IB Maths AA course map: how the five topics connect
The AA guide lists the course as five topics, and most textbooks teach it that way. But exam questions, and HL Paper 3 most of all, keep joining topics: a sequence becomes a limit, a probability becomes an area, a rotation becomes a complex number. This page takes the course apart into seven threads that run across the topics, so you can revise the connections as well as the chapters.
By Pete Bromfield, former Head of Maths with IB examining experience (independent) · Published 1 November 2026
Each topic has one big idea underneath its list of skills. The lines below are our own summaries, not IB wording. HL adds content to every topic; the main additions are listed.
Topic
The big idea
Main HL additions
1. Number and algebra
Describing quantities and patterns exactly, and growing them step by step.
Counting, complex numbers, proof by induction and contradiction, systems of equations, partial fractions
2. Functions
A rule that turns one quantity into another, and what its graph tells you.
Polynomial theorems, rational functions, odd and even functions, modulus, inequalities
3. Geometry and trigonometry
Measuring shape and turning, from triangles to the unit circle to 3D space.
Compound angles, reciprocal and inverse trig functions, vectors, lines and planes
4. Statistics and probability
Measuring uncertainty, and reasoning from data you can see to patterns you cannot.
Bayes' theorem, continuous random variables
5. Calculus
Rates of change and accumulation, and the limits that connect them.
Limits and L'Hôpital's rule, Maclaurin series, further integration, differential equations
For every syllabus point, see the AA SL and AA HL syllabus checklists.
Which threads cross which topics
A filled dot means the thread runs through that topic at SL. An open dot means it only arrives there at HL.
Each thread shows the route it takes through the course (dashed steps are HL only), one question that crosses topics, and one thing to investigate. Try the question before you open the solution. Every answer was checked in code.
1. Undoing
A lot of the course is about running a process backwards: logarithms undo exponentials, an inverse function undoes a function, integration undoes differentiation, and at HL arcsin undoes sin. The same three checks come up every time: does the undoing exist, is it unique, and on what domain?
Let \(u = e^x\). Then \(u^2 - 5u + 6 = 0\), so \((u-2)(u-3) = 0\) and \(u = 2\) or \(3\). Undo the exponential: \(x = \ln 2\) or \(x = \ln 3\). The quadratic does the algebra; the logarithm does the undoing.
Investigate this. For \(a > 1\), sketch \(y = a^x\) and its inverse \(y = \log_a x\). For which values of \(a\) do the two graphs meet, and how many times?
What you should find
They meet twice when \(1 < a < e^{1/e}\) (about 1.4447), touch once at \(x = e\) when \(a = e^{1/e}\), and never meet when \(a\) is larger. Any meeting point is on \(y = x\), so the question becomes: when does \(a^x = x\) have a solution?
2. Patterns to limits
Sequences start as lists, become sums, and then ask what happens "in the end". That question is the limit, and the limit is what calculus is built on: the derivative from first principles and the integral as a sum of thin rectangles. At HL the same idea gives Maclaurin series and L'Hôpital's rule.
Cross-topic question. Split \([0, 1]\) into \(n\) equal strips and add the areas of the rectangles whose heights are \(y = x^2\) at the right-hand end of each strip. Show that the total is \(\dfrac{(n+1)(2n+1)}{6n^2}\), and find its limit as \(n \to \infty\).
Solution
The strips have width \(\frac1n\) and heights \(\left(\frac kn\right)^2\) for \(k = 1, \dots, n\). Total \(= \dfrac{1}{n^3}\sum_{k=1}^{n} k^2 = \dfrac{1}{n^3}\cdot\dfrac{n(n+1)(2n+1)}{6} = \dfrac{(n+1)(2n+1)}{6n^2}\).
For \(n = 10\) this is \(0.385\). As \(n \to \infty\) it tends to \(\dfrac{2}{6} = \dfrac13\), which is \(\int_0^1 x^2\,dx\). A series, a limit and an integral in one question.
Investigate this. Repeat with the left-hand ends of the strips. Which estimate is too big, which is too small, and how does the gap between them shrink as \(n\) grows?
What you should find
The left sum is \(\dfrac{(n-1)(2n-1)}{6n^2}\), which is too small because \(x^2\) is increasing; the right sum is too big. The gap is exactly \(\dfrac1n\), so it halves every time you double the strips.
3. Shape and transformation
Shift, stretch and reflect a graph and you have a new function with a known shape. The same four moves explain trig graphs \(y = a\sin\big(b(x - c)\big) + d\), the standardising of any normal distribution, and what happens to an area when its graph is stretched.
Cross-topic question. \(X \sim N(50, 4^2)\). Use the transformation \(Z = \dfrac{X - 50}{4}\) to explain why \(P(X < 56) = P(Z < 1.5)\), and find the probability.
Solution
Subtracting 50 shifts the curve left so its centre is at 0; dividing by 4 squeezes it horizontally so its spread is 1. The value 56 moves to \(\frac{56 - 50}{4} = 1.5\), and the area to its left is unchanged. So \(P(X < 56) = P(Z < 1.5) \approx 0.933\).
Investigate this. Stretch the graph of a positive function horizontally by factor \(k\). What happens to the area under it? Use your answer to explain why every normal curve has total area 1, whatever its \(\sigma\).
What you should find
A horizontal stretch by factor \(k\) multiplies the area by \(k\). A normal curve with standard deviation \(\sigma\) is the standard curve stretched horizontally by \(\sigma\) and vertically by \(\frac{1}{\sigma}\), so the two effects cancel and the area stays 1.
4. Rates and accumulation
A gradient tells you how fast something changes; an integral adds the changes up. Kinematics is the cleanest place to see both at once. At HL the same pair becomes a differential equation, and a probability density becomes a rate whose area is a probability.
Cross-topic question. A particle moves with velocity \(v(t) = 6t - 3t^2\) m s\(^{-1}\) for \(0 \le t \le 3\). Find its displacement and the total distance it travels.
Solution
Displacement \(= \int_0^3 (6t - 3t^2)\,dt = \big[3t^2 - t^3\big]_0^3 = 0\) m: it ends where it started.
The velocity is zero at \(t = 2\) and negative after it. From 0 to 2 it moves \(12 - 8 = 4\) m forward; from 2 to 3 it moves 4 m back. Total distance \(= 8\) m.
Investigate this. When is the distance travelled equal to the size of the displacement? Find a velocity function where they differ by exactly 10 m over \(0 \le t \le 4\).
What you should find
They are equal exactly when the velocity never changes sign. If the overall motion is forwards, they differ by twice the distance travelled backwards, so you need 5 m of backward motion. There are many answers; checking yours with \(\int |v|\,dt\) on a GDC is the point.
5. Rotation and the circle
The unit circle defines sine and cosine, the identities come from its geometry, and differentiating \(\sin x\) turns it into \(\cos x\), which is \(\sin x\) shifted by a quarter turn. At HL, multiplying by \(i\) is also a quarter turn, and De Moivre's theorem turns repeated rotation into trig identities.
Cross-topic question (HL). Expand \((\cos\theta + i\sin\theta)^2\) in two ways to derive the double-angle formulas for \(\cos 2\theta\) and \(\sin 2\theta\).
Solution
By De Moivre, \((\cos\theta + i\sin\theta)^2 = \cos 2\theta + i\sin 2\theta\). Multiplying out instead gives \(\cos^2\theta - \sin^2\theta + 2i\sin\theta\cos\theta\).
Equate real and imaginary parts: \(\cos 2\theta = \cos^2\theta - \sin^2\theta\) and \(\sin 2\theta = 2\sin\theta\cos\theta\).
Investigate this. Differentiate \(\sin x\) four times, writing each answer as \(\sin(x + \text{something})\). Compare it with multiplying a complex number by \(i\) four times. What is the link?
What you should find
Each derivative adds \(\frac{\pi}{2}\): \(\frac{d}{dx}\sin x = \sin\left(x + \frac{\pi}{2}\right)\). After four derivatives you are back where you started, just as \(i^4 = 1\). Both are a quarter turn, repeated.
6. Chance and area
Probabilities start as counts and tree diagrams. Conditional probability reverses the tree. Distributions turn probability into a picture, and once the picture is a smooth curve (the normal distribution at SL, any density at HL) a probability is an area found by integration.
Investigate this. A test spots a condition in 90% of people who have it and wrongly flags 10% of people who do not. Suppose 2% of people have it. Find the chance that a person who tests positive has it, then vary the 2% and plot the answer. Where does the curve cross 50%?
What you should find
At 2% the answer is about 15.5%. The curve crosses 50% when the condition affects 10% of people, because then true positives (\(0.9 \times 0.1\)) and false positives (\(0.1 \times 0.9\)) are equal.
7. Proof
Proof is not a topic you visit once. "Show that" questions in algebra, trig identities, the derivative from first principles and, at HL, induction, contradiction and counterexample all ask the same thing: an argument that works every time, not just for the cases you tried.
Cross-topic question. Prove that the sum of any three consecutive positive cubes is a multiple of 9.
Solution
Write the cubes as \((n-1)^3 + n^3 + (n+1)^3 = 3n^3 + 6n = 3n(n^2 + 2)\).
If \(n\) is a multiple of 3, then \(3n\) is a multiple of 9. If not, \(n^2\) leaves remainder 1 when divided by 3, so \(n^2 + 2\) is a multiple of 3. Either way the sum is a multiple of 9.
Investigate this. What is the largest number that always divides the sum of 4 consecutive positive cubes? Of 5? Of 7? Make a conjecture, then find the case that breaks it.
What you should find
For 4, 5 and 7 cubes the answer is 4, 5 and 7, which suggests "\(k\) consecutive cubes are always divisible by \(k\)". But 6 consecutive cubes are always divisible by 9, and not always by 6: \(1^3 + 2^3 + \dots + 6^3 = 441\), which is odd.
How to use the map in your revision
Learn by topic, then revise by thread
Topics are the right way to learn something new. Once a topic is secure, revise one thread a week: work through its route in order and do its cross-topic question without notes.
Before a mixed paper, name the thread
When a question looks unfamiliar, ask which thread it belongs to. "This is undoing" or "this is a limit in disguise" usually points to the first step.
HL: Paper 3 lives on the threads
Paper 3 questions often start in one topic and finish in another. The investigation prompts above are short versions of that, and good practice for the IA too: see Paper 3 to IA for two full investigations built step by step.
Make your own connection
Add an eighth thread. Vectors and lines, or exponential growth and sequences, are good ones to try. Then write a question for it, using the steps in the next section.
Write your own cross-topic question
The fastest way to understand how an exam question is built is to build one. Work alone or with a partner, then swap questions and mark each other's work.
Pick two threads, or one thread and two topics. For example: rates and accumulation, applied to a geometric sequence.
Start with a small, concrete part that anyone in the class could do (1 to 3 marks).
Add a part that asks for a pattern: "find the first three values", then "suggest a general expression".
Add a part that makes the pattern certain: "show that" or "prove".
Finish with a "hence" part that uses the result in the second topic.
Write the mark scheme, with the method marks first. If you cannot say where the marks are, the question is not finished yet.
Check that your question has exactly one route through it. Most first attempts are either a list of unrelated parts or one long part. Look at how the two investigations on the AA HL Paper 3 page build up, part by part.
Number and algebra, functions, geometry and trigonometry, statistics and probability, and calculus. SL and HL share the same five topics; HL adds extra content in each one.
Why revise by thread instead of by topic?
Exam questions, and especially HL Paper 3, often join two or three topics in one question. Revising a thread such as inverse operations across every topic trains you to spot those joins, while revising topic by topic does not.
Is this map the official IB syllabus?
No. The five topics are the IB's, but the threads and questions are our own way of grouping the course. For the syllabus point by point, use our AA SL and AA HL syllabus checklists.
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