IB Maths AA HL syllabus

Numerical technique, sequences and series and their financial applications, and the first formal encounter with proof. At HL this extends into counting, complex numbers, and three further methods of proof.

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, salary progression and the spread of disease.

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. Exam questions expect the GDC financial package.

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

Deductive proof

Simple numerical and algebraic proof, laid out from the left-hand side to the right-hand side, and the difference between equality and identity.

SL 1.7

Laws of exponents and logarithms

Rational exponents; the laws of logarithms; change of base; and solving exponential equations using logarithms.

SL 1.8

Infinite geometric series

The sum of an infinite convergent geometric sequence, and the condition for convergence.

SL 1.9

The binomial theorem

Expansion of (a + b)ⁿ for n ∈ ℕ, using Pascal’s triangle and ⁿCᵣ — found both by formula and by technology.

AHL 1.10

Counting principles and the extended binomial theorem

Permutations and combinations, and the binomial theorem extended to fractional and negative indices (n ∈ ℚ).

AHL 1.11

Partial fractions

Decomposition where the denominator has distinct linear factors and the numerator is of lower degree. Feeds directly into AHL 5.15 integration.

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.

AHL 1.13

Polar and Euler form

Modulus–argument form r(cos θ + i sin θ) and Euler form reⁱᶿ; sums, products and quotients in all three forms and what each does geometrically.

AHL 1.14

De Moivre’s theorem and complex roots

Conjugate roots of real-coefficient polynomials; De Moivre’s theorem including rational exponents; powers and roots of complex numbers.

AHL 1.15

Induction, contradiction and counterexample

Proof by mathematical induction, proof by contradiction, and disproof by counterexample — where stating the counterexample alone earns nothing without the explanation.

AHL 1.16

Systems of linear equations

Up to three equations in three unknowns by row reduction, covering the unique-solution, infinite-solution and inconsistent cases.

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

The unifying idea of a function — as a model, a graph and an object to transform. HL adds polynomial and rational machinery, classification, and modulus and inequality work.

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

Composite and inverse functions

Composite functions, the identity function, and finding an inverse function algebraically.

SL 2.6

The quadratic function

All three forms — general, factorised and vertex — with the graph, y-intercept, axis of symmetry and x-intercepts each reads off.

SL 2.7

Quadratic equations and inequalities

The quadratic formula; solving quadratic inequalities; the discriminant and what it says about the nature of the roots.

SL 2.8

Reciprocal and rational functions

The reciprocal function and its self-inverse nature; rational functions of the form (ax + b)/(cx + d), their graphs and their asymptotes.

SL 2.9

Exponential and logarithmic functions

Their graphs, including aˣ, eˣ, log_a x and ln x — and the fact that each is the inverse of the other.

SL 2.10

Solving equations

Graphically and analytically, including equations with no appropriate analytic route, applied to real-life situations.

SL 2.11

Transformations of graphs

Translations, reflections in both axes, vertical and horizontal stretches, and composite transformations. Transformations of the form f(ax + b) are not required at SL.

AHL 2.12

Polynomial functions

Graphs, zeros, roots and factors; the factor and remainder theorems; the sum and product of the roots of a polynomial equation.

AHL 2.13

Further rational functions

Rational functions with a quadratic in the numerator or the denominator, and the behaviour of their graphs.

AHL 2.14

Odd, even and self-inverse functions

Classification of functions including periodic ones; finding an inverse with a restricted domain; self-inverse functions.

AHL 2.15

Inequalities

Solutions of g(x) ≥ f(x), both graphically and analytically.

AHL 2.16

Modulus functions and equations

The graphs of y = |f(x)|, y = f(|x|), y = 1/f(x) and y = [f(x)]², and the solution of modulus equations and inequalities.

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

Three-dimensional geometry and non-right-angled trigonometry, then the circular functions and their identities. HL is dominated by vectors — lines, planes and their intersections.

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

Radians, arcs and sectors

Radian measure of angles; the length of an arc; the area of a sector.

SL 3.5

The unit circle

Definitions of cos θ and sin θ on the unit circle; tan θ as sin θ / cos θ; exact values at 0, π/6, π/4, π/3, π/2 and their multiples; the ambiguous case of the sine rule.

SL 3.6

Trigonometric identities

The Pythagorean identity cos²θ + sin²θ = 1; the double angle identities for sine and cosine; deducing one ratio from another without finding the angle.

SL 3.7

The circular functions

Graphs of sin x, cos x and tan x — amplitude, period, periodic nature; the composite form a sin(b(x + c)) + d; transformations and real-life contexts.

SL 3.8

Trigonometric equations

Solving over a finite interval, graphically and analytically, including equations that reduce to quadratics in sin x, cos x or tan x. The general solution is not required.

AHL 3.9

Reciprocal and inverse trigonometric functions

sec θ, cosec θ and cot θ; the identities 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ; arcsin, arccos and arctan with their domains, ranges and graphs.

AHL 3.10

Compound angle identities

The compound angle identities, and the double angle identity for tan.

AHL 3.11

Symmetry of trigonometric graphs

Relationships between trigonometric functions and the symmetry properties their graphs express.

AHL 3.12

Vectors

Position and displacement vectors; directed line segments; base vectors i, j, k and components; magnitude and unit vectors; proving geometrical properties with vectors.

AHL 3.13

The scalar product

Definition and properties of the dot product; the angle between two vectors; tests for perpendicular and parallel vectors.

AHL 3.14

Vector equation of a line

In two and three dimensions — vector, parametric and Cartesian forms; the angle between two lines; simple applications to kinematics.

AHL 3.15

Coincident, parallel, intersecting and skew lines

Distinguishing between the four cases, and finding points of intersection.

AHL 3.16

The vector product

Definition and properties of the cross product, and the geometric interpretation of |v × w| as an area.

AHL 3.17

Vector equations of a plane

Parametric form r = a + λb + μc; the scalar-product form r · n = a · n; and the Cartesian equation ax + by + cz = d.

AHL 3.18

Intersections of lines and planes

Intersections of a line with a plane, of two planes and of three planes; the angle between a line and a plane, and between two planes.

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

Collecting, presenting and interpreting data, then probability and the standard distributions. The HL addition is small in hours but conceptually sharp: Bayes and continuous random variables.

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

Regression of x on y

The x on y regression line, and knowing which line is the appropriate one for the prediction being asked for.

SL 4.11

Formal conditional probability

The formal definition P(A|B) = P(A ∩ B)/P(B), and the conditional test for independence.

SL 4.12

Standardisation and z-values

Standardising a normal variable; the z-value as a count of standard deviations from the mean; inverse normal calculations where the mean or standard deviation is unknown.

AHL 4.13

Bayes’ theorem

Use of Bayes’ theorem for a maximum of three events.

AHL 4.14

Continuous random variables

Variance of a discrete random variable; probability density functions; mode and median of a continuous random variable; mean, variance and standard deviation for both cases; the effect of linear transformations of X.

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

The largest topic at HL by teaching hours. Differential and integral calculus and their applications at SL; at HL, first principles, l’Hôpital, further integration techniques, differential equations and Maclaurin series.

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 at SL.

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

Rules of differentiation

Derivatives of xⁿ (n ∈ ℚ), sin x, cos x, eˣ and ln x; differentiating sums and multiples; the chain, product and quotient rules.

SL 5.7

The second derivative

Both notations, and the graphical relationship between the graphs of f, f′ and f″.

SL 5.8

Optimisation and points of inflexion

Local maxima and minima and the tests for them; optimisation problems; points of inflexion with zero and non-zero gradient; concave-up and concave-down.

SL 5.9

Kinematics

Displacement, velocity, acceleration and total distance travelled — and the distinction between displacement and distance over an interval.

SL 5.10

Indefinite integration

Indefinite integrals of xⁿ (n ∈ ℚ), sin x, cos x, 1/x and eˣ, and their composites with a linear function; integration by inspection or substitution.

SL 5.11

Definite integrals and areas

Definite integrals analytically; areas where f(x) may be positive or negative, without technology; areas between two curves.

AHL 5.12

Continuity, differentiability and first principles

Informal continuity and differentiability at a point; limits, convergence and divergence; the definition of the derivative from first principles; higher derivatives.

AHL 5.13

L’Hôpital’s rule and limits

Evaluating limits of quotients, including as x → ∞, using l’Hôpital’s rule — with repeated application — or the Maclaurin series.

AHL 5.14

Implicit differentiation and related rates

Implicit differentiation; related rates of change; optimisation problems including those where the solution sits at an end point.

AHL 5.15

Further derivatives and integrals

Derivatives of tan x, sec x, cosec x, cot x, aˣ, log_a x, arcsin x, arccos x and arctan x, and the corresponding indefinite integrals; partial fractions to rearrange an integrand.

AHL 5.16

Integration by substitution and by parts

Integration by substitution where the integrand is not already in reverse-chain-rule form; integration by parts, including repeated application.

AHL 5.17

Areas and volumes of revolution

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

AHL 5.18

Differential equations

First order equations: numerical solution by Euler’s method, separation of variables, homogeneous equations via the substitution y = vx, and the integrating factor for y′ + P(x)y = Q(x).

AHL 5.19

Maclaurin series

Expansions for eˣ, sin x, cos x, ln(1 + x) and (1 + x)ᵖ; further series by substitution, products, integration and differentiation; series developed from differential equations.

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

Is this the current syllabus?

Everything above is the Mathematics: Analysis and Approaches — 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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