topic carries signature checks — invariants the answer must respect. A discriminant computed as negative predicts no real roots; if your formula then produced two, one of them is wrong. A percentage increase from 64 to 80 must exceed the decrease from 80 to 64, because the bases differ; equal answers expose a method error. Probabilities live in [0, 1]; a probability of 1.4 is not a near miss, it is a confession. Averages live inside the data's range — a mean of 91 for scores topping out at 88 is impossible on its face. Lengths, areas and volumes are positive in school geometry; a negative area means a sign took a wrong turn upstream. Learn the five invariants your current course uses most, and run them like a pilot's checklist: short, memorised, non-negotiable.
Inverse operations provide another family: check subtraction by adding, division by multiplying, a percentage-of by asking whether the part is plausibly smaller than the whole. For fractions, estimate against the landmarks — a product of two proper fractions must shrink below both factors; a sum that exceeds both addends' landmarks signals the common-denominator step went unwitnessed. For quadratics and simultaneous equations, substitution covers everything, which is why those topics are, paradoxically, the most checkable on the paper and the ones where unchecked errors are least forgivable.
Time strategy makes checking real instead of aspirational. Two layers: a five-second check per question as you complete it (estimate matched? invariant respected? answer actually answering the question asked — the most commonly failed check of all), and a reserved final block — ten minutes in a two-hour paper — for substitution checks on the heavy algebra and re-reads of the questions you flagged. Flag as you go: a small mark beside any answer you distrust focuses the final block on the likeliest crime scenes rather than spreading thin re-reading over everything.
What checking cannot do is rescue method errors you do not know you are making — if the wrong formula was chosen confidently, substitution (where possible) still catches its consequences, but invariant checks may not. That residual risk is what practice papers are for: they move errors from the method layer, where checking is weak, to the slip layer, where checking is nearly perfect. A student whose practice has cleaned the methods and whose checking cleans the slips is playing a different game from the student relying on either alone.
Adopt the stance, finally, that an unchecked answer is an unfinished one. The last operation in every problem is not the calculation; it is the verification — and unlike the calculation, it gets easier and faster every time it is practised, until a five-second estimate-check becomes as automatic as the equals sign.
A worked example
A worked catch. Problem: solve 2x² − 7x + 3 = 0. A student computes the discriminant as 49 − 24 = 25, then writes x = (7 ± 5)/2 — forgetting the denominator is 2a = 4, not 2 — giving x = 6 and x = 1. Signature check (estimate): roots should multiply to c/a = 1.5, but 6 × 1 = 6 ✗. Substitution of x = 1 into the original: 2 − 7 + 3 = −2 ✗. Two checks, both fail, error located: redo with ÷4 — x = 3 and x = 0.5. Product 1.5 ✓; substitution: 18 − 21 + 3 = 0 ✓ and 0.5 − 3.5 + 3 = 0 ✓. Fifteen seconds of checking recovered the full question.
Mistakes that cost marks
Re-reading your working as the 'check'. The eye sees what it wrote, not what it should have written; slips survive re-reading almost perfectly. A check must recompute, substitute, or test an invariant — anything with independent machinery.
Checking only the final answer's arithmetic. Most expensive errors live upstream — a miscopied number from the question, a wrong method chosen early. Substitute into the original question to cover the copying, and verify the method against an estimate to cover the choice.
Spending the last twenty minutes checking question 1 repeatedly. Distribute: quick checks everywhere first, deep checks where marks and suspicion concentrate. A flagged-uncertainty system beats a thorough pass over the questions that were probably fine.
Questions, answered
What if substitution is impossible, like a proof or a construction? +
Use the family's other members: special cases (does the claim hold for the triangle I can compute?), dimension and unit sense, and the invariant checks. A geometry identity tested on one concrete triangle is not a proof — but it will catch a wrong identity before you present it as one.
How much exam time should checking get? +
Roughly ten percent, protected: five-second checks continuously, plus a final block of about a tenth of the paper's length. In a two-hour paper that is twelve minutes — less than one question's value, recovering typically more than one question's worth of slips.
Is it worth checking if I might talk myself out of a right answer? +
A real check has a verdict procedure: the answer changes only when a concrete failure is produced — a substitution that does not balance, an invariant violated. 'It looks odd' is not a failure. Students who change answers on vibes lose marks; students who change them on evidence gain them.
Put it into practice
Practise the master check on a solved equation: the solver's roots, substituted back, should return zero.
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