IB Math AI HL Syllabus
Complete breakdown of topics, assessment structure, and learning resources for IB Mathematics Applications & Interpretations Higher Level. Master sophisticated mathematical applications and advanced data analysis with our expert video tutorials and comprehensive study materials.
IB Math AI HL Syllabus
Assessment Structure
30%
Paper 1
Calculator allowed
30%
Paper 2
Calculator allowed
20%
Paper 3
Extended response
20%
Internal Assessment
Mathematical exploration
Topic 1: 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 and sum of first n terms.
Sigma notation for sums of arithmetic sequences.
Applications: analysis, interpretation, prediction when models are not perfectly arithmetic.
1.3 Geometric sequences and series
Formulae for nth term and sum of first n terms.
Sigma notation for sums of geometric sequences.
Identify first term and ratio; applications.
1.4 Financial applications of geometric sequences and series
Compound interest.
Annual depreciation.
1.5 Laws of exponents and logarithms
Laws of exponents with integer exponents.
Introduction to logarithms (base 10 and e).
Numerical evaluation using technology.
1.6 Approximation
Decimal places and significant figures.
Upper and lower bounds of rounded numbers.
Percentage errors, 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(x/y) = logₐx − logₐy
logₐ(xᵐ) = m·logₐx, for a, x, y > 0
1.10 Simplifying expressions
Simplifying expressions involving rational exponents.
1.11 Sum of infinite geometric sequences
1.12 Complex numbers
Cartesian form: z = a + bi; real part, imaginary part, conjugate, modulus, argument.
Operations: sums, differences, products, quotients, powers (Cartesian).
Complex plane; solutions of quadratic equations ax² + bx + c = 0, b² − 4ac < 0.
1.13 Polar and exponential forms
Modulus–argument (polar) form: z = r(cos θ + i sin θ) = r cis θ.
Exponential form: z = r e^{iθ}.
Converting between Cartesian, polar, exponential; products, quotients, powers.
1.14 Matrices
Definition: element, row, column, order.
Algebra: equality, addition, subtraction, scalar multiplication, matrix multiplication.
Determinants, inverses (2×2 by hand, larger with technology).
Systems of equations: Ax = b, solve using inverse.
1.15 Eigenvalues and eigenvectors
Characteristic polynomial of 2×2 matrices.
Diagonalization with distinct real eigenvalues.
Applications to powers of 2×2 matrices.
Topic 2: 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) = ka^x + c, f(x) = ke^{rx} + c; horizontal asymptote.
Direct/inverse variation: f(x) = axⁿ, 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, and piecewise models
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) = L / (1 + C·e^(−k·x)), where L, C, k > 0
Piecewise models: defined for specific intervals
Topic 3: Geometry & Trigonometry
3.1 Distance, Midpoint, Volume, Surface Area, and Angles
Distance between two points in 3D space: d = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²)
Midpoint of two points: M = ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2)
Volume and surface area of 3D solids: 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: sine, cosine, tangent ratios.
Sine rule: a/sinA = b/sinB = c/sinC
Cosine rule: c² = a² + b² − 2ab cosC, cosC = (a² + b² − c²) / 2ab
Area of triangle: Area = 1/2·a·b·sinC
3.3 Applications of Trigonometry
Non-right-angled triangles, Pythagoras’ theorem.
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 = 1/2·r²·θ
3.5 Equations of Perpendicular Bisectors
Find perpendicular bisector of a line segment.
Given either two points or the equation of a line segment, calculate the midpoint.
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 and Trigonometric Identities
Definitions: cosθ, sinθ in unit circle.
Pythagorean identity: cos²θ + sin²θ = 1
Tangent: tanθ = sinθ / cosθ
Extension of sine rule to ambiguous case.
Construct graphs of f(x) = sinx and f(x) = cosx from unit circle.
Graphical methods for solving trig equations in finite intervals.
3.9 Geometric Transformations using Matrices
Transformations: reflections, horizontal/vertical stretches, enlargements, translations, rotations.
Matrix form: [a b; c d]·[x; y] + [e; f]
Compositions of transformations.
Determinant interpretation: Area of image = |detA| × area of object
3.10 Vectors – Concept and Representation
Vector vs scalar; representation using directed line segments.
Unit vectors; base vectors i, j, k.
Components: v = [v₁; v₂; v₃] = v₁i + v₂j + v₃k
Zero vector 0, negative vector −v.
Position vectors: OA = a.
Rescaling and normalizing vectors.
3.11 Vector Equation of a Line
Line in 2D and 3D: r = a + λb, where b is direction vector.
3.12 Vector Applications in Kinematics
Linear motion with constant velocity: r = r₀ + vt
Relative position: AB = B − A
Motion with variable velocity in two dimensions.
3.13 Scalar and Vector Products
Scalar product: v · w = |v||w|cosθ, angle between vectors.
Vector product: v × w = |v||w|sinθ, geometric interpretation of magnitude.
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.
Number of k-length walks between vertices.
Weighted adjacency tables.
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 algorithm (upper bound) and deleted vertex algorithm (lower bound).
Topic 4: 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, interquartile range (IQR).
Box-and-whisker diagrams.
4.3 Measures of Central Tendency and Dispersion
Mean, median, mode.
Estimation of mean from grouped data.
Modal class.
Interquartile range (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 a and b.
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
Mean and variance formulas: μ = np, σ² = np(1−p)
4.9 Normal Distribution
Properties of normal curve.
Diagrammatic representation.
Normal probability calculations using technology.
Inverse normal calculations.
4.10 Spearman’s Rank Correlation
Spearman’s rank correlation coefficient r_s.
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: 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 choosing appropriate degrees of freedom.
Reliability and validity tests.
4.13 Non-linear Regression
Least squares regression curves using technology.
Sum of square residuals (SS_res) as measure of fit.
Coefficient of determination R², evaluation using technology.
4.14 Linear Transformation of Random Variables
Expected value: E(aX + b) = a E(X) + b
Variance: Var(aX + b) = a² Var(X)
Linear combinations of n independent random variables.
Unbiased estimates: x̄ = Σx_i / n, s_n² = Σ(x_i − x̄)² / (n−1)
4.15 Normal Random Variables and CLT
Linear combination of independent normal variables is normal.
Central Limit Theorem: X̄ ~ N(μ, σ²/n)
4.16 Confidence Intervals
For mean of normal population.
4.17 Poisson Distribution
Mean and variance.
Sum of independent Poisson distributions is Poisson.
4.18 Hypothesis Testing – Advanced
Critical values and regions.
Test for population mean, proportion, Poisson.
Technology for testing population correlation ρ = 0.
Type I and II errors, probability calculations.
4.19 Markov Chains and Transition Matrices
Transition matrices, powers of matrices.
Regular Markov chains.
Initial state probabilities.
Steady state and long-term probabilities (repeated multiplication or solving linear system).
Topic 5: 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 f(x) = axⁿ + bxᵐ + …, all integer exponents.
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 f(x) = axⁿ + bxᵐ + …, n ∈ ℤ, n ≠ −1
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
d/dx (sin x) = cos x, d/dx (cos x) = −sin x, d/dx (tan x) = sec² x
d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x, d/dx (xⁿ) = nxⁿ⁻¹, n ∈ ℚ
Chain rule, product rule, quotient rule.
Related rates of change.
5.10 Second Derivative
Notation: d²y/dx² 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, 1/cos²x, eˣ, including n ∈ ℚ, n ≠ −1
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, acceleration a
v = ds/dt, a = dv/dt = d²s/dt² = v dv/ds
Displacement: ∫ₜ₁^ₜ₂ v(t) dt
Total distance: ∫ₜ₁^ₜ₂ |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: dy/dx = f(x, y)
Numerical solutions for coupled systems: dx/dt = f₁(x, y, t), dy/dt = f₂(x, y, t)
5.17 Phase Portraits
Solutions of coupled differential equations:
dx/dt = ax + by, dy/dt = cx + dy
Qualitative analysis for distinct, real, complex, imaginary eigenvalues.
Sketching trajectories, identifying equilibrium points, stable populations, and saddle points.
Solving d²x/dt² = f(x, dx/dt, t) by Euler’s method.
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IB Math AI HL covers 5 main topics: Topic 1 - Number and Algebra (advanced financial mathematics, sequences, logarithms), Topic 2 - Functions (advanced modeling, regression analysis, functions), Topic 3 - Geometry and Trigonometry (complex geometry, advanced trigonometry, graph theory), Topic 4 - Statistics and Probability (advanced descriptive statistics, probability distributions, hypothesis testing, chi-squared tests), and Topic 5 - Calculus (differential and integral calculus with applications). The HL course includes additional depth and complexity compared to SL.
Assessment consists of four components: Paper 1 (calculator allowed, 120 minutes, 30% of final grade), Paper 2 (calculator allowed, 120 minutes, 30% of final grade), Paper 3 (calculator allowed, 60 minutes, 20% of final grade), and Internal Assessment (mathematical exploration, 20% of final grade). All external papers test advanced knowledge across curriculum topics with complex problem-solving and extended response questions.
Math AI HL includes all SL content plus additional advanced topics and greater depth. HL students study more complex statistical methods, advanced calculus applications, additional geometric concepts, and sophisticated modeling techniques. The HL course has an additional Paper 3 exam and requires deeper mathematical understanding. AI HL is more suitable for students planning careers requiring advanced mathematical applications, such as economics, psychology research, or data science.
Students need proficiency with graphing calculators (TI-84, Casio fx-CG50, or similar) for all three exam papers. Essential skills include statistical calculations, regression analysis, graphing functions, solving equations, and working with matrices. Students should also be comfortable with statistical software and spreadsheet applications for the Internal Assessment. Practice with technology should begin early in the course as it's integral to success in all assessments.
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Reviewing the IB Math AI HL Syllabus helps students clearly understand all required topics, learning objectives, and assessment formats. By aligning their study plan with the syllabus, they can prioritize important areas, manage time effectively, and avoid missing key concepts tested in the final examination.
The IB Math AI HL Syllabus may be revised periodically by the IB to reflect updated teaching methods and modern mathematical applications. Students should always check the latest syllabus to ensure they are studying the most relevant material, practicing current question types, and preparing with confidence.
The IB Math AI HL Syllabus includes all the main topics students need to study, such as algebra, functions, calculus, probability, and statistics. The IB Math AI HL Syllabus also explains how exams are structured and what technology skills are required. By reviewing the IB Math AI HL Syllabus regularly, students can focus on the right content and prepare effectively for assessments.