Secondary 4 Mathematics-Cultural, Social and Technical Option Ministerial Examination - Québec common mistakes
Secondary 4 Mathematics-Cultural, Social and Technical Option Ministerial Examination - Québec · Understand the traps and how to avoid them.
When solving an equation or inequality, the student performs an operation on only one side (e.g., adds 5 only to the left of 3x − 5 = 10), or when multiplying/dividing an inequality by a negative number, keeps the inequality sign unchanged (writing −2x > 6 as x > −3).
Procedural rules are memorized as isolated steps rather than as the balance/equivalence principle; the negative-multiplication flip is a special case that is easily dropped under time pressure because it does not arise in most practice equations.
What to do instead
Before each manipulation, write the operation in the margin and apply it explicitly to both members on separate lines; for inequalities, circle the coefficient of x and, whenever it is negative, draw an arrow flipping the sign, then verify by substituting a test value into the original inequality.
Confusing a function's value with its rate of change: reading f(3) = 12 as 'the rate is 12', or computing a slope from a single point, and interpreting the y-intercept of a real-life model as a rate rather than an initial value.
The notation f(x) and the graph's visual features are learned separately from the modelling context, so students map 'big number' to 'rate' without checking units (units of y per unit of x).
What to do instead
For every relation, write the units beside the axes and beside each parameter (e.g., cost in $, time in h, so slope is $/h and intercept is $); compute rate of change only from two points using Δy/Δx and state in words what each parameter means in the situation.
Setting up proportions with mismatched or inverted ratios (e.g., writing 3/5 = x/2 instead of 3/5 = 2/x for an inversely proportional situation), or applying a direct proportion to a situation that is inverse (more workers, fewer days).
Students pattern-match on the word 'proportion' and align numbers by position on the page rather than by the quantities' units and by whether the relationship increases or decreases together.
What to do instead
Before writing any equation, build a table with a units row, decide explicitly 'when one doubles, does the other double (direct) or halve (inverse)?', and only then write the relation (y = kx or y = k/x); solve for k first and check the result against the direction of change.
Applying area and volume formulas with the wrong dimension or failing to convert units before computing (mixing cm and m in one calculation), and confusing perimeter with area or lateral area with total area of a solid.
Formulas are recalled as strings of letters without the associated picture or units, and unit conversion is treated as an afterthought performed at the end rather than as part of the given data.
What to do instead
At the start of each geometry problem, convert all lengths to a single unit and write them in a labelled sketch; state in words what is sought ('length of the boundary' vs 'surface covered' vs 'space occupied') and select the formula from that description, then check the unit of the answer (cm, cm², cm³).
Using the sine, cosine or tangent ratio on a triangle that is not right-angled, or choosing the wrong ratio because the opposite and adjacent sides were identified relative to the wrong angle.
Trigonometric ratios are practised on stereotyped diagrams where the right angle is always at the bottom right, so students do not check the 90° condition or re-identify sides when the triangle is rotated or the reference angle changes.
What to do instead
First mark the right angle and label the hypotenuse; then, at the reference angle, colour the opposite side and the adjacent side before selecting a ratio; if no angle is 90°, switch to the sine or cosine law rather than forcing a ratio.
Reporting the mean of a distribution as representative without checking for outliers or skew, or reading a measure of central tendency off a histogram at the tallest bar (confusing the mode class with the mean).
Central tendency is taught as a set of computations rather than as a choice among measures; students default to the mean and do not connect the shape of the distribution to the appropriate summary.
What to do instead
For each distribution, compute or estimate mean, median and mode, then compare them: if they diverge markedly, state which measure best represents the data and justify it in one sentence referring to outliers or asymmetry; when reading a graph, locate the balance point rather than the peak.
Computing a probability by counting favourable outcomes over all outcomes without verifying that the outcomes are equally likely, or treating dependent draws as independent (e.g., drawing two cards without replacement and using 1/4 × 1/4).
The classical ratio favourable/total is over-generalized, and the phrase 'without replacement' is not translated into a changed denominator for the second event.
What to do instead
Before computing, list the sample space or state whether outcomes are equally likely; for multi-stage experiments, draw a tree diagram and write the denominator at each branch, noting whether the total decreases (without replacement) or stays fixed (with replacement).
Confusing probability with odds (writing odds of 3:1 as a probability of 3, or 3/4), and misinterpreting expected value as the result that will occur rather than a long-run average.
The two notations share the same numbers but different reference totals, and expected value is intuitively read as a prediction of a single trial.
What to do instead
Write the conversion explicitly every time: P = favourable/(favourable + unfavourable) and odds = P/(1 − P); for expected value, state 'over many repetitions, the average gain per trial is …' and compare it to the cost of playing before concluding whether the game is favourable.
Answering the question that was asked in a modelling problem with a number only, without validating it against the context (a negative time, a non-integer number of people, a cost exceeding the budget) or without checking the requested unit and rounding.
School habits reward arriving at a numeric answer; the validation step and the return to the original question are not automatic, and constraints stated in the wording are dropped once computation begins.
What to do instead
Adopt a fixed four-step written routine - identify the unknowns and constraints, build the model, solve, then interpret and validate - and finish with a sentence answering the question with units; explicitly reject any solution violating a constraint (t ≥ 0, whole persons, budget limit) and round according to the situation.
In consumer and financial problems, applying simple interest where interest compounds, computing a percentage increase and decrease by adding and subtracting the same rate (thinking +20% then −20% returns to the original), or omitting taxes and fees from a budget comparison.
Percentage change is treated as an additive quantity rather than a multiplicative factor, and 'interest' is recalled as a single formula without checking whether it is applied to a fixed principal or to an accumulating balance.
What to do instead
Translate every percentage change into a multiplying factor (1 + r or 1 − r) and chain the factors; for interest, write the balance after each period and check whether the interest is computed on the original principal (simple) or on the current balance (compound); in budgeting, list all line items including taxes and one-time fees before comparing options.