Miyazaki Entrance Mathematics common mistakes
Miyazaki Entrance Mathematics · Understand the traps and how to avoid them.
When adding or subtracting algebraic expressions such as 3x−2−(x−5), the candidate drops the parentheses and writes 3x−2−x−5, giving 2x−7 instead of the correct 2x+3.
The candidate treats the minus sign as belonging only to the first term inside the parentheses, so the sign of the second term is never flipped. This is a carry-over from arithmetic, where a sign in front of a number usually affects just that number, and it is reinforced by rushing to combine like terms visually.
What to do instead
Before combining anything, rewrite the subtraction as addition of the opposite: 3x−2+(−x+5). Then draw a small arc from the minus sign to every term inside the parentheses. Practise five such expansions daily for a week, and after each one substitute x=1 into both the original and the answer to confirm they match.
The candidate solves an inequality such as −2x+1>7 and reports x>−3, forgetting that dividing by a negative number reverses the direction of the inequality sign.
The manipulation feels identical to solving an equation, and equations never reverse, so the habit built from equation practice carries over silently. The candidate also tends to focus on getting the number right and treats the inequality symbol as decoration rather than as the object being solved for.
What to do instead
Make the reversal a visible step: whenever dividing or multiplying by a negative, write the symbol flipped on its own line before writing the final answer. Check by testing a value, e.g. substitute x=−4 into −2x+1>7 to see it holds, then test the claimed boundary. Also practise reading the answer on a number line so the direction has a concrete meaning.
When a word problem yields two solutions to a quadratic equation, the candidate reports both without checking them against the real-world conditions, for example giving a negative length or a time before the start.
The candidate has learned that quadratic equations have two roots and treats both as equally valid answers. Word problems are read quickly to extract numbers, so the constraints (length must be positive, the ball is in the air only after t=0) are never written down as conditions.
What to do instead
Before solving, write the variable's allowed range explicitly, e.g. x>0 or 0≤t≤5. After solving, evaluate each root against that range and write one sentence explaining why the rejected root is discarded. Teachers award marks for this sentence, so treat it as part of the answer, not as optional commentary.
The candidate confuses the rate of change with the value of the function, for example reading the y-intercept of a linear graph as the speed, or claiming that a steeper line always means a larger value rather than a larger rate of change.
On a graph, height and steepness look like the same visual feature, and the candidate has not separated the two ideas. In inverse proportion, the candidate also assumes that as x increases y always decreases, which fails when the constant of proportionality is negative.
What to do instead
For every linear graph, label three things separately: the y-intercept with its meaning, the slope computed as change in y over change in x with units, and one point with its meaning. For inverse proportion, always write y=a/x and check the sign of a before stating the direction of change. Do two graph-interpretation questions per study session and state the meaning in words each time.
In similarity problems, the candidate pairs corresponding sides incorrectly, for example matching the longest side of one triangle with the shortest side of the other, or writes the ratio upside down so lengths are multiplied instead of divided.
The candidate identifies similarity from the shape but does not establish the correspondence between vertices. When the triangles are rotated or reflected in the diagram, visual position is used instead of the stated equal angles, and the vertex order in the similarity statement is ignored.
What to do instead
Always rewrite the similarity statement with matching vertices aligned, e.g. △ABC∽△DEF, then list the three side pairs directly under it before computing. Mark equal angles on the diagram with arcs and label each vertex. Verify the ratio by checking that the largest side of one triangle corresponds to the largest side of the other.
The candidate applies the inscribed angle theorem with the wrong arc, claiming an inscribed angle equals the central angle standing on the same arc, or fails to notice that angles standing on a diameter are right angles.
The theorem is remembered as a vague association between inscribed and central angles without the factor of one half or the condition of standing on the same arc. In complex diagrams, the relevant arc is not visually obvious, so the candidate guesses from the nearest-looking angle.
What to do instead
Redraw the circle alone with only the relevant points and shade the arc in question. Write the relation next to it in full: inscribed angle equals half the central angle standing on the same arc. Add a standard note that an angle inscribed in a semicircle is 90°, and check each answer against the diagram to see whether the size is plausible.
When proving two triangles congruent, the candidate writes the three equal parts but never states which congruence condition applies, or uses SSA, which is not a valid condition.
The candidate treats the proof as a list of facts rather than a logical argument, so the step that converts facts into a conclusion is omitted. SSA feels valid because three pieces of information are given, and the counterexample is rarely practised.
What to do instead
Adopt a fixed three-line template: state the three equalities with reasons, then write which condition they satisfy (SSS, SAS, or ASA and its variants), then conclude with the vertices in correct order. Memorise that SSA and AAA do not establish congruence, and draw a quick counterexample once so the exclusion is understood rather than memorised.
In probability, the candidate counts outcomes that are not equally likely as if they were, for example saying that drawing two balls from a bag gives three outcomes (both red, both white, one of each) and assigning each probability one third.
The candidate lists outcomes by their appearance rather than by distinguishable elementary events. Identical-looking objects are treated as indistinguishable, so combinations are counted instead of ordered draws, and the denominator does not match the situation.
What to do instead
Before calculating, ask whether every listed outcome has the same chance; if not, split the list until it does, e.g. label the balls R1, R2, W1 and enumerate. Alternatively use a tree diagram, where each branch is equally likely by construction, and multiply along branches and add across them. Verify that the probabilities of all outcomes sum to 1.
When asked to explain reasoning in words, the candidate writes only the calculation or only the final answer, with no statement of the property or rule used, so the explanation earns little credit even when the number is correct.
The candidate has practised computation far more than written justification and believes that showing the arithmetic is the same as explaining. There is also a fear of writing too much and making an error, so the response is kept minimal.
What to do instead
Use a two-part sentence pattern: 'Because [rule or given fact], [conclusion]'. For example, 'Because the two angles are equal, the lines are parallel'. Practise by taking a solved problem and writing one such sentence for each line of working. Read model answers to see how much detail is expected, and always name the theorem or property explicitly.
The candidate misreads the question's requirement, for example giving the value of x when the question asks for the value of 2x+3, or answering in the wrong unit or with an unsimplified fraction.
Under time pressure the candidate reads the first half of the sentence and starts calculating, and the final instruction is skipped. The candidate also stops as soon as a number appears, treating the first result as the answer rather than checking it against the question.
What to do instead
Underline the final instruction and the requested quantity before starting, and write that quantity at the top of the working space. After finishing, cover the working and answer the underlined question aloud from the result. Reserve the last few minutes of the paper to re-read each question's final line against the written answer.