IB Maths AA or AI: which to choose, and two-language study

If your son or daughter is choosing subjects for the Diploma Programme, the question about Maths comes before all the others: Analysis and Approaches (AA) or Applications and Interpretation (AI)? And straight after it: Standard Level (SL) or Higher Level (HL)? They are two different choices which together decide what kind of exam they will sit in two years’ time, how much time they will need to prepare, and which university degrees will accept it without problems.

It is not a choice made by looking at the current maths grade. It is made by looking at what comes next. I will explain how I think about it with the families I work with, and how we work when the subject has to be studied in one language and is understood in another.

AA and AI are not two difficulty levels, they are two different subjects

The first misunderstanding to clear away: AI is not “the easy maths” and AA is not “maths for the clever ones”. They are two programmes with different aims.

Analysis and Approaches (AA) is pure mathematics, built on proof and symbolic calculation: limits, derivatives and integrals handled algebraically, transcendental functions, complex numbers (at HL), sequences and series with proofs by induction. The graphing calculator is used, but a good part of the work is done by hand, without technology.

Applications and Interpretation (AI) is maths applied to real data: growth models, statistics on real datasets, financial maths, some calculus but oriented towards interpretation rather than proof. The graphing calculator (GDC) is always allowed, in both written papers, and a good part of the questions ask you to read and interpret an output, not to derive a formula from scratch.

The difference already shows in the syllabuses:

SL or HL: it is not just “how many more hours”

SL (Standard Level) involves about 150 teaching hours, HL (Higher Level) about 240. But the real difference is not the time, it is the depth: HL adds topics that do not appear at all at SL (complex numbers and more extensive vector algebra in AA HL, for example), and requires a much higher level of independence in problem solving, especially in the third paper.

Exam structure, the same in outline for AA and AI:

• Paper 1: short and extended-response questions. In AA, no calculator. In AI, calculator allowed.
• Paper 2: short and extended-response questions, calculator allowed in both subjects.
• Paper 3 (HL only): two extended problems, each developing a theme in depth through several linked points. It is the part that discriminates most between those who have only learned the techniques and those who truly understand the structure of the problem.
• Internal Assessment (IA), the “Exploration”: a piece of personal research writing on a mathematical topic chosen by the student, assessed by the school and then re-checked by an external moderator. It is worth 20% of the final grade, in both subjects and at both levels, and it is often the part where a student who is strong in the written exam loses marks through poorly written work or maths that is not deep enough.

Which one pays off, in practice

This is the question that really matters, and the answer depends on the degree you are aiming for, not on whether you are “gifted” or not.

Engineering, physics, mathematics, computer science: almost always AA HL. Many degree courses, in Italy and abroad, require it explicitly or regard it as the implicit standard for direct admission. If you are aiming at the Politecnico di Milano with an IB, admission still goes through the entrance test (the TOL for Engineering), but arriving with a solid AA HL means arriving at the test with calculus foundations already consolidated.

Economics, management, quantitative social sciences: it depends a great deal on the university. Some universities accept AI HL without problems, others prefer AA SL as a minimum base, others still require AA regardless for the more quantitative courses (e.g. economics with a strong maths/econometrics component). Here the rule is: check the requirements of the individual faculty, do not trust a general rule — they change from university to university and from year to year.

Medicine in Italy: admission goes through the TOLC-MED or the specific tests, not through the IB level itself, so the choice between AA and AI matters less than elsewhere. I have written about it separately in the guide to the TOLC-MED.

Humanities, design, art, languages: AI SL is almost always enough and leaves more time for the other Diploma subjects.

The practical rule I use with families: if there is even a suspicion of ending up on a STEM path, it is better to start on AA and, if it turns out too heavy in the first year, drop to AI in good time (schools usually allow the switch before the end of the first year) — the opposite route, moving up from AI to AA mid-course, is much harder because the foundations are missing.

How we work: the same exercise, two languages

The most common difficulty I see in Italian IB students is not the maths itself, it is the language of maths. The book, the exam questions and the language of the IA are in English, but the mathematical reasoning has often been formed in Italian — especially for those who come to the IB after Italian middle school or change school halfway through high school.

In the first lesson we do exactly this: I take an exercise from the syllabus and we solve it twice, once in Italian and once in English. If the result changes, or if in English the student slows down much more than in Italian, the problem is not the maths: it is the technical vocabulary, and it is solved with a targeted glossary, not with more hours of exercises.

Two worked examples

Example 1 — AA style: derivative and tangent line

Let $f(x)=x^3-3x^2+2$. Find the equation of the tangent line to the graph of $f$ at $x=1$.

First calculate $f(1)$:

The point of tangency is $(1,0)$. Now the derivative:

At $x=1$:

The tangent line has gradient $-3$ and passes through $(1,0)$:

In the first lesson we write this solution out twice, with the same steps and the same notation — this is exactly why the exercise works as a diagnostic. If a student gets the calculus right in Italian but stumbles on “point of tangency”, “gradient” or “point-gradient form” in English, the gap is vocabulary — not maths.

Example 2 — AI style: regression line and interpretation

A shop records weekly ice-cream sales ($y$, in hundreds of units) against the week’s average temperature ($x$, in degrees). The data give a regression line (calculated on the GDC):

with correlation coefficient $r=0.89$.

a) Estimate the sales when the average temperature is $28°C$. b) Interpret the gradient in the context of the problem. c) Is the model reliable for estimating sales at $5°C$? Why?

Solution.

a) $y=0.42(28)-3.1=11.76-3.1=8.66$, that is about 866 ice creams.

b) The gradient $0.42$ means that for every extra degree of average temperature, weekly sales rise by about $0.42$ hundred units, that is about 42 ice creams.

c) No: $5°C$ is far outside the range of the data collected (extrapolation), so the model is not reliable there — the linear relationship observed might not hold for such low temperatures.

Part (a) is a straight substitution. Part (b) asks for the gradient in context — this is the type of question AI rewards and AA rarely asks: not “what is the gradient” but “what does the gradient mean here”. Part (c) tests whether the student understands the risk of extrapolation: predicting far outside the range of the collected data is unreliable, regardless of how strong $r$ looks.

Italian ↔ English glossary, 15 terms that keep coming back

It is worth building a personal list as the syllabus goes on: most of the “careless” mistakes in English-language papers come from a technical term not recognised at a glance, not from a concept not understood.

Where to start

If the doubt is AA or AI, the first concrete step is a diagnostic lesson: we look together at an exercise from the target syllabus and work out whether the starting level holds up, in Italian and in English. For the rest of the maths syllabus you will find the detail of the paths I follow, and if the difficulty is more the language than the subject, the page on international schools explains how we work on both together.

To see how much vocabulary weighs, reread an AA topic in Italian: derivatives explained well is the starting point of Example 1, while for the statistics part of AI probability and combinatorics matter and, for those aiming at HL, complex numbers. To train calculus in exam style there is the Year 5 derivatives test.

Lessons are online, one-to-one: you will find up-to-date rates and packages on the pricing page. Payment is offline, after the lesson. The first 30-minute introductory call is free: write to me from the contact page.

In brief, for parents

If your child is choosing IB Maths, the AA/AI split matters more than the SL/HL split: Analysis and Approaches is pure, proof-based mathematics — calculus done symbolically, complex numbers at HL, limited use of the graphing calculator — and is the safer choice for engineering, physics, maths or computer science. Applications and Interpretation is data-and-modelling based, with the calculator allowed throughout, and suits economics, social sciences and design better, though some quantitative economics courses still ask for AA. Higher Level adds a third exam paper of extended, multi-step problems and roughly 90 more teaching hours than Standard Level; both levels share a 20%-weighted Internal Assessment (a personal mathematical exploration), which is often where a strong exam candidate loses marks through weak written work rather than weak maths.

For Italian-educated students, the real obstacle is rarely the mathematics itself — it is technical vocabulary in English attached to concepts already understood in Italian. Our first lesson is a short diagnostic: the same exercise, solved once in Italian and once in English, to see whether the gap is subject or language before building a study plan around it. Lessons run online, one-to-one, at the same rates as every other subject on the site — see the pricing page for single lessons and the 10-lesson package — paid offline after the lesson — and the first 30-minute call, to work out where to start, is free.