IB Maths notes: the five topics, AA and AI
Both IB Mathematics courses are built on the same five topics, but they treat them differently: AA leans on algebra, calculus and proof, while AI leans on modelling, statistics and technology. These notes summarise each topic for both courses, with the HL extensions marked.
On this page
1. Number and algebra
- Both courses: scientific notation, arithmetic and geometric sequences and series, financial applications such as compound interest, laws of exponents and logarithms, and approximation and error.
- AA adds simple proof, the binomial theorem and, at HL, proof by induction and contradiction, complex numbers and systems of linear equations.
- AI adds amortisation and annuities using technology and, at HL, complex numbers in polar and exponential form and matrices.
2. Functions
- Both courses: equations of lines, function notation, domain and range, inverse functions, graphs and their key features, and modelling.
- AA develops quadratics, rational, exponential and logarithmic functions, transformations of graphs, and at HL polynomial functions, the factor and remainder theorems, and modulus functions.
- AI focuses on modelling with linear, quadratic, exponential, direct and inverse variation, cubic and sinusoidal models, and at HL logistic models and fitting models to data.
3. Geometry and trigonometry
- Both courses: distance and midpoint in 3D, volume and surface area of solids, right-angled trigonometry, the sine and cosine rules, and areas of triangles.
- AA adds radians, arcs and sectors, the unit circle, trigonometric identities and equations, and at HL vectors, lines and planes.
- AI adds Voronoi diagrams and perpendicular bisectors, and at HL vectors, matrices for transformations and graph theory.
4. Statistics and probability
- Both courses: sampling, presenting data, measures of centre and spread, correlation and regression, probability, discrete random variables, the binomial distribution and the normal distribution.
- AA adds conditional probability, standardising normal variables and, at HL, Bayes' theorem and continuous random variables.
- AI adds Spearman's rank correlation, the chi-squared test for independence and goodness of fit, the t-test, and at HL more hypothesis testing, confidence intervals, the Poisson distribution and Markov chains.
5. Calculus
- Both courses: limits and derivatives as rates of change, differentiating polynomials, tangents and normals, maximum and minimum points, and integration as the reverse of differentiation and as area.
- AA adds the chain, product and quotient rules, derivatives of trigonometric, exponential and logarithmic functions, second derivatives, kinematics and definite integrals, and at HL integration by parts and substitution, differential equations and Maclaurin series.
- AI adds the trapezoidal rule and, at HL, related rates, volumes of revolution, differential equations and Euler's method, and slope fields.
AA: a stationary point
Find the minimum value of f(x) = x² − 8x + 3.
- f′(x) = 2x − 8 = 0 gives x = 4.
- f(4) = 16 − 32 + 3 = −13; f″(x) = 2 > 0, so it is a minimum.
AI: compound interest
5000 dollars is invested at 4% a year, compounded monthly, for 3 years. Find the final value.
- FV = 5000 × (1 + 0.04/12)36.
- (1.003333)36 = 1.1273, so FV = 5636 dollars (to the nearest dollar).
Both: a geometric series
Find the sum of the first 8 terms of 3, 6, 12, …
- u1 = 3, r = 2, so S8 = 3(28 − 1) / (2 − 1).
- S8 = 3 × 255 = 765.
How to revise with these notes
- Keep the formula booklet open: see what is given for each topic, using our formula booklet guide.
- Practise without a calculator for AA Paper 1, and with your calculator for Paper 2 and AI.
- Use past papers and Questionbank by topic, then full timed papers. See IB past papers.
- At HL, practise Paper 3 as full problems, since its extended questions need stamina.
How the papers test these topics
- AA SL: Paper 1 without a calculator and Paper 2 with one, each 40%, plus the exploration at 20%.
- AA HL and AI HL: Papers 1 and 2 at 30% each, Paper 3 (extended problem-solving) at 20%, and the exploration at 20%.
- AI SL: Papers 1 and 2, each 40%, both with a calculator, plus the exploration at 20%.
Every paper can test any of the five topics, so revise them all, and practise mixing topics in one question.
More questions
Which topics carry the most weight?
Calculus and functions are the largest parts of AA, while statistics and modelling are central to AI. All five topics appear in the exams.
Can I switch from AA to AI?
Many schools allow a switch early in the course. Check university requirements first, since some courses ask specifically for AA.
Is calculus in both AA and AI?
Yes. Both courses include differentiation and integration, but AA goes much further, with more rules, trigonometric and exponential functions and, at HL, harder integration and series.
Common mistakes
- Revising AA content for an AI course, or the reverse.
- Working in degrees when calculus or arc length needs radians.
- Relying on the calculator in AA Paper 1, where it is not allowed.
- Ignoring the internal assessment while revising for exams.
Practice questions
Try each one before opening the answer.
Q1. Differentiate x³ − 4x.
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Q2. Solve log2 x = 5.
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Q3. X ~ B(12, 0.25). Find E(X).
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Q4. Find the sum to infinity of 20 + 10 + 5 + …
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Frequently asked questions
Which is harder, AA or AI?
They are hard in different ways: AA needs strong algebra and calculus, while AI needs careful modelling, statistics and calculator use. HL is demanding in both.
Do AA and AI share topics?
Yes. Both follow the same five topics with some shared content, then diverge.
How are these notes different from the syllabus?
They summarise the main ideas and methods for revision. Your school's IB guide is the full official syllabus.
Key takeaways
- Five topics in both AA and AI.
- AA leans on algebra, calculus and proof.
- AI leans on modelling, statistics and technology.
- HL extends every topic.
- Revise with the formula booklet and past papers.