In A Level Mathematics, a surprising number of lost marks are not about not knowing the topic. The student knows how to solve a quadratic, differentiate a power or find a stationary point, and then loses the mark through a sign, a missed solution, a rounded figure or a final answer that was never checked. Those errors are cheap to fix, which is exactly why they are worth studying on their own.
This guide looks at error patterns specific to Cambridge International A Level Mathematics (9709): algebra, method, trigonometry, calculator use, notation and checking. For each one, it shows a short question in the style of 9709 Pure Mathematics, the mistake, why it tends to happen and a corrected method. It sits within our wider guide to A Level exam technique by subject, which explains why technique depends on assessment format, and it stays on Mathematics. For practical and data skills in a science, see our guide to A Level Biology practical and data questions. For numerical problem solving in Physics, see A Level Physics calculation questions.
A word on what this guide is not. It doesn't quote examiner statistics, and it doesn't claim any mistake is universal. The examples are illustrations written for this guide, not questions from a particular paper. Syllabuses and papers change, so check the current requirements on our A Level Mathematics page and in the official syllabus before relying on any detail.
What the 9709 Syllabus Says About Accuracy and Calculators
Some of the commonest losses are covered by explicit rules in the syllabus. The points below come from Cambridge's syllabus document for 9709, for examinations in 2026 and 2027, as checked on 27 October 2026. Always read the version that applies to your own exam year, which you can find on Cambridge's 9709 syllabus page.
| Topic | What the syllabus says (2026 and 2027 syllabus) |
|---|---|
| Degrees of accuracy | Non-exact numerical answers should be given correct to three significant figures, or one decimal place for angles in degrees, unless the question specifies a different level of accuracy |
| Rounding | To earn accuracy marks, candidates should avoid rounding figures until they have their final answer |
| Calculators | A scientific calculator is expected. Graphical calculators, computers and calculators capable of symbolic algebra, differentiation or integration are not permitted |
| Working | Candidates are expected to show all necessary working. No marks are given for unsupported answers from a calculator |
| Formulae | A list of formulae and statistical tables (MF19) is supplied in the examination |
Cambridge also publishes a Mathematics notation list showing the notation that may be used in examinations for this syllabus. It is worth reading once, because notation errors are among the easiest to avoid.
Algebra: Expanding, Signs and Lost Solutions
Mistake 1: Expanding a squared bracket as two squares
| Common mistake | Corrected method |
|---|---|
| Write (x + 3)² as x² + 9, so x² + 9 = 25, x² = 16 and x = ±4. | Take the square root of both sides: x + 3 = ±5, so x = 2 or x = −8. Or expand properly: x² + 6x + 9 = 25. |
Why it happens: it feels as if squaring distributes over addition, but it doesn't. The cross term 6x is missing. The check shows the problem at once: x = 4 gives (4 + 3)² = 49, not 25, whereas x = 2 gives 25 and x = −8 gives (−5)² = 25.
Mistake 2: A sign error when completing the square
| Common mistake | Corrected method |
|---|---|
| −(x² − 6x) + 1 = −(x − 3)² − 9 + 1 = −(x − 3)² − 8. | −(x² − 6x) + 1 = −[(x − 3)² − 9] + 1 = −(x − 3)² + 9 + 1 = −(x − 3)² + 10. The maximum value is 10, at x = 3. |
Why it happens: the minus sign outside the bracket has to multiply every term inside, including the −9. Check by substituting x = 3 into the original: −9 + 18 + 1 = 10, which agrees.
Mistake 3: Dividing by a variable and losing a solution
| Common mistake | Corrected method |
|---|---|
| Divide both sides by x to get x² = 4, so x = ±2. | Rearrange to x³ − 4x = 0, factorise to x(x² − 4) = 0 and so x(x − 2)(x + 2) = 0, giving x = 0, 2 or −2. |
Why it happens: dividing by x quietly assumes x isn't zero, and that throws away the solution x = 0. Collect everything on one side and factorise instead.
Mistake 4: Reversing an inequality in the wrong place, or the right place wrongly
| Common mistake | Corrected method |
|---|---|
| Factorise as (x − 1)(x − 5) < 0 and write x < 1 or x > 5. | The curve y = x² − 6x + 5 is a “u” shape that is below the axis between its roots, so 1 < x < 5. Test x = 3: 9 − 18 + 5 = −4, which is less than 0. |
Why it happens: the factorised form suggests two separate conditions. A quick sketch of the curve, or a test value, shows which region satisfies the inequality. And whenever you multiply or divide an inequality by a negative number, such as turning −2x + 3 > 11 into x < −4, the direction reverses.
Calculus and Method Errors
Mistake 5: Forgetting the chain rule
| Common mistake | Corrected method |
|---|---|
| dy/dx = 5(3x − 2)⁴. | Differentiate the outer function, then multiply by the derivative of the inner function: dy/dx = 5(3x − 2)⁴ × 3 = 15(3x − 2)⁴. |
Why it happens: the power rule is so familiar that students apply it only to the outside. Ask yourself whether the bracket contains anything other than a plain x, and if it does, there is an inner derivative to include.
Mistake 6: Stopping at the x-value, or mixing up maximum and minimum
| Common mistake | Corrected method |
|---|---|
| dy/dx = 3x² − 3 = 0 so x = ±1. Stop, or state that x = 1 is a maximum. | x = ±1. Substitute into y: when x = 1, y = −2, and when x = −1, y = 2, so the points are (1, −2) and (−1, 2). The second derivative is 6x: at x = 1 it is 6 (positive), so a minimum, and at x = −1 it is −6 (negative), so a maximum. |
Why it happens: when a question asks for coordinates, the x-value is only half the answer. And a positive second derivative means a minimum (a "smile"), which is the opposite of what intuition about "positive" suggests.
Mistake 7: Giving a negative area
| Common mistake | Corrected method |
|---|---|
| ∫₀² (x² − 4) dx = [x³/3 − 4x]₀² = −16/3, so the area is −16/3. | The integral is −16/3 because the curve is below the axis, but an area is positive, so the area is 16/3. |
Why it happens: the integral gives a signed value, and it is easy to forget to turn it into an area. Sketch the curve first, so you know which parts are above and below the axis.
Trigonometry: Ranges and Missing Solutions
Mistake 8: Forgetting to adjust the range for a multiple angle
| Common mistake | Corrected method |
|---|---|
| 2x = 45°, so x = 22.5°. One solution. | If 0° ≤ x ≤ 180° then 0° ≤ 2x ≤ 360°. Within that range, 2x = 45° or 225°, so x = 22.5° or 112.5°. |
Why it happens: the range applies to x, so it doubles for 2x. Always rewrite the range for the angle you are actually solving, then use the pattern of the function to find every solution inside it.
Calculator Use and Rounding
Mistake 9: Rounding too early
| Common mistake | Corrected method |
|---|---|
| Round 1.07¹⁰ to 1.97 first: 500 × 0.97 ÷ 0.07 = 6928.6, giving 6930. | Keep the full value, 1.07¹⁰ = 1.96715…, so S = 500 × 0.96715… ÷ 0.07 = 6908.2…, which is 6910 to 3 significant figures. |
Why it happens: rounding an intermediate value feels harmless, but it changes the third significant figure of the answer here (6930 against 6910). That is why the syllabus tells candidates to avoid rounding until the final answer. Store values in your calculator's memory, or keep exact forms, and round once at the end.
Mistake 10: The wrong angle mode
Calculators work in degrees or radians, and the answer depends on the mode. For example, sin 30 gives 0.5 in degree mode but about −0.988 in radian mode. Check the mode before every trigonometry question, because it silently changes every result. If the question uses radians, such as in circular measure, switch before you start.
Mistake 11: Relying on an unsupported calculator answer
The syllabus is explicit that no marks are given for unsupported answers from a calculator. Writing only a final value after using a calculator for a quadratic, a definite integral or a derivative may earn nothing, even if the value is right. Write the steps you would do by hand, such as the substitution or the integration, and use the calculator to check or evaluate. Check that your calculator is the permitted type too, since graphical and symbolic calculators are not allowed.
Notation and Presentation
Mistake 12: Working backwards in a “show that” question
The 9709 syllabus lists “show (that)” as a command word meaning provide structured evidence that leads to a given result. That means your working must lead to the result, not start from it.
| Common mistake | Corrected method |
|---|---|
| Write x = 2 + √3, then manipulate both sides until “0 = 0” appears, with no statement of what is being substituted. | Substitute into the left-hand side and simplify: (2 + √3)² − 4(2 + √3) + 1 = 4 + 4√3 + 3 − 8 − 4√3 + 1 = 0. So the left-hand side equals the right-hand side, and 2 + √3 is a root. |
Other common presentation slips include chaining equals signs that aren't equal, such as writing 2x + 3 = 7 = x = 2, leaving out dx in an integral and mixing up notation for inverse functions and reciprocals. Cambridge's notation list is the reference for what symbols mean.
Checking Your Solutions
Mistake 13: Not checking solutions after squaring
| Common mistake | Corrected method |
|---|---|
| Square both sides: x + 2 = x², so x² − x − 2 = 0, giving x = 2 or x = −1. Give both. | The same working gives x = 2 or x = −1, then check each in the original. If x = 2, √4 = 2, which is true. If x = −1, √1 = 1, which does not equal −1, so reject it. The only solution is x = 2. |
Why it happens: squaring can create solutions that didn't belong to the original equation. Substituting back into the original, not the squared version, is the check that catches them.
| Quick check | How long it takes | What it catches |
|---|---|---|
| Substitute the answer back into the original | 10 to 20 seconds | Algebra slips, lost signs and extraneous roots |
| Estimate the size of the answer | 5 seconds | Powers of ten and impossible values |
| Reread the question | 5 seconds | Missing parts, wrong form, requested units or accuracy |
| Check the range or domain | 5 seconds | Missing or extra solutions |
A Summary of the Patterns
| Area | Typical slip | Habit that prevents it |
|---|---|---|
| Algebra | Squaring a bracket term by term; sign errors | Write the expansion out; test with a simple value |
| Equations | Dividing by a variable; losing solutions | Collect on one side and factorise |
| Inequalities | Wrong region or direction | Sketch the curve or test a value |
| Calculus | Missing the chain rule; no y-coordinate; negative area | Ask what is inside the bracket; reread the question; sketch |
| Trigonometry | Wrong range; missing solutions; wrong mode | Rewrite the range; check the mode |
| Calculator and rounding | Early rounding; unsupported answers | Keep full values; show the steps |
| Notation | Working backwards; chained equals signs | Start from one side; one equality per line |
| Checking | No substitution back | Substitute into the original |
How to Use This List Without Just Reading It
- 1
Attempt the question yourself first. Cover the table and see whether you make the mistake.
- 2
Find your own pattern. Look back over marked work for repeats, and note them in an error log.
- 3
Turn each pattern into a habit. One line on a checklist, such as “substitute back” or “check the mode”.
- 4
Practise it on fresh questions. Try a few on our A Level practice pages or real questions from our A Level past papers pages.
- 5
Retest after a week. If the mistake has returned, make the habit more specific.
Final Thoughts
Most of the errors above are not gaps in understanding. They are lapses in habit: a bracket expanded in a hurry, a value rounded early, an answer left unchecked. The good news is that habits can be changed, quickly, with a short checklist and a record of your own mistakes.
For why technique depends on the format of each subject, read our guide to A Level exam technique by subject. Then pick a recent Mathematics paper, mark it strictly and write down your three commonest slips.
Practise the subjects you choose
Work through topic questions and past papers for your A Level subjects, and get your answers marked with the reasoning behind each mark.
