LC Ordinary Maths common mistakes
LC Ordinary Maths · Understand the traps and how to avoid them.
When solving a quadratic by the formula, candidates substitute into b² − 4ac using the coefficient values with their signs already applied twice, or drop the minus sign in front of b, so 2x² − 5x − 3 = 0 is worked with b = 5 or with −4ac evaluated as −4(−2)(−3).
The formula is memorised as a string of letters rather than as an operation on signed numbers, and students copy numbers into the discriminant before deciding the sign of each coefficient, so sign errors compound silently.
What to do instead
Before substituting, write a = 2, b = −5, c = −3 on its own line, then write the discriminant as a separate line: b² − 4ac = (−5)² − 4(2)(−3) = 25 + 24 = 49. Bracket every substituted value, including positives. Check the two roots against the graph or by multiplying them to recover c/a.
Solving simultaneous equations by substitution, candidates solve for one variable and then stop, or back-substitute into the equation they just rearranged and get 0 = 0, reporting a single value as the full answer.
The method is learned as 'find x' rather than 'find the ordered pair', and back-substitution into the rearranged equation is algebraically valid but uninformative, which looks like a completed solution.
What to do instead
Adopt a fixed routine: solve for one variable, substitute into the other original equation (never the rearranged one), then state the answer as a coordinate pair. Verify by substituting both values into both original equations. For a line and a circle, expect two pairs and say so.
Differentiating expressions such as 3/x² or √x by writing 3/x² → −6/x or √x → 1/(2√x) with the wrong power, or treating a constant multiplier as if it were a term to be differentiated separately.
Candidates apply the power rule only to expressions already written as xⁿ, and skip the rewriting step under time pressure, so fractional and negative indices are handled by pattern-matching rather than by rule.
What to do instead
Rewrite every term in the form axⁿ before differentiating: 3x⁻² → −6x⁻³, x^(1/2) → ½x^(−1/2). Then differentiate term by term, keeping constants attached. Practise a mixed set of ten terms with negative, fractional and zero powers until the rewrite is automatic.
Finding the maximum or minimum of a function by setting f(x) = 0 instead of f′(x) = 0, or finding the stationary point and declaring it a maximum without testing the sign of the derivative on either side.
The words 'maximum' and 'minimum' trigger the equation-solving reflex from algebra, and the second-derivative or sign test is treated as optional decoration rather than as the step that justifies the answer.
What to do instead
Write the three-line template every time: f′(x) = 0 gives x = …, then evaluate f″ at that x (negative = maximum, positive = minimum), then state the coordinates and the nature. If f″ is zero or awkward, use a sign table of f′ either side.
Computing a percentage change, a percentage profit, or a mark-up by dividing by the new value rather than the original, so a price rising from €80 to €100 is reported as a 20% increase instead of 25%.
The word 'percentage' brings the two numbers to mind without a fixed reference point, and candidates divide by whichever number appears first or looks larger.
What to do instead
Always write the fraction as (new − old)/old before calculating, and name the base explicitly: 'change as a percentage of the original €80'. For reverse questions, work backwards from the multiplier: a 25% increase means ×1.25, so divide by 1.25 to recover the original.
In compound-interest and loan questions, candidates use simple interest, or apply the annual rate once per year when interest is compounded monthly, or count the wrong number of periods (for example using 5 periods for 5 years of monthly compounding).
The formula A = P(1 + i)ⁿ is recalled but the meaning of i and n as matching units is not, and the difference between simple and compound growth is not checked against the wording.
What to do instead
Before substituting, write down the period length and set i and n to match it: monthly compounding over 5 years gives i = annual rate ÷ 12 and n = 60. State 'per period' and 'number of periods' explicitly on the page. Sanity-check the answer against a simple-interest estimate to confirm it is larger.
Adding probabilities of overlapping events as if they were mutually exclusive, or multiplying probabilities of events that are not independent, for example giving P(A or B) = P(A) + P(B) when A and B can both happen.
The addition and multiplication rules are memorised in their simplest form, and candidates do not check the conditions (mutually exclusive, independent) that make those simple forms valid.
What to do instead
Before any calculation, write whether the events can occur together and whether one affects the other. Use P(A ∪ B) = P(A) + P(B) − P(A ∩ B) whenever overlap is possible, and for without-replacement draws adjust the denominator for the second draw. Draw a two-circle Venn or tree diagram to make the overlap visible.
In trigonometry, candidates use the sine rule to find an angle when the given information is two sides and the included angle (an SAS case), or use the cosine rule when a matching side-and-opposite-angle pair is present (an ASA case).
Both rules are learned together and the decision is made by which formula comes to mind first rather than by matching the formula's structure to the given data.
What to do instead
Use a two-question test: is there a side with its opposite angle? If yes, sine rule. If the triangle is SAS or SSS, cosine rule. Write the given labels a, b, c and A, B, C on a sketch before choosing, and mark the knowns on the sketch so the pattern is visible.
When asked to construct or prove a result in synthetic geometry, candidates draw a freehand sketch and assert the conclusion, or quote a theorem by name without stating the hypothesis and conclusion that connect to the diagram.
Constructions and proofs are perceived as 'show your work' tasks with no marks for method, and the habit of writing only the final statement carries over from calculation questions.
What to do instead
For constructions, use compass and straight edge only, leave all arcs visible, and label the points you construct. For proofs, write each line as 'because [given or theorem], therefore [statement]', naming the theorem and the specific angles or sides involved. Practise writing out three proofs in full sentences.
In coordinate geometry, candidates find the slope of a line and then use the same slope for the perpendicular line, or compute distance with the y-difference minus the x-difference rather than the square root of the sum of squares.
The slope formula and the distance formula are both 'difference of coordinates' procedures, and the reciprocal-and-negate step for perpendicular slopes is easily dropped under exam pressure.
What to do instead
Write the perpendicular slope as m⊥ = −1/m and substitute it explicitly before forming the equation. For distance, write d = √((x₂ − x₁)² + (y₂ − y₁)²) with both squares bracketed, then evaluate. Check plausibility: a perpendicular slope should have the opposite sign.