ORT Mathematics common mistakes
ORT Mathematics · Understand the traps and how to avoid them.
Adding percentage changes instead of multiplying by factors: treating a 20% rise followed by a 20% fall as returning to the original value, or computing successive discounts by adding the percentages.
Percentages are learned as additive amounts, and the phrase 'increase by' invites addition; the compounding base is invisible.
What to do instead
Rewrite each change as a multiplier (1.20, 0.80), multiply them, and interpret the final multiplier relative to 1; check with a concrete £100 example.
Dividing or multiplying both sides of an inequality by a negative number without reversing the inequality sign, e.g. solving -2x < 6 as x < -3.
Equation-solving routines are overlearned; the sign reversal is an exception that is easy to drop under time pressure.
What to do instead
Either isolate the variable so the coefficient is positive, or after any negative multiplication/division reverse the sign and verify with a test value such as x = 0 or x = -10.
Substituting the diameter for the radius in circle area or circumference, e.g. using A = πd² or C = 2πd.
Diagrams often label the full width, and the formula's r is read quickly as 'the number on the diagram'.
What to do instead
Before substituting, write r = d/2 explicitly and carry units; do a dimension check: area must scale as length², circumference as length.
Treating a diagram as drawn to scale and reading lengths or angles from it when the question says 'not drawn to scale' or gives no scale.
Visual estimates are fast and feel reliable; candidates use the picture as data instead of the stated givens.
What to do instead
Redraw a rough sketch, mark only given values, and derive unknown lengths from stated relationships (similarity, Pythagoras, angle facts); never measure.
Cancelling a term that is added or subtracted rather than factored, e.g. simplifying (x+2)/2 to x+1 or (x²+4)/(x+2) to x+2.
Cancellation is learned with products, and the fraction bar visually invites cancelling matching symbols.
What to do instead
Factor numerator and denominator completely before cancelling; substitute a number such as x = 3 to test whether the simplified expression is equivalent.
Confusing conditional probabilities, reporting P(A|B) when the question asks for P(B|A), or treating P(A and B) as P(A)+P(B).
Natural-language probability questions reverse the conditioning subtly, and the symbols look similar.
What to do instead
Translate the question into a two-way table or a restricted sample space: 'given B' means the denominator is the number of B cases, not the total.
Averaging percentages or rates by simple mean when the groups have different sizes, e.g. averaging 80% of 100 students and 50% of 10 students to get 65%.
The arithmetic mean is the default and the unequal group sizes are easy to overlook.
What to do instead
Use a weighted mean: total successes divided by total population; check that the result lies between the two rates and closer to the larger group's rate.
Rounding intermediate results in a multi-step calculation, e.g. rounding 1/3 to 0.33 and then multiplying by 3 to get 0.99 instead of 1.
Calculator displays and handwritten working encourage early rounding; candidates treat displayed digits as exact.
What to do instead
Keep exact fractions or store full precision in calculator memory until the final answer; only round the final reported value according to the required accuracy.
Producing an algebraic answer without defining variables or units, then giving a number that answers a different question (e.g. solving for time but reporting distance).
Candidates focus on manipulation and skip the modelling setup; the final line is written quickly.
What to do instead
Write 'let t = time in hours' before equations; after solving, state the quantity and units and re-read the question's final sentence to confirm what was asked.
Accepting a mathematically valid but context-invalid solution, such as a negative time, a non-integer number of people, or a probability above 1, without checking constraints.
Algebraic solutions feel complete; domain restrictions and context constraints are not part of the manipulation.
What to do instead
After solving, test the answer against the real-world domain (t ≥ 0, integer counts, 0 ≤ p ≤ 1) and reject or reinterpret invalid roots; state the constraint.