Secondary 4 Mathematics-Technical and Scientific Option Ministerial Examination - Québec common mistakes
Secondary 4 Mathematics-Technical and Scientific Option Ministerial Examination - Québec · Understand the traps and how to avoid them.
When solving a system of two linear equations, the candidate finds one variable correctly, then substitutes it back into the equation they just rearranged (or into a derived expression) instead of the original equation, producing a wrong second variable.
The rearranged form is the most recent expression on the page, so working memory grabs it. Candidates also believe any true equation in the system will do, forgetting that a derived form can carry an algebra slip forward.
What to do instead
Adopt a fixed ritual: solve for x, then immediately substitute into the untouched original equation, and finally check the ordered pair in the other original equation. Write the check as a visible line of work, and treat a failed check as a signal to redo the substitution, not to adjust the answer.
The candidate reads a graph or table of a quadratic situation and reports the maximum or minimum as the value of x at the vertex, or reports the zeros when asked for the vertex, mixing up the roles of the coordinates.
The vertex is taught as a point, but optimization contexts ask for a quantity (area, profit, height), and candidates latch onto whichever number they computed first. The notation (h, k) encourages treating h as the answer.
What to do instead
Before computing, write a one-line statement of what is being maximized and in what units, e.g. 'maximize area in m^2'. After finding the vertex, force a labelled answer sentence that names the input and the output separately. Practise sorting vertex problems into 'find x', 'find y', and 'find both' before solving.
When simplifying or factoring algebraic expressions, the candidate cancels terms across a sum, for example reducing (x + 6)/6 to x, or (x^2 + 4x)/x to x^2 + 4 by dropping the denominator entirely after cancelling only part of the numerator.
Fraction-reduction rules from arithmetic are over-generalized to algebraic sums. The visual similarity between a common factor and a common term is strong, and the error survives because the result looks tidy.
What to do instead
Require factorization before any cancellation: rewrite numerator and denominator fully factored, then cancel only factors that appear as whole factors. Add a numerical spot-check by substituting a simple value such as x = 2 into the original and the simplified expression; if they differ, the simplification is invalid.
The candidate applies a single transformation to a figure when a composition is requested, or applies the transformations in the written order without accounting for the fact that a composition is read from the figure outward, producing a correctly shaped but wrongly placed image.
Compositions are usually written symbolically, and candidates process them left to right as if they were ordinary operations. Tracking only one vertex mentally rather than all vertices hides the ordering error.
What to do instead
Use a table with one column per vertex and one row per transformation, filling in coordinates step by step. Always start from the pre-image for the first row and from the previous row's coordinates thereafter. Verify with a single point first, then complete the table, and state the final coordinates explicitly.
The candidate confuses the scale factor of a dilation with the ratio of areas or volumes, reporting an area ratio equal to the linear scale factor (or a volume ratio equal to the square of it) in similarity problems.
The linear factor is the only number given, so it becomes the default answer. The squaring and cubing rules are memorized as phrases without being tied to a concrete figure.
What to do instead
Anchor the rule in a unit square: doubling side lengths multiplies area by four. For each similarity problem, write 'k = ..., area factor = k^2, volume factor = k^3' before substituting. When the question asks for a length, confirm the answer is in linear units; if the units come out squared, the wrong factor was used.
In a right-triangle problem, the candidate selects sine, cosine, or tangent by habit (often sine whenever an angle and a side appear) rather than by which sides are actually involved, and sets up the ratio with the known and unknown sides in the wrong positions.
The three ratios are learned as a list, and under time pressure the first one recalled is used. Candidates also label the opposite side relative to the wrong angle, especially when the triangle is rotated.
What to do instead
Follow a three-step routine every time: mark the reference angle, label the two sides that appear in the problem as opposite, adjacent, or hypotenuse, then choose the ratio that contains exactly those two labels. Write the ratio as a word equation (opposite over hypotenuse, etc.) before substituting numbers.
When solving a trigonometric equation or finding an angle, the candidate reports only the calculator's principal value and omits the second solution in the given interval, or uses degree mode when the context is in radians (or the reverse).
The inverse function returns one value, and the calculator display is accepted as complete. Mode errors are silent because the numerical output looks plausible.
What to do instead
Before solving, note the required interval and the unit, and set the calculator mode explicitly as a written step. After obtaining the principal value, use the symmetry of the situation (or the unit circle) to generate the second solution and check it lies in the interval. State the number of expected solutions before solving.
The candidate computes a mean, median, or standard deviation from grouped or summarized data by averaging the stated values without weighting by frequency, or confuses the mean with the median when describing which measure is more representative of a skewed distribution.
Summary tables hide the individual data, so the frequency column is treated as decoration. The mean is the most practised measure and is applied by default even when the distribution is skewed.
What to do instead
Always rewrite grouped data as an explicit weighted calculation (value times frequency, summed, divided by total frequency). When asked which measure is more representative, name the shape (skewed or symmetric) and the effect of extreme values before choosing. Compute both measures when the data are skewed and compare them numerically.
In probability problems involving two events, the candidate adds probabilities for 'or' situations without subtracting the overlap, or multiplies probabilities for successive events without checking whether the first event changes the second (with or without replacement).
The addition and multiplication rules are recalled as slogans, and the conditional structure of drawing without replacement is easy to overlook when the events are described in words rather than as a tree.
What to do instead
Draw a tree diagram or a two-way table before computing, and write on each branch whether the probability is conditional. Apply the general addition rule with the intersection subtracted, and for successive draws state explicitly whether the total count changes. Check that no probability exceeds 1 and that complementary events sum to 1.
The candidate produces a final answer with no units, an unrealistic precision (seven decimals from a measurement), or no validating statement, so a correct calculation is marked down and an incorrect one goes undetected.
Attention is spent on the algebra, and the justification step is treated as optional. Candidates also copy the full calculator display without considering the precision of the given data.
What to do instead
Reserve the last two lines of every solution for the answer with units and a one-sentence validation referring back to the question (does the value fit the context, the diagram, the interval?). Round to the precision supported by the data and say so. Build this into practice so the validation line is written automatically.