IB Maths AA SL: the mistakes examiners report most often

Students integrate numerator and denominator separately, lose the factor when integrating e^(kx) or sin(kx), do not know the integral of 1/x, forget the constant or never write the final function. In area questions they use the wrong limits, drop brackets when subtracting curves, or treat a region below the axis as negative area.

Why it happens. Integration by inspection needs the chain rule in reverse, which is less practised than differentiation. Area questions add a second decision about limits and signs, and students go straight to the integral without a sketch.

How to fix it. After integrating, differentiate your answer to check it, especially for e^(kx), trigonometric terms and composite functions. When a point is given, find C and then write the full function. For areas, sketch first, find the intersections, write the integral with brackets, and split it where the curve crosses the axis.

Students multiply probabilities as if events were independent, or add them as if they were mutually exclusive, when neither was stated. They confuse the union with the intersection, take P(A and not B) as P(A), miss ordered outcomes, and assume independence in a part that asks them to show it.

Why it happens. The formulas look alike and the shortcuts work in the simple cases that are practised most. Students rarely stop to ask which relationship between the events the question has actually given them.

How to fix it. Start every probability question by writing down what you know about the events: independent, mutually exclusive, or neither. Draw a Venn diagram or a tree before using a formula, and fill in every region, including the one outside both events. Check that each answer lies between 0 and 1, and that a conditional probability uses the right denominator.

Students use binomial when selection is without replacement, normal when the count is binomial, the pdf where the cdf is needed, and the wrong bound for at least or fewer than. With the normal distribution they skip the sketch, use a probability where a z-value is needed, and struggle to find an unknown mean and standard deviation.

Why it happens. The calculator menus make each distribution feel like a button to press, so the modelling decision and the inequality are done in a hurry. Inverse normal problems need a z-value linked back to the variable, a step many students have never written out.

How to fix it. Before any calculation, name the distribution and say why: fixed number of independent trials, or a continuous measurement. Translate the words into an inequality on X, then into what the calculator needs, for example at least 5 as P(X >= 5). Sketch the normal curve with the shaded region every time, and practise finding mu and sigma from two given probabilities with z-scores.

Transformations are described without the direction, the axis or the scale factor, and horizontal stretches are given as the reciprocal of the correct factor. Sketches go beyond the given domain, miss endpoints, intercepts and turning points, or show only one branch of a curve.

Why it happens. A transformation is recognised by eye but rarely put into precise words, and a horizontal stretch by 1/b feels backwards. Sketching is treated as a rough drawing rather than as an answer that is marked on its key features.

How to fix it. Describe each transformation with its type, direction and size, for example a horizontal stretch by scale factor 1/3. Check by substituting one point. When sketching, copy the window from the calculator, respect the domain, and label intercepts, turning points, endpoints and asymptotes with coordinates or equations.

Students split the log of a sum into a sum of logs, write log of 2^3 as a cube of the log, and do not spot a quadratic in e^x. With the discriminant they give an expression instead of a number, miss the sign condition, or cannot turn the result into a correct inequality. Fraction and surd slips spoil otherwise correct work on Paper 1.

Why it happens. Log laws are memorised as patterns, so the wrong patterns look just as plausible. Without a calculator on Paper 1, small arithmetic errors build up and students lose confidence in their own algebra.

How to fix it. Write the three log laws and test each one on numbers until you can see why log(a + b) cannot be split. In equations with e^x or a^x, look for a substitution that gives a quadratic. For the discriminant, write the condition first (greater than, equal to or less than zero), then solve the inequality with a sketch of the parabola. Practise non-calculator fractions and surds regularly.

The chain rule is left out on terms like sin^2 x or (x^2 + 1)^3, and the quotient rule is used where simplifying first would be quicker. With tangents and normals, students substitute into the wrong expression, read the derivative as the tangent equation, or do not link a gradient to the tangent of an angle.

Why it happens. Differentiation of single functions is secure, but composite functions need the student to see the inner function first. The tangent requires three separate values (a point, a gradient and an equation), and they get mixed up.

How to fix it. Before differentiating, rewrite the function to make the inner function visible, and simplify where you can. For a tangent, write three lines: the point, the gradient from the derivative, then y - y1 = m(x - x1). For a normal, take the negative reciprocal. Remember that the gradient of a line equals the tangent of the angle it makes with the x-axis.

Students set t = 0 or s = 0 when a particle is at rest, take a maximum of velocity as a change of direction, give the greatest velocity when the question asks for speed, and integrate velocity without the absolute value, so they find displacement instead of distance.

Why it happens. Each word (rest, speed, direction, distance) points to a different condition on s, v or a, and students react to the function they are given rather than to the word in the question.

How to fix it. Keep a one-line table: at rest means v = 0, change of direction means v changes sign, speed is |v|, distance is the integral of |v|, acceleration is the derivative of v. Before each part, write which function and which condition you are using. On Paper 2 do the integrals and the solving on the calculator, in radian mode, and keep full accuracy for t.

Students use the sector formulas with degrees, forget the triangle when finding a segment, and leave out the straight edges of a perimeter. They use the sine rule in a right-angled triangle or Pythagoras in a non-right one, miss the second angle in the ambiguous case, ignore the z-coordinate in 3D distance, and use wrong formulas for the faces or volume of a solid.

Why it happens. Diagrams in context are busy, and students start calculating before identifying the right triangle or region. The radian formulas for arcs and sectors only work in radians, which is easy to forget.

How to fix it. Redraw the triangle or region you need on its own and mark what is known. Choose the rule from what you have: right angle, two sides and an angle, or three sides. For sectors, work in radians throughout. In 3D, use all three coordinates and check whether a solid has a flat face that must be included.

Students find only one solution in the interval, pick the wrong quadrant, apply the wrong double angle identity, or do not know exact values such as sin(5pi/6). In sinusoidal models they confuse the period with the parameter b, ignore a reflection, and read the amplitude incorrectly.

Why it happens. Solving a trigonometric equation needs the unit circle or the graph to find every solution, and students stop at the first value from the calculator or the table.

How to fix it. Learn the exact values for multiples of pi/6 and pi/4, and use the unit circle to find all solutions in the given interval. Choose the form of cos 2x that leaves a single trigonometric function. For a model a sin(b(x - c)) + d, find the period as 2pi/b and read amplitude and midline from the maximum and minimum.

Students build a line from two points instead of using regression, use the y on x line to predict x, and miss that regression lines pass through the mean point. With frequency tables they ignore the frequencies or give the sample standard deviation, and comments on box plots or sampling are vague and not linked to the context.

Why it happens. The statistics menus on the calculator are used without knowing which list or which line is needed. Written comparisons are rarely practised, so students fall back on generic phrases.

How to fix it. For predictions, choose the line by what you are predicting: y on x for y, x on y for x, and mention extrapolation when you leave the data range. Enter frequencies as a second list. When comparing data sets, name a measure of centre and a measure of spread, give their values and say what they mean in context.

The calculator is left in degree mode for radian questions, or the reverse, and Paper 2 equations are solved by long algebra instead of on the calculator. Intermediate values are rounded too early, final answers are given to 2 significant figures, and leading zeros are counted as significant.

Why it happens. Students check the mode at the start of the exam and then forget it, and they trust algebra more than the calculator. Rounding rules feel like a detail, so early rounding goes unnoticed until the final answer drifts.

How to fix it. Check the angle mode at the start of every trigonometry or calculus question on Paper 2. If no exact answer is asked for, write the equation and solve it on the calculator. Store intermediate values in memory and round only at the end, to 3 significant figures unless the question says otherwise.

Students take S6 - S5 as the common difference, use the sum formula where a single term is needed, or miscount n by one. In compound interest they enter the rate as a decimal in the financial solver, use the wrong number of periods per year, and misread depreciation or monthly savings.

Why it happens. The formulas for terms and sums are close, and real contexts (a bouncing ball, a savings plan) hide which one is needed. The financial solver has its own input conventions that are easy to forget.

How to fix it. Write what is asked as either a term or a sum before choosing a formula, and check n by listing the first two or three values. For finance, set out the solver inputs one by one: N, I% as a percentage, PV, PMT, FV and periods per year, with signs for money paid out. Then check that the answer is reasonable.

Students read the inverse notation as a reciprocal or a derivative, reflect in the wrong line, and do not see that the domain of the inverse is the range of the original function. Asymptotes are swapped or written as numbers rather than equations, and range answers include infinity as an endpoint.

Why it happens. Domain, range and inverse are taught as definitions, but students do not connect them, and the notation f^-1 is easy to misread. Writing x = 2 instead of 2 feels like a formality until it costs a mark.

How to fix it. Practise finding an inverse by swapping x and y and rearranging, then state its domain from the range of the original. Write every asymptote as an equation, x = a or y = b. Read the range off a sketch, and check whether each endpoint is included.

Students verify a given result with numbers instead of proving it, work backwards from the answer, or work on both sides at once. In proofs about integers they test values instead of using n. They reject a solution without a reason, and miss the link to an earlier part when the question says hence.

Why it happens. Most questions reward reaching a number, so when the answer is given students do not know what else to write. A hence is read as an invitation, not as an instruction to use the earlier result.

How to fix it. In a show-that, start from one side and write every step until you reach the given result. For integer proofs, write general forms such as 2n + 1 and factor out the required multiple. When you reject a value, say why in one sentence. Read every hence as a hint to reuse the previous part.

Students use a sign diagram when the second derivative test is required, cannot justify a maximum or minimum, and confuse the conditions on f prime and f double prime for increasing and concave intervals. In optimisation they cannot use the constraint to remove a variable, or keep a negative value.

Why it happens. Finding where the derivative is zero is the routine part. Justifying the nature of the point and setting up the function to optimise need reasoning about the context, which is less practised.

How to fix it. Learn the conditions in a table: increasing where f prime is positive, concave up where f double prime is positive, maximum where f prime is zero and f double prime is negative. In optimisation, write the constraint, rearrange it for one variable, substitute, then differentiate. State the conclusion with the test you used and reject values that make no sense in context.

Students write (2x)^6 as 2x^6, drop a negative sign, give the whole term when the coefficient is asked for, or the binomial coefficient alone. When n is unknown, or the expansion is multiplied by another bracket, they look for the wrong power of x.

Why it happens. The general term is used mechanically, and the brackets around each part of the binomial are the first thing to disappear under time pressure.

How to fix it. Write the general term with full brackets, nCr (a)^(n-r) (b)^r, and find r from the power of x before calculating anything. Underline whether the question wants the term or the coefficient. For products, list every pair of terms that gives the required power.

Each report lists, paper by paper, the areas candidates found difficult. Every item was assigned to one of the areas above, with its session and paper recorded. An area counts once per report, however many papers mention it; each time zone has its own report.

The count says how often examiners raise something, not how many marks it costs. The explanations and the advice are mine, from teaching the course, and no wording from the reports or from exam questions is reproduced. Session by session summaries are in the examiner reports. This is independent work and is not endorsed by the International Baccalaureate. Session codes: M is May, N is November. Corrections: pietro@pietromeloni.com.