NCEA Level 2 Mathematics and Statistics common mistakes
NCEA Level 2 Mathematics and Statistics · Understand the traps and how to avoid them.
When rearranging a linear equation such as 5x - 3 = 2x + 9, the candidate changes the sign of only one term when moving it across the equals sign, or divides before collecting all x-terms, so the final solution does not satisfy the original equation.
They treat rearrangement as moving symbols around rather than applying the same inverse operation to both sides, and they skip the substitution check.
What to do instead
Require one operation per line written on both sides, collect x-terms before dividing, and always substitute the answer back into the original equation to verify it.
The candidate expands (2x + 3)^2 as 4x^2 + 9, or cancels terms across a sum in a rational expression such as (x + 2)/(x + 5) by removing x.
They overgeneralise the distributive law and treat addition as if it factors, often because they memorise patterns without checking with numbers.
What to do instead
Expand squared binomials by writing both brackets and multiplying every term, test with a simple number such as x = 1, and factor before cancelling only common factors in a product.
The candidate applies index or log laws incorrectly, for example writing (2x)^3 = 2x^3, treating x^0 as 0, or claiming log(a + b) = log a + log b.
They remember law names but not the conditions, especially that powers apply to the whole bracket and logs turn products into sums, not sums into sums.
What to do instead
Derive each law from its definition, use brackets to show what is being powered, and verify with numerical examples such as log(10 + 10) versus log 10 + log 10.
In a triangle problem the candidate uses the sine rule with a side and angle that are not opposite each other, or uses the cosine rule when a right-angled triangle method is appropriate, producing an impossible or inconsistent angle.
They match the formula to the diagram by appearance rather than by labelling which side is opposite which angle, and they do not check whether the triangle is right-angled.
What to do instead
Label every side with the opposite angle before choosing a rule, write the knowns and unknowns in a table, and check the angle sum and whether the sine rule ambiguous case applies.
The candidate states that a perpendicular gradient is the negative of the original gradient rather than the negative reciprocal, or swaps the distance and midpoint formulas when finding a length or midpoint.
They remember that perpendicular involves a sign change but forget the reciprocal, and the two coordinate formulas look similar when recalled under time pressure.
What to do instead
Write the perpendicular condition as m1 x m2 = -1 and check the product, and derive or refer to the formula sheet for distance and midpoint before substituting coordinates.
The candidate mixes centimetres and metres in a measurement calculation, uses a diameter as a radius, or rounds an intermediate result so the final area or volume is outside the acceptable range.
They focus on the formula and not on unit consistency, and they round early because they treat intermediate numbers as final answers.
What to do instead
Convert all lengths to one unit before substituting, label radius and diameter explicitly on a diagram, and keep exact values or enough decimal places until the final rounding step.
For two independent events the candidate adds the probabilities instead of multiplying them, or calculates P(A and B) when the question asks for P(A or B).
They conflate 'and' with 'or' and rely on everyday language rather than defining the events and checking whether they are independent.
What to do instead
Define A and B in words, draw a tree diagram or two-way table, use multiplication along branches for 'and' and addition across mutually exclusive branches for 'or', then check the total probability is 1.
In a two-way table the candidate reads a row total when the condition requires a column total, or calculates P(A | B) as P(A and B) without dividing by P(B).
They do not identify the given condition as the denominator, and they treat all probabilities as out of the grand total.
What to do instead
Underline the condition after the word 'given', restrict the sample space to that row or column, and write P(A | B) = P(A and B) / P(B) before substituting counts.
The candidate concludes that one variable causes another because the data are correlated, or generalises a sample result to the whole population without discussing sample size, sampling method, or variability.
They interpret association as causation and treat a single sample estimate as exact rather than variable.
What to do instead
Separate association from causation in the conclusion, name the population and sample, discuss whether the sample was random and large enough, and refer to sampling variability or margin of error.
When designing a questionnaire or experiment the candidate writes a leading question, uses a biased sampling method, or fails to control confounding variables, then analyses the results as if they were trustworthy.
They focus on the calculation stage and overlook how the data were produced, so bias and confounding are not treated as part of the statistical problem.
What to do instead
Use neutral question wording, identify the population and a random sampling method, include a control or comparison group where possible, and explicitly name likely sources of bias or confounding in the evaluation.