Nara Public High School Entrance Examination - Mathematics common mistakes
Nara Public High School Entrance Examination - Mathematics · Understand the traps and how to avoid them.
When solving a linear equation such as 3(x−2)=5x+4, the candidate distributes only to the first term, writing 3x−2 instead of 3x−6, then carries that error through to the final answer.
The distributive law is learned as a rule about parentheses but is practised mostly on single-term cases, so under exam pressure the candidate applies the multiplication to the visually adjacent term only. Working memory is devoted to the overall solving procedure, leaving no check on the expansion step.
What to do instead
Before solving, write the expansion on its own line and draw an arrow from the outside factor to each term inside the bracket. Then verify by substituting the final answer back into the original equation, not the simplified one. Practise ten expansions with a negative outside factor, since the sign error is the most common variant.
The candidate writes the solution of an inequality such as −2x > 6 as x > −3, forgetting to reverse the inequality sign when dividing by a negative number.
Equation-solving habits are far more rehearsed than inequality rules, and the sign reversal is a small procedural add-on that is easy to drop when attention is on arithmetic. Textbooks state the rule once, so it is not over-learned.
What to do instead
Adopt a fixed routine: when the coefficient of x is negative, circle it and write 'reverse' beside the line before dividing. Always test one value from the claimed solution set in the original inequality, for example x = −4, and confirm it satisfies the statement.
Asked for the range of y for a function such as y = 2x + 1 with −1 ≤ x ≤ 3, the candidate substitutes only the endpoints into y and reports the answer without checking whether the function is increasing or decreasing, giving an incorrect order when the coefficient is negative.
The candidate treats the range as a mechanical endpoint calculation rather than reasoning about how the function behaves across the domain. With a negative slope, the smallest x gives the largest y, and this reversal is not noticed.
What to do instead
Always identify the sign of the slope first and state whether the function increases or decreases over the interval. Then assign the endpoints accordingly and write the range with the smaller value on the left. Sketch a rough graph of the line over the stated x-interval to confirm the ordering.
In a proportion or inverse-proportion word problem, the candidate sets up the relationship backwards, for example writing y = k/x when the situation describes direct proportion, or misplacing the constant when using the formula for inverse proportion.
The Japanese terms 比例 and 反比例 are recognised in isolation, but in a word problem the candidate focuses on extracting numbers and skips the step of deciding which relationship the text describes. The general form y = a/x is recalled without checking whether doubling one quantity doubles or halves the other.
What to do instead
Before writing any equation, state in one sentence how the two quantities change together, for example 'as x doubles, y halves, so this is inverse proportion'. Then write the general form and substitute a known pair to find the constant. Finish by checking a second data pair from the problem against the equation.
When simplifying an expression involving square roots, the candidate writes √(a+b) = √a + √b, or simplifies √12 + √27 as √39 rather than 5√3.
The distributive-looking appearance of the radical invites the same manipulation as multiplication over addition. The candidate has memorised that √(ab) = √a·√b but over-generalises it to addition, and does not test the claim numerically.
What to do instead
Keep a written rule card distinguishing √(ab) = √a√b from the false √(a+b) = √a+√b, and disprove the false one once with numbers such as √(9+16) versus √9+√16. For addition, always reduce each radical to simplest form first, then collect like radicals as with algebraic terms.
In a geometry proof, the candidate lists given facts and the conclusion but omits the reason for each step, or uses a theorem whose conditions have not been established, such as claiming two triangles are congruent by SAS without having shown the included angle is between the two stated sides.
The candidate treats proof as a narrative restatement of what is visibly true in the figure rather than a chain in which every claim needs a stated warrant. The conditions of congruence theorems are memorised as labels, not as requirements to be checked.
What to do instead
Use a two-column format with a reason on every line, and before writing, mark the figure to show which sides and angles are known equal. Explicitly check the order condition for SAS and the correspondence of vertices. Read the finished proof backwards from the conclusion to confirm each step is supported.
When finding an angle in a circle problem, the candidate applies the inscribed angle theorem to an angle whose vertex is not on the circumference, or uses the central angle relationship without identifying which arc each angle subtends.
The theorem is remembered as 'half' or 'double' without the condition that the vertex lies on the circle and that both angles stand on the same arc. The candidate matches numbers to the diagram by appearance rather than by identifying the relevant arc.
What to do instead
For every angle in a circle problem, first label its vertex position and the arc it intercepts. State the theorem with its condition in full before substituting. Where several angles share an arc, mark them with the same symbol on the diagram so the relationship is visible.
In a probability question involving two dice or two draws, the candidate counts outcomes such as 'sum is 7' as one favourable case instead of enumerating the ordered pairs, or treats drawing without replacement as if the first item were returned.
The candidate reasons about the event rather than the sample space, and the sample space of ordered pairs is not written out. Replacement conditions are read quickly and the phrase 元に戻す is missed.
What to do instead
Always write the full sample space in a table for two-stage experiments before counting, and mark favourable cells. Underline the phrase describing replacement in the question and adjust the denominator for the second stage accordingly. Check that the total number of cells equals the product of the stage counts.
When reading a histogram or a cumulative frequency graph, the candidate reads the height of a bar as the frequency without accounting for unequal class widths, or reads a cumulative graph as if it showed the frequency in each class.
Most practice uses equal class widths, so the height equals the frequency and the distinction is never forced. The cumulative graph's y-value is a running total, but the candidate reads it as a per-class count.
What to do instead
Before reading any value, check whether all class widths are equal. If not, compute frequency density as frequency divided by class width. For cumulative graphs, recover the class frequency by subtracting consecutive cumulative values, and label the axes with what they actually measure.
On an integrated problem, the candidate answers only the first sub-question and leaves later parts blank, or restarts the whole calculation for each part instead of carrying forward the result already proved.
Sub-questions are read as independent items, and the candidate does not notice that part two typically uses the expression or result established in part one. Time pressure then makes the remaining parts feel unmanageable.
What to do instead
Read the entire problem before starting and mark which earlier result each later part depends on. Write intermediate results in a boxed summary line so they can be reused directly. If stuck on a later part, still state the result from the earlier part and explain how it would be applied, since method marks are available.