Yamaguchi Prefectural Public High School Entrance Examination - Mathematics common mistakes
Yamaguchi Prefectural Public High School Entrance Examination - Mathematics · Understand the traps and how to avoid them.
When simplifying an algebraic expression such as 3(2x−5)−2(x−4), the candidate distributes only the first term or mishandles the negative sign before the parenthesis, writing 6x−15−2x−8 instead of 6x−15−2x+8.
The minus sign in front of a parenthesis is treated as applying only to the first term inside, or the distributive law is applied mechanically without tracking each sign. This is especially common when the expression mixes numbers and letters and the candidate is rushing to reach a numeric answer.
What to do instead
Before simplifying, rewrite every subtraction of a parenthesised group as addition of the opposite: 3(2x−5)+(−2)(x−4). Then draw an arrow from the outside factor to each term inside. After simplifying, substitute x=1 into both the original and the simplified expression and check they agree.
When solving a linear equation with fractions, e.g. (x−1)/2 − (2x+1)/3 = 1, the candidate multiplies only the fraction terms by the common denominator and leaves the constant 1 unchanged, or multiplies the numerator but not every term.
The candidate focuses on 'clearing the fractions' and applies the multiplication to the visible fraction terms only, forgetting that the whole equation, including the standalone constant, must be multiplied by the least common multiple. Weak fluency with equivalent equations compounds this.
What to do instead
Write the LCM explicitly above the equation, then place a bracket around the entire left side and the entire right side before multiplying: 3(x−1) − 2(2x+1) = 6. Check by substituting the solution back into the original fractional equation, not the cleared one.
When a question asks for the range of values of a parameter (e.g. 'for what values of a does the equation have two distinct real solutions?'), the candidate finds only the boundary value and writes a single number or an inequality with the wrong direction, omitting the condition on the leading coefficient.
The candidate equates 'two solutions' with 'discriminant equals zero' or stops once a boundary is found, and does not separately check that the coefficient of x² is non-zero. Inequality direction is often decided by intuition rather than by testing a value.
What to do instead
Set up three explicit conditions before solving: leading coefficient ≠ 0, discriminant > 0 (or ≥ 0, = 0 as required), and any domain restriction. Solve each, then test one value inside and one outside the proposed range in the original equation. Write the final answer as a combined inequality.
In a proportional or inverse-proportional problem, the candidate writes y = ax for an inverse relation (or y = a/x for a direct one) because the words 比例 and 反比例 were not distinguished, or because the table looked linear at a glance.
The two relationship types are memorised as formulas rather than as behaviours: in direct proportion the ratio y/x is constant, in inverse proportion the product xy is constant. Under time pressure the candidate pattern-matches on the shape of the table instead of checking which quantity is constant.
What to do instead
For every table, compute y/x for two rows and xy for two rows before choosing a model. State in writing 'the product is constant, so y is inversely proportional to x' and only then write y = a/x. Confirm the constant a with a third row.
When finding the equation of a line parallel or perpendicular to a given line, the candidate uses the same slope for a perpendicular line, or changes the intercept instead of the slope.
The candidate remembers 'parallel means same slope' but does not have a reliable rule for perpendicular slopes, and confuses the roles of slope and intercept when only one point is given.
What to do instead
Write the given line in y = mx + b form first. For parallel use slope m; for perpendicular use slope −1/m. Then substitute the given point to find b. Always verify by checking that the new line passes through the point and that the product of the two slopes is −1.
In a circle-geometry problem, the candidate applies the inscribed-angle theorem to an angle whose vertex is not on the circumference, or uses the central angle as if it were half the inscribed angle.
The theorem is recalled as a phrase ('inscribed angle is half the central angle') without the precondition that both angles subtend the same arc and that the vertex lies on the circle. Diagrams with several chords make it easy to pick the wrong pair of angles.
What to do instead
Before using any circle theorem, mark the vertex of each angle and the arc it subtends. Write the condition next to the calculation: '∠APB and ∠AQB subtend the same arc AB, so they are equal.' If the vertex is the centre, use the central-angle relation; if it is outside the circle, use the external-angle relation instead.
When writing a proof of triangle congruence or similarity, the candidate lists three equal parts but does not state which correspondence is intended, or uses 'SSA' as a valid congruence condition.
The congruence conditions are memorised as abbreviations without the requirement that the equal parts correspond in a named order. The candidate also copies the diagram's appearance instead of justifying each equality from given information.
What to do instead
Begin the proof by naming the correspondence explicitly, e.g. '△ABC ≡ △DEF'. For each of the three equalities, write the reason in parentheses (given, common side, vertical angles, etc.). Never use SSA; if two sides and a non-included angle are known, look for a different pair of triangles or use the sine rule in a measurement context.
In a solid-geometry or measurement problem, the candidate mixes units (cm and m) or uses the diameter as the radius when computing the volume of a sphere or the area of a circle.
The formula is recalled correctly but the input values are taken directly from the diagram without checking whether the labelled length is a radius or a diameter, and unit conversion is postponed until the end, where it is forgotten.
What to do instead
At the start of the calculation, write every given length with its unit and convert all to the same unit. Label the radius r explicitly and, if a diameter is given, write r = d/2 as a separate line. Keep units in the working and check that the final unit is cubic for volume and square for area.
When asked to explain reasoning in words (記述), the candidate writes only the final formula or number, or writes 'because it is obvious from the figure' without naming the property used.
The candidate treats the explanation as a place to show the answer rather than the argument, and lacks practice in connecting a claim to a named theorem or definition. Time pressure encourages skipping the justification.
What to do instead
Use a three-part template: (1) state what is given or known, (2) name the property or theorem used, (3) state the conclusion. For example: 'In △ABC, AB = AC (given), so the base angles are equal (isosceles triangle property); therefore ∠B = ∠C.' Practise rewriting each calculation as a sentence with a 'because' clause.
In probability and data questions, the candidate counts outcomes without listing them, treats non-equally-likely outcomes as equally likely, or reads a histogram frequency as a raw count when the vertical axis is density.
Probability is approached by intuition ('there are two possibilities, so one half') rather than by enumerating the sample space. In data handling, the candidate does not check the axis label or class width before reading values.
What to do instead
For probability, write out the sample space in a table or tree diagram and confirm each outcome is equally likely before dividing. For histograms, check the vertical-axis label and multiply height by class width when the axis is frequency density. State the total number of outcomes explicitly before computing the ratio.