IB Maths AA Syllabus + Topics

Exploring the IB Mathematics Analysis & Approaches syllabus provides a clear view of the skills and knowledge you'll acquire throughout the course. Join us as we unpack the key concepts of the syllabus, giving you a better understanding of what to expect on your learning journey. 

IB Maths AA Syllabus + Topics

The IB Mathematics Analysis & Approaches syllabus is made up of five topics:

Topic 1: Number and algebra

In this topic, students explore the following concepts:

• 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 n^th term and the sum of the first n terms of the sequence
  • Use of sigma notation for sums of arithmetic sequences 
  • Applications of the skills above
  • 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 n^th term and the sum of the first n terms of the sequence
  • Use of sigma notation for the sums of geometric sequences
  • Applications of the skills above
• 1.4
  • Financial applications of geometric sequences and series (compound interest and annual depreciation)
• 1.5
  • Laws of exponents with integer exponents
  • Introduction to logarithms with base 10 and e
  • Numerical evaluation of logarithms using technology
• 1.6
  • Simple deductive proof, numerical and algebraic; how to lay out a left-hand side to right-hand side (LHS to RHS) proof
  • The symbols and notations for equality and identity
• 1.7 
  • Laws of exponents with rational exponents
  • Laws of logarithms
  • Change of base of a logarithm
  • Solving exponential equations, including using logarithms. 
• 1.8
  • Sum of infinite convergent geometric sequences
• 1.9
  • The binomial theorem
  • Use of Pascal's triangle and ⁿC_r
• 1.10 (HL only)
  • Counting principles, including permutations and combinations
  • Extension of the binomial theorem to fractional and negative indices
• 1.11 (HL only)
  • Partial fractions 
• 1.12 (HL only)
  • Complex numbers: the number i, where i² = − 1
  • Cartesian form z = a + bi; the terms real part, imaginary part, conjugate, modulus and argument
  • The complex plane
• 1.13 (HL only)
  • Modulus-argument (polar) form
  • Euler form
  • Sums, products and quotients in Cartesian, polar or Euler forms and their geometric interpretation
• 1.14 (HL only)
  • Complex conjugate roots of quadratic and polynomial equations with real coefficients
  • De Moivre’s theorem and its extension to rational exponents
  • Powers and roots of complex numbers
• 1.15 (HL only)
  • Proof by mathematical induction 
  • Proof by contradiction 
  • Use of a counterexample to show that a statement is Example: Consider the set P of numbers of the form not always true
• 1.16 (HL only)
  • Solutions of systems of linear equations (a maximum of three equations in three unknowns) including cases where there is a unique solution, an infinite number of solutions or no solution

Topic 2: Functions

In this topic, students explore the following concepts:

• 2.1 
  • Different forms of the equation of a straight line
  • Gradient; intercepts
  • Lines with gradients m₁ and m₂
  • Parallel lines m₁ = m₂
  • Perpendicular lines m₁ × m₂ = − 1
• 2.2
  • Concept of a function, domain, range and graph
  • Function notation
  • The concept of a function as a mathematical model
  • Informal concept that an inverse function reverses or undoes the effect of a function
  • Inverse function as a reflection in the line y = x, and the notation f−¹ (x)
• 2.3
  • The graph of a function
  • Creating a sketch from information given or a context, including transferring a graph from screen to paper
  • Using technology to graph functions including their sums and differences
• 2.4
  • Determine key features of a graph
  • Finding the point of intersection of two curves or lines using technology
  • Composite functions
  • Identity function. Finding the inverse function
• 2.6
  • The quadratic function, its graph, and y-intercept 
  • The quadratic function f(x) = ax² + bx + c: its graph, y-intercept (0, c). Axis of symmetry
  • The form f(x) = a(x − p)(x − q), x-intercepts (p, 0) and (q, 0)
  • The form f(x) = a (x − h)² + k, vertex (h, k)
• 2.7
  • Solution of quadratic equations and inequalities. The quadratic formula
  • The discriminant Δ = b² − 4ac and the nature of the roots, that is, two distinct real roots, two equal real roots, no real roots
  • The reciprocal function f(x) = 1/x , x ≠ 0: its graph and self-inverse nature
  • Rational functions of the form f(x) = ax+b/cx+d and their graphs
  • Equations of vertical and horizontal asymptotes
• 2.9
  • Exponential functions and their graphs: f(x) = a^x, a > 0, f(x) = e^x
  • Logarithmic functions and their graphs: f(x)=log_ax, x>0, f(x)=lnx, x>0
• 2.10
  • Solving equations, both graphically and analytically
  • Use of technology to solve a variety of equations, including those where there is no appropriate analytic approach
  • Applications of graphing skills and solving equations that relate to real-life situations
• 2.11
  • Transformations of graphs
  • Transaltions
  • Reflections
  • Verticle and horizontal stretch 
  • Composite transformations
• 2.12 (HL only)
  • 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 (HL only)
  • Rational functions
• 2.14 (HL only)
  • Odd and even functions 
  • Finding the inverse function, including domain restriction 
  • Self-inverse functions
• 2.15 (HL only)
  • Solutions of g(x) ≥ f (x), both graphically and analytically
• 2.16 (HL only)
  • Solution of modulus equations and inequalities. 

Topic 3: Geometry and Trigonometry

In this topic, students explore the following concepts:

• 3.1
  • The distance between two points in three- dimensional space, and their midpoint
  • Volume and surface area of three-dimensional solids including right-pyramid, right cone, sphere, hemisphere and combinations of these solids
  • The size of an angle between two intersecting lines or between a line and a plane
• 3.2
  • Use of sine, cosine and tangent ratios to find the sides and angles of right-angled triangles 
  • The sine and cosine rule
  • Area of a triangle
• 3.3
  • Applications of right and non-right angled trigonometry, including Pythagoras’s 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; area of a sector 
• 3.5
  • Definition of cosθ, sinθ in terms of the unit circle
  • Definition of tanθ as sinθ/cosθ 
  • Extension of the sine rule to the ambiguous case
• 3.6
  • The Pythagorean identity cos²θ + sin²θ = 1
  • Double angle identities for sine and cosine
  • The relationship between trigonometric ratios
• 3.7
  • The circular functions sinx, cosx, and tanx; amplitude, their periodic nature, and their graphs 
  • Composite functions of the form f(x) = asin(b(x + c)) + d
  • Transformations
• 3.8
  • Solving trigonometric equations in a finite interval, both graphically and analytically
  • Equations leading to quadratic equations in sinx, cosx or tanx
• 3.9 (HL only)
  • Definition of the reciprocal trigonometric ratios secθ, cosecθ and cotθ
  • Pyhtagorean identities
  • The inverse functions f(x) = arcsinx, f(x) = arccosx, f(x) = arctanx; their domains and ranges; their graphs
• 3.10 (HL only)
  • Compound angle identities
  • Double angle identity for tan
• 3.11 (HL only)
  • Relationships between trigonometric functions and the symmetry properties of their graphs
• 3.12 (HL only)
  • Concept of a vector; position vectors; displacement vectors 
  • Representation of vectors using directed line segments
  • Base vectors i,j,k
  • Components, magnitude, position, and displacement of a vector
  • Proofs of geometrical properties using vectors
• 3.13 (HL only)
  • The definition of the scalar product of two vectors
  • The angle between two vectors
  • Perpendicular vectors; parallel vectors
• 3.14 (HL only)
  • Vector equation of a line in two and three dimensions: r = a + λb
  • The angle between two lines
  • Simple applications to kinematics
• 3.15 (HL only)
  • Coincident, parallel, intersecting and skew lines, distinguishing between these cases 
  • Points of intersection
• 3.16 (HL only)
  • The definition of the vector product of two vectors
  • Properties of the vector product
  • Geometric interpretation of | v × w |
• 3.17 (HL only)
  • Vector equations of a plane: r = a + λb + μc, where b and c are non-parallel vectors within the plane; r · n = a · n, where n is a normal to the plane and a is the position vector of a point on the plane
  • Cartesian equation of a plane ax + by + cz = d
• 3.18 (HL only)
  • Intersections of: a line with a plane; two planes; three planes
  • Angle between: a line and a plane; two planes