LGCSE Mathematics (0178) common mistakes
LGCSE Mathematics (0178) · Understand the traps and how to avoid them.
When adding or subtracting fractions such as 2/3 + 1/4, the candidate adds numerators and denominators separately to get 3/7 instead of finding a common denominator.
The visual pattern of fraction notation suggests two independent whole-number operations, and learners carry over the additive rule for integers without checking whether the parts are the same size.
What to do instead
Have the candidate draw both fractions as slices of the same bar split into twelfths, so 8/12 + 3/12 = 11/12 becomes visible; then drill the rule 'same denominator before you add' and always estimate the answer (near 1) to catch 3/7.
When converting between units of area or volume, the candidate multiplies or divides by 10 or 100 instead of 100 or 1000, for example writing 1 m² = 100 cm².
They memorise a single 'move the decimal point' routine from length conversions and apply the same factor to squared and cubed units without thinking about how many small squares or cubes fit inside.
What to do instead
Make them build the conversion from a sketch: a 1 m by 1 m square holds 100 by 100 cm squares, so 1 m² = 10 000 cm². Practise writing the factor as 10² or 10³ before calculating, and check with a real-world sanity estimate (a floor tile is not 100 cm²).
The candidate expands a bracket with a negative sign, such as 5 − 2(x − 3), as 5 − 2x − 6 rather than 5 − 2x + 6.
They apply the minus only to the first term inside the bracket, or they treat the subtraction sign as belonging to the 2 and lose track of the double negative when multiplying −2 by −3.
What to do instead
Require a middle line where every term is written with its sign before simplifying: 5 + (−2)(x) + (−2)(−3). Have them say aloud 'negative times negative is positive' at each term, and check by substituting x = 1 into the original and the simplified form.
When solving an equation, the candidate moves a term across the equals sign without changing its sign, e.g. going from 3x + 4 = 19 to 3x = 19 + 4.
The shortcut 'move it over' is remembered as a physical relocation rather than as subtracting the same quantity from both sides, so the inverse operation is skipped.
What to do instead
Insist on writing the operation on both sides explicitly (3x + 4 − 4 = 19 − 4) for several weeks before allowing shortcuts. Pair each solution with a substitution check back into the original equation, which exposes the sign error immediately.
The candidate reads a gradient from a straight-line graph by dividing the horizontal change by the vertical change, or ignores the negative slope of a line falling from left to right.
They recall 'rise over run' only vaguely and confuse which difference goes on top, and the sign of the gradient is treated as cosmetic rather than as a description of direction.
What to do instead
Have them mark two clear points, label the vertical difference first and the horizontal second, and write gradient = vertical ÷ horizontal each time. Reinforce with a physical walk: moving right and down gives a negative gradient, and check against the equation y = mx + c.
When finding the equation of a line through two points, the candidate substitutes one point and the gradient into y = mx + c but solves for c incorrectly, often by adding instead of subtracting mx.
Rearranging y = mx + c to isolate c is a two-step algebraic manipulation, and under time pressure candidates do it mentally and drop the sign of a negative coordinate.
What to do instead
Make them write the full substitution line, e.g. 7 = 2(3) + c, then 7 = 6 + c, then c = 1, one step per line. Finish by plotting the two original points and checking the intercept visually.
In problems combining bearings and right-angled triangles, the candidate measures the angle from the wrong north line or uses the bearing directly as an interior angle of the triangle.
Bearings are measured clockwise from north at the point of departure, but diagrams show several north arrows and the candidate loses track of which vertex the angle belongs to.
What to do instead
Require a fresh north arrow at every named point before any calculation, and have them write the bearing beside each arrow. Then identify the interior angle as 180° minus a bearing (or a difference of bearings) explicitly before choosing sine, cosine or tangent.
The candidate uses the slant height in place of the perpendicular height when calculating the volume of a cone or pyramid.
Formula lists show l and h as similar-looking letters, and the slant edge is the length that is labelled on many diagrams, so it is grabbed first.
What to do instead
Have them shade the perpendicular height inside the solid and mark the right angle at its foot, then state 'volume uses h, surface area uses l'. For cones, drill the link l² = r² + h² so the correct height can always be recovered by Pythagoras.
When finding the mean from a grouped frequency table, the candidate divides the sum of the frequencies by the number of classes, or uses class boundaries instead of midpoints.
The structure of a frequency table looks like a list of values, so the candidate treats each row as one data item and forgets that each midpoint must be weighted by its frequency.
What to do instead
Make them add a midpoint column and a frequency × midpoint column to the table, then write mean = (sum of f×x) ÷ (sum of f) with both totals shown. Compare with an estimate read off a cumulative frequency curve to confirm the answer is sensible.
In probability, the candidate adds the probabilities of independent events, or fails to subtract from 1 when asked for 'at least one' or 'not' outcomes.
The word 'or' in everyday speech suggests adding, and the complement rule is not linked to the idea that all probabilities sum to 1, so the candidate never considers the shorter route.
What to do instead
Have them classify each question first as 'and' (multiply, for independent events) or 'or' (add, for mutually exclusive events), writing the label before any numbers. For 'at least one', practise the complement: P(at least one) = 1 − P(none), and check every answer lies between 0 and 1.