DSE Maths common mistakes and misconceptions

Measures, Shape & Space

13 pitfalls

Mensuration (3-D)

Confusing volume with surface area (and similar-solid ratios)

The trap: Using a volume formula where surface area is asked (or vice versa), and scaling both by the same factor for similar solids.
The idea: Volume has units³, surface area units². For similar solids with length ratio k, areas scale as k² and volumes as k³.
How to avoid it: Check the units the answer should have (cm² vs cm³) to pick the right formula; for similar solids use k : k² : k³.

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Mensuration (2-D)

Confusing perimeter with area of a sector

The trap: Finding the arc length or area when the other is asked, and forgetting a sector's perimeter includes the two radii, not just the arc.
The idea: For a sector of angle θ (in a circle of radius r): arc = (θ⁄360)·2πr, area = (θ⁄360)·πr², perimeter = arc + 2r.
How to avoid it: Say out loud whether you need a length (perimeter/arc) or an area, and remember to add the two radii for a sector's perimeter.

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Tangents to Circles

Not using tangent properties

The trap: Overlooking that a tangent is perpendicular to the radius at the point of contact, and that two tangents from an external point are equal.
The idea: These two facts (radius ⊥ tangent; equal tangents from a point) unlock most tangent problems and tangent proofs.
How to avoid it: Mark the right angle at the point of contact and the equal tangent segments on the diagram before writing anything.

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Angles in Circles

Misapplying angle-in-circle relationships

The trap: Treating the angle at the centre as equal to (rather than twice) the angle at the circumference on the same arc, or missing that angles in the same segment are equal.
The idea: Angle at centre = 2 × angle at circumference on the same arc; angles in the same segment are equal; opposite angles of a cyclic quadrilateral sum to 180°.
How to avoid it: Identify which arc each angle stands on before pairing them, and name the theorem you're using at each step.

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Trigonometry (3-D)

Not isolating the correct right-angled triangle in 3-D

The trap: Assuming a triangle inside a 3-D figure is right-angled when it isn't, and confusing the angle between a line and a plane with the angle between two planes.
The idea: The angle between a line and a plane uses the line's projection onto the plane; the angle between two planes is measured perpendicular to their line of intersection. Only some faces/sections are right-angled.
How to avoid it: Redraw the relevant triangle flat on its own, mark which angle is 90°, and confirm perpendicularity before using sin/cos/tan.

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Trigonometry (2-D)

Rounding too early in trigonometry

The trap: Rounding an intermediate angle or length to a few figures and carrying it forward, so the final answer drifts off — and forgetting the sine rule's ambiguous (two-solution) case.
The idea: Rounding errors compound through a multi-step calculation. For the sine rule, an obtuse angle can also satisfy the equation, giving a second valid triangle.
How to avoid it: Keep full precision in the calculator's memory and round only the final answer; check whether an obtuse angle also fits.

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Angle Relations

Assuming angles are equal when they are complementary or supplementary

The trap: Reading two marked angles as equal because they look similar, when the figure actually makes them complementary (sum 90°) or supplementary (sum 180°).
The idea: Equal, complementary and supplementary are three different relationships; a diagram is not proof — the angle relation has to be justified.
How to avoid it: Justify each equality/relationship by a named reason (parallel lines, angle sum, isosceles…) rather than trusting the picture.

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Angles & Polygons

Forgetting the exterior-angle sum is always 360°

The trap: Using the interior-angle-sum formula (n − 2)·180° when the exterior angles are what matter, or thinking the exterior sum depends on the number of sides.
The idea: The exterior angles of ANY convex polygon sum to 360°, regardless of n; each interior + its exterior = 180°.
How to avoid it: For a regular n-gon, one exterior angle = 360°⁄n. Decide whether the problem is about interior or exterior angles first.

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Straight Lines

Overlooking a stated constraint on a line

The trap: Missing a condition such as ‘the lines meet at a point on the x-axis’ (which forces y = 0), or forgetting that perpendicular lines have slopes multiplying to −1.
The idea: A point on the x-axis has y = 0 and a point on the y-axis has x = 0. Perpendicular lines satisfy m₁ · m₂ = −1 (neither vertical).
How to avoid it: Underline every geometric phrase in the question and turn it into an equation before you start solving.

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Coordinate Geometry Basics

Treating an equidistant point as the midpoint

The trap: Given AP = BP, taking P to be the midpoint of AB — when P only has to lie on the perpendicular bisector of AB.
The idea: The set of points equidistant from A and B is the whole perpendicular bisector of AB; the midpoint is just one point on it. If P must also lie on another line, solve the two conditions together.
How to avoid it: Write AP² = BP² as an equation and intersect it with any other given line — don't jump to the midpoint.

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Congruence & Similarity

Forgetting areas scale as the square of the length ratio

The trap: Using the same ratio for corresponding lengths and for areas of similar figures, and assuming two figures are similar without checking the condition.
The idea: For similar figures with length ratio k, areas are in ratio k². Similarity needs equal corresponding angles AND proportional corresponding sides (or a valid AA/SAS/SSS argument).
How to avoid it: State the similarity reason first, then square the length ratio for areas (and cube it for volumes of similar solids).

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Transformation & Symmetry

Mis-forming a transformed function

The trap: Confusing which transformation acts inside vs outside the function — e.g. writing −f(x) + 4 (reflect then shift up) when a reflection in a horizontal line or a horizontal shift was intended.
The idea: f(x) + a shifts up; f(x + a) shifts left; −f(x) reflects in the x-axis; f(−x) reflects in the y-axis. Inside-the-bracket changes affect x (horizontal) and are often ‘opposite’ to intuition.
How to avoid it: Decide first whether the change is horizontal (inside the bracket, on x) or vertical (outside, on the whole function), then apply it.

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Centres of Triangles

Which triangle centres are collinear (and which lie inside)

The trap: Grouping the in-centre with the centroid, orthocentre and circumcentre as collinear, and assuming every centre always lies inside the triangle.
The idea: The centroid, orthocentre and circumcentre lie on the Euler line; the in-centre generally does not. The centroid and in-centre are always inside, but the orthocentre and circumcentre can fall outside an obtuse triangle.
How to avoid it: Remember the trio on the Euler line (centroid–orthocentre–circumcentre) and treat the in-centre separately; sketch to see which centres fall outside.

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Data Handling

6 pitfalls

Scatter Diagrams & Correlation

Reading a scatter diagram: correlation is not causation

The trap: Concluding one variable causes the other from a strong scatter trend, and confusing the strength (how tight the points are) with the direction (positive/negative slope).
The idea: A scatter diagram shows how two variables move together, not that one causes the other. Direction is the sign of the trend; strength is how closely the points hug a line.
How to avoid it: Describe a scatter as ‘positive/negative’ and ‘strong/weak’ separately, and never claim causation from correlation alone.

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Standard Scores & Normal Curve

Misreading a standard score

The trap: Forgetting a standard score can be negative (below the mean), and comparing raw marks across two tests instead of standardising first.
The idea: z = (x − mean) ⁄ (standard deviation); z < 0 means below average. Only standard scores are comparable across distributions with different means/spreads.
How to avoid it: Compute z with the correct mean and standard deviation, keep its sign, and compare z-values (not raw marks) across tests.

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Measures of Dispersion

How shifting vs scaling data changes the spread

The trap: Thinking that adding the same constant to every value changes the standard deviation or range, and forgetting that multiplying scales the spread.
The idea: Adding a constant c to all data shifts the mean by c but leaves range, standard deviation and IQR unchanged. Multiplying all data by k multiplies these spreads by |k|.
How to avoid it: Separate the two effects: + c moves the centre only; × k stretches both centre and spread. Removing an outlier usually shrinks the spread.

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Mean, Median & Mode

How an extra or changed datum affects mean, median and mode

The trap: Assuming a new value changes all three averages equally, and mixing up the median position for an even vs odd number of data.
The idea: The mean uses every value (so an extreme value pulls it), the median depends only on the middle position, and the mode only on frequency. For n data the median position is (n + 1) ⁄ 2.
How to avoid it: Ask which measure the change actually touches; re-sort and re-find the middle position rather than assuming.

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Permutations & Combinations

Miscounting ‘at least’ selections

The trap: Treating ‘at least one’ as ‘exactly one’, and confusing when order matters (permutation) with when it doesn't (combination).
The idea: ‘At least one’ is easiest as total − (none). Committees/groups are combinations (order irrelevant); arrangements in a row/order are permutations.
How to avoid it: For ‘at least’, subtract the complement from the total; decide order-matters (permutation) vs not (combination) before counting.

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Combined Probability

Adding when you should multiply (and replacement)

The trap: Adding probabilities for events that must all happen (should multiply), and ignoring whether a draw is with or without replacement.
The idea: Use × for ‘A and B’ (independent), + for mutually exclusive ‘A or B’. Without replacement, the second probability uses the reduced totals.
How to avoid it: Read ‘and’ → multiply, ‘or’ (exclusive) → add; redraw the pool after each pick if there's no replacement, and enumerate the separate cases.

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The mistakes that cost the most marks

What this is (and isn't)

Number & Algebra

Dividing only part of an expression by the coefficient

Losing a root when taking a square root

Confusing irrational roots with non-real roots

Assuming a complex expression must be real or rational

Forgetting a geometric ratio can be negative — and mixing up AP/GP sums

Confusing marked price, selling price and cost

Not flipping the sign, and listing instead of counting

Splitting log of a sum

Reading the wrong optimal vertex (least vs greatest)

Mixing up direct and inverse variation

Mishandling negative and fractional indices

Measures, Shape & Space

Confusing volume with surface area (and similar-solid ratios)

Confusing perimeter with area of a sector

Not using tangent properties

Misapplying angle-in-circle relationships

Not isolating the correct right-angled triangle in 3-D

Rounding too early in trigonometry

Assuming angles are equal when they are complementary or supplementary

Forgetting the exterior-angle sum is always 360°

Overlooking a stated constraint on a line

Treating an equidistant point as the midpoint

Forgetting areas scale as the square of the length ratio

Mis-forming a transformed function

Which triangle centres are collinear (and which lie inside)

Data Handling

Reading a scatter diagram: correlation is not causation

Misreading a standard score

How shifting vs scaling data changes the spread

How an extra or changed datum affects mean, median and mode

Miscounting ‘at least’ selections

Adding when you should multiply (and replacement)