IB Topics and subtopics for all courses

Specific to AI SL and AI HL, this content covers the practical numerical skills that underpin the course. Students work with approximation techniques including rounding to decimal places and significant figures, calculate upper and lower bounds of rounded numbers, and determine percentage errors. Estimation is emphasised throughout, with students expected to assess the reasonableness of results in context.

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Shared across all four courses, arithmetic and geometric sequences and series are studied using the formulae for the nth term and the sum of the first n terms, with sigma notation used to express sums compactly. Applications include real-life contexts such as population growth, salary changes and the spread of disease. The sum of infinite convergent geometric series is also addressed, requiring the condition |r| < 1. In AA the treatment is more formal and algebraically driven; in AI the emphasis falls on interpretation, modelling and the use of technology to generate and display sequences.

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Covered in AA SL, AA HL and AI HL, the laws of exponents and logarithms are developed and applied to solve equations and model real-world phenomena. In AA, students work with integer and rational exponents, logarithms in base 10 and e, the full suite of logarithm laws, the change of base rule, and analytical solutions of exponential equations. In AI HL the treatment is more focused, covering the laws of logarithms and rational exponents with an emphasis on their role in scaling data and supporting modelling tasks.

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Specific to AI SL and AI HL, financial mathematics addresses the practical applications of geometric sequences and series. Students work with compound interest, annual depreciation, and the real value of investments accounting for inflation. AI HL extends this to amortization and annuities, with students using technology — including built-in financial packages on graphic display calculators — to solve problems involving loans, repayments and retirement planning.

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Covered in AA SL and AA HL only, the binomial theorem is introduced for expanding (a + b)^n where n is a positive integer, using both Pascal's triangle and the combination formula nCr. Students apply the theorem to find specific terms in an expansion and connect counting principles to its development. The AI courses do not include the binomial theorem in their syllabus.

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Exclusive to AA HL, the binomial theorem is extended to the case where n is a rational number. Students work with expansions of the form (a + b)^n for n ∈ ℚ, understanding that these produce infinite series valid within a restricted domain. The connection to Maclaurin series and power series expansions is noted explicitly in the syllabus guidance.

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Exclusive to AA HL, counting principles introduce the fundamental tools of combinatorics. Students work with the product and sum principles, factorial notation, permutations and combinations, applying these to count arrangements and selections in a variety of contexts. Circular arrangements and permutations of identical objects are not required.

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Covered in AA HL and AI HL, complex numbers are introduced in Cartesian form z = a + bi, with students learning the terminology of real and imaginary parts, conjugates, modulus and argument, and representing complex numbers on the Argand diagram. Operations — addition, subtraction, multiplication, division and powers — are performed both by hand and with technology. In AI HL, complex numbers also appear as solutions to quadratic equations with negative discriminant, and the geometric interpretation of operations on the Argand diagram is emphasised.

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Covered in AA HL and AI HL, the treatment of complex numbers is extended to modulus-argument (polar) form and Euler's form. Students convert between Cartesian, polar and Euler representations and perform products, quotients and powers in these forms. In AA HL, De Moivre's theorem is introduced and applied to find powers and roots of complex numbers, with proof by induction for the case n ∈ ℤ⁺. In AI HL the focus is on geometric interpretation and applications such as adding sinusoidal functions with the same frequency but different phase shifts.

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Exclusive to AI HL, matrices are introduced with students learning to define and perform operations including addition, subtraction, scalar multiplication and matrix multiplication, understanding why multiplication is non-commutative. Determinants and inverses are calculated by hand for 2×2 matrices and with technology for larger ones, and used to solve systems of linear equations. The content extends to eigenvalues and eigenvectors for 2×2 matrices, including diagonalisation and its application to computing powers of matrices — connecting to real-life contexts such as population modelling and Markov chains.

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Covered in AA SL and AA HL, deductive proof introduces students to the nature and structure of mathematical reasoning. Students construct simple proofs, both numerical and algebraic, working from a left-hand side to right-hand side format using correct notation for equality and identity. At AA HL the scope broadens to include proof by equivalence, proof by exhaustion, disproof by counterexample, and proof by contrapositive, with students expected to communicate their reasoning with full logical rigour.

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Exclusive to AA HL, proof by contradiction develops students' ability to establish mathematical statements by assuming the negation and deriving a logical contradiction. Classic examples include proving the irrationality of √3 and the infinitude of primes. Where a counterexample is used to disprove a statement, students must not only identify it but also explain clearly why it invalidates the claim.

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Exclusive to AA HL, proof by mathematical induction provides a formal technique for establishing results that hold for all positive integers. Students follow the structured process of verifying a base case, assuming the result holds for n = k, and proving it for n = k + 1. Applications span a wide range of contexts including divisibility, summation formulae, inequalities, and results involving complex numbers and differentiation.

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Addressed in AA HL, AI HL and AI SL, systems of linear equations in up to three unknowns are solved using a range of methods depending on the course. In AI SL, technology is used to find unique solutions. In AI HL and AA HL, students work with cases involving unique, infinite and no solutions, interpreting these geometrically as intersecting, coincident or parallel planes. AA HL requires algebraic methods including row reduction, while AI HL connects this content to matrices and the inverse matrix method.

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Common to all four courses, linear functions and graphs establish the foundational language of straight-line relationships. Students work with multiple forms of the equation of a line — gradient-intercept, general and point-gradient — and apply conditions for parallel and perpendicular lines. In AI, piecewise linear models are also addressed with practical contexts such as mobile phone charges and conversion graphs, reflecting the modelling-first philosophy of that course.

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Shared across all four courses, functions are introduced formally with students developing fluency in domain, range, function notation and graphical representation. Composite functions are formed and inverse functions found, with students understanding that inverses exist only for one-to-one functions and are reflected in the line y = x. In AA the treatment is more analytical; in AI HL the content extends to include domain restriction when finding inverses and the formal composition f ∘ f⁻¹ = x.

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Covered across all four courses, quadratic functions are studied through multiple representations — standard, factored and vertex form. Students solve quadratic equations using factoring, completing the square and the quadratic formula, and use the discriminant to determine the nature of roots. In AA the algebraic treatment is more rigorous; in AI the emphasis is on using quadratics as models for real-life situations such as projectile motion and cost functions, with technology used to find roots and key features.

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Covered across all four courses at the level of simple rational functions, students work with the reciprocal function f(x) = 1/x and functions of the form f(x) = (ax + b)/(cx + d), identifying vertical and horizontal asymptotes and sketching graphs. In AA HL the treatment extends to more complex rational functions with quadratic denominators and oblique asymptotes, covered under the Further functions subtopic, alongside partial fractions as a decomposition technique with applications to integration.

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Addressed across all four courses, exponential and logarithmic functions are studied as inverses of each other, with students working with their graphs, equations and key properties. In AA, students solve equations analytically and link this to the laws of exponents and logarithms developed in Topic 1. In AI, exponential models covering growth and decay are central; AI HL extends this to natural logarithmic models, logistic models and the use of logarithmic scaling for data with wide-ranging values.

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Covered in AA SL, AA HL and AI HL, transformations of functions address translations, reflections, vertical and horizontal stretches, and their algebraic representations. Students apply these to a variety of function types and consider the significance of the order in which composite transformations are performed. In AI HL the emphasis is on applying transformations across all functions in the course, including in real-life modelling contexts; in AA the treatment is more formal with explicit links to composite functions.

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Exclusive to AA HL, polynomial functions are studied in depth through their graphs, zeros, roots and factors. Students apply the factor and remainder theorems to factorise and solve polynomial equations, and use the sum and product of roots relationships. Cubic and quartic functions are sketched and analysed, and complex roots of polynomial equations with real coefficients are addressed, linking directly to the complex numbers subtopics.

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Exclusive to AA HL, this content extends the functions material covered at SL into more sophisticated territory. Students investigate even and odd functions, work with the graphs of y = |f(x)| and y = f(|x|), and solve modulus equations and inequalities both analytically and graphically. Rational functions of more complex forms — including those with quadratic denominators and oblique asymptotes — are analysed and sketched, and partial fractions are introduced as a decomposition technique with direct applications to integration later in the course.

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Covered in AA SL, AA HL and AI HL, radian measure is introduced as an alternative to degrees and the conversion between the two systems is established. Students calculate arc length and sector area using radians and work with exact multiples of π. In AA this content appears at SL and underpins the remainder of the trigonometry topic; in AI it is an HL extension, with AI SL students working exclusively in degrees throughout.

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Common to all four courses, this content covers the foundational trigonometric ratios applied to right-angled triangles, as well as distances, midpoints and volumes and surface areas of standard three-dimensional solids. The unit circle is introduced to extend the definitions of cosθ and sinθ beyond acute angles, covering exact values at standard angles and the Pythagorean identity. In AA the unit circle and its quadrant relationships are developed at SL with formal treatment of the ambiguous case of the sine rule; in AI HL this material is an AHL extension.

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Shared across all four courses, this content develops the tools needed to solve triangles that are not right-angled. Students apply the sine rule, the cosine rule, and the formula for the area of a triangle (½ab sinC) in two- and three-dimensional contexts, including problems involving angles of elevation and depression and bearings. Applications such as triangulation and map-making are highlighted across both AA and AI.

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Covered in AA SL, AA HL and AI HL, this content addresses the solution of trigonometric equations in a finite interval, both graphically and analytically, including equations that reduce to quadratics in sinx, cosx or tanx. The Pythagorean identity and double angle identities for sine and cosine are applied. In AA this is SL content; in AI it is an HL extension where the unit circle provides the foundation and graphical methods are emphasised. The general solution of trigonometric equations is not required in either course.

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Common to all four courses, this content applies trigonometric functions to real-world periodic phenomena. Students work with functions of the form f(x) = a sin(b(x + c)) + d and f(x) = a cos(b(x + c)) + d, identifying amplitude, period and principal axis, and fitting sinusoidal models to data. Contexts include tidal patterns, Ferris wheels and temperature cycles. In AI HL the treatment is more detailed, incorporating phase shifts, radian measure and regression to fit sinusoidal models; AI SL students work only in degrees and are not required to translate between sin and cos forms.

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Exclusive to AA HL, this content extends the trigonometry covered at SL into more advanced territory. Students are introduced to the reciprocal trigonometric functions — secant, cosecant and cotangent — along with the associated Pythagorean identities, and the inverse trigonometric functions arcsin, arccos and arctan are defined with their domains, ranges and graphs. Compound and double angle identities are developed and applied, including the double angle identity for tangent, and students explore the symmetry properties and relationships between trigonometric functions across different quadrants.

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Addressed at HL in both AA and AI, this content introduces the concept of a vector and establishes the foundational language and operations of vector algebra. Students work with vectors in two and three dimensions, represented as directed line segments and in component form using base vectors i, j and k, covering addition, subtraction, scalar multiplication, magnitude and unit vectors. The scalar product is defined and used to find angles between vectors and to establish perpendicularity. In AA HL the vector product is also introduced, with its geometric interpretation in terms of area and applications to proof using vector geometry.

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Addressed at HL in both AA and AI, this content extends vector knowledge to lines and planes in two and three dimensions. Students work with vector, parametric and Cartesian equations of lines, find angles between lines, and determine whether lines are coincident, parallel, intersecting or skew. In AA HL the scope extends further to include vector and Cartesian equations of planes, angles between lines and planes, and the intersection of two or three planes — linking closely to systems of linear equations. In AI HL the focus is on modelling linear motion with constant velocity, finding positions and intersections, and working with variable velocity in two dimensions, including projectile and circular motion as special cases.

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Exclusive to AI HL, geometric transformations are represented and applied using matrix multiplication. Students perform reflections, stretches, enlargements, translations and rotations on points in two dimensions, and combine these as composite transformations. The geometric interpretation of the determinant as a scale factor for area is developed, and iterative matrix techniques for generating fractals are introduced, reflecting the broader AI emphasis on technology and real-world structural applications.

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Exclusive to AI HL, graph theory introduces the mathematical study of networks. Students work with simple, complete, weighted and directed graphs, and understand key terminology including degree, subgraphs and trees. Adjacency matrices connect this content directly to the matrices subtopic. Algorithms are central: students apply Kruskal's and Prim's algorithms for minimum spanning trees, and tackle the Chinese postman and travelling salesman problems using the nearest neighbour and deleted vertex algorithms to find upper and lower bounds on optimal routes.

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Common to all four courses, this content establishes the foundational concepts and vocabulary of statistical analysis. Students distinguish between populations and samples, discrete and continuous data, and explore sampling techniques including simple random, systematic, stratified, quota and convenience sampling. Data is presented and interpreted using frequency tables, histograms, cumulative frequency graphs and box-and-whisker diagrams, with outliers identified and discussed in context. Measures of central tendency and dispersion — mean, median, mode, interquartile range, standard deviation and variance — are calculated and interpreted throughout.

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Shared across all four courses, this content develops students' ability to analyse relationships between two quantitative variables. Pearson's product-moment correlation coefficient r is calculated using technology and interpreted in terms of the strength and direction of linear association, with the critical distinction between correlation and causation emphasised throughout. Students find and use the regression line of y on x for prediction, interpreting parameters in context. In AI, Spearman's rank correlation coefficient is also introduced as an alternative for non-linear monotonic relationships, and AI HL extends the treatment to non-linear regression models including quadratic, cubic, exponential and power forms, using the coefficient of determination R² to evaluate model fit.

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Common to all four courses, this content develops students' understanding of probability from first principles. Students work with sample spaces, combined and mutually exclusive events, and calculate probabilities using Venn diagrams, tree diagrams and tables of outcomes. Conditional probability and independence are central, with students applying P(A|B) = P(A∩B)/P(B) and the independence condition P(A∩B) = P(A)P(B). Probabilities with and without replacement are also covered. In AA HL, Bayes' theorem is introduced as a further extension for a maximum of three events.

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Shared across all four courses, this content introduces discrete random variables and their probability distributions. Students calculate the expected value E(X) and apply it in contexts such as fair game problems. In AA HL the variance of a discrete random variable is formally defined and properties of linear transformations E(aX + b) = aE(X) + b and Var(aX + b) = a²Var(X) are derived. The subtopic provides the theoretical foundation for the specific named distributions — binomial, Poisson and normal — covered in subsequent subtopics.

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Addressed at HL only in both AA and AI, this content introduces probability density functions and uses integration to calculate probabilities, mean, median, mode and variance for continuous distributions. Properties of linear transformations are established, and in AI HL the central limit theorem is introduced — showing that the sample mean approaches normality for sufficiently large n — alongside unbiased estimators for population mean and variance. The connection between this content and the normal distribution and hypothesis testing subtopics is explicit in both courses.

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Common to all four courses at SL, the binomial distribution is introduced as a model for the number of successes in a fixed number of independent Bernoulli trials. Students identify situations where the binomial model is appropriate, calculate probabilities using technology, and work with the mean and variance of the distribution. The link to counting principles and expected value is noted, and the use of technology to find binomial probabilities is expected in examinations across all courses.

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Exclusive to AI HL, the Poisson distribution is introduced as a model for events that occur independently at a uniform average rate. Students learn to identify contexts where the Poisson model is appropriate, work with its mean and variance (both equal to λ), and apply the result that the sum of two independent Poisson distributions is also Poisson. A key skill is selecting between the normal, binomial and Poisson distributions given a real-world context.

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Shared across all four courses, the normal distribution is introduced as a model for continuous data, covering its key properties: symmetry about the mean, the 68-95-99.7 rule, and the bell-curve shape. Students calculate normal probabilities and perform inverse normal calculations using technology. In AA HL, standardisation using z-values is developed and applied to find unknown means or standard deviations when probabilities are known. In AI HL, the normal distribution also appears in the context of confidence intervals and hypothesis testing.

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Exclusive to AI HL, this content introduces confidence intervals for the mean of a normal population. Students use the normal distribution when the population standard deviation σ is known, and the t-distribution when σ is unknown, regardless of sample size. Results are interpreted in context, with an emphasis on understanding what a confidence interval communicates rather than just computing it. The subtopic connects directly to the broader statistical inference theme running through AI HL.

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Covered in AI SL and AI HL, hypothesis testing introduces the formal framework for making statistical inferences. Students formulate null and alternative hypotheses, interpret p-values against significance levels, and apply the χ² test for independence in contingency tables, the χ² goodness of fit test, and the t-test for comparing means of two populations. AI HL extends this substantially to include critical regions, tests for population mean using normal and t-distributions, tests for proportion using the binomial distribution, tests using the Poisson distribution, hypothesis testing for the correlation coefficient ρ, and Type I and Type II errors with associated probability calculations.

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Shared across all four courses at SL, limits are introduced informally with students estimating values from tables and graphs and interpreting the derivative as the gradient of a curve approached as a limiting process. In AA HL the treatment becomes formal: continuity and differentiability are defined precisely, the derivative is developed from first principles using the definition f'(x) = lim[f(x+h)-f(x)]/h, higher derivatives are introduced, and L'Hôpital's rule is applied to evaluate limits of indeterminate forms. For AI SL and AA SL the focus remains on intuitive understanding and interpreting the gradient as a rate of change.

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Covered across all four courses, differentiation rules are developed progressively depending on the course. All students begin with the power rule and interpret increasing and decreasing functions using the sign of f'(x). In AA SL, AA HL and AI HL, students extend to the chain rule, product rule and quotient rule, differentiating trigonometric, exponential and logarithmic functions. AA HL adds derivatives of inverse and reciprocal trigonometric functions and introduces higher derivatives. For AI SL, only the basic power rule and differentiation of simple polynomial functions is required.

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Common to all four courses, this content connects the behaviour of a function to the behaviour of its derivative. Students identify intervals where a function is increasing or decreasing, locate stationary points and determine their nature using first and second derivative tests, and identify points of inflection through a change in concavity. The relationships between the graphs of f, f' and f'' are explored using both analytic and graphical approaches. In AA HL, L'Hôpital's rule extends the analysis to function behaviour near asymptotes and indeterminate forms.

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Shared across all four courses, optimization applies differential calculus to find maximum and minimum values of functions in practical contexts. Students set up and solve problems involving areas, volumes, profit, cost and other real-life quantities, confirming the nature of stationary points using derivative tests. In AI the emphasis is on constructing models from context and interpreting results meaningfully; in AA the algebraic setup tends to be more demanding and may involve more complex functions.

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Exclusive to AA HL, implicit differentiation and related rates extend the techniques of differential calculus to functions that are not explicitly defined in terms of x. Students differentiate implicitly using the chain rule and apply this to find gradients of curves defined implicitly. Related rates problems require connecting rates of change of different quantities through the chain rule, and optimisation problems are also addressed here, including cases where the optimum occurs at an endpoint of the domain.

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Shared across all four courses at SL, integration is introduced as the reverse process of differentiation. Students find indefinite integrals of polynomial functions of the form axⁿ, determine constants of integration using boundary conditions, and evaluate definite integrals to find areas under curves where f(x) > 0. In AI SL the trapezoidal rule is also introduced as a numerical method for approximating areas when an analytic approach is unavailable. All courses share this foundational content, with AA and AI HL extending to more sophisticated techniques in subsequent subtopics.

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Covered in AA SL, AA HL and AI HL, integration by substitution extends the basic integration toolkit to a wider class of functions. In AA SL students integrate by inspection and substitution for expressions of the form ∫kg'(x)f(g(x))dx, alongside standard integrals of sinx, cosx, eˣ and 1/x and composites with linear functions. AA HL and AI HL apply substitution to more complex integrands, with substitutions provided in examinations when the integral is not of the standard recognisable form.

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Exclusive to AA HL, integration by parts provides a technique for integrating products of functions using the formula ∫u dv = uv − ∫v du. Students apply this to integrands such as x sinx, ln x and arcsin x, and use repeated integration by parts where necessary — for example for ∫x²eˣdx and ∫eˣsinx dx.

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Exclusive to AA HL, trigonometric integrals apply identities and substitutions to integrate powers and products of trigonometric functions. Students evaluate integrals of powers of sine and cosine — for example ∫sin³x cos²x dx — using the Pythagorean identity sin²x + cos²x = 1 and, for even powers, the double-angle identities; they integrate powers of tangent and secant, such as ∫tan³x sec³x dx, using the identity 1 + tan²x = sec²x. The section also covers trigonometric substitution, where substituting x = a sinθ, x = a tanθ or x = a secθ transforms integrands containing √(a² − x²), a² + x² or √(x² − a²) into trigonometric integrals that standard identities then simplify.

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Exclusive to AA HL, integration by partial fractions uses the decomposition of rational functions — with a maximum of two distinct linear factors in the denominator — to evaluate integrals that would otherwise be intractable. The technique connects directly to the partial fractions content introduced in the Further functions subtopic, and its application to integration is explicitly noted in the syllabus guidance.

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Shared across all four courses, the applications of integration extend beyond finding areas under curves to a range of practical contexts. Students calculate areas enclosed between curves and the x-axis where f(x) may be positive or negative, and areas between two curves. In AA HL and AI HL, volumes of revolution about the x- and y-axes are also addressed. In AI SL, the trapezoidal rule provides a numerical alternative for estimating areas when an exact analytic result is not available.

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Covered in AA SL, AA HL and AI HL, kinematics applies calculus to the motion of a particle. Students work with displacement s, velocity v and acceleration a as related functions of time, differentiating to move from displacement to velocity to acceleration and integrating to reverse the process. Total distance travelled is distinguished from net displacement using the integral of |v(t)|. In AI HL, kinematics also connects to the vectors subtopic through position vectors and variable velocity in two dimensions, including projectile and circular motion as special cases.

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Exclusive to AA HL, Maclaurin series provides a powerful method for representing functions as infinite power series centred at zero. Students develop expansions for standard functions including eˣ, sinx, cosx, ln(1+x) and (1+x)^p, and use substitution, multiplication, differentiation and integration to obtain series for related functions. Convergence is discussed, and Maclaurin series can also be developed as solutions to differential equations, linking this subtopic to both calculus and the broader course.

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Exclusive to AA HL and AI HL, differential equations are approached quite differently across the two courses. In AA HL, students solve first order differential equations analytically using separation of variables, the integrating factor method for equations of the form y' + P(x)y = Q(x), and homogeneous equations via the substitution y = vx; Euler's method is also covered for numerical approximation. In AI HL the emphasis is on modelling and numerical methods: students set up differential equations from context, solve separable equations, interpret slope fields, apply Euler's method using spreadsheets, and analyse systems of coupled differential equations through phase portraits — identifying equilibrium points and long-term behaviour using eigenvalue analysis.

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