IB Maths AI HL: the mistakes examiners report most often

When separating variables, students lose the minus sign, integrate 1/(2x) wrongly, drop the constant or never use the initial condition to find it. Euler's method is copied from the booklet but not run on the calculator, especially for second order and coupled systems. Phase portraits are guessed or drawn as the wrong type, and equilibrium points are taken where the two derivatives are equal.

Why it happens. This is HL-only content that appears late in the paper and was sometimes not taught at all. Each method is learnt as a recipe, so the link between the eigenvalues and the shape of the phase portrait, or between a second order equation and a coupled system, is missing.

How to fix it. For separable equations write the separated form, integrate both sides with one constant, and substitute the initial condition before rearranging. Set up Euler's method as a recursive sequence or table on your calculator and practise until you can fill a table quickly for first and second order equations. Learn the link between the eigenvalues and the phase portrait: real with the same sign, real with opposite signs, complex, and purely imaginary.

Hypotheses are written about the sample, about counts or about the p-value instead of a population parameter, and the alternative is the wrong way round or the wrong tail. Students run an unpaired test when the data are paired, use the contingency table rule for degrees of freedom in a goodness of fit test, forget to subtract one for an estimated parameter, and end with no conclusion in context. Type I and type II errors are defined but their probabilities cannot be calculated.

Why it happens. AI HL has many tests and each has its own small rules. Students learn the calculator menus well, so the write-up around the output becomes the weak point, and error probabilities need the test to be understood rather than just run.

How to fix it. Write every test in the same five lines: hypotheses with the parameter named, the test and its distribution, the p-value or critical value, the comparison, and a conclusion in context. Keep a one-page sheet of degrees of freedom for each chi-squared test. For error probabilities, first write the critical region, then find the probability of landing in it under the null for type I and under the alternative for type II.

Students miss a logarithmic integral, apply a substitution wrongly and drop the constant of integration. The trapezoidal rule is set up with the wrong strip width or not recognised as a way to estimate the integral of y squared for a volume. In volumes of revolution about the y-axis they integrate y squared with respect to x instead of rearranging for x in terms of y, use the wrong limits, or forget pi, and composite solids are not split into the right pieces.

Why it happens. Most integrals in the course are done on the calculator, so the few that must be done by hand come as a surprise. The y-axis volume needs a rearrangement before any integration, and that step is skipped under time pressure.

How to fix it. Practise the short list of integrals the course expects by hand: powers, exponentials, the reciprocal giving a logarithm, and simple substitutions, always with + C. For a volume, sketch the region, mark the axis of rotation and write the integral as pi times the square of the radius, with the variable of the axis. Then let the calculator evaluate it.

Matrices are multiplied in the wrong order for a composite transformation or applied to the point the wrong way round, and the determinant is not used as the area scale factor. In Markov chains the transition matrix is built with inconsistent entries, columns do not sum to one, the wrong power is used for the number of steps, and the steady state is read from a high power instead of the eigenvector method asked for.

Why it happens. Matrix multiplication is not commutative, but most other operations students meet are. In Markov chains the convention of columns or rows differs between textbooks, so the set-up step is where errors start.

How to fix it. Write a composite transformation from right to left, the first transformation next to the point. Check every transition matrix by summing its columns. Count the steps from the initial state before choosing the power. For the steady state, solve for the eigenvector with eigenvalue 1 and scale it so the entries sum to one.

Final answers are given to one or two significant figures, intermediate values are rounded early and the error carries through, and decimal places are confused with significant figures. Calculator syntax such as E notation or function names appears in place of mathematics, and Euler or finance answers arrive with no method, so a single slip costs every mark. Upper bounds are taken from an already rounded value.

Why it happens. In a course built around technology the calculator does the hard part, and students forget that the examiner only sees what is written. Rounding rules feel like a detail until they cost a mark on almost every question.

How to fix it. Keep full calculator values in memory and round only the final answer, to three significant figures unless the question says otherwise. Before using the calculator, write the equation, integral or distribution you are about to evaluate in standard notation. For bounds, work out the half-unit either side of the stated value before calculating anything.

Selections without replacement are treated as with replacement and the order of outcomes is ignored. Students do not spot that a normal probability must feed into a binomial model, add standard deviations instead of variances, confuse 2X with X1 + X2, and forget to divide the standard deviation by the square root of n for a sample mean.

Why it happens. Each distribution is taught on its own, so a question that moves from one to another, or combines several variables, has no ready template. The variance rules look similar and are easy to mix up.

How to fix it. Before calculating, define the random variable in words and state its distribution with parameters. Learn the variance rules as a short list: Var(aX) = a squared Var(X), and variances add for independent variables, even when subtracting. For a sample mean, write its distribution explicitly with sigma over root n.

The inner derivative is forgotten in the chain rule, the product rule is skipped or misapplied for a second derivative, and values are substituted before differentiating. Inflexion points are sought by setting the first derivative to zero, normals are found without differentiating, and related rates problems are not recognised as such.

Why it happens. AI students differentiate on the calculator for most of the course, so hand differentiation is underpractised. When a model has a parameter or a composite function, the rules are needed in combination and gaps show.

How to fix it. Do a short set of hand derivatives every week that combine the chain rule with the product or quotient rule. Always differentiate first and substitute second. For rates of change, write the chain of derivatives with the units of each, so you can see which ones cancel.

Students struggle to turn a context into a function, treat a constant parameter as a variable, or ignore the domain when sketching. Horizontal stretches and translations are applied the wrong way or to the wrong variable, inverse and derivative notation are confused, and endpoints, asymptotes and intercepts are left off sketches.

Why it happens. Modelling is usually practised on familiar contexts with the model given. When the question asks students to build the function or change it, they have to reason about inputs and outputs, which routine practice does not train.

How to fix it. For every model, state in words what the input and output represent, with units, and the domain. Learn transformations through points: track where two or three key points go and check against the graph. When sketching, label endpoints, intercepts, turning points and asymptotes, and keep within the stated domain.

Frequencies are left out of mean and variance calculations, outliers are judged with a non-standard rule and tied ranks are not averaged in Spearman's coefficient. When data are linearised with logarithms, the logs are left out of the final equation, the model is written in x and y instead of the named variables, and r is compared where r squared was asked.

Why it happens. Students run regression on the calculator without checking what the variables are. Linearising needs log laws and a clear idea of what is plotted on each axis, and both are often vague.

How to fix it. Always enter the frequency list with the data. For a power or exponential model, write which variable goes on each axis before running the regression, then rewrite the line in the original variables. Rank tied values with the average of their positions, and state the meaning of r and R squared in context.

Students multiply complex numbers in Cartesian form but stall when two sinusoids have to be added through complex exponentials, when a number must be converted to Euler form on the calculator, or when multiplication by a complex number has to be read as a rotation and enlargement. Real and imaginary parts are not equated, and degree mode quietly ruins the arguments.

Why it happens. In AI HL complex numbers appear mainly as a tool for something else, such as alternating currents or geometric transformations, so the link between the algebra and the application is never practised on its own.

How to fix it. Learn the conversion between a cos(t) + b sin(t), the real part of a complex exponential and a single sinusoid, and do it on the calculator in radian mode. When multiplying by a complex number, state the rotation angle and the scale factor in words. After expanding a product, collect real and imaginary parts explicitly before equating anything.

Position vectors are used where direction vectors are needed, a length is read as a vector and the triangle area from the cross product is not halved. In motion problems speed is not recognised as the magnitude of velocity, the constant of integration is left out, and students have no plan for the closest approach or the shortest distance between paths.

Why it happens. Vector questions in AI HL are almost always set in a context such as boats or aircraft, so students must translate the story into vectors before using any formula. Without a sketch, the order of steps is lost.

How to fix it. Draw a rough diagram and label which vectors are positions and which are directions. Learn the standard plans: a general point on the line in terms of the parameter, a displacement vector between two moving objects, then minimise its length or use a scalar product of zero. When integrating velocity, add the constant and fix it with the starting position.

Students ignore guidance in the stem, give a general result instead of the specific value, compare the wrong quantities, and interpret results without context. When a part has no suggested method they stall, miss the link with earlier parts, or run out of time before the final HL questions.

Why it happens. Most practice is on structured questions that say what to do. Paper 3 and the end of Papers 1 and 2 reward planning and reading, and students rush through the stem to reach the calculation.

How to fix it. Underline the command term, the variable and the units before starting. In a long question, ask how each part could use the previous one. Practise unstructured problems by writing a plan of two or three steps before calculating, and end each answer with one sentence in the language of the context.

Brackets are dropped when rearranging, a negative is squared without brackets, and fractions are not simplified. Log laws are misapplied or not known, a difference of logs is not combined into a single log, and solutions are lost when square rooting.

Why it happens. The calculator handles most numerical work in AI, so manual algebra gets little practice. It then becomes the step that blocks the final part of a long question, where the method marks depend on it.

How to fix it. Keep a regular routine of short rearranging exercises: making a variable the subject, especially in exponential and logarithmic models. Write brackets around anything substituted. Learn the three log laws and practise turning an equation in logs into one without them and back again.

Three-figure bearings are misread, the cosine rule is replaced by right-angled methods, and the angle at the wrong vertex is used. In three-dimensional and mensuration problems the right angle is not found or unit conversions for area and volume are linear. In Voronoi diagrams sites are joined instead of drawing perpendicular bisectors, and the bisector gradient or equation is wrong.

Why it happens. These look like easy topics, so they are revised least. Contexts with bearings and Voronoi diagrams need a careful diagram, and errors in the first part carry through the whole question.

How to fix it. Always draw a north line at every point before working with bearings. Label the triangle and decide which rule you need from what is known. For Voronoi questions, practise finding a perpendicular bisector from two sites in three lines: midpoint, negative reciprocal gradient, equation. Remember that areas scale by the square of a length factor and volumes by the cube.

Loans with regular payments are worked with the compound interest formula, present value and payment are given the same sign, and months and years are mixed in N. In sequences, n and n - 1 are confused, the first term is wrong, and a term is used where a sum is needed or the reverse.

Why it happens. The finance solver has many fields, and the sign convention is not intuitive. Sequence questions in context rarely say which term is the first, so students start counting from the wrong place.

How to fix it. Learn the finance solver with a fixed rule: money you receive is positive, money you pay is negative, and N counts payments, not years. Write a small table of the first three terms in context before choosing a formula. Ask whether the question wants one value (a term) or an accumulated total (a sum).

Prim's algorithm is done by eye with no order of edges shown, or confused with nearest neighbour. In travelling salesman problems the route does not return to the start, the wrong bounds are chosen as best, and the deleted vertex method adds only one edge. With adjacency matrices, students use one power where a sum of powers is needed or miss double edges.

Why it happens. The answer to a graph question can often be found by inspection, so students do not see why the method matters. The marks, however, are for showing the algorithm step by step.

How to fix it. Write the order in which edges are added for every algorithm, not just the final tree. Learn which bound is better: the lowest upper bound and the highest lower bound. For walks of up to n steps, add the matrix powers from 1 to n.

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.