Course hub · Mathematics: Analysis and Approaches HL

IB Maths AA HL: free cheatsheets, paper analysis and study guides

Everything for IB Maths AA HL in one place: a free cheatsheet for each of the 20 topics, the marks-weighted analysis of every recent paper, the guidebook, IA help and predicted papers.

20 topics
3 exam papers
IA worth 20%
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Free20 topic cheatsheetsEvery formula, method and common trap. FreePast-paper analysisWhere the marks have gone since 2021. GuidebookIB Maths AA HL e-bookEvery question type, with a free chapter. IA helpMaths IA guideCriteria in plain words, ideas and outlines. Set 1 freePredicted papersFull papers with markschemes.

All 20 IB Maths AA HL topics

Grouped by theme. Each one opens its free cheatsheet.

Number & Algebra

The biggest theme by topic count, covering everything from polynomial structure through complex numbers, partial fractions, and the AHL extensions of proof, induction, and the binomial theorem. Mostly examined on Paper 1 (no calculator).

Topic 01PolynomialsPolynomial structure, factor & remainder theorems, polynomial division, complex roots, Vieta's formulas, quartic graph cases.Topic 02Systems of Linear Equations2×2 and 3×3 systems, parametric solutions, geometric interpretation as planes intersecting in a line or a point.Topic 05Sequences & SeriesArithmetic and geometric sequences, infinite GP convergence, compound interest, GDC TVM solver, Σ-notation.Topic 06Counting PrinciplesFundamental counting principle, permutations, combinations, block method, complementary counting, residue classes.Topic 07Binomial TheoremBinomial expansion, the general term, generalised binomial series for |x| < 1, links to partial fractions and Maclaurin.Topic 08ProofDirect proof, contradiction, parity, irrationality templates, counterexamples, IB mark-scheme code conventions.Topic 09Proof by InductionUniversal template, sums, divisibility, inequality, trigonometric, derivative and recurrence induction patterns.Topic 10Partial FractionsDecomposition forms (linear, repeated, irreducible quadratic), with applications to integration and logistic DEs.Topic 13Complex NumbersCartesian, polar, and Euler forms; conjugate root theorem; De Moivre's theorem; nth roots of unity; Argand diagrams.

Functions

The foundation theme — function notation, transformations, asymptotes, inverses, and inequalities. Heavily examined on both Paper 1 and Paper 2.

Topic 03FunctionsDomain & range, composite and inverse functions, transformations, asymptotes, oblique asymptotes (HL), modulus.Topic 04Exponentials & LogarithmsLaws of exponents and logs, change of base, natural log, exponential equations, hidden quadratics in log form.

Geometry & Trigonometry

Trigonometry from the basic right triangle through compound angle identities and 3D geometry, plus the AHL vectors content (lines, planes, intersections). High-yield for Paper 3 problem-solving.

Topic 11TrigonometryExact values, ASTC quadrants, identities, equations, sine and cosine rule, 3D trigonometry, inverse trig, identity proofs.Topic 12VectorsDot & cross product, vector lines and planes, classifying intersections, three-plane systems, angles between objects.

Calculus

The largest theme on Paper 3 and the most heavily examined on Papers 1 and 2 combined. From basic derivatives through L'Hôpital's rule, integration techniques, separable differential equations, and Maclaurin series.

Topic 14DifferentiationAll derivative rules, implicit differentiation, L'Hôpital's rule, optimisation, related rates, kinematics, GDC syntax.Topic 15IntegrationStandard integrals, substitution, integration by parts, definite integrals, areas, volumes of revolution.Topic 16Differential EquationsSeparable, integrating factor, Euler's method, homogeneous DEs, with logistic and population modelling examples.Topic 17Maclaurin SeriesStandard Maclaurin expansions, derivation by repeated differentiation, multiplication and substitution applications.

Statistics & Probability

Sampling, summary statistics, regression, correlation, conditional probability, Bayes' theorem (HL), and discrete & continuous random variables. Almost entirely examined on Paper 2 (calculator).

Topic 18StatisticsData types, sampling, central tendency & dispersion, regression and correlation, scatter plots, GDC stats syntax.Topic 19ProbabilityConditional probability, independence, mutually exclusive events, tree diagrams, Venn diagrams, Bayes' theorem (HL).Topic 20Probability DistributionsDiscrete and continuous random variables, expected value & variance, Binomial distribution, Normal distribution, z-scores.

The exam

How IB Maths AA HL is assessed

Paper 1 No calculator. Short and long questions across the course.
Paper 2 Calculator allowed. Short and long questions across the course.
Paper 3 Calculator allowed. Two extended problem-solving questions (HL only).
Internal assessment 20% of the grade. Read the IA guide.

See how the marks are spread across topics in recent papers.

The guidebook

Mathematics Analysis and Approaches HL for the IB Diploma

Two volumes and 19 chapters for the full Higher Level course, with Paper 3 and exam technique in Volume 2.

See the book

HL Paper 3 guidebook · Free sample chapters

Where marks are lost

The most-dropped marks across the AA HL syllabus

Across thousands of marked mock papers, these five errors account for more lost marks than every other mistake combined. Read them once before you start revising — they alone can lift you a grade.

Total distance ≠ net displacement. Integrating velocity gives the signed displacement; for total distance you must split at every t where v(t) = 0 and sum |Δs| on each segment, or compute nInt(abs(v(t)),t,a,b) on the GDC. This is the single biggest dropped mark in HL kinematics.
Dropping lim notation in L'Hôpital's rule. The IB mark scheme requires you to write the lim symbol at every step. Doing the algebra perfectly but omitting the notation costs the final A1 mark — a 100% solve scoring 75%.
f''(a) = 0 claimed as inflexion without sign check. A second derivative being zero is necessary but NOT sufficient for an inflexion. You must verify f'' changes sign across x = a (counter-example: f(x) = x⁴ at x = 0).
Forgetting dy/dx when differentiating a y-term implicitly. d/dx[y²] = 2y · dy/dx, NOT just 2y. Every implicit-differentiation question we have ever seen examined hinges on this rule.
Substituting numerical values BEFORE differentiating in related rates. Differentiate the formula with respect to t first, THEN substitute the numerical values. Substituting first turns variables into constants, so the derivative is zero and the question collapses.

How to revise

How top IB Math AA HL students revise

From students who scored 7s, here is the exact revision sequence we coach. Follow it topic by topic for the 4–8 weeks before mock or final exams.

1. Open the formula booklet

Always have the official IB Math formula booklet open while you revise. Cross-reference every formula in each cheatsheet — anything NOT in the booklet must be memorised (reciprocal trig derivatives, key limits, R sin/cos derivation, Bayes' theorem statement).

2. Read the TRAPS first

Skim the red "TRAP" boxes in every section before reading anything else. These are the marks examiners report being lost on year after year. Knowing them up front stops you making the same errors in your next mock.

3. Drill past papers by syllabus reference

Find one past-paper question per topic on the same syllabus reference (e.g. SL 5.8 for kinematics). The cheatsheet's exam-attack plan tells you which technique to reach for. Repeat until the trigger-to-technique link is automatic.

Compare the IB maths courses

IB Math AA HL vs AA SL vs AI HL vs AI SL

IB students often choose between AA HL, AA SL, AI HL and AI SL. Here is the difference at a glance — choose the row that matches your maths background and university plans.

Course	Best For	Papers	Hardest Topics
AA HL	Engineering, Maths, Physics, Computer Science, Economics (top universities). Loves abstract proofs.	P1 (no GDC), P2 (GDC), P3 (extended)	Proof by Induction, Vectors at AHL, Maclaurin Series, Differential Equations
AA SL	Business, Life Sciences, Psychology — needs strong maths but not the AHL depth.	P1, P2 only	Trigonometric equations, Functions transformations, Integration
AI HL	Data science, Statistics, Economics, Business Analytics. Likes real-world modelling and tech.	P1, P2, P3	Matrices & transformations, Graph theory, Markov chains
AI SL	Humanities, Languages, Visual Arts students keeping maths broad and applied.	P1, P2 only	Voronoi diagrams, Statistical inference, Optimisation

IB Maths AA HL questions

How many topics are in IB Maths AA HL?

This hub covers 20 topics, grouped into the course's five themes. Each has its own free cheatsheet.

Which papers do IB Maths AA HL students sit?

Paper 1: No calculator. Short and long questions across the course. Paper 2: Calculator allowed. Short and long questions across the course. Paper 3: Calculator allowed. Two extended problem-solving questions (HL only). The internal assessment makes up the rest of the grade.

Is the formula booklet allowed in every paper?

Yes, the IB provides its formula booklet in every maths paper. Each cheatsheet flags the results that are not in it and must be learned.

Are the cheatsheets free?

Yes, every cheatsheet and the paper analysis are free to read without signing in.

Who writes these pages?

Ejaz Ahmad and the Photon Academy team, IB teachers since 2015. Every example and question is original.

In Singapore?

Learn IB Maths AA HL live

The same teachers run small classes in Singapore. See live classes at Photon Academy in Singapore.