Nagasaki High School Entrance Exam - Mathematics common mistakes
Nagasaki High School Entrance Exam - 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 inside the second parenthesis, producing 6x−15−2x−4 and then 4x−19 instead of 4x−7.
The candidate treats the subtraction sign as belonging to the 2 only, not to the whole bracket. This is a habit left over from arithmetic, where a minus before a number rarely precedes a bracket, so the rule that every term inside must change sign has never been made automatic.
What to do instead
Before simplifying, rewrite the expression with the subtraction converted to adding the negative of the whole bracket: 3(2x−5)+(−2)(x−4). Then multiply term by term. Practise ten mixed expressions per session and check by substituting x=1 into both the original and the simplified form, which must give the same value.
The candidate solves 5x−3=2x+9 by moving terms but forgets to change signs, writing 5x+2x=9−3 and getting x=6/7, or moving a term across the equals sign without reversing its sign.
The candidate memorised 'move it to the other side' as a physical action rather than as adding or subtracting the same quantity on both sides. Without the underlying operation, the sign change feels optional and is dropped under time pressure.
What to do instead
Write the intermediate line explicitly each time: subtract 2x from both sides, then add 3 to both sides, showing the operation on both sides before simplifying. Only after several fully written solutions begin skipping lines. Verify the answer by substitution into the original equation.
When solving an inequality such as −2x+1>7, the candidate divides by −2 and writes x>−3, failing to reverse the inequality sign.
The candidate applies the same transformation rules as for equations, where the direction of the relation never changes. The reversal rule is easy to state but is forgotten precisely when attention is on the arithmetic of negative numbers.
What to do instead
Adopt a fixed routine: after any multiplication or division by a negative number, pause and physically rewrite the inequality symbol reversed before continuing. Cross-check by testing a value, for example x=−4 in the original inequality, to confirm which side of the boundary satisfies it.
Given a proportional relation y=ax², the candidate computes the constant a from one point but then substitutes the y-value instead of the x-value, or confuses the rate of change of a linear function with the constant of a quadratic one, giving answers off by a square factor.
The candidate carries over the linear model y=ax, where the constant is found by dividing y by x. In the quadratic model the constant requires dividing by x², and the two procedures look superficially identical.
What to do instead
Always write the general form first, substitute the known pair, and solve for the constant explicitly: 12=a·3², so a=12/9=4/3. Then restate the completed equation y=(4/3)x² before answering anything else, and substitute the new x to get y. Keep linear and quadratic worked examples side by side to contrast the two constants.
When asked for the coordinates of the intersection of a line and a parabola, the candidate reads the crossing point off a sketched graph and gives approximate values, or solves only for x and reports x alone as the answer.
The candidate stops when the value 'looks right' on the sketch and does not treat the algebraic solution as the authoritative answer. Also, coordinates are a pair, and the habit of reporting a single number from equation solving carries over.
What to do instead
Set the two expressions equal, solve the resulting quadratic fully, and for each x substitute back to obtain the matching y. Present the answer as a pair of coordinates. Use the sketch only to check plausibility, never to produce the value, and state explicitly that the graph is a check.
In an angle-chasing problem involving parallel lines and a transversal, the candidate labels alternate interior angles as equal when the lines are not stated to be parallel, or uses corresponding angles without citing the parallel condition.
The candidate recognises the visual pattern of parallel-line angle pairs and applies it whenever the diagram looks parallel. Diagrams in this exam are schematic, and the parallel condition must come from the text or from a marked equal angle.
What to do instead
Before writing any equality of angles, write down the reason in words: 'because l is parallel to m, alternate interior angles are equal'. If no parallel condition is given, look for another route such as the exterior angle of a triangle or the angle sum. Train by annotating each equal angle in the working with its justification.
When proving two triangles congruent, the candidate lists three equal parts in an order that does not match a valid condition, for example two sides and a non-included angle, and then concludes congruence.
The candidate remembers 'three conditions' but not that the angle must be included between the two sides for the side-angle-side condition. The non-included case is visually convincing in many diagrams, so it survives self-checking.
What to do instead
Write the congruence statement with vertices in corresponding order first, then check the three pairs against one named condition: three sides, two sides and the included angle, two angles and the included side, or the right-triangle hypotenuse-leg case. Explicitly mark which angle lies between which two sides in the diagram before concluding.
In a probability question drawing two items without replacement, the candidate uses the same denominator for the second draw as for the first, computing 2/5 × 2/5 instead of 2/5 × 1/4.
The candidate treats each draw as an independent event because the two-stage structure looks like the coin and dice problems practised earlier. The phrase 'without replacement' is read but not converted into a change in the total count.
What to do instead
Before calculating, write the total number of remaining items after the first draw next to the second-stage fraction. Draw a two-branch tree diagram and label each branch with its denominator. Contrast a with-replacement and a without-replacement version of the same question in one sitting so the difference in denominators becomes salient.
When asked to compare two data sets, the candidate reports only the mean and concludes one set is larger or more consistent, ignoring the range or the spread of the distributions.
The candidate equates 'average' with 'typical value' and has not internalised that consistency is a statement about dispersion. Textbooks often present mean first, so it becomes the default summary.
What to do instead
For every comparison question, compute and state at least two measures: a representative value (mean or median) and a measure of spread (range, or the spread of the box in a box plot). Then write a sentence that names both, for example 'the means are equal but the range of B is smaller, so B is more consistent'.
When asked to explain reasoning in words, the candidate writes only the final numerical answer, or writes 'because it is' without naming the property, theorem, or condition used.
The candidate treats explanation questions as calculation questions with extra space, since most prior practice rewarded only the answer. Naming a reason requires a separate vocabulary that has not been rehearsed in writing.
What to do instead
Use a fixed sentence frame for every explanation: state what is known, name the property or theorem, then state the conclusion. For example: 'In triangle ABC, the two base angles are equal, so by the converse of the isosceles triangle theorem the two sides are equal.' Practise converting each solved problem into one such sentence.