Eigenvalues, eigenvectors, matrices and Markov chains
Number and Algebra includes eigenvalues, eigenvectors, matrix algebra and Markov chains.
IB HL Mathematics
IB HL Math Syllabus: Topics, Assessment and Planning in the IB Maths Curriculum
IB higher level math is offered as two courses within the wider IB maths curriculum: Analysis and Approaches HL and Applications and Interpretation HL. This guide covers the topics, the assessment, the Internal Assessment and a practical way to plan your two years.
Where IB HL Maths Sits in the IB Maths Curriculum
The IB Diploma Programme offers two Higher Level mathematics courses: Mathematics: Analysis and Approaches HL, usually shortened to AA, and Mathematics: Applications and Interpretation HL, usually shortened to AI. Both belong to the same IB maths curriculum and both build on a shared core of algebra, functions, geometry, statistics and calculus.
The HL courses extend the Standard Level content with additional sub-topics, greater rigour and more teaching hours. That means an HL student covers everything an SL student covers, then goes further. The right course for you depends on your university requirements and your personal strengths, so it helps to understand both pathways before you commit.
Quick Overview of IB HL Mathematics
IB HL suits students who are comfortable with mathematics and willing to work with depth, detail and longer problems. The five shared core topics are Number and Algebra, Functions, Geometry and Trigonometry, Statistics and Probability, and Calculus. Each HL pathway then adds its own specialised material on top of that core.
IB higher level math is not simply more of the same content. It asks for more rigour, more independent problem solving and steady practice across the whole course. University prerequisites vary, so check what your intended programmes ask for before you finalise your choice. Use the list below as a quick scan.
Two HL pathways: AA and AI
Shared core of five topics
AA HL: theory and proof focus
AI HL: modelling and statistics focus
More teaching hours for HL
University prerequisites vary
Which HL Math Course Should You Choose: AA or AI?
Analysis and Approaches HL emphasises algebraic manipulation, proof and theoretical calculus. It tends to suit students who enjoy mathematics for its own sake, or who are heading toward mathematics, engineering or the physical sciences. Applications and Interpretation HL focuses on mathematical modelling, statistics and technology use, and suits students heading toward social sciences, natural sciences or business.
If you are comparing courses and searching for IB higher maths advice, start with two questions. First, what do your target university programmes require? Second, which style of work do you enjoy more: building arguments and proofs, or modelling real situations with data and technology? Talk to your maths teacher as well, because they know how your school delivers each course.
- Check university requirements first
- AA suits proof, algebra and theory lovers
- AI suits modelling, data and technology users
- Consider your own strengths honestly
- Ask your teacher how your school runs each course
AA HL Syllabus Guide: Topics and Depth
This AA HL syllabus guide starts with the extra material the pathway adds to the shared core. Expect complex numbers and De Moivre's theorem, proof by induction and contradiction, vector lines, planes and products, Bayes' theorem and continuous distributions, plus advanced calculus including limits, L'Hopital's rule and differential equations.
The common thread is theory. You are expected to justify steps, not only reach answers, so writing clear mathematical arguments matters as much as calculation. Integration techniques and their applications also take real practice. Build the habit of reviewing each worked solution line by line, and ask why every step is allowed, not just what it produces.
Complex numbers and De Moivre's theoremProof by induction and contradictionVector lines, planes, and productsBayes' theorem and continuous distributionsAdvanced calculus: limits, L'Hopital, ODEsIntegration techniques and applications
AI HL Syllabus: Topics and Modelling Focus
AI HL adds specialised material to each core topic.
Modelling with trigonometric, logarithmic and logistic functions
Functions extends to composite functions, transformations and modelling with trigonometric, logarithmic and logistic functions.
Graph theory and Voronoi diagrams
Geometry and Trigonometry adds vector applications, Voronoi diagrams and graph theory concepts.
Hypothesis testing, chi-squared tests and the central limit theorem
Statistics and Probability covers non-linear regression, chi-squared tests, confidence intervals, hypothesis testing and the central limit theorem.
Euler's method and phase portraits
Calculus covers kinematics, optimisation, related rates, Euler's method for differential equations and phase portraits.
Technology is central here, so learn your calculator thoroughly and practise explaining what a model tells you in plain words.
How the IB HL Math Syllabus Is Organised Across the Two Years
Schools sequence the course differently, so treat this as a general picture and follow your own teacher's plan. Most students meet the shared core topics early, because later HL material depends on them.
- Learn core algebra and functions securely earlyAlgebra and functions usually come first, since nearly everything else uses them.
- Build on the core with HL extensionsGeometry, trigonometry, statistics and calculus then build on that base, with the HL extensions layered on as the course moves forward.
- Revisit older topics on a regular rhythmA useful habit is to revisit older topics regularly rather than leaving them behind.
- Keep a checklist of secure and shaky topicsKeep a topic checklist, mark each item as secure, shaky or not yet started, and use it to decide what to practise each week.
- Start Internal Assessment thinking well before deadlinesThe second year typically brings more demanding HL material, consolidation of earlier topics and work on the Internal Assessment.
Assessment Structure for IB Higher Level Maths
Paper 1: non-calculator for AA, calculator for AI
Paper 1 is a non-calculator paper for AA HL and a calculator-allowed paper for AI HL.
Paper 2: calculator allowed for both courses
Paper 2 is calculator-allowed for both HL courses.
Paper 3: HL only, two compulsory questions
Paper 3 is HL only: an extended problem-solving paper with two compulsory questions.
Internal Assessment: mathematical investigation
The Internal Assessment, a mathematical investigation, contributes a portion of the final grade alongside the written papers.
Check the subject guide for weightings
Check your school's subject guide for exact weightings and timings rather than relying on second-hand figures.
Because calculator rules differ, your preparation should too. AA students need strong non-calculator fluency in algebra and calculus, while AI students need confident, accurate technology use. For Paper 3, practise sustained problems where one idea leads into the next.
Internal Assessment: Mathematical Exploration Guidance
The Exploration is a mathematical investigation on a topic of your choice. It is assessed on communication, mathematical presentation, personal engagement, reflection and use of mathematics. Choose a topic that genuinely interests you and that connects to the syllabus, because engagement is hard to fake and the mathematics needs enough substance to work with.
Common pitfalls include picking a topic that is too broad, copying methods without understanding them, and describing results without reflecting on them. Write for a reader who is your classmate, explain your notation, and say why each step matters. Draft early, ask your teacher for feedback within your school's rules, and leave time to revise your reflection.
Past Papers and Practice Resources for HL Math
Past papers, mark schemes and question banks are most useful once you have learned a topic, not before. Your school or teacher can tell you which materials are available to you.
Practise single topics before full papers
Start by practising individual questions on one topic at a time.
Use the right calculator rules for your course
Later, move to full papers under timed conditions, using the correct calculator rules for your course.
Mark honestly against the mark scheme
After each attempt, mark honestly against the mark scheme and note why you lost marks: a knowledge gap, a careless slip or unclear working.
Keep an error log and revisit it weekly
Keep a short error log and revisit it every week.
Rehearse extended Paper 3 style problems separately
Practise Paper 3 style problems separately, since the extended format rewards patience and careful setup.
Consistent review beats occasional marathon sessions.
Frequently Asked Questions
What is the IB maths curriculum at Higher Level?
At Higher Level, the IB Diploma Programme offers two courses: Analysis and Approaches HL and Applications and Interpretation HL. Both share a core of algebra, functions, geometry, statistics and calculus, and each adds specialised topics, greater depth and more teaching hours than the Standard Level course.
How do I compare courses when looking at IB higher maths?
Compare the two HL pathways by focus. AA emphasises proof, algebraic manipulation and theoretical calculus, while AI emphasises modelling, statistics and technology. Then check what your university programmes require, think about which style you enjoy, and ask your teacher how your school delivers each course.
Is a calculator allowed in IB HL maths exams?
It depends on the paper and course. Paper 1 is non-calculator for AA HL and calculator-allowed for AI HL. Paper 2 is calculator-allowed for both. Paper 3 is HL only and focuses on extended problem solving. Confirm the details in your school's official subject guide.
What does the IB mathematics HL Internal Assessment involve?
The Internal Assessment is called the Exploration. It is a mathematical investigation on a topic of your choice, assessed on communication, mathematical presentation, personal engagement, reflection and use of mathematics. Choosing a topic you care about and explaining your reasoning clearly makes the work easier to manage.
How should I use past papers when studying HL math?
Use them after learning a topic. Begin with single questions, then move to timed full papers with the correct calculator rules. Mark against the mark scheme, record why you lost marks, and revisit that error log regularly so the same mistakes do not repeat.
Plan Your HL Math Course with Support
If you want help choosing between AA and AI, or building a study plan for your two years, use the consultation form to get started with IB Tutoring Academy.