Okayama Public High School Entrance Examination - Mathematics common mistakes
Okayama Public High School Entrance Examination - Mathematics · Understand the traps and how to avoid them.
When simplifying an expression such as 3(2x−5)−2(x−4), the candidate distributes the minus sign only to the first term and writes 6x−15−2x−8 instead of 6x−15−2x+8.
The bracket is treated as a container to be removed rather than as a quantity multiplied by −2; the candidate applies the distributive law mechanically to the first term and then reverts to left-to-right deletion of brackets, a habit carried over from primary-school arithmetic where a minus before a bracket rarely appeared.
What to do instead
Before simplifying, rewrite every subtraction of a bracket explicitly as addition of the negative: 3(2x−5)+(−2)(x−4). Then multiply term by term and check by substituting x=1 into both the original and the simplified expression; the two values must agree. Do ten such items with the intermediate line always shown.
The candidate solves an equation or inequality correctly but then answers the wrong quantity - for example, finding x=4 when the question asked for the value of 2x−3, or giving x<3 when the question asked for the range of 5−2x.
Attention is fixed on the mechanics of solving, and the final sentence of the question is read once at the start and not revisited. In inequality work the candidate also forgets that multiplying or dividing by a negative number reverses the sign, so the endpoint is right but the direction is wrong.
What to do instead
Adopt a two-line habit: underline what is being asked before starting, and after obtaining the solution write a short 'therefore' sentence naming the quantity. For inequalities, circle any negative coefficient that will be divided out and physically flip the inequality sign at that step, then test one value from the answer range in the original inequality.
Given a proportional relationship y=ax², the candidate computes the constant a from one point but then answers a question about the rate of change as if the relationship were linear, using a constant gradient between two points.
The word 'proportional' is associated with the linear case y=ax learned earlier, and the squared variable is treated as a computational detail rather than as a change of shape. The candidate has not internalised that for y=ax² the ratio y/x² is constant while y/x is not.
What to do instead
For every function question, first classify the relationship by testing whether y/x or y/x² is constant across the given table. Then sketch the graph and mark the two points in question, drawing the actual curve between them. Compare the average rate of change over successive equal intervals to see the acceleration explicitly.
In a linear function problem set in a word context, the candidate identifies the gradient and the intercept but assigns them to the wrong roles - for example, treating a fixed monthly fee as the gradient and the per-unit charge as the intercept.
The candidate matches numbers to letters by position in the formula y=ax+b rather than by meaning. Without a physical interpretation of what happens when x=0 and what one extra unit of x adds, the two parameters look interchangeable.
What to do instead
For each context, answer two questions in writing before writing the formula: what is the value when the quantity is zero (that is b), and how much does the total rise when the quantity increases by one (that is a). Then substitute a concrete pair of values back into the context and check the sentence reads sensibly.
When asked to prove a geometrical statement, the candidate writes the conclusion first and then lists angle equalities without stating which lines and which transversal make them equal, or uses 'corresponding angles' for angles that are alternate interior angles.
The figure is read holistically and the equal angles are seen rather than derived. School proof requires naming the pair of parallel lines and the transversal for each equality, and candidates have often memorised the names of angle relationships without the conditions attached to them.
What to do instead
Write each line of the proof in the fixed form: because [lines] are parallel with transversal [line], angle [name] equals angle [name] (reason). Train by covering the figure and reconstructing it from the proof text alone; if the text is ambiguous, the proof is incomplete.
The candidate uses a property of a figure that the question does not give - for example, assuming a quadrilateral is a parallelogram because it looks like one, or assuming a point is the midpoint because it is drawn centrally.
Diagrams in entrance examinations are drawn to scale for clarity, and the visual impression is unconsciously promoted to a given condition. Candidates also confuse the hypothesis and the conclusion of a theorem, using the conclusion as if it were available.
What to do instead
Before writing, list on the page the given conditions and the conditions to be proved, in two columns. Every statement in the proof must cite a given, a definition, or a previously proved theorem. Any property taken from the drawing must be justified or discarded.
In a probability question drawing two items without replacement, the candidate computes the second draw's probability with the original total, or counts ordered and unordered outcomes inconsistently between numerator and denominator.
The two draws are compressed into a single act in the candidate's mental model, so the change in the sample space after the first draw is not registered. Counting inconsistency arises because tree diagrams naturally produce ordered outcomes while the candidate lists unordered pairs.
What to do instead
Always draw a two-stage tree diagram and write the reduced denominator on each branch. State explicitly whether order matters, then apply the same convention to numerator and denominator. Verify by checking that the probabilities of all final branches sum to 1.
The candidate reads a histogram as if bar height were frequency, rather than frequency density, when the class widths are unequal.
In earlier work all histograms had equal class widths, where height and frequency coincide. When widths differ, the area represents frequency, and the candidate has never had to distinguish the two.
What to do instead
Before reading any value, check whether the class widths are equal. If not, write 'frequency = width × height' at the top of the page and compute areas rather than reading heights. Practise by reconstructing a frequency table from a histogram with unequal widths and checking the total equals the stated sample size.
Asked to explain reasoning in words, the candidate writes only the calculation or only the final answer, omitting the justification the question awards marks for - for example, writing 'x=5' when asked why the value must be 5.
The candidate treats explanation items as calculation items and believes a correct number speaks for itself. Explanation questions in this examination carry marks for the reasoning, and candidates have had little practice in producing written mathematical argument under time pressure.
What to do instead
Use a three-part template for every explanation item: state what is known, state the principle used, state the conclusion. Write at least one sentence per part. Practise on past-style items by writing the explanation first and the calculation second, so the reasoning is not crowded out.
The candidate solves a problem correctly but does not check whether the answer is admissible in the context - giving a negative length, a non-integer number of people, or a probability greater than 1.
The algebraic solution is treated as complete once the equation is solved, and the domain of the variable in the original context is never re-examined. Where a quadratic yields two roots, the candidate reports both without testing.
What to do instead
End every contextual problem with a one-line admissibility check: substitute the answer into the context sentence and confirm it is physically possible. When a quadratic gives two roots, explicitly reject the inadmissible one with a written reason, since the rejection itself often carries a mark.