IB Math AA HL Syllabus
Complete breakdown of topics, assessment structure, and learning resources for IB Mathematics Analysis & Approaches Higher Level. Master complex mathematical concepts with our expert video tutorials and comprehensive study materials.
IB Math AA HL Syllabus
Assessment Structure
30%
Paper 1
Non-calculator
30%
Paper 2
Calculator allowed
20%
Paper 3
Extended response
20%
Internal Assessment
Mathematical exploration
Topic 1: Number & Algebra
1.1 Operations with Numbers in the Form a × 10^k
Operations with numbers in the form a × 10^k, where 1 ≤ a < 10 and k is an integer.
1.2 Arithmetic Sequences and Series
Use of the formulae for the nth term and the sum of the first n terms of the sequence.
Use of sigma (Σ) notation for sums of arithmetic sequences.
Applications: analysis, interpretation, and prediction where a model is not perfectly arithmetic in real life.
1.3 Geometric Sequences and Series
Use of the formulae for the nth term and the sum of the first n terms of the sequence.
Use of sigma (Σ) notation for the sums of geometric sequences.
Applications of the topics mentioned above.
1.4 Financial Applications of Geometric Sequences and Series
Compound interest: A = P(1 + r)^n
Annual depreciation.
1.6 Simple Deductive Proof (Numerical and Algebraic)
How to lay out a Left-Hand Side (LHS) to Right-Hand Side (RHS) proof.
Symbols and notation for equality (=) and identity (≡).
1.7 Laws of Exponents with Rational Exponents
Laws of logarithms:
logₐ(xy) = logₐx + logₐy
logₐ(x/y) = logₐx − logₐy
logₐ(x^m) = m·logₐx
Change of base formula: logₐx = log_bx / log_ba
Solving exponential equations, including using logarithms.
1.8 Sum of Infinite Convergent Geometric Sequences
Formula: S = a / (1 − r), for |r| < 1
1.9 The Binomial Theorem
Expansion of (a + b)^n, n ∈ ℕ
Use of Pascal’s triangle and nCr.
1.10 Counting Principles, Permutations, and Combinations
Counting principles for arrangement and selection problems.
Permutations and combinations.
Extension of the binomial theorem to fractional and negative indices: (a + b)^n, where n ∈ Q.
1.11 Partial Fractions
Decomposition of rational expressions into partial fractions for integration and simplification.
1.12 Complex Numbers
The imaginary unit i, where i^2 = −1.
Cartesian form: z = a + b·i.
Terms: real part, imaginary part, conjugate, modulus, and argument.
Representation of complex numbers in the complex plane.
1.13 Modulus, Argument, and Polar Form
Polar form: z = r(cos θ + i·sin θ) = r·cis θ.
Equivalent exponential form: z = r·e^(iθ).
Operations: sum, product, and quotient in Cartesian or polar form.
Geometric interpretation of complex number operations.
1.15 Proof Methods
Proof by mathematical induction.
Proof by contradiction.
Use of counterexamples to show that a statement is not always true.
1.16 Solutions of Systems of Linear Equations
Solving systems of linear equations with a maximum of three equations in three unknowns.
Cases considered:
Unique solution
Infinite number of solutions
No solution
Topic 2: Functions
2.1 Different forms of equations of a straight line
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 as a reflection in y = x: f⁻¹(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 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) = a·x² + b·x + c, y-intercept (0, c), axis of symmetry
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 Solution of quadratic equations and inequalities
Quadratic formula: x = (−b ± √Δ)/(2a)
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) = 1/x, x ≠ 0, self-inverse nature
Rational: f(x) = (ax + b)/(cx + d), graphs, vertical and horizontal asymptotes
2.9 Exponential and logarithmic functions
Exponential: f(x) = a^x, f(x) = e^x
Logarithmic: f(x) = logₐx, f(x) = ln x
2.10 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
Rational functions of the form f(x) = (ax + b)/(cx² + dx + e) and f(x) = (ax² + bx + c)/(dx + e).
Analysis of graphs, asymptotes, and key features.
2.14 Odd and Even Functions
Definition and identification of odd and even functions.
Finding the inverse function, f⁻¹(x), including domain restriction.
Self-inverse functions.
2.15 Solutions of Inequalities
Solving g(x) ≥ f(x) both graphically and analytically.
2.16 Special Function Graphs and Transformations
Graphs of y = |f(x)| and y = f(|x|).
Transformations: y = 1/f(x), y = f(ax + b), y = [f(x)]²
Topic 3: 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 to find sides and angles
Sine rule: a/sinA = b/sinB = c/sinC
Cosine rule: c² = a² + b² − 2ab·cosC
Area of a triangle: (1/2)·a·b·sinC
3.3 Applications of trigonometry
Right-angled and non-right-angled problems, including Pythagoras’ theorem
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 = (1/2)·r²·θ
3.5 Unit circle definitions
cosθ and sinθ in terms of the unit circle
tanθ = sinθ/cosθ
Extension of the sine rule to the ambiguous case
3.6 Trigonometric identities
Pythagorean identity: sin²θ + cos²θ = 1
Double-angle formulas for sine and cosine
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
Definition of secθ, cosecθ, and cotθ
Pythagorean identities: 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ
Inverse functions: arcsin x, arccos x, arctan x; domains, ranges, and graphs
3.10 Compound angle identities
Compound angle formulas for sine, cosine, and tangent
Double-angle identity for tan
3.11 Symmetry and trigonometric graphs
sin(π − θ) = sinθ
cos(π − θ) = −cosθ
tan(π − θ) = −tanθ
Relationships between trigonometric functions and symmetry properties of their graphs
3.12 Vectors: basic concepts
Concept of a vector; position vectors; displacement vectors
Representation using directed line segments
Base vectors i, j, k
Components of a vector: v = v₁i + v₂j + v₃k
Algebraic and geometric approaches to: sum and difference of vectors, zero vector, −v, scalar multiplication, parallel vectors, magnitude |v|, unit vectors v/|v|
Position vectors OA = a, OB = b; displacement vector AB = b − a
Proofs of geometrical properties using vectors
3.13 Scalar product
Definition of scalar product: a · b
Angle between two vectors
Perpendicular and parallel vectors
3.14 Vector equation of a line
Vector equation of a line 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
Points of intersection
3.16 Vector product
Definition of the vector product of two vectors: v × w
Properties of the vector product
Geometric interpretation of |v × w|
3.17 Vector equations of a plane
r = a + λb + μc, where b and c are non-parallel vectors in the plane
r · n = a · n, where n is a normal to the plane, a is the position vector of a point on the plane
Cartesian equation of a plane: 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
Topic 4: 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 = ax + 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) and applications
4.8 Binomial distribution
Mean and variance of the binomial distribution
4.9 Normal distribution
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 definition)
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
z-values and inverse normal calculations when mean and standard deviation are unknown
4.13 Use of Bayes’ theorem
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
Topic 5: Calculus
5.1 Introduction
Concept of a limit: lim x → 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 polynomial functions
f(x) = a·xⁿ + b·xᵐ → derivative: f′(x) = a·n·xⁿ⁻¹ + b·m·xᵐ⁻¹
5.4 Tangents and normals at a given point
Equations of tangent and normal lines
5.5 Introduction to integration
Anti-differentiation of functions: f(x) = a·xⁿ + b·xᵐ
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, e^x, ln x
Derivative of sums and multiples
Chain rule for composite functions: sin (3x − 1)
Product and quotient rules
5.7 Second derivative
Graphical behavior: relationships between f, f′, f″
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), 1/x, e^x, composites with linear functions a·x + b
Integration by inspection, reverse chain rule, or substitution
5.11 Definite integrals
Analytical approach: ∫ f(x) dx from a to b
Areas of regions enclosed by curves y = f(x) and x-axis (f(x) positive or negative)
Areas between curves
5.12 Informal understanding of continuity and differentiability
Understanding of limits (convergence and divergence)
Definition of derivative from first principles: f′(x) = (f(x + h) − f(x)) / h as h → 0
Higher derivatives
5.13 Evaluation of limits
Limits of the form lim x → a f(x)/g(x) and lim x → ∞ 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 Derivatives and integrals of special functions
Derivatives of tan x, sec x, cosec x, cot x, a^x, log a x, arcsin x, arccos x, arctan x
Indefinite integrals of the derivatives of the above functions
Composites of the above with linear functions
Use of partial fractions to rearrange the integrand
5.16 Integration techniques
Integration by substitution
Integration by parts
Repeated integration by parts
5.17 Areas and volumes
Area of the region enclosed by a curve and the y-axis in a given interval
Volumes of revolution about the x-axis or y-axis
5.18 First-order differential equations
Numerical solution of dy/dx = f(x, y) using Euler’s method
Variables separable
Homogeneous differential equation dy/dx = f(y/x), using substitution y = v x
Solution of y′ + P(x)y = Q(x) using the integrating factor
5.19 Maclaurin series
Expansions for e^x, sin x, cos x, ln(1 + x), (1 + x)^p, p rational
Use of simple substitution, products, integration, and differentiation to obtain other series
Maclaurin series developed from differential equations
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The curriculum covers five main areas: Number and Algebra (advanced sequences, complex numbers, proof techniques), Functions and Equations (advanced polynomial functions, exponential and logarithmic models, trigonometric functions), Geometry and Trigonometry (advanced coordinate geometry, vectors in 3D, complex trigonometric identities), Statistics and Probability (advanced statistical methods, hypothesis testing, regression analysis), and Calculus (differential calculus, integral calculus, differential equations).
Assessment consists of four components: Paper 1 (non-calculator, 120 minutes, 30% of final grade), Paper 2 (calculator allowed, 120 minutes, 30% of final grade), Paper 3 (calculator allowed, 60 minutes, 20% of final grade), and Internal Assessment (mathematical exploration, 20% of final grade). All external papers test advanced knowledge across curriculum topics with complex problem-solving and extended response questions.
Effective preparation involves consistent practice with past papers, mastering fundamental concepts before advanced topics, developing strong algebraic manipulation skills, and practicing both calculator and non-calculator problems. Students should focus on understanding mathematical reasoning, work on time management during exams, and regularly review all topic areas. Creating a structured study schedule and practicing problem-solving techniques are essential for success.
Math 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. AA HL is essential for students planning mathematics, physics, engineering, or computer science at university level.
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The IB Math AA HL syllabus is important because it defines exactly what students need to study and master. By understanding the syllabus, learners can focus on essential topics like calculus, algebra, probability, and statistics, instead of wasting time on irrelevant material. A clear roadmap through the IB Math AA HL syllabus helps improve efficiency, preparation, and exam confidence.
To study effectively, students should break down the IB Math AA HL syllabus into smaller sections and set weekly goals. Using past papers and exam-style questions aligned with the syllabus ensures stronger practice. With consistent revision and guidance, following the IB Math AA HL syllabus step by step helps students build solid foundations and score higher in assessments.
The IB Math AA HL syllabus acts as a roadmap for students, outlining all the topics required for exams. By following the IB Math AA HL syllabus, learners can focus on essential concepts like calculus, algebra, functions, and probability in a structured way. Teachers also use the syllabus to design lessons, so students who align their preparation with the IB Math AA HL syllabus are better equipped to score higher and avoid missing important content.