MATSEC SEC Mathematics common mistakes
MATSEC SEC Mathematics · Understand the traps and how to avoid them.
When solving a linear equation such as 5x + 3 = 2x + 12, the candidate adds or subtracts a term from only one side, or moves a term across the equals sign without changing its sign (writing 5x - 2x = 12 + 3).
The equals sign is treated as an instruction ('do something') rather than a statement of balance. Under pressure, candidates apply an operation to the side they are looking at and forget the other side, and the 'change side, change sign' shortcut is memorised without the underlying balance idea.
What to do instead
Train a written two-column routine: write the equation, then on the next line write the same operation applied to both sides explicitly (subtract 3 from both sides, then subtract 2x from both sides) before collecting. Do ten equations where the working must show each operation on both sides; only then allow the shortcut, and check by substituting the answer back into the original equation.
Expanding a bracketed product incorrectly, typically writing (x + 3)^2 as x^2 + 9, or mishandling a negative sign in -(2x - 5) as -2x - 5, and failing to multiply every term of one bracket by every term of the other.
The square is read as 'square each thing inside' because that rule works for multiplication, and the minus sign in front of a bracket is processed as belonging only to the first term. Candidates also run out of working space and drop the outer and inner products in a four-term expansion.
What to do instead
Have candidates expand (x + 3)^2 by writing (x + 3)(x + 3) and drawing the four arrows every time, then compare with the wrong answer numerically by substituting x = 2 into both expressions. For a leading minus, rewrite the subtraction as adding negative one times the bracket before expanding. Insist on four terms written out before collecting like terms.
Getting a negative or reversed inequality when multiplying or dividing by a negative number, for example solving -2x > 6 as x > -3, and drawing the wrong direction of shading on a number line.
The rule that the inequality sign flips is learned as an arbitrary exception and is easily forgotten mid-calculation, especially when the division happens as a quick final step. Candidates also confuse the open and closed circle conventions.
What to do instead
Always solve by keeping the coefficient of x positive where possible (move x terms to the side that makes them positive). When a negative division is unavoidable, write the flip as an explicit step on its own line with the reason in words. Test the answer by substituting a value from the claimed solution set and one from outside it into the original inequality.
Applying the wrong index law, such as writing 2^3 x 2^4 as 2^12, treating (3x^2)^2 as 3x^4, or simplifying x^5 / x^2 as x^2.5 or x^3 incorrectly by subtracting in the wrong order.
The laws for multiplying powers (add indices), raising a power to a power (multiply indices) and dividing (subtract indices) blur together because they look similar. The coefficient inside a bracket is also forgotten because attention is fixed on the letter.
What to do instead
Build a one-page index-law table with a numerical example beside each rule (2^3 x 2^4 = 2 x 2 x 2 x 2 x 2 x 2 x 2 = 2^7). When a bracket contains a coefficient, expand it fully first, e.g. (3x^2)^2 = 3x^2 times 3x^2. Have candidates verify a simplification by substituting x = 2 into the original and the answer.
When factorising a quadratic such as x^2 - 5x + 6, the candidate picks two numbers that multiply to 6 but ignores the sign requirement for the sum, giving (x + 2)(x + 3), or abandons the question and uses the formula with arithmetic slips instead.
Candidates search only for factor pairs of the constant term and forget that the same pair must add to the coefficient of x, and sign handling in the pair (both negative here) is the step most often skipped. The formula route then fails because b and c are substituted with wrong signs.
What to do instead
Require the pair to be written down explicitly with both conditions checked: 'product = +6, sum = -5, so both numbers are negative: -2 and -3.' Then expand the factorised answer back as a compulsory final check. Practise ten quadratics where the constant is positive and the middle term negative, and ten where the constant is negative.
In percentage and ratio problems, applying a percentage change to the wrong base, for example working out a 15% reduction by subtracting 15% of the reduced price, or reversing a percentage increase by subtracting the same percentage instead of dividing by the multiplier.
The phrase 'reduce by 15%' is processed as a sequence of operations rather than as a single multiplier, and reverse percentage problems are not recognised as a different type. Ratio sharing is also done by dividing by the number of parts incorrectly when the ratio has more than two terms.
What to do instead
Convert every percentage change to a multiplier first (increase 15% means multiply by 1.15; decrease 15% means multiply by 0.85) and write the multiplier above the question before calculating. For reverse problems, set up original x multiplier = final value and solve by dividing. For ratios, state the total number of parts explicitly and find the value of one part before sharing.
Using Pythagoras' theorem where trigonometry is needed, or vice versa, and labelling the opposite and adjacent sides relative to the wrong angle, so that sin, cos and tan are assigned to the wrong side pairs in a right-angled triangle.
Both tools involve the three sides of a right-angled triangle, and candidates pattern-match on 'triangle with two sides given' rather than asking whether an angle is involved. The opposite side is identified relative to the angle at the top of the diagram rather than the angle actually marked.
What to do instead
Enforce a decision step written at the top of the working: 'Pythagoras only if no angle is involved; SOHCAHTOA if an angle is involved.' Then mark the angle in question, trace the side opposite it and the side touching it, and write the ratio equation before substituting. Use a right triangle drawn in three orientations so the hypotenuse is not always horizontal.
Calculating the area or volume of a compound shape by adding the dimensions or using the slant height as the perpendicular height, for example using the slant edge of a cone or pyramid in the volume formula, or counting only the visible faces when finding surface area.
Diagrams show the slant height prominently and the perpendicular height is often not drawn, so the more visible number is substituted. Surface area questions are answered by imagining the object rather than by listing each face or curved surface.
What to do instead
Draw and label the perpendicular height on every 3D diagram, and if only the slant height is given, find the height with Pythagoras first. For surface area, require a written list of faces (base, four triangles, curved surface) with the area of each calculated separately before summing. Keep a formula sheet with a sketch beside each volume and surface-area formula.
Reading values from a cumulative frequency curve or histogram at the wrong axis or scale, for example reading the median from the cumulative frequency axis instead of the variable axis, or treating unequal class widths in a histogram as if each bar's height were the frequency.
Cumulative frequency graphs have the variable on the horizontal axis and the running total on the vertical, which is the reverse of the habit formed with ordinary frequency diagrams. In histograms, frequency density is a new idea and the bar height is read as the count.
What to do instead
Annotate the axes in words before reading anything: 'horizontal = marks, vertical = cumulative frequency'. For the median, start at half the total on the vertical axis, go across to the curve, then down. For histograms, write frequency density = frequency / class width in the margin and, when asked for a frequency, multiply the bar height by the class width.
Comparing distributions by giving the mean and range of one set without a like-for-like comparison, or reporting a probability from a two-way table or tree diagram without adjusting for the without-replacement condition on the second pick.
Candidates describe each data set separately and never write a comparative sentence, so no marks are earned for comparison. In probability, the tree diagram is drawn with the same denominator on both sets of branches because the reduced total after the first pick is not tracked.
What to do instead
For comparisons, require the sentence frame 'the mean of A is greater than the mean of B, but the range of A is greater, so A is more spread out', naming both averages and both spreads. For without-replacement probability, write the remaining totals beside each second-stage branch and check that the probabilities on each pair of branches sum to 1 at every stage.