IB Math AA Syllabus & Topics
The following content outlines the syllabus for Mathematics: Analysis and Approaches (AA), a subject within the International Baccalaureate (IB) Diploma Programme.
This course is designed for students who enjoy exploring abstract mathematics, focusing on algebraic methods, calculus, and rigorous problem-solving. It emphasizes mathematical reasoning, formal proofs, and theoretical understanding over practical application.
The syllabus is structured around five main topic groups: Number and Algebra, Functions, Geometry and Trigonometry, Statistics and Probability, and Calculus.
If you're struggling with any topics in the Math AA syllabus, feel free to reach - our team of IB experts has extensive experience helping students from all around the world.
IB Mathematics Analysis and Approaches is offered at both Standard Level (SL) and Higher Level (HL). The syllabus is divided below to reflect the key differences in content and depth between the two levels.
Number and Algebra
SL
1.1
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.
Sigma notation.
Applications and real-life modeling.
1.3
Geometric Sequences and Series
Formulae for nth term and sum.
Sigma notation.
Applications.
1.4
Financial Applications: Compound interest and depreciation.
1.5
Laws of exponents (integer exponents).
Introduction to logarithms (base 10 and e).
Numerical evaluation using technology.
1.6
Simple Deductive Proof
LHS to RHS structure.
Equality and identity notation.
1.7
Exponents and Logarithms
Rational exponents.
Laws of logarithms.
Change of base.
Solving exponential equations.
1.8
Sum of infinite convergent geometric sequences.
1.9
Binomial Theorem
Pascal’s triangle.
Combinations (nCr).
HL only
1.10
Counting principles: permutations and combinations.
Binomial theorem for fractional or negative indices.
1.11
Partial fractions.
1.12
Complex Numbers (Introduction)
Definition of i.
Cartesian form z = a + bi
Real and imaginary parts, conjugate, modulus, argument.
Complex plane.
1.13
Modulus-argument (polar) form.
Euler form.
Sums, products, and quotients in all forms with geometric interpretation.
1.14
Complex conjugate roots.
De Moivre’s theorem.
Powers and roots of complex numbers.
1.15
Proof by induction and contradiction.
Use of counterexamples.
1.16
Systems of linear equations (up to 3 variables).
Unique, infinite, or no solutions.
2. Functions
SL
2.1
Different forms of the equation of a straight line.
Gradient; intercepts.
Parallel (m₁ = m₂) and perpendicular lines (m₁ × m₂ = −1)
2.2
Concept of a function: domain, range, and graph
Function notation.
Modeling with functions.
Inverse functions and reflection in y = x
2.3
Graphing functions
Sketches from context.
Graphs using technology.
Graphing function sums and differences.
2.4
Key features of graphs.
Intersections using technology.
Composite and inverse functions.
Identity function.
2.6
Quadratic Functions and Graph Forms
Standard form: f(x) = ax² + bx + c
Factor form: f(x) = a(x − p)(x − q)
Vertex form: f(x) = a(x − h)² + k
2.7
Solving quadratic equations and inequalities.
Discriminant Δ = b² − 4ac
Reciprocal function and its self-inverse nature
Rational functions: f(x) = (ax+b)/(cx+d)
Asymptotes.
2.9
Exponential and Logarithmic Functions:
f(x) = a^x, f(x) = e^x
f(x) = logax, f(x) = ln(x)
2.10
Solving equations graphically and analytically.
Use of technology.
Real-life applications.
2.11
Graph Transformations
Translations and reflections.
Vertical and horizontal stretches.
Composite transformations.
HL only
2.12
Polynomial Functions
Graphs, equations, zeros, and roots.
Factor and remainder theorems.
Sum and product of roots.
2.13
Rational Functions
2.14
Odd and even functions.
Inverse functions with domain restrictions.
Self-inverse functions.
2.15
Solving inequalities of the form g(x) ≥ f(x)
Graphical and analytical methods.
2.16
Modulus Equations and Inequalities
3. Geometry and Trigonomentry
SL
3.1
Distance and midpoint in 3D space.
Volume and surface area of 3D solids (pyramids, cones, spheres, hemispheres, and combinations).
Angle between lines and between a line and a plane.
3.2
Sine, cosine, and tangent ratios.
Sine and cosine rules.
Area of a triangle.
3.3
Applications of trigonometry in right and non-right triangles.
Pythagoras’s theorem.
Angles of elevation and depression.
Constructing labeled diagrams.
3.4
The circle.
Radian measure of angles.
Arc length and sector area.
3.5
Definitions of sinθ and cosθ using the unit circle
Definition of tanθ = sinθ / cosθ
Ambiguous case in the sine rule.
3.6
Pythagorean identity: cos²θ + sin²θ = 1
Double angle identities for sine and cosine.
Relationships between trigonometric ratios.
3.7
Circular functions: sinx, cosx, tanx
Amplitude, periodic nature, and graphs.
f(x) = a·sin(b(x + c)) + d
Transformations.
3.8 Solving Trigonometric Equations
Solving trigonometric equations in a finite interval.
Graphical and analytical methods.
Quadratic forms involving sinx, cosx, or tanx
HL only
3.9
Reciprocal trig ratios: secθ, cosecθ, cotθ
Pythagorean identities.
Inverse trig functions: arcsin, arccos, arctan
Graphs, domains, and ranges.
3.10
Compound angle identities.
Double angle identity for tan.
3.11
Trig function relationships
Symmetry properties of graphs
3.12
Vectors: position and displacement
Representation with line segments
Base vectors i, j, k
Components, magnitude
Geometric proofs with vectors
3.13
Scalar Product (Dot Product)
Angle between vectors.
Perpendicular and parallel vectors.
3.14
Vector equation of a line: r = a + λb
Angles between lines.
Applications in kinematics.
3.15
Coincident, parallel, intersecting, and skew lines.
Points of intersection.
3.16
Vector product (cross product) .
Properties and geometric meaning of |v × w|
3.17 Vector and Cartesian Equations of a Plane
Vector equation of a plane:
r = a + λb + μc
r · n = a · n
Cartesian form: ax + by + cz = d
3.18
Intersections of:
Line and plane.
Two planes.
Three planes.
Angles between:
Line and plane.
Two planes.
4. Statistics and Probability
SL
4.1
Population, sample, and random sample.
Discrete and continuous data.
Bias and reliability.
Outliers and sampling techniques.
4.2
Data presentation: frequency tables, histograms
Cumulative frequency and graphs.
Finding the median, quartiles, percentiles, range, and IQR.
Box and whisker diagrams.
4.3
Central tendency: mean, median, mode.
Estimation from grouped data.
Modal class.
Dispersion: IQR, standard deviation, variance.
Effects of constant changes
Quartiles of discrete data.
4.4
Linear correlation and Pearson’s
Scatter plots and lines of best fit.
Regression line of y on x.
Prediction and interpretation of parameters a and b in y = ax + b
4.5
Trial, outcome, and relative frequency.
Sample space and event.
Complementary events.
Expected occurrences.
4.6
Venn, tree, sample space diagrams
Combined events.
Mutually exclusive, conditional, and independent events.
4.7
Discrete random variables and distributions.
Expected value (mean).
Applications.
4.8
Binomial distribution.
Mean and variance.
4.9
Normal distribution curve.
Properties and diagrams.
Normal and inverse normal calculations.
4.10
Regression line of x on y.
Use for predictions.
4.11 Conditional Probability
Conditional probabilities.
Independent events.
4.12 Standardization and the Normal Distribution
Standardization (z-values).
Inverse normal with unknown mean/standard deviation
HL only
4.13
Bayes’ theorem (up to three events).
4.14
Variance of discrete random variables.
Continuous random variables and density functions.
Mode and median of continuous random variables.
Mean, variance, standard deviation (discrete & continuous)
Effects of linear transformations of X
5. Calculus
SL
5.1
Concept of a limit.
Derivative as gradient function and rate of change.
5.2
Increasing and decreasing functions.
Interpretation of f′(x) > 0, f′(x) = 0, f′(x) < 0
5.3
Derivative of f(x) = axⁿ is f′(x) = anxⁿ⁻¹, n ∈ ℤ
Derivatives of polynomial functions with integer exponents.
5.4
Tangents and normals at a point.
Equations of tangents and normals.
5.5
Integration as anti-differentiation.
Use of boundary conditions.
Definite integrals with technology.
Area under curve y = f(x), f(x) > 0
5.6
Derivatives of xⁿ (n ∈ ℚ), sinx, cosx, eˣ, ln x
Sum, multiple, chain rule, product and quotient rules
5.7
Second derivative.
Relationship between f, f′, f′′ and graphical behavior
5.8
Local maxima and minima.
Tests for extrema.
Optimization.
Points of inflection.
5.9
Kinematics: displacement, velocity, acceleration
Total distance traveled.
5.10
Indefinite integrals.
Composite functions.
Reverse chain rule.
5.11
Definite integrals (analytically).
Area under curves (positive or negative).
Area between curves.
HL only
5.12
Continuity and differentiability.
Convergence and divergence of limits.
Higher-order derivatives.
5.13
Evaluation of limits.
Repeated use of l’Hôpital’s rule.
5.14
Implicit differentiation.
Related rates of change.
Optimization problems.
5.15
Derivatives of tanx, secx, cosecx, cotx, aˣ, logₐx, arcsinx, arccosx, arctanx
Indefinite integrals of the above functions.
Composites with linear functions.
Partial fractions in integrals.
5.16
Integration by substitution.
Integration by parts (including repeated).
5.17
Area under a curve to the y-axis.
Volumes of revolution about the x-axis or y-axis.
5.18
First-order differential equations.
Euler’s method.
Separable and homogeneous differential equations.
Integrating factor method.
5.19 Maclaurin Series
Maclaurin series expansions for eˣ, sinx, cosx, ln(1+x), (1+x)^p, p ∈ ℚ
Use of substitution, integration, and differentiation to find other series.
