UCL Centre for Languages & International Education (CLIE)


Mathematics (UPCH)

Find out more about the Mathematics foundation module on the UPCH. Learn about what you will study, teaching methods, assessments and recommended reading.

Key information 

Module Title: Mathematics

UPC Pathway: Undergraduate Preparatory Certificate for Humanities (UPCH) - optional module

Module Leader: Dr Amelie Berger

Number of students on module in 2021/22 academic year: 30

Module description

The Mathematics foundation module is an optional part of the UPCH course. You’ll gain an in-depth coverage of topics usually taught in A-Level Mathematics and Further Mathematics.

The Mathematics foundation module will prepare you thoroughly with the skills you need to study at UK universities. This module’s students usually go on to study degrees such as Economics BSc, Economics and Statistics BSc, Mathematics BSc and Art and Sciences BASc.

The syllabus covers the following topics:


  • Differentiation from first principles
  • Derivatives of polynomial, trigonometric, hyperbolic, exponential and logarithmic functions, parametric equations
  • The chain rule, product rule and quotient rule
  • Implicit differentiation
  • Tangent and normal lines
  • Stationary points and curve sketching
  • Taylor series and Maclaurin series


  • Indefinite integrals
  • Integration techniques: substitution, integration by parts, trigonometric and hyperbolic 
  • Definite integrals
  • Applications to area and volume of revolution
  • Separable differential equations

Algebra and Hyperbolic Functions

  • Quadratic equations and their roots. Sum and product of roots
  • The remainder and factor theorems
  • The generalised binomial theorem for all real powers, with applications
  • Hyperbolic functions and their inverses and calculus


  • Convergence and divergence
  • Arithmetic and geometric sequences and series
  • Tests for convergence: nth term test, comparison test, limit comparison test, ratio test
  • Compound interest. Applications to investments and mortgages
  • Proof by induction

Complex Numbers

  • Basic arithmetic and notation
  • Argand diagrams
  • Polar form
  • de Moivre’s Theorem and applications
  • Complex roots of numbers
  • Exponential form
  • Loci


  • Matrix arithmetic
  • Inverse matrices
  • Systems of linear equations, Gaussian elimination and echelon forms


  • Counting: permutations and combinations, distinct and identical objects
  • Probability: and, or and conditional probability
  • Discrete random variables. Mean, standard deviation and variance
  • The binomial distribution
  • Continuous random variables. Mean, standard deviation, variance and mode

How we teach Mathematics

You’ll have two lectures every week. Both lectures will be available as recordings. Each week, you’ll attend two in-person small-group tutorials. In these classes, you’ll work on mathematics exercises and activities based on the lectures. You’ll have one additional weekly face-to-face class with the whole group to discuss the lectures.

We expect you to carry out independent study and problem-solving outside of your classes.

Office hours will be available for additional support.

Intended learning outcomes

By the end of the module, you should be able to:

  • state and prove the main theoretical results the syllabus is based on
  • apply the theory to problem-solving
  • present well-written solutions to problems with good attention to detail and a logical structure
  • work independently on mathematics exercises, taking responsibility for your own understanding and asking for help when necessary
  • confidently discuss mathematical ideas in English, using correct vocabulary.

Overview of Summative Assessment

OneTest (2 hours)10%
TwoTest (2 hours)10%
ThreeTest (2 hours)10%
 Final Exam (3 hours)30%
Throughout the yearClass Tests (45 mins each)30%
 Coursework (8 assignments)10%
  100% total

Assessment weighting is for the current academic year; this may be subject to change for 2022/23 entry. 

Recommended reading

Before starting the course, you should prepare by revising high school work. You’ll need to be comfortable with logarithms and have strong trigonometry knowledge.

You should be able to work with all six trigonometric functions (sine, cosine, tangent, cotangent, secant, cosecant) and the inverse trigonometric functions. You should also be familiar with standard trigonometric identities and using them in proofs. Any standard high school mathematics text should cover this material.

The set text (‘Mathematics - The Core Course for A-level’, by L Bostock and S Chandler, published by Stanley Thornes, ISBN 0859503062) covers this material in chapters 6 and 7.

Additional costs

All students should buy the set course textbook and calculator.

Calculator: Casio fx-83GT X, cost is approximately £10. (Older models are acceptable.)

Textbook: 'Mathematics- The Core Course for A-level’, by L Bostock and S Chandler, published by Stanley Thornes, ISBN 0859503062. This book costs approximately £35, if bought new. Cheaper second-hand copies are available.

Students may wish to buy the follow-on book to this text, although it’s not compulsory. This book contains additional topics and covers certain topics in more depth: ‘Further Pure Mathematics’, by L Bostock, S Chandler and C Rourke, published by Stanley Thornes, ISBN 0859501035. This also costs around £35 if bought new.

Module selection guide

Please check out our optional module selection guide for information on suitable modules for your future degree plans.