Elementary 6 Mathematics Ministerial Examination - Québec common mistakes
Elementary 6 Mathematics Ministerial Examination - Québec · Understand the traps and how to avoid them.
When converting between units of length, area and volume, the student applies the same ×10 or ×100 factor to all three, for example writing 1 m² = 100 cm² or 1 m³ = 1 000 cm³ instead of 10 000 cm² and 1 000 000 cm³.
The student memorises a single ladder of prefixes (km, m, cm, mm) and transfers the ×10 step of linear measure directly to squared and cubed units without squaring or cubing the factor. The notation m² and m³ is read as a label rather than as a repeated multiplication, so the exponent is never used in the conversion.
What to do instead
Rebuild the conversion from the definition each time: write 1 m = 100 cm, then 1 m² = 100 cm × 100 cm = 10 000 cm², and 1 m³ = 100 cm × 100 cm × 100 cm = 1 000 000 cm³. Have the student draw a 1 m by 1 m square subdivided into 100 rows of 100 squares to see the 10 000 count, and a 1 m cube built from 100 layers of 10 000 small cubes. Drill with a two-column table (linear factor, area factor, volume factor) filled in from scratch before any exercise.
The student adds or subtracts fractions by operating on numerators and denominators separately, producing results such as 1/2 + 1/3 = 2/5, and cannot explain why the answer should be close to 1.
The student treats the fraction bar as a separator between two independent whole numbers rather than as a single number (a part of a whole). Whole-number addition has been over-learned, and no mental estimate is made, so an answer smaller than either addend is accepted without suspicion.
What to do instead
Anchor every operation in a concrete model: two paper strips, one folded into halves and one into thirds, laid end to end on a common strip folded into sixths, so the student sees 3/6 + 2/6 = 5/6. Establish the habit of estimating first (both fractions are near 1/2, so the sum must be near 1) and rejecting any result outside that range. Then formalise the common-denominator procedure as a shortcut for what the strips already showed.
The student confuses the place value of digits after the decimal point, reading 0.7 as smaller than 0.25 because 25 is larger than 7, or writing 0.5 as 0.05 when a zero is added during a calculation.
The student applies whole-number comparison rules (longer numeral = larger) to decimals and does not treat tenths, hundredths and thousandths as positions with decreasing value. The decimal point is seen as a full stop rather than as a marker separating whole units from fractions of a unit.
What to do instead
Use a place-value chart with labelled columns (unités, dixièmes, centièmes, millièmes) and place both numbers in it, then compare column by column from left to right. Support with a hundredths grid or a number line from 0 to 1 where 0.7 and 0.25 are plotted and their order becomes visible. Insist on writing 0.70 beside 0.7 to show that adding a zero on the right changes nothing.
In percentage problems the student cannot decide which quantity is the whole, for example computing 15 % of a price and then subtracting the result from the percentage instead of from the original price, or answering that a 20 % discount followed by a 20 % increase returns to the starting price.
The percentage is treated as an operation to perform rather than as a ratio describing a part relative to a base. The student has memorised 'multiply and subtract' without identifying the base of each percentage, and does not realise that the second percentage in a chain is taken on a different whole.
What to do instead
Force a written statement of the base before any calculation: 'the whole is the original price of 80 $; 15 % of 80 $ is 12 $; the new price is 80 $ − 12 $'. For chained changes, compute each step explicitly with its own base (80 $ → 64 $ → 76.80 $) and compare with the starting value. Link percentages to fractions of 100 and to the 0.15 multiplier so the student sees the same relationship in three forms.
When computing the area or perimeter of a composite figure, the student adds all the side lengths to find the area, or multiplies length by width for a triangle or a trapezoid, and does not decompose the figure into known shapes.
Perimeter and area are both taught with formulas that look similar, and the student has not consolidated the distinction between a boundary measured in linear units and a surface measured in square units. Composite figures are attacked as a single unknown shape rather than broken into rectangles and triangles.
What to do instead
Require the unit to be written with every answer and ask the student to say what is being measured: 'I am covering the surface, so square units' versus 'I am walking around the edge, so linear units'. Train decomposition explicitly: draw the cutting line, label each simple shape, compute each area, then add or subtract. Include a counter-example where two figures have the same perimeter but different areas, drawn on grid paper.
The student computes the volume of a prism by adding the three dimensions or by using the area formula, and mixes up the net, the volume and the total surface area of a solid such as a cube or a rectangular prism.
The three quantities are taught close together and all involve the same three numbers, so the student selects a formula by guesswork. The meaning of a cubic unit (a small cube filling the solid) has not been built, so volume is not distinguished from the surface that wraps the solid.
What to do instead
Build the solid from unit cubes on the desk or in a drawing: fill the base layer, count the cubes in one layer, then multiply by the number of layers, which produces length × width × height from the counting itself. Contrast with the net, which is unfolded and measured in square units. Give mixed questions where the student must first decide 'am I filling it or wrapping it?' and justify the choice in one sentence.
The student measures or draws an angle with a protractor by reading the wrong scale, placing the centre of the protractor away from the vertex, or aligning the baseline with one side but reading the outer scale on an acute angle, and reports 150° for an angle of 30°.
The protractor carries two numbered scales and the student has no strategy for choosing between them. The physical alignment steps (centre on vertex, zero line on one side) are skipped under time pressure, and the student has no expectation about whether the angle is acute or obtuse before measuring.
What to do instead
Impose a fixed procedure: estimate the angle type first (acute < 90°, right = 90°, obtuse > 90°), then place the centre mark exactly on the vertex, align the zero line with one side, and choose the scale whose zero touches that side. Check the reading against the estimate and reject readings that contradict it. Practise on angles drawn in both orientations, including one side vertical, so the alignment is not tied to a single picture.
When reading a bar graph, pictograph or circle graph, the student reads the height of a bar rather than its value when the axis has a scale of 2, 5 or 10, forgets that one symbol in a pictograph may represent several units, or compares sectors of two circle graphs with different totals.
The student reads graphs as pictures rather than as representations with a stated scale and legend. The scale, the legend and the total are treated as decoration, and the same visual height is assumed to mean the same quantity regardless of the axis or the whole.
What to do instead
Before answering any question, have the student write down three things: the scale of each axis, what one symbol or one degree represents, and the total. Then read values by counting intervals, not bar tops. For circle graphs, compute the number represented by a sector (percentage of the total) before comparing two graphs. Include a deliberately misleading pair of graphs with different scales to make the point concrete.
The student states a probability as a comparison of raw counts ('there are 3 red and 5 blue, so red is more likely') or gives a probability as a count rather than as a fraction or percentage between 0 and 1, and does not recognise that a probability cannot exceed 1.
Probability is confused with counting and with data interpretation. The student has not built the idea of favourable outcomes over all equally likely outcomes, and the range 0 to 1 has not been connected to the impossible-to-certain scale.
What to do instead
Have the student list the complete set of equally likely outcomes (the sample space) before computing, then write the probability as a fraction of that total: 3 red out of 8 gives 3/8, and blue gives 5/8. Place results on a 0-to-1 line labelled impossible, equally likely, certain. Verify with a small experiment (drawing from a bag with replacement, 40 trials) and compare the observed frequency with the predicted fraction.
In a multi-step problem the student performs one operation and stops, leaves the answer without a unit or a sentence, does not check whether the result is plausible, and cannot explain the steps taken when asked to justify the approach.
The student reads the problem once, grabs the numbers that appear, and applies the most recent operation learned. There is no planning stage and no validation stage, and written communication of the reasoning has never been modelled or practised as a required part of the answer.
What to do instead
Impose a four-step written routine on every problem: identify what is asked and the given data, plan the steps in order, carry out the calculation, then verify (estimate, inverse operation, or reread the question to confirm the answer matches what was asked). Require a final sentence with the unit. Regularly give problems with a superfluous number or missing data so that selecting information becomes an explicit skill, and ask the student to explain a solution aloud to a partner before writing it.