IB Math AA SL and HL Syllabus: What's in Each Topic, How It's Assessed and What HL Adds

The five analysis and approaches topics, one by one, with what Higher Level adds and how the exams work.

AA stands for Analysis and Approaches. The full name is Mathematics: analysis and approaches (in Spanish, Matemáticas: Análisis y Enfoques), and it can be taken at Standard Level (SL) or Higher Level (HL). It is the IB maths course with the most algebra, analysis and proof.

This article sums up what is in each topic, what HL adds and how the course is assessed. It is meant for students starting the IB, for anyone torn between SL and HL, and for parents who want to know what their child is facing. If you haven't chosen between AA and AI yet, start with AA vs AI: which should you choose?

The five topics and their hours

The course has five topics plus a block of inquiry skills linked to the exploration. These are the IB's recommended teaching hours:

The biggest HL jumps are in geometry, which doubles because vectors are HL-only, and in calculus. In statistics the gap is small.

What's in each topic, and what HL adds

1 · Number and algebra

SL: scientific notation, arithmetic and geometric sequences and series (including infinite sums), compound interest and depreciation, exponents and logarithms and their laws, simple proofs and the binomial theorem for natural exponents.

HL adds: counting principles (permutations and combinations), the binomial theorem with negative and fractional exponents, partial fractions, complex numbers (Cartesian, polar and Euler forms, De Moivre's theorem, roots of polynomials), proof by induction, by contradiction and by counterexample, and systems of linear equations.

2 · Functions

SL: parallel and perpendicular lines, domain, range, composite and inverse functions, key features of graphs using technology, quadratics and the discriminant, simple rational functions and their asymptotes, exponentials and logarithms, and transformations of graphs.

HL adds: the factor and remainder theorems, sum and product of the roots of a polynomial, more general rational functions, odd, even and self-inverse functions, inequalities and graphs with the modulus function.

3 · Geometry and trigonometry

SL: volumes, areas, distances and angles in 3D, the sine and cosine rules, angles of elevation and bearings, radians, arcs and sectors, the unit circle and exact values, the Pythagorean identity and double angles, graphs of circular functions and trigonometric equations.

HL adds: secant, cosecant and cotangent, inverse trigonometric functions, compound angle identities and the whole vectors block: scalar and vector products, equations of lines and planes, and their intersections. Note: AA SL has no vectors.

4 · Statistics and probability

SL: sampling, histograms, cumulative frequency and box plots, mean, median, standard deviation and quartiles, Pearson's correlation and regression lines, conditional probability and independence, discrete random variables and the binomial and normal distributions (including inverse normal).

HL adds: Bayes' theorem and the properties of discrete and continuous random variables (probability density functions, expectation and variance).

5 · Calculus

SL: the derivative as a rate of change, increasing and decreasing functions, tangents and normals, the chain, product and quotient rules, the second derivative, maxima, minima, points of inflexion and optimisation, kinematics, indefinite integrals (including substitution) and definite integrals, and areas under and between curves.

HL adds: differentiation from first principles, limits and L'Hôpital's rule, implicit differentiation and related rates, derivatives and integrals of more functions, integration by parts, areas with respect to the y-axis, volumes of revolution, first-order differential equations (Euler's method, separation of variables, homogeneous equations and integrating factors) and Maclaurin series.

How it's assessed

80% of the grade comes from the final exams, sat in May or November depending on your school's session, and 20% from the exploration.

Papers 1 and 2 have two compulsory sections: short-response questions (A) and extended-response problems (B). Paper 3, HL only, consists of two compulsory extended problems that investigate a new situation step by step. It is not about knowing more content but about reasoning with what you know in a context you haven't seen.

The exploration: the 20% that depends on you

The exploration (the Internal Assessment) is a written piece on a maths topic you choose. The guide recommends around 12 to 20 pages, and it is marked out of 20 against five criteria:

• A · Presentation (4 marks): it reads as a coherent piece, with introduction, development and conclusion.
• B · Mathematical communication (4): correct notation, terminology and graphs.
• C · Personal engagement (3): the topic is clearly yours and the decisions are your own.
• D · Reflection (3): evaluating results, limitations and possible improvements.
• E · Use of mathematics (6): the heaviest criterion, judged against the level of your course: at HL, HL-level maths is expected.

Choosing the topic well is half the job. We cover it in how to choose a good Internal Assessment topic.

So what does HL really add?

Ninety more hours, but it is not "the same, only harder". Whole blocks don't exist at SL (vectors, complex numbers, differential equations, series), there is more proof, and there is an extra exam, Paper 3, which rewards reasoning in unfamiliar situations. That makes the decision easier:

• If algebra is already a struggle at SL, HL shows from the first term: complex numbers, induction and vectors come early.
• If you are heading for engineering, physics, maths or computer science, HL is the foundation of first-year university maths. More on this in AA vs AI.
• If in doubt, decide early and with evidence: your marks in the first HL tests say far more than gut feeling. More on this in Is IB Maths HL hard?

Heads up: the IB changes the syllabus from 2027

The IB has announced a new mathematics curriculum, taught from 2027 and examined for the first time in 2029. The current course, the one described here, is examined for the last time in the November 2028 session. If you start the IB in 2026, you will sit this syllabus. If you start in 2027, you will follow the new one, so check it with your school.

How to prepare each part

• Paper 1: algebra by hand, exact values and clean working. You train it by doing, not by reading solutions.
• Paper 2: master the graphic calculator (intersections, distributions, equations) so you don't lose time, and use it to check your hand work.
• Paper 3 (HL): practise with past papers and follow the thread of each part: every question sets up the next.
• Exploration: choose the topic early and write a first draft with time left to improve it.

I have taught IB Mathematics analysis and approaches HL at an international school in Valencia, and this is exactly what we work on in online IB Maths tutoring.