GCSE maths · Twelve mistakes · Foundation & Higher
Common GCSE maths mistakes, and how to fix them
Most marks lost in GCSE maths are lost on topics the student already knows. A squared negative, a percentage taken from the wrong starting value, a “show that” with no working: twelve slips like these account for a surprising share of dropped marks, and every one of them has a simple fix.
A mistake caught on the line it was written — not three questions later
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GCSE maths marks are mostly lost on topics students already know
The questions that cost the most marks are rarely the ones a student has never seen. They are routine questions answered with one small slip: a negative number squared without brackets, a value rounded halfway through, a reason written as “Z angles”.
The twelve mistakes below come up again and again when GCSE papers are marked line by line. Each has a wrong version, a right version and the habit that prevents it. Unless a mistake is marked Higher, it applies to both tiers. The questions are written for this page in GCSE style, not copied from past papers.
Number
Four number mistakes that cost marks on every paper
1. Squaring a negative without brackets
Powers are done before multiplication, and a calculator follows that order exactly. Typed as −4², most calculators give −16, because the square happens before the minus sign is applied.
Fix: every substituted negative goes in brackets, on paper and on the calculator.
2. Rounding in the middle of a calculation
Rounding an intermediate value can push the final answer outside the range a mark scheme accepts, even when the method is perfect.
A sphere has surface area 200 cm². Find its volume to 3 s.f.
Wrong: r ≈ 4.0, so V = ⁴⁄₃π × 4³ = 268 cm³ Right: r = 3.98942… kept in memory, so V = 266 cm³
Fix: keep the full value with the Ans or memory key, write four or five figures in the working, and round only on the final line.
3. Percentage change from the wrong base
Percentage change is always measured against the original value. Reverse percentages trip up even more students: a price after a 20% cut is 80% of the original, so the original is found by dividing, not by adding 20% back on.
A price rises from £80 to £92. Find the percentage increase.
Fix: write “original = ?” before calculating and underline which number it is. For the reverse case, £48 after a 20% cut came from 48 ÷ 0.8 = £60, not 48 × 1.2.
4. Bounds and error intervals
A length of 7.3 cm rounded to one decimal place could be anything from 7.25 up to, but not including, 7.35. A truncated 7.3 is different again: 7.3 ≤ l < 7.4. On Higher, the largest value of a division uses the upper bound on top and the lower bound underneath.
Error interval for l = 7.3 (1 d.p.)
Wrong: 7.2 ≤ l ≤ 7.4, or an upper bound of 7.34 Right: 7.25 ≤ l < 7.35
Fix: go half a unit either side, then check the signs: ≤ on the left, < on the right.
Algebra
Three algebra slips that look like the right answer
5. Expanding (x + 3)²
Squaring a bracket means multiplying it by itself, and that always produces a middle term.
Fix: write the bracket out twice, then test with x = 1. (1 + 3)² is 16; 1 + 6 + 9 is 16; 1 + 9 is only 10.
6. Dividing an inequality by a negative
Multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign. Students who treat an inequality exactly like an equation lose this mark almost every time.
Solve −3x > 12
Wrong: x > −4 Right: x < −4
Fix: test a value. x = −5 gives 15, which is greater than 12, so the answers lie below −4.
7. Dividing by x and losing a solution
Cancelling an x from both sides of a quadratic quietly throws away the solution x = 0.
Solve x² = 5x
Wrong: divide by x, so x = 5 only Right: x² − 5x = 0, so x(x − 5) = 0, giving x = 0 or x = 5
Fix: rearrange a quadratic to “= 0” and factorise. Never divide by an expression that could be zero.
Geometry and measures
Three geometry mistakes a quick check would catch
8. The sine rule’s obtuse-angle trap (Higher)
When the sine rule is used to find an angle, the calculator only ever returns the acute answer. If the diagram shows an obtuse angle, the right answer is 180° minus the calculator’s value.
a = 9 cm, b = 12 cm, A = 40°, and B is obtuse. Find B.
Wrong: sin B = 12 × sin 40° ÷ 9 = 0.857…, so B = 59.0° Right: 180° − 59.0° = 121.0°
Fix: after any sine-rule angle, look back at the diagram and ask whether the obtuse partner fits better.
9. A calculator left in radians
If sin 30 comes out as −0.988 rather than 0.5, the calculator is in radians, and every trigonometry answer on the paper will be wrong in the same way.
Thirty-second check
Look for a small “D” on the screen, then type sin 30. The answer should be 0.5.
Fix: do this at the start of both calculator papers, before the first trigonometry question.
10. Angle reasons that earn nothing
“Z angles” and “F angles” are not accepted. Mark schemes want the proper wording: alternate angles are equal, corresponding angles are equal, angles on a straight line add up to 180°.
Where it usually goes wrong A correct angle with a slang reason often keeps the answer mark but drops the reasoning mark. Learn the six or seven standard reasons word for word.
Probability and statistics
Two statistics errors that come from rushing the setup
11. Adding when the question needs multiplying
“And” means multiply along the branches of a tree diagram. “Without replacement” means the top and the bottom of the second fraction both go down by one.
3 red and 5 blue counters; two taken without replacement. P(both red)?
Fix: sketch the tree first, even when the question does not ask for one.
12. A grouped mean divided by the wrong number
An estimated mean from a grouped frequency table is the total of midpoint × frequency, divided by the total frequency. Dividing by the number of groups, or using group widths instead of midpoints, gives a number that looks plausible and scores nothing.
Where it usually goes wrong Add a midpoint column and an fx column to the table before doing any arithmetic.
Exam technique
Two exam habits lose marks on almost every paper
Missing working on “show that” questions. The answer is printed in the question, so every mark is for the steps. Writing (2x + 1)(x + 3) = 2x² + 6x + x + 3 = 2x² + 7x + 3 earns them; writing the last line alone earns nothing. Never work backwards from the given result.
Reading a calculator time as hours and minutes. 2.3 hours is 2 hours 18 minutes, because 0.3 × 60 = 18. It is not 2 hours 30.
Prevention
A five-part routine stops most of these before the exam
None of this is about learning more maths. It is about catching the maths a student already knows before a slip turns it into a lost mark.
Keep an error log. After each practice paper, sort every lost mark into careless, method, or didn’t know. The careless column is nearly always longer than the student expected, and it is the quickest one to shrink.
Re-read the final line of the question before writing an answer: units, accuracy, “in terms of π”, “give a reason”.
Check with a second method. Substitute the answer back in, or estimate before calculating.
Practise in exam conditions. One timed paper at a time, with the non-calculator paper done genuinely without one.
Know the formulae sheet. Ofqual has confirmed that GCSE maths candidates get a formulae sheet in the exam for the remaining life of the current specifications. Knowing what is on it, and what is not, saves time in the room.
Where ClassArc fits
Tutoring catches these slips because the teacher sees every line
In a ClassArc class the student and teacher write on the same live worksheet, so a missing bracket or an early rounding is spotted on the line where it happens rather than discovered on a marked paper a week later.
The teacher is the same person every class, which means they learn which of the twelve your child keeps making and bring it back until it stops. Homework is matched to that day’s lesson, submitted inside ClassArc and reviewed before the next class opens by talking it through.
Two classes a week suits most GCSE students, rising to three before the exams. GCSE-year classes are from £14 for 55 minutes, one to one.
“
A wrong final answer tells you something went wrong. Watching the working tells you which line.
Missing or unclear working, especially on “show that” questions and multi-step problems. A correct answer with no working can still drop method marks, while a wrong answer with clear working often keeps some of them.
Do careless mistakes really matter at GCSE?
Yes. Two or three slips on each of the three papers add up, and close to a grade boundary that can decide the grade. They are also the easiest marks to win back, because the maths itself is already known.
How can my child stop rounding too early?
Keep the unrounded value in the calculator with the Ans or memory key, write at least four or five significant figures in the working, and round only in the final answer to the accuracy the question asks for.
Do students still have to learn formulae for GCSE maths?
A formulae sheet is provided in the exam, and Ofqual has decided this continues for as long as the current specifications run. Students still need to know when each formula applies, and several standard results are not on the sheet at all.
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