Érettségi Mathematics common mistakes
Érettségi Mathematics · Understand the traps and how to avoid them.
When a task asks for an approximate value or an estimate, the candidate computes the exact result with full precision, then rounds only at the very end, and treats the rounded number as 'the estimate'.
Hungarian school practice drills exact computation, so 'becslés' is read as 'calculate, then round'. The candidate never separates the estimation step (choosing a convenient nearby value, bounding the error) from the exact calculation, and loses the points awarded for the reasoning of the estimate.
What to do instead
Before computing, write one line of the form 'a ≈ 20, b ≈ 30, so ab ≈ 600, error at most …' and only then compute exactly. Practise with tasks where the exact value is deliberately awkward (e.g. 4.87·19.6) so that estimating first is the only sensible route. In the exam, if the wording is 'becsülje meg' or 'körülbelül', the marker expects the bounding argument, not just the final number.
Uses a calculator result such as 0.3333333 or 1.4142136 as if it were exact, then substitutes it back into an equation or compares it with a fraction and declares an equality or inequality that is false.
The display cuts off digits silently; the candidate forgets that √2 ≠ 1.414 and that 1/3 cannot be written as a finite decimal. Rounding error is then carried into a proof step where exactness is required.
What to do instead
Keep irrationals and fractions symbolic until the final answer line. When a decimal is unavoidable, write the rounding explicitly (≈, and to how many decimals). For comparisons, square or cross-multiply instead of evaluating: to test √2 < 3/2, compare 2 < 9/4.
When a problem contains a set-operation or logic question, the candidate answers with an example ('e.g. A = {1,2}, B = {2,3}, so it works') and treats the example as a proof of a universal statement, or conversely rejects a true universal statement because one example fails.
The distinction between 'van olyan' (there exists) and 'minden' (for all) is not practised enough; the candidate reads the quantifier loosely and substitutes a single concrete case for a general argument.
What to do instead
Before answering, underline the quantifier in the task. For 'minden', a single counterexample is a complete disproof - so search for one deliberately. For 'van olyan', one correct construction is a complete proof - so give it explicitly. Never present a generic example as proof of a 'minden' claim.
In a combinatorics task, the candidate cannot decide whether order matters and whether repetition is allowed, so applies n! or a binomial coefficient to a situation that needs the other one (e.g. counts a handshake problem with permutations, or counts a password without repetition when repetition is allowed).
The four counting schemes (permutation, variation, combination, and their repetition-allowed versions) are memorised as formulas rather than as answers to two yes/no questions, so the candidate guesses from keywords in the text.
What to do instead
Write the two questions at the top of every solution: 'Számít a sorrend?' and 'Ismétlődhet?'. Then pick the scheme from the answer pair, not from the wording. Check with a tiny case (n = 3) by hand: if the hand count and the formula disagree, the scheme is wrong.
In a graph problem the candidate draws the graph, reads off a degree sequence or an Euler path by eye, and does not check the handshake lemma (degree sum is even) or the connectivity/parity conditions.
Graphs are taught visually, so the drawing feels like the object. The candidate does not translate the picture into degree data, and misses that an Euler circuit requires every vertex to have even degree and that a graph with two odd-degree vertices has an Euler trail but no circuit.
What to do instead
Always tabulate the degrees next to the drawing and add them up: an odd sum proves a drawing error. Before claiming an Euler path, list the odd-degree vertices; 0 odd → circuit, 2 odd → trail, more → none. For trees, check edges = vertices − 1.
When finding the domain of a function or solving an equation with a square root, logarithm, or fraction, the candidate solves the equation and gives all roots without checking the domain conditions (radicand ≥ 0, argument > 0, denominator ≠ 0), and reports extraneous roots.
The algebra is done as a sequence of manipulations, and the domain is treated as a formality to be stated at the start and then forgotten. Squaring both sides in particular introduces roots silently.
What to do instead
Write the domain conditions first as inequalities and keep them until the end. After solving, substitute every root into the original equation (not the squared one) and strike out any that fail. State the final answer set explicitly, e.g. x ∈ {3}, not x = 3 or x = −1.
For an arithmetic or geometric sequence the candidate confuses the nth term with the sum of the first n terms, or uses a1 where the formula needs a1 and d (or q), giving answers off by one index (e.g. computing aₙ as a₁ + nd).
The formulas aₙ = a₁ + (n−1)d and Sₙ = n(a₁+aₙ)/2 look similar and are often written from memory under time pressure; the (n−1) is dropped and the index shift goes unnoticed.
What to do instead
Anchor every sequence problem with the first three terms written out explicitly (a₁, a₂, a₃) and verify the formula against them. Check the index: the nth term uses n−1 steps, the sum has n terms. For geometric sums, confirm |q| ≠ 1 before using the closed formula.
In a triangle problem the candidate applies the sine rule where the cosine rule is needed (or vice versa), and when using the sine rule to find an angle, takes the calculator's acute answer without considering the obtuse possibility.
Both rules are remembered as 'a/sin A = …' style formulas; the decision rule (two sides and the included angle, or three sides → cosine rule; a side and its opposite angle → sine rule) is not applied. The ambiguous case of the sine rule is rarely flagged.
What to do instead
Classify the given data first: SSS or SAS → cosine rule; AAS or SSA → sine rule. For SSA, immediately check whether the angle could be obtuse (sin θ = sin(180° − θ)) and test both against the angle sum. Verify the result with the angle sum = 180° and with a rough sketch to scale.
In coordinate geometry the candidate computes a distance, midpoint, or line equation correctly in isolation but does not connect it to the geometric question (e.g. proves two sides equal but never states that the triangle is isosceles, or finds the intersection point but does not check it lies on the segment).
The task is read as a list of computations rather than as a geometric claim to be established; the final interpretive sentence is skipped, and conditions such as 'on the segment AB' or 'in the first quadrant' are ignored.
What to do instead
End every coordinate-geometry solution with an explicit sentence naming the geometric conclusion and the data that supports it. Check positional conditions by comparing coordinates (betweenness, sign of coordinates) before declaring the answer. Keep a sketch with labelled points.
In a probability task the candidate computes favourable outcomes over total outcomes using the wrong sample space - e.g. treats drawing without replacement as with replacement, or adds probabilities of non-mutually-exclusive events without subtracting the intersection.
The sample space is assumed rather than written down; the words 'visszatevés nélkül' (without replacement) and the dependence between successive draws are not translated into a changing denominator. The addition rule is recalled as P(A)+P(B).
What to do instead
Write the sample space and its size before counting anything, and mark whether each draw changes the total. Use P(A∪B) = P(A)+P(B)−P(A∩B) and check independence before multiplying. Sanity-check that the probability lies in [0,1] and that complementary events sum to 1.
When asked for a mean, median, or standard deviation from a frequency table, the candidate averages the class midpoints without weighting by frequency, or reports the median as the middle class midpoint rather than locating the position (n+1)/2 in the cumulative frequency.
The table is read as a list of values rather than as grouped data with weights; the distinction between mean of the data and mean of the class marks is not made, and the median's positional definition is replaced by a visual 'middle row'.
What to do instead
Build a cumulative frequency column before anything else and find the position (n+1)/2 for the median. For the mean, multiply each value (or midpoint) by its frequency, sum, and divide by n. For standard deviation, use the deviation formula and keep a table of squared deviations; check that the result is non-negative and plausible relative to the range.
When solving an inequality the candidate multiplies or divides by a negative number (or by an expression of unknown sign) without flipping the inequality, and gives the complementary interval.
The manipulation is copied from equation-solving, where multiplying by a negative is harmless; the sign change of the inequality is a separate rule that is easily skipped, especially when the negative factor is a parameter or a bracketed expression.
What to do instead
Never multiply an inequality by an expression containing the unknown unless its sign is known; instead bring everything to one side and use a sign chart. When dividing by a known negative number, write '⟹ flip' explicitly. Test the final interval with a concrete value from it and one from outside.
In a word problem the candidate writes an equation using the numbers in the order they appear in the text, without defining the variable and its unit, so the equation models the wrong relationship (e.g. speed = distance × time, or mixing percentages by simple averaging).
The text is translated phrase by phrase into symbols; the underlying relation (rate × time = distance, or the weighted nature of a mixture) is not identified first, and the variable is left undefined.
What to do instead
Before writing any equation, write 'Let x = … (unit)' and list the known quantities with units. Identify the physical or logical relation in words first ('the two parts sum to the whole', 'the times are equal'). Then translate. Check the answer against the story: units, plausibility, and whether it satisfies every sentence.
On a history-of-mathematics question the candidate attributes a result to the wrong person or century - e.g. credits the Pythagorean theorem to Pythagoras as its discoverer, or places Bolyai's appendix in the wrong decade - and gives a name without the result it belongs to.
Names are memorised as a list detached from results and dates; exam tasks often ask for the result, the person, and the era together, and the candidate supplies only the familiar name.
What to do instead
Build a two-column table pairing each name with one specific result and one date or century (e.g. Bolyai János, Appendix, 1832; Erdős Pál, combinatorics and the probabilistic method, 20th century). Learn one concrete fact per name rather than the name alone, and be ready to state what the result says in one sentence.