GCE O/L Mathematics common mistakes
GCE O/L Mathematics · Understand the traps and how to avoid them.
When adding or subtracting fractions, the candidate adds the numerators and adds the denominators (e.g., 1/2 + 1/3 = 2/5).
The candidate treats fractions as two separate whole numbers and applies the whole-number addition algorithm to the parts. This is often a habit carried from earlier work with integers, reinforced by the visual similarity between the fraction bar and a division sign.
What to do instead
Teach the meaning of a common denominator: the parts must be the same size before they can be combined. Have students draw bar models or use fraction circles to show that 1/2 and 1/3 cannot be added until both are expressed in sixths. Require them to write the conversion step explicitly (1/2 = 3/6, 1/3 = 2/6) before adding, and give mixed practice that includes whole numbers and improper fractions.
When expanding a bracket preceded by a negative sign, the candidate changes the sign of the first term only, e.g., writing 5 − 2(x + 3) = 5 − 2x + 6.
The candidate applies the minus sign only to the term immediately following the bracket and does not treat the bracket as a single quantity multiplied by −2. This is a common overgeneralisation from arithmetic, where a minus sign usually affects just the next number.
What to do instead
Have students rewrite the expression as 5 + (−2)(x + 3) before expanding, so the multiplication by −2 is explicit. Use a substitution check with a simple value such as x = 1 to verify the expanded form. Provide worked examples where the bracket is multiplied by a negative number, and include error-analysis tasks where students identify and correct the sign error.
The candidate solves an equation such as 3x + 4 = 19 by subtracting 4 from the left-hand side only, writing 3x = 19, or by dividing only one side by 3.
The candidate sees the equation as a sequence of operations to undo rather than as a balance. The equals sign is interpreted as 'the answer follows' rather than 'the two sides have equal value', so operations are applied to individual terms instead of to both sides.
What to do instead
Use a balance model or a two-pan diagram for the first lessons on solving equations. Require students to record each operation on both sides explicitly, e.g., 'subtract 4 from both sides' written above the equation. Follow every solution with a substitution check back into the original equation, and give equations where the unknown appears on both sides to reinforce the balance idea.
When finding the area of a triangle, the candidate multiplies base by height and forgets to halve, or uses a slanted side as the height.
The formula is recalled as 'base times height' because that is the rectangle formula, and the factor of 1/2 is dropped. The candidate also confuses the perpendicular height with a side length, often because the diagram shows the triangle in an orientation where the perpendicular height is not drawn.
What to do instead
Have students derive the triangle area formula by drawing a rectangle around a triangle and showing that two congruent triangles fill the rectangle. Insist that the height is identified as the perpendicular distance from the base to the opposite vertex, and practise finding the height when it lies outside the triangle. Use a checklist: identify base, identify perpendicular height, multiply, halve.
When calculating the mean from a frequency table, the candidate adds the frequencies and divides by the number of rows instead of using the total frequency.
The candidate treats each row as a single data value and overlooks that the frequency tells how many times the value occurs. The structure of a frequency table is new, and the candidate does not connect the frequency column to the idea of repeated values.
What to do instead
Have students expand a small frequency table into a raw list of values and calculate the mean from the list, then compare with the frequency-table method. Require them to write the formula as 'sum of (value × frequency) divided by sum of frequencies' and to compute the total frequency explicitly. Give examples where the frequencies are large so that the difference between the two methods is obvious.
When finding the probability of a combined event, the candidate adds the probabilities of the individual outcomes instead of multiplying, e.g., for two heads in two coin tosses giving 1/2 + 1/2 = 1.
The candidate confuses 'or' with 'and' and does not have a model of independent events. The addition rule is overlearned from single-event probability, and the candidate has not seen a sample space diagram that shows why the combined probability is smaller than either individual probability.
What to do instead
Use sample space diagrams and tree diagrams to list all outcomes for two-stage experiments. Have students count favourable outcomes and divide by the total number of outcomes, then compare with the multiplication rule. Give contrasting questions using 'or' and 'and' in the same lesson so that students choose the correct operation deliberately. Include a check that a probability cannot exceed 1.
When solving a ratio problem such as 'divide £60 in the ratio 2:3', the candidate divides £60 by 2 and by 3 separately, giving £30 and £20.
The candidate treats the ratio numbers as fractions of the whole rather than as parts of a total number of parts. The candidate does not identify the total number of parts (2 + 3 = 5) and so does not find the value of one part.
What to do instead
Teach the unitary method explicitly: add the ratio parts to find the total number of parts, divide the amount by that total to find one part, then multiply by each ratio number. Use bar models where the whole is split into equal boxes labelled with the ratio numbers. Check that the sum of the answers equals the original amount.
When using SOHCAHTOA to find a missing side, the candidate substitutes the hypotenuse and the opposite side into the wrong ratio, or uses the adjacent side where the opposite is required.
The candidate labels the sides relative to the angle incorrectly, often because the triangle is not in the standard orientation. The mnemonic is recalled but not connected to the actual sides of the given triangle, so the wrong ratio is selected.
What to do instead
Require students to mark the right angle, mark the given angle, and label the three sides as opposite, adjacent and hypotenuse before choosing a ratio. Practise with triangles in several orientations, including rotated ones. Have students state the ratio in words first ('sine = opposite over hypotenuse') and then substitute, and check that the answer is sensible relative to the hypotenuse.
When converting between units of area or volume, the candidate multiplies or divides by 100 or 1000 instead of by 100² or 100³.
The candidate applies the linear conversion factor to a two- or three-dimensional measure. The squared and cubed notation is treated as decoration rather than as a signal that the conversion factor must also be squared or cubed.
What to do instead
Derive the conversion by drawing a 1 m by 1 m square and showing that it contains 100 × 100 = 10,000 cm². Do the same for a cube to show 1 m³ = 1,000,000 cm³. Give conversion problems in context (paint coverage, tank volume) and require students to write the unit conversion as a calculation, not as a remembered number.
When finding a percentage increase or decrease, the candidate calculates the amount of change and stops, or divides by the new value instead of the original value.
The candidate does not distinguish between the change and the percentage change, and does not identify the original value as the base for the calculation. The word 'increase' is associated with addition only, so the division step is omitted.
What to do instead
Use the formula 'percentage change = change ÷ original × 100' and have students write down the original value before calculating. Give paired questions where the same change is expressed as an increase and as a decrease so that the base is always the starting value. Include reverse problems (finding the original given the percentage change) to reinforce the role of the original amount.