IB Math AA HL Syllabus (2026)
Complete topic outline with assessment structure, expanded formulas, and where students consistently lose marks. Built from 6,500+ hours of teaching this course.
Need help this term? Apply for tutoring →Assessment Structure
The Internal Assessment is a mathematical exploration worth 20% of the final grade. It is marked against five criteria that most students misunderstand without guidance.
Not sure if AA HL is right for you? AA focuses on pure mathematics, algebraic reasoning, and proof. AI focuses on applied mathematics, statistics, and modelling. Both come in SL and HL. See the full comparison between all four IB Math courses →
Number & Algebra
1.1 Operations with Numbers
- Operations with numbers in the form a × 10k
- Where 1 ≤ a < 10 and k is an integer
1.2 Arithmetic Sequences
- Formulae for the nth term: un = u1 + (n − 1)d
- Sum of first n terms: Sn = n2(2u1 + (n − 1)d)
- Use of sigma (Σ) notation, e.g. Σk=1n (3k + 2)
- Applications: analysis and prediction in real models
1.3 Geometric Sequences
- Formulae for the nth term: un = u1 rn−1
- Sum of first n terms: Sn = u1(rn − 1)r − 1
- Use of sigma (Σ) notation for sums
- Applications of the topics mentioned above
1.4 Financial Applications
- Compound interest formula: A = P(1 + r)n
- Annual depreciation calculations
1.6 Proofs
- Simple deductive proof: numerical and algebraic
- Layout of LHS to RHS proof
- Symbols: equality (=) and identity (≡)
1.7 Exponents & Logarithms
- Laws of exponents with rational exponents
- Laws of logarithms:
- loga(xy) = logax + logay
- loga(xy) = logax − logay
- loga(xm) = m · logax
- Change of base: logax = logbxlogba
- Solving exponential equations using logarithms
1.8 Infinite Geometric Sequences
- Sum of infinite convergent sequences
- Formula: S∞ = a1 − r where |r| < 1
1.9 The Binomial Theorem
- Expansion of (a + b)n, where n ∈ ℕ
- General term formula: nCr an−r br
- Use of Pascal’s triangle and nCr values
1.10 Counting & Binomial Extension
- Counting principles for arrangement and selection
- Permutations: nPr = n!(n − r)!
- Combinations: nCr = n!r!(n − r)!
- Extension of binomial theorem to fractional/negative indices: (a + b)n, where n ∈ ℚ
1.11 Partial Fractions
- Decomposition of rational expressions for integration and simplification
- Notation: P(x)Q(x) = Ax − a + Bx − b
1.12 Complex Numbers
- The imaginary unit i, where i2 = −1
- Cartesian form: z = a + b·i
- Terms: real part, imaginary part, conjugate, modulus, and argument
- Representation in the complex plane (Argand diagram)
1.13 Polar Form & Operations
- Polar form: z = r(cos θ + i·sin θ) = r·cis θ
- Exponential (Euler) form: z = r·eiθ
- Operations: sum, product, and quotient in Cartesian or polar form
- Geometric interpretation of operations
1.15 Proof Methods
- Proof by mathematical induction
- Proof by contradiction
- Use of counterexamples to disprove statements
1.16 Systems of Equations
- Solving systems with max 3 equations and 3 unknowns
- Cases: unique solution, infinite solutions, or no solution
Functions
2.1 Equations of a Straight Line
- Equation: y = mx + c
- Lines with gradients m1 and m2:
- Parallel: m1 = m2
- Perpendicular: m1 × m2 = −1
2.2 Concept of a Function
- Domain, range, and graph
- Function notation: f(x), v(t)
- Concept of a function as a mathematical model
- Inverse function f−1(x) as a reflection in y = x
2.3 The Graph of a Function
- Equation y = f(x)
- Creating a sketch from given information or a context
- Using technology to graph functions, including sums and differences
2.4 Key Features of Graphs
- Determine key features of graphs
- Finding intersections of two curves or lines using technology
2.5 Composite Functions
- (f ∘ g)(x) = f(g(x))
- Identity function and inverse function: f−1(x), (f ∘ f−1)(x) = x
2.6 The Quadratic Function
- Standard form: f(x) = ax2 + bx + c, y-intercept (0, c), axis of symmetry x = −b2a
- Factor form: f(x) = a(x − p)(x − q), x-intercepts (p, 0) and (q, 0)
- Vertex form: f(x) = a(x − h)2 + k, vertex (h, k)
2.7 Quadratic Equations
- Solution of quadratic equations and inequalities
- Quadratic formula: x = −b ± √Δ2a
- Discriminant: Δ = b2 − 4ac
- Nature of roots: two distinct real roots, two equal real roots, or no real roots
2.8 Reciprocal and Rational Functions
- Reciprocal: f(x) = 1x, x ≠ 0, self-inverse nature
- Rational: f(x) = ax + bcx + d, graphs, vertical and horizontal asymptotes
2.9 Exponential and Logarithmic
- Exponential: f(x) = ax, f(x) = ex
- Logarithmic: f(x) = logax, f(x) = ln x
2.10 Solving Equations
- Solving equations graphically and analytically, use of technology
- Applications to real-life situations
2.11 Transformations of Graphs
- Translations: y = f(x) + b, y = f(x − a)
- Reflections: y = −f(x), y = f(−x)
- Vertical stretch: y = p·f(x), horizontal stretch: y = f(qx)
- Composite transformations
2.12 Polynomial Functions
- Polynomial functions, their graphs and equations
- Zeros, roots, and factors
- The factor and remainder theorems
- Sum and product of the roots of polynomial equations
2.13 Rational Functions
- Forms: f(x) = ax + bcx2 + dx + e and f(x) = ax2 + bx + cdx + e
- Analysis of graphs, asymptotes (vertical, horizontal, oblique), and key features
2.14 Odd and Even Functions
- Definition and identification:
- Odd: f(−x) = −f(x)
- Even: f(−x) = f(x)
- Finding the inverse function, f−1(x), including domain restriction
- Self-inverse functions: f−1(x) = f(x)
2.15 Solutions of Inequalities
- Solving g(x) ≥ f(x) both graphically and analytically
2.16 Special Function Graphs
- Graphs of y = |f(x)| and y = f(|x|)
- Transformations: y = 1f(x), y = f(ax + b), y = [f(x)]2
Geometry & Trigonometry
3.1 3D Geometry
- Volume and surface area of three-dimensional solids, including right pyramids, cones, spheres, hemispheres, and combinations of these solids
- Size of an angle between two intersecting lines or between a line and a plane
3.2 Trigonometry in Right-Angled Triangles
- Use of sine, cosine, and tangent ratios:
- sin θ = opphyp, cos θ = adjhyp, tan θ = oppadj
- Sine rule: asin A = bsin B = csin C
- Cosine rule: c2 = a2 + b2 − 2ab · cos C
- Area of a triangle: 12 · a · b · sin C
3.3 Applications of Trigonometry
- Right-angled and non-right-angled problems, including Pythagoras’ theorem: a2 + b2 = c2
- Angles of elevation and depression
- Construction of labelled diagrams from written statements
3.4 The Circle
- Radian measure of angles
- Length of an arc: s = r · θ
- Area of a sector: A = 12 · r2 · θ
3.5 Unit Circle Definitions
- cos θ and sin θ in terms of the unit circle
- tan θ = sin θcos θ
- Extension of the sine rule to the ambiguous case
3.6 Trigonometric Identities
- Pythagorean identity: sin2 θ + cos2 θ = 1
- Double-angle formulas:
- sin 2θ = 2 sin θ cos θ
- cos 2θ = cos2 θ − sin2 θ
- cos 2θ = 2cos2 θ − 1
- cos 2θ = 1 − 2sin2 θ
- Relationships between trigonometric ratios
3.7 Circular Functions
- sin x, cos x, tan x, amplitude, periodic nature, and graphs
- Composite functions: f(x) = a · sin (bx + c) + d
- Transformations and real-life contexts
3.8 Solving Trigonometric Equations
- Graphically and analytically in finite intervals
- Equations leading to quadratic equations in sin x, cos x, or tan x
3.9 Reciprocal Trigonometric Ratios
- Definitions:
- sec θ = 1cos θ
- cosec θ = 1sin θ
- cot θ = 1tan θ
- Pythagorean identities:
- 1 + tan2 θ = sec2 θ
- 1 + cot2 θ = cosec2 θ
- Inverse functions: arcsin x, arccos x, arctan x; domains, ranges, and graphs
3.10 Compound Angle Identities
- Compound angle formulas:
- sin(A ± B) = sin A cos B ± cos A sin B
- cos(A ± B) = cos A cos B ∓ sin A sin B
- tan(A ± B) = tan A ± tan B1 ∓ tan A tan B
- Double-angle identity for tan:
- tan 2θ = 2tan θ1 − tan2 θ
3.11 Symmetry & Trig Graphs
- sin(π − θ) = sin θ
- cos(π − θ) = −cos θ
- tan(π − θ) = −tan θ
- Relationships between trigonometric functions and symmetry properties of their graphs
3.12 Vectors: Basic Concepts
- Concept of a vector; position vectors; displacement vectors
- Base vectors i, j, k
- Components: v = v1i + v2j + v3k
- Algebraic and geometric approaches to: sum/difference, zero vector, −v, scalar multiplication, parallel vectors, magnitude |v|, unit vectors v|v|
- Position vectors OA = a, OB = b; displacement AB = b − a
- Proofs of geometrical properties using vectors
3.13 Scalar Product
- Definition: a · b = |a| |b| cos θ
- Angle between two vectors
- Perpendicular (dot product = 0) and parallel vectors
3.14 Vector Equation of a Line
- Vector equation in 2D and 3D: r = a + λb
- Angle between two lines
- Simple applications to kinematics
3.15 Lines in Space
- Coincident, parallel, intersecting, and skew lines
- Distinguishing between these cases and points of intersection
3.16 Vector Product
- Definition: v × w
- Properties of the vector product
- Geometric interpretation of |v × w| (Area of parallelogram)
3.17 Vector Equations of a Plane
- r = a + λb + μc (parametric form)
- r · n = a · n (scalar/normal form)
- Cartesian equation: ax + by + cz = d
3.18 Intersections and Angles
- Intersections of: a line with a plane, two planes, three planes
- Angle between: a line and a plane, two planes
Statistics & Probability
4.1 Concepts of Data
- Population, sample, random sample, discrete and continuous data
- Reliability of data sources and bias in sampling
- Interpreting outliers, sampling techniques
4.2 Presentation of Data
- Frequency distributions, histograms
- Cumulative frequency and cumulative frequency graphs
- Using graphs to find median, quartiles, percentiles, range, and interquartile range (IQR)
- Box-and-whisker diagrams
4.3 Measures of Central Tendency
- Mean, median, and mode
- Estimation of mean from grouped data
- Modal class
- Measures of dispersion: IQR, standard deviation, variance
- Effect of constant changes on original data
- Quartiles of discrete data
4.4 Linear Correlation of Bivariate Data
- Pearson’s correlation coefficient r
- Scatter diagrams, lines of best fit by eye through mean point
- Regression line of y on x: y = a · x + b, interpretation of a and b
4.5 Probability Concepts
- Trial, outcome, equally likely outcomes, relative frequency, sample space U, and event
- Probability of event A: P(A) = n(A)n(U)
- Complementary events: A and A′
- Expected number of occurrences
4.6 Using Diagrams for Probability
- Venn diagrams, tree diagrams, sample space diagrams, and tables of outcomes
- Combined events: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
- Mutually exclusive events: P(A ∩ B) = 0
- Conditional probability: P(A|B) = P(A ∩ B)P(B)
- Independent events: P(A ∩ B) = P(A) · P(B)
4.7 Discrete Random Variables
- Probability distributions
- Expected value (mean) E(X) and applications
4.8 Binomial Distribution
- General notation: X ~ B(n, p)
- Mean and variance of the binomial distribution
4.9 Normal Distribution
- General notation: X ~ N(μ, σ2)
- Properties and diagrammatic representation
- Normal probability calculations and inverse normal calculations
4.10 Regression Line of x on y
- Use for prediction purposes
4.11 Conditional Probability (Formal)
- P(A|B) = P(A ∩ B)P(B) for conditional probabilities
- Independent events: P(A|B) = P(A) = P(A|B′)
4.12 Standardization of Normal Variables
- Formula: z = x − μσ
- z-values and inverse normal calculations when mean and standard deviation are unknown
4.13 Use of Bayes’ Theorem
- Formula: P(A|B) = P(B|A) · P(A)P(B)
- Extended: P(A|B) = P(B|A) · P(A)P(B|A)P(A) + P(B|A′)P(A′)
- For a maximum of three events
4.14 Variance of a Random Variable
- Variance of a discrete random variable
- Continuous random variables and their probability density functions
- Mode and median of continuous random variables
- Mean, variance, and standard deviation of both discrete and continuous random variables
- Effect of linear transformations of X
Calculus
5.1 Introduction to Calculus
- Concept of a limit: limx → a f(x)
- Derivative as gradient function and rate of change
5.2 Increasing and Decreasing Functions
- Graphical interpretation: f′(x) > 0, f′(x) = 0, f′(x) < 0
5.3 Derivatives of Polynomials
- f(x) = a · xn + b · xm ⇒ f′(x) = a · n · xn−1 + b · m · xm−1
5.4 Tangents and Normals
- Equations of tangent and normal lines at a given point
5.5 Introduction to Integration
- Anti-differentiation of functions: ∫ (a · xn + b · xm) dx
- Anti-differentiation with boundary condition to determine constant
- Definite integrals using technology: area under y = f(x), f(x) ≥ 0
5.6 Derivatives of Elementary Functions
- xn, sin x, cos x, ex, ln x
- Derivative of sums and multiples
- Chain rule for composite functions: e.g. sin(3x − 1) or ln(2x + 5)
- Product and quotient rules
5.7 Second Derivative
- Graphical behavior: relationships between f(x), f′(x), f″(x)
5.8 Local Maxima and Minima
- Testing for maxima and minima, optimization
- Points of inflection: zero and non-zero gradient
5.9 Kinematics Problems
- Displacement (s), velocity (v), acceleration (a)
- Total distance traveled
5.10 Indefinite Integrals
- xn (n rational), 1x, ex
- Composites with linear functions: ∫ f(ax + b) dx
- Integration by inspection (e.g. ∫ cos(3x) dx), reverse chain rule, or substitution
5.11 Definite Integrals
- Analytical approach: ∫ab f(x) dx
- Areas of regions enclosed by curves y = f(x) and x-axis (f(x) positive or negative)
- Areas between curves: ∫ab (f(x) − g(x)) dx
5.12 Continuity & Differentiability
- Understanding of limits (convergence and divergence)
- Definition of derivative from first principles: f′(x) = limh → 0 f(x + h) − f(x)h
- Higher derivatives
5.13 Evaluation of Limits
- Limits of the form limx → a f(x)g(x) and limx → ∞ f(x)g(x) using l’Hôpital’s rule or Maclaurin series
- Repeated use of l’Hôpital’s rule
5.14 Implicit Differentiation
- Related rates of change
- Optimization problems
5.15 Special Functions
- Derivatives of tan x, sec x, cosec x, cot x, ax, logax, arcsin x, arccos x, arctan x
- Indefinite integrals of the derivatives of the above functions
- Composites with linear functions and use of partial fractions to rearrange integrand
5.16 Integration Techniques
- Integration by substitution
- Integration by parts, including repeated integration by parts
5.17 Areas and Volumes
- Area of region enclosed by a curve and y-axis
- Volumes of revolution about the x-axis or y-axis
5.18 Differential Equations
- Numerical solution of dydx = f(x, y) using Euler’s method
- Variables separable
- Homogeneous: dydx = f(yx) using y = vx
- Linear: y′ + P(x)y = Q(x) using integrating factor
5.19 Maclaurin Series
- Expansions for ex, sin x, cos x, ln(1 + x), (1 + x)p (p rational)
- Use of simple substitution, products, integration, and differentiation
- Series developed from differential equations
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Where AA HL Students Lose Marks
AA HL is one of the most demanding courses in the IB Diploma. Students often underestimate how much depth is required and how precisely examiners expect mathematical reasoning to be communicated.
What Separates a 5 from a 7 in AA HL
Understands the content but loses marks on execution. Can start a proof but misses a logical step. Can differentiate correctly but forgets to justify behaviour at a critical point. Finishes the paper but leaves marks on the table through imprecise algebraic reasoning.
Does what the examiner expects at every step. Defines variables. States assumptions. Completes proofs with clear logical connectives. Shows enough working to earn method marks even when the final answer is wrong. The mathematics is the same. The difference is rigour under pressure.
Most students who improve from a 5 to a 7 do not learn new content. They learn how to present their working in the format examiners actually mark for.
How Tutoring Helps in AA HL
At HL level, covering content is not enough. Students need fluency with abstract reasoning, proof techniques, and the precise communication that examiners reward. Work is adapted to each student’s weak points and adjusted as those improve.
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See student results →Frequently Asked Questions
Common questions about the IB Math AA HL course, exams, and preparation.
AA HL covers all SL content in greater depth, plus additional HL-only material including proof by induction, complex numbers, advanced calculus, and extended work on vectors and series.
The course demands strong algebraic skills and the ability to reason abstractly across connected topics.
Assessment consists of three written papers and an Internal Assessment. Paper 1 is a non-calculator exam focusing on algebraic manipulation and exact reasoning. Paper 2 allows a calculator and assesses application and problem solving. Paper 3 presents extended problems in unfamiliar contexts, requiring students to apply mathematics flexibly.
Together, the papers reward not just correct answers but clear, structured mathematical communication.
Paper 3 is a 75-minute calculator paper worth 20 percent of the final grade. It typically contains two extended problems set in unfamiliar contexts, requiring students to interpret, model, and apply mathematics they have not seen in that exact form before.
Preparation involves practising with past papers and developing the ability to read carefully, identify relevant techniques, and structure responses clearly under time pressure.
The Internal Assessment is a mathematical exploration worth 20 percent of the final grade. At HL, examiners expect greater sophistication in the mathematics used and a deeper level of analysis throughout the exploration.
The same five criteria apply as at SL, but the standard for top marks is higher.
AA HL includes all SL content plus additional advanced topics and greater depth. HL students study complex numbers, advanced calculus techniques, 3D vectors, additional statistical methods, and mathematical proof.
The HL course has an additional Paper 3 exam and requires stronger mathematical foundations. Students considering HL should be comfortable with abstract reasoning and extended problem solving. See the full comparison between all four IB Math courses.
Effective preparation involves consistent practice with past papers, mastering fundamental concepts before advancing to HL-only material, and developing strong algebraic manipulation skills for the non-calculator paper.
Students who improve most focus on understanding why methods work, not just how to apply them. They also practise presenting proofs and extended reasoning clearly, since this is where many marks are gained or lost.
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