Understand Mathematics
Choose Analysis and Approaches or Applications and Interpretation
IB Mathematics has two distinct courses. Analysis and Approaches emphasizes algebraic fluency, proof and calculus, while Applications and Interpretation emphasizes modelling, statistics, technology and mathematics in context.
Know the papers.
Prepare with purpose.
Both pathways combine external papers with an individual mathematical exploration worth 20%. Paper formats differ by pathway and level, particularly in calculator access and the additional HL paper.
AA Paper 1
Short and extended questions emphasize algebraic manipulation, exact reasoning and analytical methods.
AA Paper 2
Problems use a graphing calculator alongside analytical work; HL students also complete Paper 3 problem solving.
AI Papers
Both principal papers expect confident use of technology, modelling and interpretation; HL includes an additional Paper 3.
Mathematical exploration
An individual investigation communicates mathematics around a focused question, with reflection and appropriate use of notation and technology.
Every area,
explained clearly.
Number and algebra
Both pathways develop numerical sense and symbolic reasoning, with AA going further into algebraic structure.
Sequences and seriesArithmetic and geometric patterns lead to recurrence, sigma notation and financial applications.＋
Arithmetic and geometric patterns lead to recurrence, sigma notation and financial applications. HL work develops more sophisticated series and proof.
Exponents and logarithmsLaws of indices and logarithms support equations, growth models and scale comparisons.＋
Laws of indices and logarithms support equations, growth models and scale comparisons. Students choose exact or numerical methods appropriately.
Binomial theorem and proofExpansion techniques connect coefficients, combinations and approximation.＋
Expansion techniques connect coefficients, combinations and approximation. AA places stronger emphasis on proof and formal algebraic reasoning.
Complex numbers · HLCartesian, polar and exponential forms extend algebra beyond the real numbers.＋
Cartesian, polar and exponential forms extend algebra beyond the real numbers. Roots, transformations and geometric interpretation support HL analysis.
Functions
Representations of relationships support prediction, transformation and equation solving.
Function languageDomain, range, inverse and composite functions formalize relationships between variables.＋
Domain, range, inverse and composite functions formalize relationships between variables. Graphs and algebra are used together to inspect behaviour.
Models and transformationsLinear, quadratic, exponential, logarithmic, rational and polynomial models describe different behaviours.＋
Linear, quadratic, exponential, logarithmic, rational and polynomial models describe different behaviours. Parameters are interpreted rather than treated as decoration.
Equations and inequalitiesAnalytical, graphical and numerical methods solve equations and systems.＋
Analytical, graphical and numerical methods solve equations and systems. Solutions are checked against domain and context.
Further functions · HLPolynomial structure, rational behaviour and advanced transformations deepen the analysis.＋
Polynomial structure, rational behaviour and advanced transformations deepen the analysis. AA emphasizes symbolic development while AI emphasizes modelling choices.
Geometry and trigonometry
Spatial relationships are explored through diagrams, vectors and periodic models.
TrigonometryRight-triangle and non-right-triangle methods lead to identities, equations and periodic functions.＋
Right-triangle and non-right-triangle methods lead to identities, equations and periodic functions. Radian measure supports calculus and modelling.
GeometryCoordinate and circle geometry connect algebra to space.＋
Coordinate and circle geometry connect algebra to space. Bearings, arcs, sectors and three-dimensional problems develop visualization.
VectorsVector components, scalar products and equations of lines describe magnitude, direction and geometry.＋
Vector components, scalar products and equations of lines describe magnitude, direction and geometry. HL work extends to more demanding spatial reasoning.
Statistics and probability
Data are summarized, modelled and evaluated with attention to variation and uncertainty.
Describing dataSampling, visualization and summary statistics reveal centre, spread and possible bias.＋
Sampling, visualization and summary statistics reveal centre, spread and possible bias. Technology supports calculation but interpretation remains central.
Probability modelsConditional probability, distributions and expected value quantify uncertainty.＋
Conditional probability, distributions and expected value quantify uncertainty. Assumptions are checked before using a model.
InferenceCorrelation, regression, confidence and hypothesis tests support conclusions from samples.＋
Correlation, regression, confidence and hypothesis tests support conclusions from samples. AI develops broader statistical modelling, especially at HL.
Calculus
Rates of change and accumulation connect local behaviour to global quantities.
DifferentiationDerivatives describe gradients, rates and optimization.＋
Derivatives describe gradients, rates and optimization. Graphical, numerical and analytical viewpoints reinforce one another.
IntegrationAntiderivatives and definite integrals determine accumulated change, displacement, area and volume.＋
Antiderivatives and definite integrals determine accumulated change, displacement, area and volume. Methods expand substantially at HL.
Differential equationsSimple models describe growth, decay and changing systems.＋
Simple models describe growth, decay and changing systems. HL students solve and interpret a broader range of equations.
Students should choose AA or AI by considering university prerequisites, preferred style of mathematics and their intended degree—not simply perceived difficulty.
Content is presented as an original student-friendly explanation. Always use the official syllabus for the examination year as the final authority.