IB Maths Topics

As an IB DP maths student, it can be tricky to find information about the content that you need to know. The IB DP offers two different maths options: Applications & Interpretations and Analysis & Approaches. Each of these two options is available to study at either standard level (SL) or higher level (HL), giving a total of four different courses. 

Some mathematical content is common to all four courses, and each HL course includes all of the content from the same option at SL.

In this article, you’ll find a breakdown of all the IB maths topics you'll need to cover for your specific course. As a result, you’ll know exactly what you need to study and where to find the best revision resources to help you with this.

1. Applications & Interpretation Standard Level Topics

2. Applications & Interpretation Higher Level Topics

3. Analysis & Approaches Standard Level Topics

4. Analysis & Approaches Higher Level Topics

Applications & Interpretation Standard Level Topics

The Applications & Interpretation SL course is divided into 5 different topic areas:

1. Number & Algebra

2. Functions

3. Geometry & Trigonometry

4. Statistics & Probability

5. Calculus

1. Number & Algebra

This topic covers fundamental numerical concepts and their real-world applications. It introduces problem-solving techniques including an emphasis on using your graphical display calculator (GDC) for efficiency. 

Topics include:

Number Toolkit

• Standard form

• Exponents & logarithms

• Approximation & estimation

• GDC: Solving equations

Sequences & Series

• Language of sequences & series

• Arithmetic sequences & series

• Geometric sequences & series

• Applications of sequences & series

Financial Applications

• Compound interest & depreciation

• Amortisation & annuities

2. Functions

This topic explores functions and their graphs, including linear, quadratic, cubic, exponential and sinusoidal models. It covers key properties of graphs, how to graph functions and how to model real-world situations using functions. It also includes direct and inverse variation and strategies for choosing appropriate mathematical models.

Topics include:

Linear Functions & Graphs

• Equations of a straight line

Further Functions & Graphs

• Functions

• Graphing functions

• Properties of graphs

Modelling with Functions

• Linear & piecewise models

• Quadratic & cubic models

• Exponential models

• Direct & inverse variation

• Sinusoidal models

• Strategy for modelling functions

3. Geometry & Trigonometry

This topic explores fundamental concepts in geometry and trigonometry, focusing on shapes, measurements and spatial relationships. It includes coordinate geometry, 3D shapes and trigonometric principles for solving real-world problems. Additionally, it introduces Voronoi diagrams, a mathematical tool used in spatial analysis for applications like optimising locations and resource distribution.

Topics include:

Geometry Toolkit

• Coordinate geometry

• Arcs & sectors

Geometry of 3D Shapes

• 3D coordinate geometry

• Volume & surface area

Trigonometry

• Pythagoras & right-angled trigonometry

• Non right-angled trigonometry

• Applications of trigonometry & Pythagoras

Voronoi Diagrams

• Voronoi diagrams

• Toxic waste dump problem

4. Statistics & Probability

This topic covers key principles in statistics and probability, focusing on data collection, representation and interpretation. It explores measures of central tendency, variability, correlation and regression to identify patterns in data. Probability concepts and distributions help model uncertainty, while hypothesis testing methods, such as chi-squared and t-tests, assess statistical significance in real-world contexts.

Topics include:

Statistics Toolkit

• Sampling & data collection

• Statistical measures

• Frequency tables

• Linear transformations of data

• Outliers

• Univariate data

• Interpreting data

Correlation & Regression

• Bivariate data

• Correlation coefficients

• Linear regression

Probability

• Probability & types of events

• Conditional probability

• Sample space diagrams

Probability Distributions

• Discrete probability distributions

• Expected values

Binomial Distribution

• The binomial distribution

• Calculating binomial probabilities

Normal Distribution

• The normal distribution

• Calculations with normal distribution

Hypothesis Testing

• Hypothesis testing

• Chi-squared test for independence

• Goodness of fit test

• The t-test

5. Calculus

This topic explores differentiation and integration, two fundamental concepts in calculus. Differentiation examines how functions change and integration is used to calculate areas, accumulations and totals. Real-life applications of these tools are also explored.

Topics include:

Differentiation

• Introduction to differentiation

• Applications of differentiation

• Modelling with differentiation

Integration

• Trapezoid rule: numerical integration

• Introduction to integration

• Applications of integration 

Revision Resources for Applications & Interpretation Standard Level

At Save My Exams, we have course-specific revision notes, exam questions, flashcards and practice papers for the IB Applications & Interpretation Standard Level course.

Applications & Interpretation Higher Level Topics

The Applications & Interpretation HL course is divided into 5 different topic areas:

1. Number & Algebra

2. Functions

3. Geometry & Trigonometry

4. Statistics & Probability

5. Calculus

1. Number & Algebra

This topic covers fundamental numerical concepts and their real-world applications. It introduces problem-solving techniques, including an emphasis on using your graphical display calculator (GDC) for efficiency. 

The HL course includes additional number topics beyond the SL course, such as the laws of logarithms, sum to infinity, complex numbers, matrices and eigenvalues and eigenvectors.

Topics include:

Number Toolkit

• Standard form

• Approximation & estimation

• GDC: Solving equations

Exponentials & Logs

• Exponents

• Logarithms

Sequences & Series

• Language of sequences & series

• Arithmetic sequences & series

• Geometric sequences & series

• Applications of sequences & series

Financial Applications

• Compound interest & depreciation

• Amortisation & annuities

Complex Numbers

• Introduction to complex numbers

• Modulus & argument

• Introduction to Argand diagrams

Further Complex Numbers

• Geometry of complex numbers

• Forms of complex numbers

• Applications of complex numbers

Matrices

• Introduction to matrices

• Operations with matrices

• Determinants & inverses

• Solving systems of linear equations with matrices

Eigenvalues & Eigenvectors

• Eigenvalues & eigenvectors

• Applications of matrices

2. Functions

This topic explores functions and their graphs, including linear, quadratic, cubic, exponential and sinusoidal models. It covers key properties of graphs, how to graph functions and how to model real-world situations using functions. It also includes direct and inverse variation and strategies for choosing appropriate mathematical models.

In addition to the SL course, the HL course includes sinusoidal models with phase shift, composite functions, transformations of graphs and additional types of models (logarithmic, logistic and non-linear piecewise).

Topics include:

Linear functions & Graphs

• Equations of a straight line

Further Functions & Graphs

• Functions

• Graphing functions

• Properties of graphs

Modelling with Functions

• Linear models

• Quadratic & cubic models

• Exponential models

• Direct & inverse variation

• Sinusoidal models

• Strategy for modelling functions

Functions Toolkit

• Composite & inverse functions

Transformations of Graphs

• Translations of graphs

• Reflections of graphs

• Stretches of graphs

• Composite transformations of graphs

Further Modelling with Functions

• Properties of further graphs

• Natural logarithmic models

• Logistic models

• Piecewise models 

3. Geometry & Trigonometry

This topic explores fundamental concepts in geometry and trigonometry, focusing on shapes, measurements and spatial relationships. It includes coordinate geometry, 3D shapes and trigonometric principles for solving real-world problems. Additionally, it introduces Voronoi diagrams, a mathematical tool used in spatial analysis for applications like optimising locations and resource distribution.

There is a lot of additional content within geometry and trigonometry at HL. As well as introducing radians and the ambiguous sine rule, new areas include further trigonometry, matrices, vectors and graph theory.

Topics include:

Geometry Toolkit

• Coordinate geometry

• Radian measure

• Arcs & sectors

Geometry of 3D Shapes

• 3D coordinate geometry

• Volume & surface area

Trigonometry

• Pythagoras & right-angled trigonometry

• Non right-angled trigonometry

• Applications of trigonometry & Pythagoras

Further Trigonometry

• The unit circle

• Simple identities

• Solving trigonometric equations

Voronoi Diagrams

• Voronoi diagrams

• Toxic waste dump problem

Matrix Transformations

• Matrix transformations

• Determinant of a transformation matrix

Vector Properties

• Introduction to vectors

• Position & displacement vectors

• Magnitude of a vector

• The scalar product

• The vector product

• Components of vectors

• Geometric proof with vectors

Vector Equations of Lines

• Vector equations of lines

• Shortest distances with lines

Modelling with Vectors

• Kinematics with vectors

• Constant & variable velocity

Graph Theory

• Introduction to graph theory

• Walks & adjacency matrices

• Minimum spanning trees

• Chinese postman problem

• Travelling salesman problem

• Bounds for travelling salesman problem

4. Statistics & Probability

This topic covers key principles in statistics and probability, focusing on data collection, representation and interpretation. It explores measures of central tendency, variability, correlation and regression to identify patterns in data. Probability concepts and distributions help model uncertainty, while hypothesis testing methods, such as chi-squared and t-tests, assess statistical significance in real-world contexts.

Non-linear regression and logarithmic scales are introduced at HL as well as linear combinations of random variables, the distribution of sample means, confidence intervals, the Poisson distribution and more in-depth hypothesis testing, including errors and paired t-tests. The HL course also covers Markov chains and transition matrices.

Topics include:

Statistics Toolkit

• Sampling

• Data collection

• Statistical measures

• Frequency tables

• Linear transformations of data

• Outliers

• Univariate data

• Interpreting data

Correlation & Regression

• Bivariate data

• Correlation coefficients

• Linear regression

Further Correlation & Regression

• Non-linear regression

• Logarithmic scales

• Linearising using logarithms

Probability

• Probability & types of events

• Conditional probability

• Sample space diagrams

Probability Distributions

• Discrete probability distributions

• Expected values

Random Variables

• Linear combinations of random variables

• Unbiased estimates

Binomial Distribution

• The binomial distribution

• Calculating binomial probabilities

Normal Distribution

• The normal distribution

• Calculations with normal distribution

Further Normal Distribution

• Sample mean distribution

• Confidence interval for the mean

Poisson Distribution

• Poisson distribution

• Calculating Poisson probabilities

Hypothesis Testing

• Hypothesis testing

• Chi-squared test for independence

• Goodness of fit test 

Further Hypothesis Testing

• Hypothesis testing for mean (one sample)

• Hypothesis testing for mean (two sample)

• Binomial hypothesis testing

• Poisson hypothesis testing

• Hypothesis testing for correlation

• Type I & Type II errors

Transition Matrices & Markov Chains

• Markov chains

• Transition matrices

5. Calculus

This topic explores differentiation and integration, two fundamental concepts in calculus. Differentiation examines how functions change and integration is used to calculate areas, accumulations and totals. Real-life applications of these tools are also explored.

The HL course builds on the SL course by differentiating and integrating special functions, and by introducing useful techniques for differentiating and integrating more complex functions. Additionally, kinematics and differential equations are covered.

Differentiation

• Introduction to differentiation

• Applications of differentiation

• Modelling with differentiation

Further Differentiation

• Differentiating special functions

• Techniques of differentiation

• Related rates of change

• Second order derivatives

• Further applications of differentiation

• Concavity & points of inflection

Integration

• Trapezoid rule: numerical integration

• Introduction to integration

• Applications of integration

Further Integration

• Integrating special functions

• Techniques of integration

• Further applications of integration

• Volumes of revolution

Kinematics

• Kinematics toolkit

• Calculus for kinematics

Differential Equations

• Modelling with differential equations

• Separation of variables

• Slope fields

• Approximate solutions to differential equations

Further Differential Equations

• Coupled differential equations

• Second order differential equations

Revision Resources for Applications & Interpretation Higher Level

At Save My Exams, we have course-specific revision notes, exam questions, flashcards and practice papers for the IB Applications & Interpretation Higher Level course.

Analysis & Approaches Standard Level Topics

The Analysis & Approaches SL course is divided into 5 different topic areas:

1. Number & Algebra

2. Functions

3. Geometry & Trigonometry

4. Statistics & Probability

5. Calculus

1. Number & Algebra

This unit explores fundamental number concepts, including standard form, indices, exponentials and logarithms, focusing on their properties and applications. It covers sequences and series, including arithmetic and geometric progressions, with real-world uses like compound interest and depreciation. The unit also introduces mathematical proof and the binomial theorem for algebraic expansion.

Topics include:

Number Toolkit

• Standard form

• Laws of indices

Exponentials & Logs

• Introduction to logarithms

• Laws of logarithms

• Solving exponential equations

Sequences & Series

• Language of sequences & series

• Arithmetic sequences & series

• Geometric sequences & series

• Applications of sequences & series

• Compound interest & depreciation

Proof & Reasoning

• Proof

Binomial Theorem

• Binomial theorem

2. Functions

This topic explores functions and their graphical representations, including linear, quadratic, reciprocal, exponential and logarithmic functions. It covers solving equations, function transformations and inequalities, as well as composite and inverse functions. Emphasis is placed on modelling real-world situations and understanding key properties like discriminants and function transformations through reflections, translations and stretches.

Topics include:

Linear Functions & Graphs

• Equations of a straight line

Quadratic Functions & Graphs

• Quadratic functions

• Factorising & completing the square

• Solving quadratics

• Quadratic inequalities

• Discriminants

Functions Toolkit

• Language of functions

• Composite & inverse functions

• Graphing functions

Further Functions & Graphs

• Reciprocal & rational functions

• Exponential & logarithmic functions

• Solving equations

• Modelling with functions

Transformations of Graphs

• Translations of graphs

• Reflections of graphs

• Stretches of graphs

• Composite transformations of graphs

3. Geometry & Trigonometry

This unit covers foundational geometry and trigonometry, including coordinate geometry, measurement of angles in radians, and properties of 2D and 3D shapes. It emphasises trigonometric principles, such as Pythagoras' theorem, right-angled and non-right-angled trigonometry, and it introduces advanced concepts like the unit circle, graph transformations and solving trigonometric equations and identities, applicable in real-world modelling.

Topics include:

Geometry Toolkit

• Coordinate geometry

• Radian measure

• Arcs & sectors

Geometry of 3D Shapes

• 3D coordinate geometry

• Volume & surface area

Trigonometry

• Pythagoras & right-angled trigonometry

• Non right-angled trigonometry

• Applications of trigonometry & Pythagoras

Further Trigonometry

• The unit circle

• Exact values

Trigonometric Functions & Graphs

• Graphs of trigonometric functions

• Transformations of trigonometric functions

• Modelling with trigonometric functions

Trigonometric Equations & Identities

• Simple identities

• Double angle formulae

• Relationship between trigonometric ratios

• Linear trigonometric equations

• Quadratic trigonometric equations

4. Statistics & Probability

This unit covers statistics and probability, focusing on data collection, organisation and interpretation. It explores statistical measures, correlation and regression, as well as probability concepts, including conditional probability and probability distributions. Key distributions such as the binomial and normal distributions are introduced, along with methods for calculating probabilities, expected values and standardisation. 

Topics include:

Statistics Toolkit

• Sampling & data collection

• Statistical measures

• Frequency tables

• Linear transformations of data

• Outliers

• Univariate data

• Interpreting data

Correlation & Regression

• Bivariate data

• Correlation & regression

Probability

• Probability & types of events

• Conditional probability

• Sample space diagrams

Probability Distributions

• Discrete probability distributions

• Expected values

Binomial Distribution

• The binomial distribution

• Calculating binomial probabilities

Normal Distribution

• The normal distribution

• Calculations with normal distribution

• Standardisation of normal variables

5. Calculus

This topic explores differentiation and integration, key concepts in calculus. It covers differentiation techniques, special functions, second-order derivatives and applications such as concavity, optimisation and graph analysis. Integration is introduced with definite and indefinite integrals, special functions and real-world applications. The topic also includes kinematics, using calculus to model motion and solve problems involving velocity and acceleration.

Topics include:

Differentiation

• Introduction to differentiation

• Applications of differentiation

Further Differentiation

• Differentiating special functions

• Techniques of differentiation

• Second order derivatives

• Further applications of differentiation

• Concavity & points of inflection

• Derivatives & graphs

Integration

• Introduction to integration

• Applications of integration

Further Integration

• Integrating special functions

• Techniques of integration

• Definite integrals

• Further applications of integration

Optimisation

• Modelling with differentiation

Kinematics

• Kinematics toolkit

• Calculus for kinematics

Revision Resources for Analysis & Approaches Standard Level

At Save My Exams, we have course-specific revision notes, exam questions, flashcards and practice papers for the IB Analysis & Approaches Standard Level course.

Analysis & Higher Level Topics

The Analysis & Approaches HL course is divided into 5 different topic areas:

1. Number & Algebra

2. Functions

3. Geometry & Trigonometry

4. Statistics & Probability

5. Calculus

1. Number & Algebra

This unit explores fundamental number concepts, including standard form, indices, exponentials and logarithms, focusing on their properties and applications. It covers sequences and series, including arithmetic and geometric progressions, with real-world uses like compound interest and depreciation. The unit also introduces mathematical proof and the binomial theorem for algebraic expansion.

The HL course includes additional number topics beyond the SL course, such as partial fractions, proof by induction and by contradiction, as well as permutations & combinations, complex numbers and systems of linear equations.

Topics include:

Number & Algebra Toolkit

• Standard form

• Laws of indices

• Partial fractions

Exponentials & Logs

• Introduction to logarithms

• Laws of logarithms

• Solving exponential equations

Sequences & Series

• Language of sequences & series

• Arithmetic sequences & series

• Geometric sequences & series

• Applications of sequences & series

• Compound interest & depreciation

Simple Proof & Reasoning

• Proof 

Further Proof & Reasoning

• Proof by induction

• Proof by contradiction

Binomial Theorem

• Binomial theorem

• Extension of the binomial theorem

Permutations & Combinations

• Counting principles

• Permutations & combinations

Complex Numbers

• Intro to complex numbers

• Modulus & argument

• Introduction to Argand diagrams

Further Complex Numbers

• Geometry of complex numbers

• Forms of complex numbers

• Complex roots of polynomials

• De Moivre's theorem

• Roots of complex numbers

Systems of Linear Equations

• Systems of linear equations

• Algebraic solutions

2. Functions

This topic explores functions and their graphical representations, including linear, quadratic, reciprocal, exponential and logarithmic functions. It covers solving equations, function transformations and inequalities, as well as composite and inverse functions. Emphasis is placed on modelling real-world situations and understanding key properties like discriminants and function transformations through reflections, translations and stretches.

In addition to the SL course, the HL course includes symmetry of functions, and covers some areas in more detail such as reciprocal and rational functions and composite transformations of graphs. The HL course also covers polynomial functions, solving inequalities graphically and modulus functions.

Topics include:

Linear Functions & Graphs

• Equations of a straight line

Quadratic Functions & Graphs

• Quadratic functions

• Factorising & completing the square

• Solving quadratics

• Quadratic inequalities

• Discriminants

Functions Toolkit

• Language of functions

• Composite & inverse functions

• Symmetry of functions

• Graphing functions

Other Functions & Graphs

• Exponential & logarithmic functions

• Solving equations

• Modelling with functions

Reciprocal & Rational Functions

• Reciprocal & rational functions

Transformations of Graphs

• Translations of graphs

• Reflections of graphs

• Stretches of graphs

• Composite transformations of graphs

Polynomial Functions

• Factor & remainder theorem

• Polynomial division

• Polynomial functions

• Roots of polynomials

Inequalities

• Solving inequalities graphically

• Polynomial inequalities

Further Functions & Graphs

• Modulus functions

• Modulus transformations

• Modulus equations & inequalities

• Reciprocal & square transformations

3. Geometry & Trigonometry

This unit covers foundational geometry and trigonometry, including coordinate geometry, measurement of angles in radians and properties of 2D and 3D shapes. It emphasises trigonometric principles, such as Pythagoras' theorem, right-angled and non-right-angled trigonometry, and it introduces advanced concepts like the unit circle, graph transformations and solving trigonometric equations and identities, applicable in real-world modelling.

There is additional content within geometry and trigonometry at HL. As well as introducing more trigonometric formulae and inverse and reciprocal trig functions, new areas include vector properties, vector equations of lines and vector planes.

Topics include:

Geometry Toolkit

• Coordinate geometry

• Radian measure

• Arcs & sectors

Geometry of 3D Shapes

• 3D coordinate geometry

• Volume & surface area

Trigonometry Toolkit

• Pythagoras & right-angled trigonometry

• Non right-angled trigonometry

• Applications of trigonometry & Pythagoras

Trigonometry

• The unit circle

• Exact values

Trigonometric Functions & Graphs

• Graphs of trigonometric functions

• Transformations of trigonometric functions

• Modelling with trigonometric functions

Trigonometric Equations & Identities

• Simple identities

• Compound angle formulae

• Double angle formulae

• Relationship between trigonometric ratios

• Linear trigonometric equations

• Quadratic trigonometric equations

Inverse & Reciprocal Trig Functions

• Reciprocal trig functions

• Inverse trig functions

Further Trigonometry

• Trigonometric proof

• Strategy for trigonometric equations

Vector Properties

• Introduction to vectors

• Position & displacement vectors

• Magnitude of a vector

• The scalar product

• The vector product

• Geometric proof with vectors

Vector Equations of Lines

• Vector equations of lines

• Applications to kinematics

• Pairs of lines in 3D

• Shortest distances with lines

Vector Planes

• Vector equations of planes

• Intersections of lines & planes

• Angles between lines & planes

• Shortest distances with planes

4. Statistics & Probability

This topic covers statistics and probability, focusing on data collection, organisation and interpretation. It explores statistical measures, correlation and  regression, as well as probability concepts, including conditional probability and probability distributions. Key distributions such as the binomial and normal distributions are introduced, along with methods for calculating probabilities, expected values and standardisation. 

Bayes’ theorem and the probability density function are additions to the HL course.

Topics include:

Statistics Toolkit

• Sampling & data collection

• Statistical measures

• Frequency tables

• Linear transformations of data

• Outliers

• Univariate data

• Interpreting data

Correlation & Regression

• Bivariate data

• Correlation & regression

Probability

• Probability & types of events

• Conditional probability

• Bayes' theorem

• Sample space diagrams

Probability Distributions

• Discrete probability distributions

• Mean & variance 

Binomial Distribution

• The binomial distribution

• Calculating binomial probabilities

Normal Distribution

• The normal distribution

• Calculations with normal distribution

• Standardisation of normal variables

Further Probability Distributions

• Probability density function

5. Calculus

This topic explores differentiation and integration, key concepts in calculus. It covers differentiation techniques, special functions, second-order derivatives and applications such as concavity, optimisation and graph analysis. Integration is introduced with definite and indefinite integrals, special functions and real-world applications. The topic also includes kinematics, using calculus to model motion and solve problems involving velocity and acceleration.

The HL course builds on the SL course by introducing the idea of limits and continuity, as well as more advanced differentiation and integration techniques. Additionally, differential equations, the Maclaurin series and l’Hôpital’s rule are covered.

Topics include:

Differentiation

• Introduction to differentiation

• Applications of differentiation

Further Differentiation

• Differentiating special functions

• Techniques of differentiation

• Higher order derivatives

• Further applications of differentiation

• Concavity & points of inflection

• Derivatives & graphs

Integration

• Introduction to integration

• Applications of integration

Further Integration

• Integrating special functions

• Techniques of integration

• Definite integrals

• Further applications of integration

Optimisation

• Modelling with differentiation

Kinematics

• Kinematics toolkit

• Calculus for kinematics

Basic Limits & Continuity

• Basic limits & continuity

Advanced Differentiation

• First principles differentiation

• Applications of chain rule

• Implicit differentiation

• Differentiating further functions

Advanced Integration

• Integrating further functions

• Further techniques of integration

• Integrating with partial fractions

• Advanced applications of integration

• Modelling with volumes of revolution

Differential Equations

• Numerical solutions to differential equations

• Analytical solutions to differential equations

• Modelling with differential equations

Maclaurin Series

• Maclaurin series

• Maclaurin series from differential equations

Further Limits (including L'Hôpital's Rule)

• Further limits

Revision Resources for Analysis & Approaches Higher Level

At Save My Exams, we have course-specific revision notes, exam questions, flashcards and practice papers for the IB Analysis & Approaches Higher Level course.

Improve Your Grades with Save My Exams

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