Toyama Prefectural Public High School Entrance Examination - Mathematics common mistakes
Toyama Prefectural Public High School Entrance Examination - Mathematics · Understand the traps and how to avoid them.
When simplifying an expression such as 3(x−2)−2(2x−5), the student expands only the first bracket correctly and drops the minus sign into the second, writing 3x−6−4x−10 instead of 3x−6−4x+10.
The student treats the subtraction sign as belonging only to the first term inside the bracket rather than distributing it across every term; this is a fragile rule that breaks under time pressure.
What to do instead
Have the student rewrite each subtraction of a bracket as addition of the negative, e.g. 3(x−2)+(−2)(2x−5), and physically draw arrows from the multiplier to each term before collecting. Practise five such expansions daily for a week with the arrows shown.
When solving 5x−3 = 2x+9 the student moves the 2x to the left but forgets to change its sign, writing 5x+2x = 9+3 and obtaining x = 12/7.
The student memorised 'move it to the other side' as a physical relocation instead of applying the same operation to both sides, so the sign change has no meaning and is easily omitted.
What to do instead
Require the student to write the intermediate line showing the operation applied to both sides (subtract 2x from both sides, add 3 to both sides) for every equation for two weeks, and check the answer by substitution every time.
When solving an inequality such as −2x+1 > 7, the student divides by −2 and writes x > −3, leaving the inequality sign unchanged.
The reversal rule is taught as an exception to memorise rather than a consequence of multiplying by a negative number, so it is forgotten in the middle of a multi-step calculation.
What to do instead
Have the student verify each answer by testing a number (e.g. try x = 0 and x = −5) against the original inequality; also derive the rule once by adding 2x to both sides instead of dividing by a negative, so the reversal is seen to be optional and the logic is visible.
When asked for the range of a function such as y = (x−2)² − 3 on the domain 0 ≤ x ≤ 5, the student answers −3 ≤ y ≤ 6 by substituting only the endpoints x = 0 and x = 5.
The student conflates 'substitute the endpoints' with 'find the maximum and minimum', overlooking that a quadratic has a turning point inside the interval.
What to do instead
Train a fixed three-step routine: complete the square or note the vertex, mark the vertex x-value on the domain, then compare the vertex value with the endpoint values. Do this for both upward and downward parabolas with the vertex inside and outside the interval.
Given a graph of a straight line, the student reads the y-intercept correctly but computes the gradient as Δx/Δy (run over rise) or as (y₂−y₁)/(x₁−x₂), producing the reciprocal or the negative of the correct slope.
The formula is recalled as a jumble of symbols without the underlying idea of 'change in y per unit change in x', so the order of subtraction is arbitrary.
What to do instead
Have the student state the gradient in words first ('for every 1 to the right, how far up?') and read it directly off the grid before writing any formula; then confirm with the formula, always subtracting in the same direction for both coordinates.
In a geometry problem the student uses a length given in centimetres together with one given in metres without converting, for example adding 1.2 m and 35 cm to get 36.2 of some unit.
Students scan for numbers and apply a formula without first listing the units of every quantity; unit conversion is treated as an optional preliminary rather than part of the problem.
What to do instead
Enforce a habit of writing the unit next to every number as it is copied from the question, and converting all lengths to one unit in a single 'unit line' before any calculation begins.
When finding the volume of a composite solid (for example a cone sitting on a cylinder), the student adds the two volumes but uses the slant height of the cone as its perpendicular height, or uses the diameter as the radius in the formula.
The student matches numbers to formula slots by position rather than by meaning, and does not distinguish slant height from perpendicular height in a cone.
What to do instead
Before substituting, have the student label each given length on a quick sketch as r, h or l, and compute the missing perpendicular height with Pythagoras where only the slant height is given. Require the labelled sketch to be handed in with the answer.
When asked for the probability of drawing two red balls without replacement from a bag, the student computes (3/7)×(3/7), reusing the first fraction for the second draw.
The student treats successive draws as independent by default because most textbook examples involve replacement, and does not update the counts after the first event.
What to do instead
Have the student draw a tree diagram and write the actual remaining counts on the second branch (e.g. 2/6 rather than 3/7) before multiplying. Contrast a with-replacement and a without-replacement version of the same problem side by side.
When computing a mean from a frequency table, the student averages the class values (e.g. 10, 20, 30) instead of weighting each by its frequency, or divides the total by the number of classes rather than the total frequency.
The student applies the simple mean procedure learned for raw lists without recognising that a frequency table encodes repeated values.
What to do instead
Have the student expand the table into a full raw list once (writing 10, 10, 20, 20, 20, 30 …) to see where the weights come from, then compute the weighted mean and compare. Always write 'total ÷ total frequency' explicitly.
On a multi-part word problem the student answers only the final question and never states what was found, so an intermediate result such as 'the number of students is 12.5' is left unremarked and the answer is submitted as a valid count.
Students treat the last computed number as the answer without checking it against the real-world constraints of the problem, and skip the habit of reviewing whether the result is sensible.
What to do instead
Require a final 'sanity line': restate the question in one sentence, insert the answer, and check that it is a whole number where a count is required and within the stated range. Build this into every practice problem so it becomes automatic.