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.
Topic 1: 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 = n/2 (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(x/y) = logax − logay
- loga(xm) = m · logax
- Change of base: logax = logbx / logba
- Solving exponential equations using logarithms
1.8 Infinite Geometric Sequences
- Sum of infinite convergent sequences
- Formula: S∞ = a / (1 − 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) = A/(x − a) + B/(x − 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
Topic 2: 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 = −b / (2a)
- 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 ± √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
- 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 + b) / (cx2 + dx + e) and f(x) = (ax2 + bx + c) / (dx + 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 = 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:
- sin θ = opp / hyp
- cos θ = adj / hyp
- tan θ = opp / adj
- Sine rule: a / sin A = b / sin B = c / sin C
- Cosine rule: c2 = a2 + b2 − 2ab · cos C
- Area of a triangle: 1/2 · 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 = 1/2 · r2 · θ
3.5 Unit Circle Definitions
- cos θ and sin θ in terms of the unit circle
- tan θ = sin θ / cos 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 (b x + 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 = 1 / cos θ
- cosec θ = 1 / sin θ
- cot θ = 1 / tan 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 B) / (1 ∓ tan A tan B)
- Double-angle identity for tan:
- tan 2θ = 2tan θ / (1 − 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
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 = 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 − μ) / σ
- 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
Topic 5: Calculus
5.1 Introduction to Calculus
- 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 Polynomials
- f(x) = a · xn + b · xm → derivative: 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), 1/x, 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) = lim h → 0 (f(x + h) − f(x)) / h
- 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 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 dy/dx = f(x, y) using Euler’s method
- Variables separable
- Homogeneous: dy/dx = f(y/x) 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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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.