IB Math AI HL Syllabus (2026)
Complete topic outline with assessment structure, expanded formulas, and where students consistently lose marks. Built from 6,500+ hours of teaching this course.
Need help this term? Apply for tutoring →Assessment Structure
The Internal Assessment is a mathematical exploration worth 20% of the final grade. It is marked against five criteria that most students misunderstand without guidance.
Not sure if AI HL is right for you? AI focuses on applied mathematics, statistics, and modelling. AA focuses on pure mathematics and theory. Both come in SL and HL. See the full comparison between all four IB Math courses →
Number & Algebra
1.1 Operations with Numbers
- Numbers in the form a × 10k
- Where 1 ≤ a < 10 and k is an integer
1.2 Arithmetic Sequences and Series
- Formulae for nth term: un = u1 + (n − 1)d
- Sum of first n terms: Sn = n2(2u1 + (n − 1)d)
- Use of sigma (Σ) notation, e.g. Σk=1n (3k + 2)
- Applications: analysis, interpretation, prediction when models are not perfectly arithmetic
1.3 Geometric Sequences and Series
- Formulae for nth term: un = u1 rn−1
- Sum of first n terms: Sn = u1(rn − 1)r − 1
- Sigma notation for sums of geometric sequences
- Identify first term and ratio; applications
1.4 Financial Applications
- Compound interest: FV = PV × (1 + r100k)nk
- Annual depreciation
1.5 Laws of Exponents and Logarithms
- Laws of exponents with integer exponents
- Introduction to logarithms: log10 x and ln x
- Numerical evaluation using technology
1.6 Approximation
- Decimal places and significant figures
- Upper and lower bounds of rounded numbers
- Percentage errors: ε = |vA − vEvE| × 100%, estimation
1.7 Amortization and Annuities
- Use of technology to calculate payments and balances
1.8 Solving Equations with Technology
- Systems of linear equations (up to 3 variables)
- Polynomial equations
1.9 Laws of Logarithms
- loga(xy) = logax + logay
- loga(xy) = logax − logay
- loga(xm) = m · logax
- Condition: a, x, y > 0
1.10 Simplifying Expressions
- Simplifying expressions involving rational exponents
1.11 Infinite Geometric Sequences
- Sum of infinite geometric sequences
- Formula: S∞ = u11 − r, where |r| < 1
1.12 Complex Numbers
- Cartesian form: z = a + bi
- Terms: real part, imaginary part, conjugate z*, modulus |z|, argument
- Operations: sums, differences, products, quotients, powers (Cartesian)
- Complex plane and solutions of quadratics ax2 + bx + c = 0 where Δ < 0
1.13 Polar and Exponential Forms
- Polar form: z = r(cos θ + i sin θ) = r cis θ
- Exponential form: z = r eiθ
- Converting between Cartesian, polar, and exponential forms
- Products, quotients, and powers in polar/exponential forms
1.14 Matrices
- Definition: element, row, column, order
- Algebra: equality, addition, subtraction, scalar multiplication, matrix multiplication
- Determinants and inverses (2×2 by hand, larger with technology)
- Solving systems of equations Ax = b using inverse x = A−1b
1.15 Eigenvalues and Eigenvectors
- Characteristic polynomial of 2×2 matrices: det(A − λI) = 0
- Diagonalization with distinct real eigenvalues: A = PDP−1
- Applications to powers of 2×2 matrices
Functions
2.1 Equations of a Straight Line
- Different forms: gradient, intercepts
- Parallel lines: m1 = m2
- Perpendicular lines: m1 × m2 = −1
2.2 Concept of a Function
- Domain, range, graph, notation (f(x), v(t), C(n))
- Inverse function as reflection in y = x, notation f−1(x)
2.3 Graphing Functions
- Sketch from context or data; transfer from screen to paper
- Graph sums and differences using technology
2.4 Key Features of Graphs
- Determine points of intersection of curves/lines using technology
2.5 Modelling with Functions
- Linear: f(x) = mx + c
- Quadratic: f(x) = ax2 + bx + c, a ≠ 0; axis, vertex, zeros, intercepts
- Exponential: f(x) = k · ax + c, f(x) = k · erx + c; horizontal asymptote
- Direct/inverse variation: f(x) = a · xn, n ∈ ℤ
- Cubic: f(x) = ax3 + bx2 + cx + d
- Sinusoidal: f(x) = a sin(bx) + d, f(x) = a cos(bx) + d
2.6 Modelling Skills
- Develop, fit, test, reflect, and use models
- Select a reasonable domain
- Justify the choice of a model based on data, curve shape, and context
2.7 Composite and Inverse Functions
- Composite functions: (f ∘ g)(x) = f(g(x))
- Inverse functions with domain restriction; find inverse
2.8 Transformations
- Translations: y = f(x) + b, y = f(x − a)
- Reflections: y = −f(x), y = f(−x)
- Vertical stretch: y = p · f(x)
- Horizontal stretch: y = f(qx)
- Composite transformations
2.9 Logarithmic, Sinusoidal, Logistic & Piecewise
- Logarithmic model: f(x) = a + b · ln(x)
- Sinusoidal models: f(x) = a · sin(b·x − c) + d, f(x) = a · cos(b·x − c) + d
- Logistic model: f(x) = L1 + C · e−kx, where L, C, k > 0
- Piecewise models: defined for specific intervals
Geometry & Trigonometry
3.1 Distance, Midpoint, and Solids
- Distance between two points: d = √((x2−x1)2 + (y2−y1)2 + (z2−z1)2)
- Midpoint: M = (x1+x22, y1+y22, z1+z22)
- Volume and surface area: right pyramid, right cone, sphere, hemisphere, and combinations
- Angle between two intersecting lines or between a line and a plane
3.2 Trigonometry in Triangles
- Right-angled triangles: sin θ = opphyp, cos θ = adjhyp, tan θ = oppadj
- Sine rule: asin A = bsin B = csin C
- Cosine rule: c2 = a2 + b2 − 2ab cos C
- Area of triangle: Area = 12 · a · b · sin C
3.3 Applications of Trigonometry
- Applications including Pythagoras’ theorem: a2 + b2 = c2
- Angles of elevation and depression
- Constructing labelled diagrams from statements
3.4 The Circle
- Radian measure of angles
- Length of an arc: L = r · θ
- Area of a sector: A = 12 · r2 · θ
3.5 Equations of Perpendicular Bisectors
- Find perpendicular bisector of a line segment
- Determine slope of perpendicular bisector
- Equation using point-slope form: y − y0 = m(x − x0)
3.6 Voronoi Diagrams
- Sites, vertices, edges, cells
- Adding a site to an existing diagram
- Nearest neighbour interpolation
- Applications such as the “toxic waste dump” problem
3.7 Radian Measure
- Definition of a radian
- Conversion between degrees and radians
- Using radians for sector area and arc length calculations
3.8 Unit Circle & Identities
- Definitions: cos θ, sin θ in unit circle
- Pythagorean identity: cos2 θ + sin2 θ = 1
- Tangent: tan θ = sin θcos θ
- Extension of sine rule to ambiguous case
- Graphs of f(x) = sin x and f(x) = cos x
- Graphical methods for solving trig equations in finite intervals
3.9 Geometric Transformations
- Transformations: reflections, stretches, enlargements, translations, rotations
- Matrix form: [a b; c d] · [x; y] + [e; f]
- Compositions of transformations
- Determinant interpretation: Area of image = |det A| × Area of object
3.10 Vectors – Concepts
- Vector vs scalar; representation using directed line segments
- Unit vectors; base vectors i, j, k
- Components: v = v1i + v2j + v3k
- Zero vector, negative vector, position vectors
- Rescaling and normalizing vectors
3.11 Vector Equation of a Line
- Line in 2D and 3D: r = a + λb, where b is the direction vector
3.12 Kinematics (Vectors)
- Linear motion with constant velocity: r = r0 + vt
- Relative position and motion with variable velocity in two dimensions
3.13 Scalar and Vector Products
- Scalar product: v · w = |v||w| cos θ (angle between vectors)
- Vector product: v × w = |v||w| sin θ (geometric interpretation)
- Components of vectors
3.14 Graph Theory – Basic Concepts
- Graphs, vertices, edges, adjacent vertices/edges
- Degree of a vertex
- Types: simple, complete, weighted, directed (in/out degree)
- Subgraphs, trees
3.15 Adjacency Matrices and Walks
- Adjacency matrices and weighted adjacency tables
- Number of k-length walks between vertices
- Transition matrices for strongly-connected, undirected, or directed graphs
3.16 Tree and Cycle Algorithms
- Walks, trails, paths, circuits, cycles
- Eulerian trails and circuits
- Hamiltonian paths and cycles
- Minimum spanning tree: Kruskal’s and Prim’s algorithms
- Chinese postman problem: shortest route covering all edges
- Travelling salesman problem: nearest neighbour (upper bound) and deleted vertex (lower bound)
Statistics & Probability
4.1 Population and Sampling
- Concepts of population, sample, random sample, discrete and continuous data
- Reliability of data sources and bias in sampling
- Interpretation of outliers
- Sampling techniques and their effectiveness
4.2 Presentation of Data
- Frequency distributions (tables) for discrete and continuous data
- Histograms
- Cumulative frequency and cumulative frequency graphs; find median, quartiles, percentiles, range, IQR
- Box-and-whisker diagrams
4.3 Measures of Central Tendency
- Mean, median, mode
- Estimation of mean from grouped data
- Modal class
- IQR, standard deviation, variance
- Effect of constant changes on data
4.4 Linear Correlation
- Scatter diagrams; lines of best fit (by eye through mean point)
- Pearson’s correlation coefficient r
- Regression line of y on x: y = ax + b; interpret parameters
- Use of regression for prediction
4.5 Probability – Basic Concepts
- Trial, outcome, equally likely outcomes, relative frequency, sample space U, event
- Probability: P(A) = n(A)n(U)
- Complementary events: P(A′) = 1 − P(A)
- Expected number of occurrences
4.6 Probability – Diagrams and Rules
- Venn diagrams, tree diagrams, sample space diagrams, tables of outcomes
- Combined events: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
- Mutually exclusive: P(A ∩ B) = 0
- Conditional probability: P(A|B) = P(A ∩ B)P(B)
- Independent events: P(A ∩ B) = P(A) · P(B)
4.7 Discrete Random Variables
- Probability distributions
- Expected value (mean) E(X)
- Applications
4.8 Binomial Distribution
- General notation: X ~ B(n, p)
- Mean: μ = np
- Variance: σ2 = np(1 − p)
4.9 Normal Distribution
- General notation: X ~ N(μ, σ2)
- Properties of normal curve and diagrammatic representation
- Normal probability calculations using technology
- Inverse normal calculations
4.10 Spearman’s Rank Correlation
- Spearman’s rank correlation coefficient rs
- Awareness of Pearson vs Spearman correlation
- Effect of outliers
4.11 Hypothesis Testing
- Null and alternative hypotheses: H0 and H1
- Significance levels, p-values
- Chi-square test (χ2): independence, goodness of fit
- t-test: comparing two population means
- One-tailed and two-tailed tests
4.12 Data Collection and Design
- Surveys, questionnaires, selecting relevant variables
- Categorizing data in χ2 tables and degrees of freedom
- Reliability and validity tests
4.13 Non-linear Regression
- Least squares regression curves using technology
- Sum of square residuals as measure of fit
- Coefficient of determination R2
4.14 Linear Transformation of Variables
- Expected value: E(aX + b) = aE(X) + b
- Variance: Var(aX + b) = a2Var(X)
- Unbiased estimates: sample mean and sample variance s2n−1
4.15 Normal Variables and CLT
- Linear combination of independent normal variables is normal
- Central Limit Theorem: X̄ ~ N(μ, σ2n)
4.16 Confidence Intervals
- Confidence intervals for the mean of a normal population
4.17 Poisson Distribution
- General notation: X ~ Po(m)
- Mean and variance: E(X) = Var(X) = m
- Sum of independent Poisson distributions is Poisson
4.18 Hypothesis Testing – Advanced
- Critical values and regions
- Test for population mean, proportion, Poisson mean
- Testing population correlation ρ = 0
- Type I and II errors, probability calculations
4.19 Markov Chains
- Transition matrices, powers of matrices
- Regular Markov chains, initial state probabilities
- Steady state and long-term probabilities
Calculus
5.1 Introduction to Limits and Derivatives
- Concept of a limit
- Derivative interpreted as gradient function and rate of change
5.2 Increasing and Decreasing Functions
- Graphical interpretation: f′(x) > 0, f′(x) = 0, f′(x) < 0
5.3 Derivatives of Polynomial Functions
- f(x) = axn ⇒ f′(x) = anxn−1, n ∈ ℤ
- Derivative of sums of power terms
5.4 Tangents and Normals
- Tangent and normal lines at a given point
- Equations of tangents and normals
5.5 Introduction to Integration
- Anti-differentiation of axn + bxm + …, n ≠ −1
- Notation: ∫ f(x) dx
- Anti-differentiation with boundary condition
- Definite integrals using technology
- Area of a region enclosed by y = f(x) and the x-axis where f(x) > 0
5.6 Stationary Points
- Values of x where gradient is zero: f′(x) = 0
- Local maximum and minimum points
5.7 Optimisation Problems
- Solving context-based optimisation problems
5.8 Approximating Areas
- Trapezoidal rule for approximating areas under curves
5.9 Derivatives of Standard Functions
- ddx(sin x) = cos x, ddx(cos x) = −sin x
- ddx(tan x) = sec2 x, ddx(ex) = ex, ddx(ln x) = 1x
- Rules: Chain rule, product rule, quotient rule
- Related rates of change
5.10 Second Derivative
- Notation: d2ydx2 and f″(x)
- Second derivative test for maxima and minima
5.11 Integration of Standard Functions
- Definite and indefinite integrals of xn, sin x, cos x, 1cos2 x, ex
- Integration by inspection or substitution: ∫ f(g(x)) g′(x) dx
5.12 Areas and Volumes
- Area of region enclosed by a curve and the x– or y-axis
- Volumes of revolution about the x-axis or y-axis
5.13 Kinematics and Calculus
- Displacement s, velocity v = dsdt, acceleration a = dvdt
- Displacement: ∫t1t2 v(t) dt
- Total distance: ∫t1t2 |v(t)| dt
- Speed = magnitude of velocity
5.14 Modelling with Differential Equations
- Setting up models/differential equations from context
- Solving by separation of variables
5.15 Slope Fields
- Sketching slope fields and diagrams for differential equations
5.16 Euler’s Method
- Numerical solutions for first-order differential equations: dydx = f(x, y)
- Numerical solutions for coupled systems
5.17 Phase Portraits
- Coupled differential equations: dxdt = ax + by, dydt = cx + dy
- Qualitative analysis for distinct, real, complex, imaginary eigenvalues
- Sketching trajectories, identifying equilibrium points, stable populations, and saddle points
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Where AI HL Students Lose Marks
AI HL is the most statistics-intensive course in the IB Mathematics curriculum. Students often underestimate the depth required and the precision expected in interpreting and communicating results.
What Separates a 5 from a 7 in AI HL
Understands the content but loses marks on execution. Can set up a hypothesis test but writes a vague conclusion. Can run a regression but doesn’t justify the model choice. Finishes the paper but leaves marks on the table through imprecise communication.
Does what the examiner expects at every step. Defines variables. States assumptions. Interprets results in context. Shows enough working to earn method marks even when the final answer is wrong. The mathematics is the same. The difference is precision under pressure.
Most students who improve from a 5 to a 7 do not learn new content. They learn how to present their working in the format examiners actually mark for.
How Tutoring Helps in AI HL
At HL level, covering content is not enough. Students need fluency with interpretation, structured written communication, and strategies specific to how AI HL papers are marked. Work is adapted to each student’s weak points and adjusted as those improve.
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See student results →Frequently Asked Questions
Common questions about the IB Math AI HL course, exams, and preparation.
AI HL covers all SL content in greater depth, plus substantial HL-only material including advanced statistics, differential equations, Voronoi diagrams, and graph theory.
The course places heavy emphasis on modelling, data analysis, and the interpretation of mathematical results in real-world contexts.
Assessment consists of three written papers and an Internal Assessment. All three papers allow calculator use. Paper 1 contains shorter questions. Paper 2 features longer, context-based problems. Paper 3 presents extended modelling or problem-solving tasks in unfamiliar settings.
Marks are awarded for correct use of mathematics, clear interpretation, and well-structured communication.
Paper 3 is a 60-minute calculator paper worth 20 percent of the final grade. It typically contains two extended problems requiring students to apply mathematics to unfamiliar real-world contexts.
Preparation involves practising with past papers and developing the ability to model situations, interpret results, and communicate reasoning clearly under time pressure.
The Internal Assessment is a mathematical exploration worth 20 percent of the final grade. At HL, examiners expect more sophisticated use of mathematics and deeper critical analysis.
Many AI HL students choose topics involving data collection, statistical modelling, or real-world applications, but the same five criteria apply as at SL.
AI HL includes all SL content plus significant additional material in statistics, calculus, and discrete mathematics. The depth of analysis expected is considerably higher.
HL exams are longer and include Paper 3, which tests modelling and problem solving in unfamiliar contexts. Students considering HL should be comfortable working with data, interpreting results, and reasoning through extended problems. See the full comparison between all four IB Math courses.
Effective preparation involves working consistently with exam-style questions, especially those requiring interpretation, modelling, or extended written responses.
Students who improve most focus on understanding the reasoning behind statistical methods, not just how to perform calculations. They also practise explaining their conclusions clearly in context, since this is where many marks are gained or lost.
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