Saitama Public High School Entrance Examination - Mathematics common mistakes
Saitama 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 only the first factor or drops the minus sign on the second term, writing 6x−15−2x−8 instead of 6x−15−2x+8.
The minus sign in front of a bracket is read as belonging to the first term inside rather than to the whole bracket; mental arithmetic skips the intermediate step of writing −2·(−4)=+8, so the sign error is never made visible.
What to do instead
Force a written intermediate line: first rewrite as 3(2x−5)+(−2)(x−4), then distribute to get 6x−15−2x+8, then collect. Train with five sign-critical items per session and check by substituting x=1 into both the original and the simplified form.
The candidate solves 5x−7=2x+8 correctly but then answers a word problem with the wrong quantity, for example giving the value of x when the question asked for a length or a price, or forgetting to state units.
Attention collapses onto the algebra once an equation is obtained; the candidate stops re-reading the question after setting up and treats 'solve the equation' as the goal rather than the means.
What to do instead
Adopt a fixed three-step routine: underline what is asked before starting, write 'let x be ...' explicitly, and after solving write one sentence answering the underlined quantity with units. Practise on problems where the equation's solution is deliberately not the answer.
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 reversal rule is memorised as a sentence rather than as a consequence of order, so it is recalled only when the candidate happens to think of it; under time pressure the mechanical division step overrides it.
What to do instead
Either always isolate the positive coefficient (move −2x to the right to get 1>7+2x, avoiding division by a negative) or verify every answer by testing a number: substitute x=−4 into −2x+1>7 to see it fails. Do ten mixed items where only some require reversal.
Given a linear function through two points, the candidate computes the slope correctly but then misreads the y-intercept from the graph, or confuses the roles of x and y when substituting a point to find the intercept.
The candidate treats the graph as decoration and reads coordinates approximately by eye instead of using the stated points; the meaning of b in y=ax+b as the value at x=0 is not firmly linked to the graph.
What to do instead
Always derive b algebraically by substituting one known point into y=ax+b, and only afterwards check visually that the line crosses the y-axis at that value. Practise converting between graph, table, and equation for the same function.
In a 'rate of change' or 'meeting point' problem involving two linear functions, the candidate finds the intersection coordinates but reports them without interpreting what the intersection means in the situation, or sets up the two expressions with the variables swapped.
The candidate has practised intersection problems only as simultaneous equations, so the geometric meaning (equal values at the same time) is not connected to the context; variable definitions were never written down.
What to do instead
Before solving, write what each function represents and in what unit x is measured; after solving, state 'at x=..., the two quantities are both ...'. Use problems where the intersection is outside the physically meaningful range and must be rejected.
In an angle-chasing problem, the candidate uses a theorem whose conditions are not satisfied - for example applying the inscribed-angle theorem to an angle whose vertex is not on the circle, or assuming a line is tangent because it looks tangent.
Theorems are recalled by their conclusion ('half the central angle') rather than with their hypotheses; diagrams in textbooks are drawn in the special configuration, so the conditions feel automatic.
What to do instead
For each theorem used, write the hypothesis explicitly next to the step, e.g. '∠APB = ½∠AOB because P lies on the circle and A, B, O are fixed'. Redraw the figure in a distorted way to test whether the claimed relation still holds.
When proving triangle congruence, the candidate lists three equal parts in a different order from the triangle names, or uses SSA (two sides and a non-included angle) as a valid condition.
The correspondence between the order of letters in ΔABC and ΔDEF and the order of the stated equal parts is treated as a formality; SSA looks like SAS on a rough diagram.
What to do instead
Write the target congruence first, then list equal parts in exactly that letter order, and name the condition (SSS, SAS, ASA, or the hypotenuse-leg case for right triangles) explicitly. Include one counterexample drawing showing SSA gives two different triangles.
In a proof, the candidate writes the conclusion as a reason ('because AB=AC, therefore the base angles are equal, so the triangle is isosceles') or cites 'definition' where a theorem is needed, producing circular reasoning.
Proof is treated as a narrative of true statements rather than a chain from hypotheses to conclusion; the candidate has not separated 'what is given' from 'what is to be shown'.
What to do instead
Structure every proof as: Given / To prove / Proof, and annotate each line with the reason (definition, hypothesis, or named theorem). Have the candidate mark which lines are hypotheses and which is the goal before writing anything.
When asked for a probability, the candidate counts outcomes incorrectly - treating two identical-looking objects as distinguishable, or using 'number of red balls' over 'number of colours' instead of over the total number of balls.
The sample space is never written out, so the denominator is chosen by intuition; equally-likely outcomes are assumed without checking, which fails when objects are not symmetric.
What to do instead
Always enumerate or tabulate the sample space (a two-way table for two-stage experiments) and count the total before counting favourable cases. Verify that all listed outcomes are equally likely, and check that probabilities of all outcomes sum to 1.
When reading a histogram or relative-frequency table, the candidate compares frequencies across classes of different widths directly, or reads a cumulative-frequency graph as if the vertical axis were a frequency per class.
The candidate equates the height of a bar with the count regardless of class width, and has not met the idea that frequency density is needed when widths differ; cumulative graphs are read point-by-point without understanding the accumulation.
What to do instead
Before answering, write the class width under each bar and compute frequency density where widths differ. For cumulative graphs, read a value as 'number up to this point' and obtain a class frequency by subtraction of consecutive readings. Practise with one table where widths are unequal.