IB Maths AI Syllabus + Topics

Taking a look at the IB Mathematics Applications and Interpretaton (AI) syllabus gives you a good idea of the skills and knowledge you’ll pick up in the course. Come along as we break down the main concepts, helping you get a better feel for what to expect on your learning journey. 

IB Maths AI Syllabus + 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
  • Approximation: decimal places, significant figures 
  • Upper and lower bounds of rounded numbers
  • Percentage errors
  • Estimation 
• 1.7
  • Amortization and annuities using technology
• 1.8
  • Use technology to solve: 
    • Systems of linear equations in up to 3 variables
    • Polynomial equations
• 1.9 (HL only)
  • Laws of logarithms: 
    • log_axy = log_ax + log_ay
    • log_a x/y = logax - log_ay 
    • log_ax^m = mlog_ax  
    • for a, x, y > 0
• 1.10 (HL only)
  • Simplifying expressions, both numerically and algebraically, involving rational exponents
• 1.11  (HL only)
  • The sum of infinite geometric sequences
• 1.12  (HL only)
  • Complex numbers: the number i such that i² = - 1
  • Cartesian form: z = a + bi; the terms real part, imaginary part, conjugate, modulus and argument
  • Calculate sums, differences, products, quotients, by hand and with technology. Calculating powers of complex numbers, in Cartesian form, with technology 
  • The complex plane
  • Complex numbers as solutions quadratic equations of the form ax² + bx + c = 0, a ≠ 0, with real coefficients where b² - 4ac < 0 
• 1.13  (HL only)
  • Modulus–argument (polar) form: z=r(cosθ+isinθ) =rcisθ 
  • Exponential form: z = re^iθ 
  • Convrsion between Cartesian, polar, and exponential form, by hand and with technology
  • Calculate products, quotients and integer powers in polar or exponential forms 
  • Adding sinusoidal functions with the same frequencies but different phase shift angles
  • Geometric interpretation of complex numbers
• 1.14  (HL only)
  • Definition of a matrix: the terms element, row, column and order for m × n matrices
  • Algebra of matrices: equality; addition; subtraction; multiplication by a scalar for m × n matrices
  • Multiplication of matrices. Properties of matrix multiplication: associativity, distributivity and non-commutativity
  • Identity and zero matrices. Determinants and inverses of n × n matrices with technology, and by hand for 2 × 2 matrices
  • Awareness that a system of linear equations can be written in the form Ax = b
  • Solution of the systems of equations using inverse matrix
• 1.15  (HL only)
  • Eigenvalues and eigenvectors 
  • Characteristic polynomial of 2 × 2 matrices 
  • Diagonalization of 2 × 2 matrices (restricted to the case where there are distinct real eigenvalues)
  • Applications to powers of 2 × 2 matrices

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, for example f(x), v(t), C(n)
  • 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; its equation y = f (x)
  • 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 graphs
  • Finding the point of intersection of two curves or lines using technology
• 2.5
  • Modelling with the following functions:
    • Linear models: f(x) =mx + c
    • Quadratic models: f(x) = ax2 + bx + c ; a ≠ 0. Axis of symmetry, vertex, zeros, and roots., intercepts on the x - axis and y - axis.
    • Exponential growth and decay models. 
      • f(x) = ka^x + c 
      • f(x)=ka^-x + c (for a>0) 
      • f(x) = ke^rx + c
    • Equation of a horizontal asymptote
    • Direct/inverse variation: 
      • f(x) = axⁿ, n ∈ Z 
      • The y - axis as a vertical asymptote when n < 0
    • Cubic models: f(x) =ax³ + bx² + cx + d
    • Sinuisodal models:
      • f(x) = asin(bx) + d 
      • f(x) = acos (bx) + d
• 2.6
  • Modelling skills: Use the modelling process described in the “mathematical modelling” section to create, fit and use the theoretical models in section 2.5 and their graphs
  • Develop and fit the model: Given a context recognize and choose an appropriate model and possible parameters. Determine a reasonable domain for a model
  • Find the parameters of a model
  • Test and reflect upon the model: 
    • Comment on the appropriateness and reasonableness of a model
    • Justify the choice of a particular model, based on the shape of the data, properties of the curve and/or on the context of the situation.
  • Use the model: Reading, interpreting and making predictions based on the model
• 2.7 (HL only)
  • Composite functions in context
  • The notation (f ∘ g)(x) = f(g(x)) 
  • Inverse function f^-1, including domain restriction
  • Finding an inverse function 
• 2.8 (HL only)
  • Transformations of graphs 
  • Translations: y = f(x) + b; y = f(x - a) 
  • Reflections: in the x - axis y = - f(x), and in the y - axis y = f( - x)
  • Vertical stretch with scale factor p: y = pf(x) 
  • Horizontal stretch with scale factor 1/q : y = f(qx)
  • Composite transformations
• 2.9 (HL only)
  • In addition to the models covered in the SL content the AHL content extends this to include modelling with the following functions: 
    • Exponential models to calculate half-life
    • Natural logarithmic models: f(x) = a + blnx
    • Sinusoidal models: fx = asin(b(x - c)) + d
    • Logistic models: f(x) = L/1 + Ce^-kx ;L, C, k>0
    • Piecewise models
• 2.10 (HL only)
  • Scaling very large or small numbers using logarithms
  • Linearizing data using logarithms to determine if the data has an exponential or a power relationship using best-fit straight lines to determine parameters
  • Interpretation of log-log and semi-log graphs

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 rule 
  • The cosine rule 
  • Area of a triangle 
• 3.3
  • Applications of right and non-right angled trigonometry, including Pythagoras’ theorem 
  • Angles of elevation and depression 
  • Construction of labelled diagrams from written statements
• 3.4
  • The circle: length of an arc; area of a sector
• 3.5
  • Equations of perpendicular bisectors
• 3.6
  • Voronoi diagrams: sites, vertices, edges, cells 
  • Addition of a site to an existing Voronoi diagram 
  • Nearest neighbour interpolation 
  • Applications of the “toxic waste dump” problem
• 3.7 (HL only)
  • The definition of a radian and conversion between degrees and radians
  • Using radians to calculate area of sector, length of arc.
• 3.8 (HL only)
  • The definitions of cosθ and sinθ in terms of the unit circle
  • The Pythagorean identity: cos²θ + sin²θ = 1
  • Definition of tanθ as sinθ/cosθ
  • Extension of the sine rule to the ambiguous case
  • Graphical methods of solving trigonometric equations in a finite interval
• 3.9 (HL only)
  • Geometric transformations of points in two dimensions using matrices: reflections, horizontal and vertical stretches, enlargements, translations and rotations
  • Compositions of the above transformations
  • Geometric interpretation of the determinant of a transformation matrix
• 3.10 (HL only)
  • Concept of a vector and a scalar 
  • Representation of vectors using directed line segments
  • Unit vectors; base vectors i, j, k 
  • Components of a vector; column representation 
  • The zero vector 0, the vector - y
  • Position vectors
  • Rescaling and normalizing vectors
• 3.11 (HL only)
  • Vector equation of a line in two and three dimensions: r = a + λb, where b is a direction vector of the line
• 3.12 (HL only)
  • Vector applications to kinematics
  • Modelling linear motion with constant velocity in two and three dimensions
  • Motion with variable velocity in two dimensions
• 3.13 (HL only)
  • Definition and calculation of the scalar product of two vectors
  • The angle between two vectors; the acute angle between two lines
  • Definition and calculation of the vector product of two vectors
  • Geometric interpretation of | v × w |
  • Components of vectors
• 3.14 (HL only)
  • Graph theory: Graphs, vertices, edges, adjacent vertices, adjacent edges
  • Degree of a vertex
  • Simple graphs; complete graphs; weighted graphs
  • Directed graphs; in degree and out degree of a directed graph
  • Subgraphs; trees
• 3.15 (HL only)
  • Adjacency matrices 
  • Walks 
  • Number of k-length walks (or less than k-length walks) between two vertices
  • Weighted adjacency tables 
  • Construction of the transition matrix for a strongly-connected, undirected or directed graph
• 3.16 (HL only)
  • Tree and cycle algorithms with undirected graphs 
  • Walks, trails, paths, circuits, cycles
  • Eulerian trails and circuits 
  • Hamiltonian paths and cycles 
  • Minimum spanning tree (MST) graph algorithms: Kruskal’s and Prim’s algorithms for finding minimum spanning trees
  • Chinese postman problem and algorithm for solution, to determine the shortest route around a weighted graph with up to four odd vertices, going along each edge at least once
  • Travelling salesman problem to determine the Hamiltonian cycle of least weight in a weighted complete graph
  • Nearest neighbour algorithm for determining an upper bound for the travelling salesman problem 
  • Deleted vertex algorithm for determining a lower bound for the travelling salesman problem