Fukushima Entrance Mathematics common mistakes
Fukushima Entrance Mathematics · Understand the traps and how to avoid them.
When simplifying an expression such as 3(x−2)−2(2x−5), the candidate distributes only the first term or drops the minus sign into the second bracket, writing 3x−6−4x−10 and then −x−16 instead of −x+4.
The candidate treats the bracket as a container to be removed rather than applying the distributive law to every term, and the double negative created by −2×(−5) is processed too quickly to be checked. Under time pressure the minus sign in front of a bracket is the most frequently lost symbol.
What to do instead
Write the distribution as a separate line before collecting: 3x−6−4x+10, explicitly computing −2×(−5)=+10. Then collect like terms on a third line. Adopt the habit of drawing an arrow from the factor to each term inside the bracket, and finish by substituting x=1 into both the original and simplified expressions to confirm they agree.
The candidate solves an inequality such as −3x+1>7 and reports x>−2, forgetting that dividing both sides by a negative number reverses the inequality sign.
The procedure for equations is over-learned and transfers automatically to inequalities; the reversal rule is memorised as a sentence rather than connected to the number line. Because most practice inequalities have positive coefficients, the negative-division case is rarely rehearsed.
What to do instead
Before dividing, say explicitly 'I am dividing by a negative, so the sign flips' and write the reversal on its own line. Verify by testing a value: if x>−2 were correct then x=0 should satisfy −3x+1>7, but 1>7 is false, so the direction must be x<−2. Always test one point from the proposed solution set in the original inequality.
In a word problem the candidate sets up an equation using the wrong quantity as the unknown, for example letting x be the number of 50-yen stamps when the question asks for the number of 80-yen stamps, and then answers with x without converting back.
The candidate focuses on making an equation that can be solved rather than on defining the variable in terms of what the question asks. The final line of working is treated as the answer, so the relationship between the variable and the requested quantity is never re-examined.
What to do instead
Begin every word problem by writing 'Let x = ...' in a full sentence that names the requested quantity, and if a different variable is more convenient, label it clearly and write the conversion at the end. After solving, reread the question sentence and check that the value reported answers exactly that sentence, including units.
Given a quadratic function the candidate reads the vertex from the completed-square form y=(x−2)²+3 as (−2, 3), or reads the axis of symmetry of y=ax²+bx+c as x=b/2a rather than x=−b/2a.
The sign inside the bracket is confused with the sign of the coordinate because the transformation x−2 shifts the graph right, which feels like a negative movement. The formula for the axis of symmetry is recalled as a pattern of letters without the sign structure.
What to do instead
Derive the vertex once by completing the square from y=x²−4x+7 and see that it becomes (x−2)²+3, so the minimum occurs where the squared term is zero, at x=2. Anchor the rule as 'the bracket shows the opposite sign to the coordinate'. For the general form, remember that the axis is the midpoint of the two roots, x=(α+β)/2=−b/2a, and check with a quick sketch.
When finding the intersection of a line and a parabola the candidate solves the quadratic, obtains two x-values, and stops, or pairs the x-values with the wrong y-values by substituting into the wrong equation.
The candidate treats the algebra as the goal and forgets that an intersection is a point, so a y-coordinate is required. Substituting into the line is easier arithmetic, but if the line was misremembered the pairing silently breaks.
What to do instead
Write the answer as coordinate pairs from the start, leaving blanks for both coordinates. Substitute each x into the simpler equation (usually the line) and immediately write the pair next to the x-value before moving to the next root. As a check, substitute the pair into the parabola as well; if it fails, the pairing or the line equation is wrong.
In a circle problem the candidate applies the inscribed-angle theorem to an angle whose vertex is not on the circumference, or claims that the angle at the centre equals the inscribed angle standing on the same arc instead of being twice it.
The diagrams in textbook figures usually place the centre conveniently, so the candidate learns the picture rather than the condition 'vertex on the circle, both rays cutting the circle'. The factor of two is remembered as a vague relationship rather than a direction.
What to do instead
Before using any circle theorem, state the condition aloud: where is the vertex, and which arc do both angles subtend? Write the relation with the factor explicit, central angle = 2 × inscribed angle on the same arc, and test it on a semicircle where the inscribed angle must be 90°. If the vertex is inside the circle, use the intersecting-chords relation instead.
When asked to prove that a quadrilateral is a parallelogram, the candidate writes 'the opposite sides are equal, therefore it is a parallelogram' without first establishing that the figure is a quadrilateral, or uses the conclusion as a premise in the middle of the proof.
The candidate lists properties of parallelograms instead of selecting one sufficient condition and proving it. Because the properties are all true of the target figure, they feel like valid reasons, and circular reasoning is hard to notice when the statements are individually correct.
What to do instead
Write the proof backwards: state the condition to be established (for example, 'one pair of opposite sides is both equal and parallel') and list exactly what must be shown to use it. Check that every reason cites a definition or a previously proved result, and scan the proof for any line that assumes the figure already has the property being proved.
When comparing two data sets the candidate concludes that the set with the larger mean also has the larger spread, or reads the median from a histogram by taking the middle bar rather than locating the position within the cumulative frequency.
Mean and spread are both summarised as 'the average size' in everyday language, so the two ideas merge. Histograms show bar heights, and the candidate reads the tallest bar as the centre because it is visually dominant.
What to do instead
Keep the two questions separate: centre is answered by mean or median, spread by range, interquartile range, or variance. For the median from a histogram, build a cumulative frequency column and find the class containing the middle position before interpolating. When comparing, write one sentence for centre and a separate sentence for spread, each citing a number.
In a probability problem with two draws the candidate computes 2/5 × 2/5 for a without-replacement situation, or adds probabilities for events that can occur together without subtracting the overlap.
The multiplication rule is learned as 'multiply the fractions' without the condition that the second denominator changes when the first item is not replaced. For combined events, 'or' is mapped to addition and 'and' to multiplication without checking whether the events are mutually exclusive.
What to do instead
Draw a tree diagram and write the denominator on each branch explicitly, so the second branch shows 4 remaining items rather than 5. For 'or' questions, list the outcomes in the sample space and count them, or use P(A∪B)=P(A)+P(B)−P(A∩B), checking whether the overlap is empty before dropping the last term.
On an integrated problem the candidate writes a formula immediately after reading the question, substitutes the given numbers, and produces an answer to a question that was not asked, without checking whether the result is physically or geometrically possible.
Integrated problems combine several domains, and the candidate reduces them to the most recently practised technique. The reading stage is skipped because it feels unproductive, and no plausibility check is made at the end.
What to do instead
Spend the first minute restating the question in your own words and listing the quantities given and required, with units. After obtaining a numerical answer, run three checks: units, order of magnitude, and whether the value lies in the permitted range (a length cannot be negative, a probability cannot exceed 1). If a check fails, return to the setup rather than adjusting the arithmetic.