Notes – Maths & Further Maths

These excellent summary A level notes for students from Mathsbox were made available during lockdown, my students really like them; see Pure Mathematics A Level Notes. The notes have a very comprehensive set of clear and well-explained examples.

See also: Worked Examples – A Level  (Sources of fully worked examples by topic)

Mathsbox – Pure summary notes

Also available are Statistics and Mechanics Notes

Download Here

Mathsbox – Statistics summary notes
Mathsbox – Mechanics summary notes

Helm Notes

I have referred to the HELM (Helping Engineers Learn Mathematics) notes many times over the years on this blog. They are referred to on the Notes and Examples page for Further Mathematics as well as in many individual blog posts, for example, Mechanics – Dimensional Analysis, Differential Equations, and Further Calculus.

HELM Notes - Basic Algebra
HELM (2008): Workbook 1: Basic Algebra

If you have not come across the HELM Project before, the project was designed to support the mathematical education of engineering students and includes an extensive collection of notes which include very clear worked examples. Whilst the workbooks cover the basic engineering mathematics and statistics teaching for first and second year students in a typical UK undergraduate engineering degree many of the workbooks include content appropriate for A Level Mathematics and particularly, Further Mathematics. For easy access to these resources, the HELM Project Workbooks are hosted by Loughborough University’s Mathematics Learning Support Centre. Alternatively, the complete set is hosted by the Open University. To access the Open University resources you will need to create an account (easy and free), this will also give you access to the numerous free online courses.

Looking at Loughborough University’s Mathematics Learning Support Centre as well as providing access to the workbooks, we can additionally read about the past, present and future of the resources and see the details of the HELM consortium members and their roles.

You will see from Loughborough’s description that each workbook varies in length from 25 pages to 75 pages (average 50 pages), and includes Mathematics (and Statistics) for engineering simply explained, worked examples, tasks and exercises with answers provided. Note the last workbooks in the series, Workbook 49 is a Student’s Guide and Workbook 50 a Tutor’s Guide.

The Student guide includes a description of the format of the workbooks and a comprehensive list of contents.

HELM - Student Guide
HELM Project – Workbook 49, Student’s Guide

Plymouth University Workbooks

Calculus workbook from Plymouth University

Try these very clear notes with exercises from Plymouth University. Sections are available on Basic Algebra, Graphs, Trigonometry, Calculus, Vectors, Logic and proof, Matrices and Determinants, Vector calculus, Units, and Applications including Forces.

The aim of this project was to produce a library of portable, interactive, web-based support packages to help you learn various mathematical ideas and techniques and to support classroom teaching. 

Forces Workbook Plymouth University


The excellent Math Centre site includes extensive resources. The quick reference leaflets which are available on numerous topics are very clearly written and succinct, see these for example on the Product Rule and the Quotient Rule. There are also teach-yourself booklets, revision booklets, videos and diagnostic tests.

The Math Centre – Product and quotient rules
Mathscentre – Quick Reference – Product and Quotient Rules

For students revising or perhaps preparing for university, then to make sure A Level knowledge is secure – perhaps check the Algebra Refresher from The Mathcentre which has many questions and the answers are at the end of the document.

Just the Maths

AJ Hobson – Just the Maths

AJ Hobson’s Just the Maths (individual pdfs hosted by UEA) (or a complete pdf from the Math Centre: AJ Hobson’s ‘Just the Maths’is very useful. 19 units provide a very comprehensive set of notes and examples for students studying Mathematics for A Level and beyond. The units cover Algebra, Series, Trigonometry, Hyperbolic Functions, Geometry, Complex Numbers, Determinants, Vectors, Matrices, Differentiation, Integration, Partial Differentiation, Ordinary Differential Equations, Laplace Transforms, Z-Transforms, Numerical Mathematics, Statistics and Probability.

Just the Maths” does not have the format of a traditional
text-book or a course of programmed learning; but it is written in a
traditional pure-mathematics style with the minimum amount of
formal rigor. By making use of the well-worn phrase, “it can be
shown that”, it is able to concentrate on the core mathematical
techniques required by any scientist or engineer. The techniques
are demonstrated by worked examples and reinforced by
exercises that are few enough in number to allow completion, or
near-completion, in a one-hour tutorial session. Answers to
exercises are supplied at the end of each unit of work.

AJ Hobson
AJ Hobson – Just the Maths, Complex Numbers 5