The course and the exams

⚖️AA or AI — what the split really is

Every Diploma student takes one mathematics subject, and there are two to choose from, each available at standard and higher level. They are not a hard course and an easy course. They are different mathematics, aimed at different appetites.

• Analysis and Approaches (AA) is for students who enjoy building and justifying mathematical arguments. It leans on algebraic fluency, keeps calculus at its centre, and includes proof — deductive proof at SL, and induction, contradiction and counterexample at HL. It is the course where you are regularly asked to work without technology.
• Applications and Interpretation (AI) is for students who want mathematics aimed at describing the world — modelling, statistics, and the deliberate use of technology to explore problems that would be intractable by hand. Every AI paper allows a calculator.

The honest test is not “which is easier” but “which do I want to spend 150 or 240 hours doing”. If manipulating an expression until it yields is satisfying, you are an AA student. If you would rather fit a model to real data and argue about what it means, look at AI. Check your university course requirements too: some engineering, physics and mathematics degrees name AA specifically, and a few name HL.

📚The five topics

• 1 — Number and algebra. Sequences and series, exponents and logarithms, proof, the binomial theorem; at HL complex numbers, induction and systems of equations.
• 2 — Functions. Lines, domain and range, composites and inverses, quadratics, rational and exponential graphs, transformations; at HL polynomials, inequalities and the modulus function.
• 3 — Geometry and trigonometry. Solids, the sine and cosine rules, radians, the unit circle, identities, trigonometric equations; at HL compound angles and the whole vectors strand.
• 4 — Statistics and probability. Sampling, summary statistics, regression, probability, the binomial and normal distributions; at HL Bayes’ theorem and continuous random variables.
• 5 — Calculus. Differentiation, optimisation, kinematics, integration and area; at HL first principles, further integration techniques, differential equations and Maclaurin series.

The first few sub-topics of each topic are shared word for word with AI, which is why a friend on the other course can genuinely help you with straight lines and the sine rule — and cannot help you at all with proof by induction.

🎓SL or HL

Standard level is 150 teaching hours; higher level is 240. The difference is not a separate HL syllabus bolted on the end — the additional higher level material sits inside the same five topics, and both levels are examined on the same shared content in the same papers. Roughly speaking, HL adds complex numbers and formal proof to Topic 1, polynomials and the modulus function to Topic 2, an entire vectors strand to Topic 3, continuous random variables to Topic 4, and about as much calculus again to Topic 5.

Topic 5 is where the levels diverge most sharply: 28 hours at SL against 55 at HL. If you are deciding late, look honestly at how you found differentiation.

📝How you are assessed

Standard level — three hours of examinations plus the Exploration.

• Paper 1 — 40%, 90 minutes, 80 marks, no calculator. Section A is short-response, Section B is extended-response. Both cover the whole syllabus.
• Paper 2 — 40%, 90 minutes, 80 marks, calculator required. Same two-section structure.
• The Exploration — 20%, marked out of 20 across five criteria, internally marked and externally moderated.

Higher level — five hours of examinations plus the same Exploration.

• Paper 1 — 30%, 2 hours, 110 marks, no calculator.
• Paper 2 — 30%, 2 hours, 110 marks, calculator required.
• Paper 3 — 20%, 1 hour, 55 marks, calculator required. Two long problem-solving questions, each developing a single theme towards a generalisation. This paper exists only at HL.
• The Exploration — 20%.

🔢The formula booklet

You are given a clean copy of the mathematics formula booklet in every paper, including Paper 1. It carries the sequence and series formulae, the binomial expansion, the trigonometric identities, the standard derivatives and integrals, and the statistical formulae. So do not spend revision time memorising what is printed for you.

Spend it instead on knowing which formula applies and why, and on the things the booklet does not give you: the exact values of the trigonometric ratios, the laws of logarithms as manipulation habits, the shapes of the standard graphs, and how to lay out a proof. Work with the booklet open from the first week so that finding a result in it under time pressure is automatic rather than a search.

📝Command terms are instructions, not decoration

Examiners choose these words carefully and the markscheme follows them exactly.

• Write down — little or no working expected; you are extracting an answer.
• Find / Calculate — show the relevant stages of working.
• Show that — the answer is given to you; the marks are entirely for the route to it. Never work backwards from the printed result.
• Prove — a formal sequence of logical steps, laid out properly.
• Hence — you must use the previous part. A fresh method scores nothing. Hence or otherwise means you may start again if you prefer.
• Sketch — general shape with the important features labelled. Draw — accurate, to scale, ruler for straight lines.
• Justify — give the reason, not just the conclusion.

Marks are awarded for method, accuracy, answers and reasoning. A correct answer with no working does not automatically earn full marks, and a wrong answer with correct method usually earns several — which is the whole argument for writing things down.

✍️Accuracy conventions worth fixing now

Unless a question says otherwise, give final answers either exactly or to three significant figures. Keep full accuracy in your calculator through the middle of a question and round only at the end: rounding at every step is how a correct method produces an answer the markscheme will not accept. On Paper 1 “exact” usually means leaving \( \pi \), a surd, a logarithm or a fraction in the answer rather than converting it to a decimal.