Additional Higher Level (AHL)

AHL Syllabus Topics (25 Extension Guides)

Choose a topic to open its notes, worked proofs, exam exercises, and Paper 3 investigations.

AHL Topic 1

Number & Algebra (HL)

Mathematical induction, complex numbers in Cartesian, polar and Euler forms, De Moivre's theorem, roots of unity, 3-variable linear systems, and Vieta's formulas.

  • Mathematical Induction
  • Complex Numbers & Euler
  • De Moivre & Roots of Unity
  • Systems & Matrices
  • Vieta's Formulas
Open Number & Algebra HL (8 Guides)
AHL Topic 2

Functions (HL)

Modulus functions and inequalities, rational functions with oblique/slant asymptotes, polynomial long division, and reciprocal graphs $y = \frac{1}{f(x)}$ and $y = [f(x)]^2$.

  • Modulus Functions & Inequalities
  • Oblique Asymptotes
  • Composite Transformations
  • Reciprocal Curves
Open Functions HL (3 Guides)
AHL Topic 3

Geometry & Trigonometry (HL)

Reciprocal and inverse trig functions, compound and double angle identities, 3D vectors, dot and cross products, vector equations of lines and planes in 3D.

  • Reciprocal & Inverse Trig
  • Compound Angles
  • 3D Vectors & Dot Product
  • Skew Lines & Cross Product
  • Equations of Planes
Open Geometry & Trig HL (5 Guides)
AHL Topic 4

Statistics & Probability (HL)

Continuous random variables, probability density functions (PDF), cumulative distribution functions (CDF), expected value, variance, and linear combinations of variables.

  • Continuous Random Variables
  • PDF & CDF Integration
  • Expected Value & Variance
  • Linear Combinations
Open Statistics & Probability HL (3 Guides)
AHL Topic 5

Calculus (HL)

L'Hôpital's rule, implicit differentiation, related rates, integration by parts and partial fractions, first-order differential equations, volumes of revolution, and Maclaurin series.

  • L'Hôpital's Rule
  • Implicit Differentiation
  • Integration by Parts
  • Differential Equations (Euler/IF)
  • Volumes of Revolution
  • Maclaurin Series
Open Calculus HL (6 Guides)
IB Assessment Model

AA HL Examination Structure

Higher Level includes three external written examinations plus the internally assessed Mathematical Exploration.

Paper 1 (30%)

120 Minutes · Non-Calculator

Section A (short response, ~50 marks) and Section B (extended response, ~60 marks). Pure algebraic manipulation, analytic proofs, and exact values without technology.

Paper 2 (30%)

120 Minutes · Graphic Display Calculator

Section A and Section B. Questions require efficient calculator use for polynomial root finding, numerical differential equation solving, and continuous probability integrals.

Paper 3 (20%)

60 Minutes · Two Extended Investigations

Two compulsory multi-part sustained inquiry questions (27 marks each). Tests mathematical generalization, pattern discovery, conjecture formation, and proof.

Internal Assessment (20%)

Mathematical Exploration

Individual 12–20 page written report on a topic chosen by the student, assessed against five IB criteria: Communication, Mathematical Presentation, Personal Engagement, Reflection, and Use of Mathematics.

Questions & Advice

Frequently Asked Questions

What makes Paper 3 different from Papers 1 & 2?
Paper 3 does not test routine textbook problems. Instead, each of the two questions introduces an unfamiliar mathematical scenario, guiding the candidate from a basic initial example through scaffolded parts toward a general proof or formula. Practicing sustained problem-solving and conjecture verification is critical.
Can I take AA HL if I didn't take extended mathematics in MYP or IGCSE?
AA HL moves rapidly through abstract algebraic manipulation and proofs. While possible, students coming from standard-level prerequisites should thoroughly review the 30 AA SL foundation units, especially advanced trigonometry, functions, and calculus.
Which university courses require IB Math AA HL?
Leading global universities typically require or strongly recommend AA HL for Engineering, Computer Science, Pure Mathematics, Physics, and highly quantitative Economics or Data Science degree programmes.