IB Maths AI Syllabus Checklist

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This Interactive syllabus checklist for IB Maths AI is broken down by topics, then subtopics. Your progress is saved site-wide and you can update it as you move throughout the IB Maths AI syllabus.

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1: Number & Algebra 2: Functions 3: Geometry & Trigonometry 4: Stats & Probability 5: Calculus

1 - Number & Algebra

Mark Complete

Master IB Maths AI Topic 1: Number & Algebra. Learn to apply mathematics to real-world scenarios with in-depth notes on financial mathematics, sequences, series, and scientific notation. Elevate your HL knowledge by exploring advanced exponents, complex numbers, and matrix algebra for solving complex systems.

Subtopic

Understanding

1.1: Scientific notation

Operations with numbers in the form

a \times10^{k}

where

1 \leq a < 10

and

k

is an integer

1.2: Arithmetic sequences

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

1.3: Geometric sequences

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

The sum of infinite geometric sequences

1.4: Interest

Simple interest and applications of arithmetic sequences and series

Financial applications of geometric sequences and series (compound interest and annual depreciation)

Analysis, interpretation and prediction where a model is not perfectly arithmetic in real life

1.5: Introduction to exponents & logs

Laws of exponents with integer exponents

Introduction to logarithms with base

10

and

e

Numerical evaluation of logarithms using technology

1.6: Handling numbers

Approximation: decimal places, significant figures

Upper and lower bounds of rounded numbers

Percentage errors

Estimation

1.7: Use of technology

Amortization and annuities using technology

Use technology to solve systems of linear equations in up to

3

variables and polynomial equations

1.8: Advanced exponents & logs (HL)

Laws of logarithms for

a, x, y > 0

\log_{a}{xy} = \log_{a}{x} + \log_{a}{y}

\log_{a}{\frac{x}{y}} = \log_{a}{x} - \log_{a}{y}

\log_{a}{x^{m}} = m\log_{a}{x}

Simplifying expressions, both numerically and algebraically, involving rational exponents

1.9: Introduction to complex numbers (HL)

Complex numbers: the number

i

such that

i^{2} = -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 to quadratic equations of the form

ax^{2} + bx + c = 0

a \neq 0

, with real coefficients where

b^{2} - 4ac < 0

Modulus–argument (polar) form:

z = r(\cos{\theta} + i\sin{\theta}) = r\text{cis}\theta

Exponential form:

z = re^{i\theta}

Conversion between Cartesian, polar and exponential forms, by hand and with technology

Calculate products, quotients and integer powers in polar or exponential forms

1.10: Advanced complex numbers (HL)

Adding sinusoidal functions with the same frequencies but different phase shift angles

Geometric interpretation of complex numbers

1.11: Introduction to matrices (HL)

Definition of a matrix: the terms element, row, column and order for

m \times n

matrices

Algebra of matrices: equality; addition; subtraction; multiplication by a scalar for

m \times n

matrices

Multiplication of matrices. Properties of matrix multiplication: associativity, distributivity and non-commutativity

Identity and zero matrices. Determinants and inverses of

n \times n

matrices with technology, and by hand for

2 \times 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

1.12: Advanced matrices (HL)

Eigenvalues and eigenvectors

Characteristic polynomial of

2 \times 2

matrices

Diagonalization of

2 \times 2

matrices (restricted to the case where there are distinct real eigenvalues)

Applications to powers of

2 \times 2 matrices

2 - Functions

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Excel in IB Maths AI Topic 2: Functions. Focus on practical mathematical modelling with comprehensive guides on linear, exponential, and sinusoidal functions. For HL students, master advanced modelling techniques, graph transformations, composite functions, and the interpretation of log-log and semi-log graphs.

Subtopic

Understanding

2.1: Forms of lines

Different forms of the equation of a straight line

Gradient; intercepts

Lines with gradients

m_{1}

and

m_{2}

Parallel lines

m_{1} = m_{2}

Perpendicular lines

m_{1} \times m_{2} = -1

2.2: Introduction to functions

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)

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

Determine key features of graphs

10:

Finding the point of intersection of two curves or lines using technology

2.3: Introduction to modelling functions

Modelling with the following functions:

Linear models:

f(x) = mx + c

Quadratic models:

f(x) = ax^{2} + bx + c

;

a \neq 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:

10:

f(x) = ax^{n}

n \in \mathbb{Z}

11:

The

y

-axis as a vertical asymptote when

n < 0

12:

Cubic models:

f(x) = ax^{3} + bx^{2} + cx + d

13:

Sinuisodal models:

14:

f(x) = a\sin(bx) + d

15:

f(x) = a\cos(bx) + d

2.4: Modelling applications

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

2.5: Further functions (HL)

Composite functions in context

The notation

(f \circ g)(x) = f(g(x))

Inverse function

f^{-1}

, including domain restriction

Finding an inverse function

2.6: Graph transformations (HL)

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

\frac{1}{q}

y = f(qx)

Composite transformations

2.7: Advanced modelling functions (HL)

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 + b\ln{x}

Sinusoidal models:

f(x) = a\sin(b(x - c)) + d

Logistic models:

f(x) = \frac{L}{1 + Ce^{-kx}}

;

L, C, k > 0

Piecewise models

2.8: Logarithmic graphs (HL)

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

3 - Geometry & Trigonometry

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Navigate IB Maths AI Topic 3: Geometry & Trigonometry. Apply spatial mathematics to real-life contexts with study materials covering 3D geometry, applied trigonometry, and Voronoi diagrams. Unlock advanced HL topics including geometric transformations, vectors, kinematics, and graph theory algorithms.

Subtopic

Understanding

3.1: Fundamentals of geometry

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: Introduction to trigonometry

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: Applied trigonometry

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: Circles

The circle: length of an arc; area of a sector

3.5: Perpendicular bisectors

Equations of perpendicular bisectors

3.6: Voronoi diagrams

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: Unit circle (HL)

The definition of a radian and conversion between degrees and radians

Using radians to calculate area of sector, length of arc

The definitions of

\cos{\theta}

and

\sin{\theta}

in terms of the unit circle

The Pythagorean identity:

\cos^{2}{\theta} + \sin^{2}{\theta} = 1

Definition of

\tan{\theta}

\frac{\sin{\theta}}{\cos{\theta}}

Extension of the sine rule to the ambiguous case

Graphical methods of solving trigonometric equations in a finite interval

3.8: Geometric transformations (HL)

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

3.9: Vectors (HL)

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

Definition and calculation of the scalar product of two vectors

The angle between two vectors; the acute angle between two lines

10:

Definition and calculation of the vector product of two vectors

11:

Geometric interpretation of

|v \times w|

12:

Components of vectors

3.10: Vector lines (HL)

Vector equation of a line in two and three dimensions:

r = a + \lambda b

, where

b

is a direction vector of the line

Vector applications to kinematics

Modelling linear motion with constant velocity in two and three dimensions

Motion with variable velocity in two dimensions

3.11: Graph theory (HL)

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

Tree and cycle algorithms with undirected graphs

Walks, trails, paths, circuits, cycles

3.12: Adjacency matrices (HL)

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

3.13: Algorithms & problems (HL)

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

4

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

4 - Stats & Probability

Mark Complete

Ace IB Maths AI Topic 4: Statistics & Probability. Develop rigorous data analysis skills with notes on descriptive statistics, probability distributions, and hypothesis testing. Master HL concepts including confidence intervals, the Poisson distribution, non-linear regression, and Markov chain transition matrices.

Subtopic

Understanding

4.1: Fundamentals of statistics

Concepts of population, sample, random sample, discrete and continuous data

Reliability of data sources and bias in sampling

Interpretation of outliers

Sampling techniques and their effectiveness

Presentation of data (discrete and continuous): frequency distributions (tables)

Measures of central tendency (mean, median and mode)

Estimation of mean from grouped data

Modal class

Quartiles of discrete data

4.2: Visualising statistics

Histograms

Cumulative frequency; cumulative frequency graphs; use to find median, quartiles, percentiles, range and interquartile range (IQR)

Production and understanding of box and whisker diagrams

4.3: Distribution properties

Measures of dispersion (interquartile range, standard deviation and variance)

Effect of constant changes on the original data

4.4: Introduction to linear statistics

Linear correlation of bivariate data

Pearson's product-moment correlation coefficient,

r

Scatter diagrams; lines of best fit, by eye, passing through the mean point

Equation of the regression line of

y

x

Use of the equation of the regression line for prediction purposes

Interpret the meaning of the parameters,

a

and

b

, in a linear regression

y = ax + b

Spearman's rank correlation coefficient,

r_{s}

Awareness of the appropriateness and limitations of Pearson's product moment correlation coefficient and Spearman's rank correlation coefficient, and the effect of outliers on each

4.5: Probability

Concepts of trial, outcome, equally likely outcomes, relative frequency, sample space (

U

) and event

The probability of an event

A

P(A) = \frac{n(A)}{n(U)}

The complementary events

A

and

A'

(not

A

)

Expected number of occurrences

Use of Venn diagrams, tree diagrams, sample space diagrams and tables of outcomes to calculate probabilities

Combined events

Mutually exclusive events

Conditional probability

Independent events

4.6: Probability distributions

Concept of discrete random variables and their probability distributions

Expected value (mean),

E(X)

for discrete data

Applications of the concepts above

Binomial distribution

Mean and variance of the binomial distribution

The normal distribution and curve

Properties of the normal distribution

Diagrammatic representation

Normal probability calculations

10:

Inverse normal calculations

4.7: Hypotheses & tests

Formulation of null and alternative hypotheses

Significance levels

p

-values

Expected and observed frequencies

The

\chi^{2}

test for independence: contingency tables, degrees of freedom, critical value

The

\chi^{2}

goodness of fit test

The

t

-test

Use of the

p

-value to compare the means of two populations

Using one-tailed and two-tailed tests

4.8: Data collection & categorisation (HL)

Design of valid data collection methods, such as surveys and questionnaires

Selecting relevant variables from many variables

Choosing relevant and appropriate data to analyse

Categorizing numerical data in a

\chi^{2}

table and justifying the choice of categorisation

Choosing an appropriate number of degrees of freedom when estimating parameters from data when carrying out the

\chi^{2}

goodness of fit test

Definition of reliability and validity

Reliability tests

Validity tests

4.9: Advanced linear statistics (HL)

Non-linear regression

Evaluation of least squares regression curves using technology

Sum of square residuals (

SS_{\text{res}}

) as a measure of fit for a model

The coefficient of determination (

R^{2}

)

Evaluation of

R^{2}

using technology

4.10: Linear transformations & combinations (HL)

Linear transformation of a single random variable

Expected value of linear combinations of

n

random variables

Variance of linear combinations of

n

independent random variables

\bar{x}

as an unbiased estimate of

\mu

s^{2}_{n-1}

as an unbiased estimate of

\sigma^{2}

A linear combination of

n

independent normal random variables is normally distributed

Central limit theorem

4.11: Confidence intervals (HL)

Confidence intervals for the mean of a normal population

4.12: Poisson distribution (HL)

Poisson distribution, its mean and variance

Sum of two independent Poisson distributions has a Poisson distribution

4.13: Distribution tests (HL)

Critical values and critical regions

Test for population mean for normal distribution

Test for proportion using binomial distribution

Test for population mean using Poisson distribution

Use of technology to test the hypothesis that the population product moment correlation coefficient (

\rho

) is

0

for bivariate normal distributions

Type I and II errors including calculations of their probabilities

4.14: Transition matrices (HL)

Transition matrices

Powers of transition matrices

Regular Markov chains

Initial state probability matrices

Calculation of steady state and long-term probabilities by repeated multiplication of the transition matrix or by solving a system of linear equations

5 - Calculus

Mark Complete

Conquer IB Maths AI Topic 5: Calculus. Understand rates of change and accumulation with practical study guides on derivatives, integration, and optimization. Advance your HL calculus expertise with notes on kinematics, phase portraits, coupled systems, and solving differential equations using Euler's method.

Subtopic

Understanding

5.1: Introduction to derivatives

Introduction to the concept of a limit

Derivative interpreted as gradient function and as rate of change

Increasing and decreasing functions

Graphical interpretation of

f'(x) > 0

f'(x) = 0

f'(x) < 0

Derivative of

f(x) = ax^{n}

f'(x) = anx^{n-1}

n \in \mathbb{Z}

The derivative of functions of the form

f(x) = ax^{n} + bx^{n-1} + \dots

where all exponents are integers

5.2: Applying derivatives

Tangents and normals at a given point, and their equations

Values of

x

where the gradient of a curve is zero

Solution of

f'(x) = 0

Local maximum and minimum points

Optimisation problems in context

5.3: Fundamentals of integration

Introduction to integration as anti-differentiation of functions of the form

f(x) = ax^{n} + bx^{n-1} + \dots

, where

n \in \mathbb{Z}

n \neq -1

Anti-differentiation with a boundary condition to determine the constant term

Definite integrals using technology

Area of a region enclosed by a curve

y = f(x)

and the

x

-axis, where

f(x) > 0

Approximating areas using the trapezoidal rule

5.4: Advanced derivatives (HL)

The derivatives of

\sin{x}

\cos{x}

\tan{x}

e^{x}

\ln{x}

x^{n}

where

n \in \mathbb{Q}

The chain rule, product rule and quotient rules

Related rates of change

The second derivative

Use of second derivative test to distinguish between a maximum and a minimum point

5.5: Further integration (HL)

Definite and indefinite integration of

x^{n}

where

n \in \mathbb{Q}

, including

n = -1

\sin{x}

\cos{x}

\frac{1}{\cos^{2}{x}}

, and

e^{x}

Integration by inspection, or substitution of the form

\int (g(x))g'(x)\,dx

5.6: Applying integration (HL)

Area of the region enclosed by a curve and the

x

y

-axes in a given interval

Volumes of revolution about the

x

-axis or

y

-axis

5.7: Kinematics (HL)

Kinematic problems involving displacement

s

, velocity

v

and acceleration

a

5.8: Introduction to differential equations (HL)

Setting up a model/differential equation from a context

Solving by separation of variables

Slope fields and their diagrams

Euler's method for finding the approximate solution to first order differential equations

Numerical solution of

\frac{dy}{dx} = f(x, y)

Numerical solution of the coupled system

\frac{dx}{dt} = f_{1}(x, y, t)

and

\frac{dy}{dt} = f_{2}(x, y, t)

5.9: Advanced differential equations (HL)

Phase portrait for the solutions of coupled differential equations of the form:

\frac{dx}{dt} = ax + by

\frac{dy}{dt} = cx + dy

Qualitative analysis of future paths for distinct, real, complex, and imaginary eigenvalues

Sketching trajectories and using phase portraits to identify key features such as equilibrium points, stable populations, and saddle points

Solutions of

\frac{d^{2}x}{dt^{2}} = f\left(x, \frac{dx}{dt}, t\right)

by Euler's method

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