IB Maths AI HL syllabus

Numerical technique and sequences aimed squarely at financial application, plus the accuracy and error work that underpins every model in this course. HL adds complex numbers and — distinctively for AI — matrices and eigenvalues.

SL 1.1

Standard form

Operations with numbers written as a × 10ᵏ, where 1 ≤ a < 10 and k is an integer. Calculator notation such as 5.2E30 is not accepted in exams.

SL 1.2

Arithmetic sequences and series

The nth term and sum formulae, sigma notation, and applications including simple interest — plus interpreting data that is only approximately arithmetic.

SL 1.3

Geometric sequences and series

The nth term and sum formulae, sigma notation, and applications such as population growth and compound decay.

SL 1.4

Financial applications

Compound interest computed yearly through monthly, annual depreciation, and the real value of an investment once inflation is accounted for.

SL 1.5

Exponents and logarithms — introduction

Laws of exponents with integer exponents; logarithms to base 10 and base e, evaluated numerically with technology.

SL 1.6

Approximation and error

Decimal places and significant figures; upper and lower bounds of rounded numbers; percentage error; and estimation as a check that an answer is reasonable. Distinctive to AI — there is no equivalent in AA.

SL 1.7

Amortization and annuities

Loan repayment and annuity calculations using the GDC or a spreadsheet. In exams payments are made at the end of each period; the annuity formula itself is not examined.

SL 1.8

Solving equations with technology

Using technology to find the zeros or roots of polynomial equations and the solution to a system of linear equations.

AHL 1.9

Laws of logarithms

The product, quotient and power laws for logarithms, applied numerically and algebraically.

AHL 1.10

Rational exponents

Simplifying numerical and algebraic expressions involving rational exponents.

AHL 1.11

Infinite geometric series

The sum of an infinite geometric sequence, feeding into transformation matrices and Markov chains later in the course.

AHL 1.12

Complex numbers — Cartesian form

The number i where i² = −1; z = a + bi; real part, imaginary part, conjugate, modulus and argument; the complex plane; complex roots of real quadratics.

AHL 1.13

Polar and exponential form

Modulus–argument and exponential forms and conversion between all three; products, quotients and integer powers; adding sinusoids of equal frequency but different phase — the AC-circuit application.

AHL 1.14

Matrices

Order, equality, addition, scalar multiplication and matrix multiplication with its properties; identity and zero matrices; determinants and inverses; solving linear systems as Ax = b.

AHL 1.15

Eigenvalues and eigenvectors

The characteristic polynomial of a 2 × 2 matrix, diagonalization for distinct real eigenvalues, and using Mⁿ = PDⁿP⁻¹ for powers — the engine behind Markov chains and phase portraits.

Codes marked AHL are Higher Level only. The SL codes are shared with AI SL — AI HL covers both.

The largest SL topic in AI, and the heart of the course: not function theory for its own sake but choosing, fitting and justifying a model against real data. HL adds logistic and piecewise models and log-linearization.

SL 2.1

The straight line

Different forms of the equation of a line; gradient and intercepts; the conditions for parallel and perpendicular lines.

SL 2.2

Concept of a function

Domain, range and graph; function notation; a function as a mathematical model; and the informal idea of an inverse as a reflection in y = x.

SL 2.3

Graphing functions

The graph of y = f(x); sketching from given information or a context, including transferring a graph from screen to paper; graphing with technology.

SL 2.4

Key features of graphs

Intercepts, symmetry, vertex, zeros and roots, vertical and horizontal asymptotes; finding intersections of curves using technology.

SL 2.5

Modelling with functions

The AI model library: linear, quadratic, exponential growth and decay, direct and inverse variation, cubic, and sinusoidal models — with amplitude, period and principal axis.

SL 2.6

Modelling skills

The full modelling cycle: choose an appropriate model and a reasonable domain, find its parameters, comment on whether it is appropriate, then use it to predict. The subtopic that defines AI.

AHL 2.7

Composite and inverse functions in context

Composite functions and the notation (f ∘ g)(x); finding an inverse function and the domain restriction it may require.

AHL 2.8

Transformations of graphs

Translations, reflections in both axes, vertical and horizontal stretches and composite transformations — applied to every model in the course, where order matters.

AHL 2.9

Further models

Exponential models for half-life, natural logarithmic models, sinusoidal models with a phase shift, logistic models with a carrying capacity, and piecewise models made continuous.

AHL 2.10

Linearizing data with logarithms

Scaling very large or small numbers logarithmically; using log-log and semi-log plots with a line of best fit to decide whether data is exponential or a power relationship, and to extract the parameters.

Codes marked AHL are Higher Level only. The SL codes are shared with AI SL — AI HL covers both.

A modest SL core — 3-D solids, non-right-angled triangles and Voronoi diagrams — followed by the single largest HL extension in either course: vectors, matrix transformations and a full graph-theory strand with named algorithms.

SL 3.1

Three-dimensional geometry

Distance and midpoint between two points in 3-D; volume and surface area of solids including the right pyramid, cone, sphere and hemisphere; the angle between two lines or between a line and a plane.

SL 3.2

The sine and cosine rules

Sine, cosine and tangent ratios in right-angled triangles; the sine rule; the cosine rule; and the area of a triangle as ½ab sin C.

SL 3.3

Applications of trigonometry

Right and non-right-angled applications including Pythagoras; angles of elevation and depression; constructing labelled diagrams from written statements.

SL 3.4

Arcs and sectors

The length of an arc and the area of a sector. At SL these are worked in degrees — radians arrive only at HL.

SL 3.5

Perpendicular bisectors

Equations of perpendicular bisectors, built from the gradient of a segment and its midpoint. The prerequisite for Voronoi diagrams.

SL 3.6

Voronoi diagrams

Sites, vertices, edges and cells; adding a site to an existing diagram; nearest neighbour interpolation; and the “toxic waste dump” problem. Unique to AI, with contexts in urban planning, epidemiology and resource management.

AHL 3.7

Radian measure

The definition of a radian, conversion to and from degrees, and arc length and sector area in radians.

AHL 3.8

The unit circle and trigonometric equations

Definitions of cos θ and sin θ on the unit circle; the Pythagorean identity; the ambiguous case of the sine rule; and solving trigonometric equations graphically over a finite interval.

AHL 3.9

Transformations using matrices

Reflections, stretches, enlargements, translations and rotations of points in two dimensions as matrices; composition of transformations; and the geometric meaning of the determinant.

AHL 3.10

Vectors

Vectors and scalars, directed line segments, magnitude and components, base vectors i, j, k, the resultant of two or more vectors, position vectors, and rescaling and normalizing.

AHL 3.11

Vector equation of a line

The line r = a + λb in two and three dimensions, where b is a direction vector.

AHL 3.12

Vector kinematics

Modelling linear motion at constant velocity as r = r₀ + vt in two and three dimensions; relative position; motion with variable velocity; projectile and circular motion as special cases.

AHL 3.13

Scalar and vector products

Definition and calculation of both products; the angle between two vectors and the acute angle between two lines; the geometric meaning of |v × w|; and resolving a vector into parallel and perpendicular components.

AHL 3.14

Graph theory — foundations

Graphs, vertices, edges, adjacency and the degree of a vertex; simple, complete, weighted and directed graphs; in-degree and out-degree; subgraphs and trees.

AHL 3.15

Adjacency and transition matrices

Adjacency matrices and walks; counting k-length walks between two vertices; weighted adjacency tables; and constructing a transition matrix — with Google’s PageRank as the worked example.

AHL 3.16

Tree and cycle algorithms

Walks, trails, paths, circuits and cycles; Eulerian trails and circuits; Hamiltonian paths and cycles; Kruskal’s and Prim’s minimum spanning tree algorithms; the Chinese postman problem; and the travelling salesman problem with nearest-neighbour upper and deleted-vertex lower bounds.

Codes marked AHL are Higher Level only. The SL codes are shared with AI SL — AI HL covers both.

The biggest topic in AI SL by teaching hours, and the clearest break from AA: formal hypothesis testing sits in the Standard Level course. HL extends into estimation, the Poisson distribution, Type I and II errors and Markov chains.

SL 4.1

Sampling and data

Population, sample, random sample, discrete and continuous data; reliability of sources and bias; outliers by the 1.5 × IQR rule; sampling techniques and their effectiveness.

SL 4.2

Presenting data

Frequency distributions, histograms, cumulative frequency graphs, and box-and-whisker diagrams — including judging normality from their symmetry.

SL 4.3

Central tendency and dispersion

Mean, median, mode and modal class; estimating the mean from grouped data; IQR, variance and standard deviation; the effect of constant changes on each.

SL 4.4

Correlation and regression

Linear correlation of bivariate data; Pearson’s r; scatter diagrams and lines of best fit; the regression line of y on x and its use — and its misuse — for prediction.

SL 4.5

Probability basics

Trial, outcome, equally likely outcomes, sample space and event; P(A) = n(A)/n(U); complementary events; the expected number of occurrences.

SL 4.6

Combined and conditional probability

Venn diagrams, tree diagrams and tables of outcomes; combined and mutually exclusive events; conditional probability; independence; with and without replacement.

SL 4.7

Discrete random variables

Probability distributions and expected value, including E(X) = 0 as the condition for a fair game.

SL 4.8

The binomial distribution

Its mean and variance, with probabilities found using technology. Formal proof of the mean and variance is not required.

SL 4.9

The normal distribution

Properties and diagrammatic representation; normal probability calculations; inverse normal calculations where the mean and standard deviation are given.

SL 4.10

Spearman’s rank correlation

Spearman’s rₛ found with technology, averaging tied ranks; and knowing when it is the right measure — it tolerates outliers better than Pearson’s r, which only tests for linearity.

SL 4.11

Hypothesis testing

Null and alternative hypotheses, significance levels and p-values; the χ² test for independence with contingency tables and degrees of freedom; the χ² goodness of fit test; and the two-sample t-test, one- and two-tailed. Sits at SL in AI — in AA there is no hypothesis testing at all.

AHL 4.12

Data collection, reliability and validity

Designing valid surveys and questionnaires; selecting and categorizing variables; choosing degrees of freedom when parameters are estimated; and reliability tests (test-retest, parallel forms) against validity tests (content, criterion-related).

AHL 4.13

Non-linear regression

Least squares regression curves via technology; the sum of square residuals as a measure of fit; the coefficient of determination R² and its connection to Pearson’s r — and why R² alone is a poor way to choose between models.

AHL 4.14

Linear transformations and unbiased estimates

E(aX + b) and Var(aX + b); expected value and variance of linear combinations of independent random variables; and the sample mean and sₙ₋₁² as unbiased estimates of μ and σ².

AHL 4.15

The central limit theorem

A linear combination of independent normal variables is normal; the distribution of the sample mean; and the central limit theorem, with n > 30 taken as sufficient in exams.

AHL 4.16

Confidence intervals

Confidence intervals for the mean of a normal population — the normal distribution when σ is known, the t-distribution when it is not, regardless of sample size.

AHL 4.17

The Poisson distribution

Its mean and variance, the sum of two independent Poisson variables, the conditions under which it models a situation, and choosing between the normal, binomial and Poisson distributions in context.

AHL 4.18

Further hypothesis testing

Critical values and critical regions; tests for a population mean, for a proportion using the binomial, and for a Poisson mean; testing whether ρ = 0 for bivariate normal data; and Type I and Type II errors with their probabilities.

AHL 4.19

Markov chains

Transition matrices and their powers; transition diagrams for discrete dynamical systems; regular Markov chains and initial state matrices; and steady-state long-term probabilities by repeated multiplication or by solving a linear system.

Codes marked AHL are Higher Level only. The SL codes are shared with AI SL — AI HL covers both.

The smallest SL topic in AI, kept to optimisation and areas — kinematics is explicitly excluded at Standard Level. HL more than doubles it, ending in coupled differential equations and phase portraits.

SL 5.1

Limits and the derivative

The informal concept of a limit; the derivative read as a gradient function and as a rate of change. Formal analytic methods for limits are not required.

SL 5.2

Increasing and decreasing functions

Graphical interpretation of f′(x) > 0, f′(x) = 0 and f′(x) < 0.

SL 5.3

Differentiating polynomials

The derivative of axⁿ and of sums of such terms, where all exponents are integers.

SL 5.4

Tangents and normals

Tangents and normals at a given point, and their equations.

SL 5.5

Introduction to integration

Integration as anti-differentiation; the constant determined from a boundary condition; definite integrals using technology; area between a curve and the x-axis where f(x) > 0.

SL 5.6

Stationary points

Values of x where the gradient is zero; solving f′(x) = 0; local maximum and minimum points — and the awareness that a local extremum need not be the greatest or least value on the domain.

SL 5.7

Optimisation in context

Applied optimisation such as maximizing volume for a given surface area. Note that kinematics questions are not set at AI SL.

SL 5.8

The trapezoidal rule

Approximating the area under a curve using intervals of equal width, and recognising whether the estimate is an over- or under-estimate.

AHL 5.9

Rules of differentiation

Derivatives of sin x, cos x, tan x, eˣ, ln x and xⁿ for n ∈ ℚ; the chain, product and quotient rules; and related rates of change.

AHL 5.10

The second derivative

Both notations; the second derivative test to distinguish a maximum from a minimum; points of inflexion; and the terms concave-up and concave-down.

AHL 5.11

Further integration

Definite and indefinite integration of xⁿ for n ∈ ℚ including n = −1, plus sin x, cos x and 1/cos²x; integration by inspection or by substitution.

AHL 5.12

Areas and volumes of revolution

Area enclosed by a curve and the x- or y-axis over an interval; volumes of revolution about either axis.

AHL 5.13

Kinematics

Displacement, velocity and acceleration; displacement against total distance travelled over an interval; speed as the magnitude of velocity. HL only in AI.

AHL 5.14

Differential equations

Setting up a differential equation from a context and solving it by separation of variables, including the general solution.

AHL 5.15

Slope fields

Slope field diagrams and reading the shape of solution curves from them.

AHL 5.16

Euler’s method

Numerical solution of first order differential equations, and of coupled systems, using Euler’s method — normally set up in a spreadsheet.

AHL 5.17

Phase portraits

Phase portraits for coupled linear differential equations; qualitative analysis of future paths for distinct real, complex and imaginary eigenvalues; sketching trajectories and identifying equilibrium points, stable populations and saddle points.

AHL 5.18

Second order differential equations

Solutions of d²x/dt² equations, including by reduction to a coupled system and investigation with the phase-portrait method.

Codes marked AHL are Higher Level only. The SL codes are shared with AI SL — AI HL covers both.

Is this the current syllabus?

Everything above is the Mathematics: Applications and Interpretation — Higher Level guide for first exams 2021, which remains the live syllabus for every candidate sitting through May 2028. Pages that advertise a “2026 syllabus” are describing this same document.

The IB is redeveloping the DP mathematics courses. The new guides launch in February 2027, with first teaching in August 2027 and first assessment in May 2029. The IB describes the change as refinement rather than reinvention — improved coherence and reduced content overload, with no new content areas — so if you are studying now, this is your syllabus and it is not about to move.

Source: IB curriculum updates — mathematics: analysis and approaches.

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