Resources to self-learn A-level Further Mathematics
Last year, I took up the challenge to self-learn Cambridge A-level Further Mathematics and in the process I came across a lot of interesting free high-quality online resources which have helped me achieve an A*. The goal of this post is to compile all these resources in one place.
I am not affiliated in any way with the resources mentioned below. The order in which the resources appear is arbitrary and some of the links provided may contain copyrighted material.
Books
| Resource | Comment |
|---|---|
| EdExcel books | Google Drive folder containing PDF books of A-level Pure, Mechanics, Statistics, Decision mathematics, Core Pure for the EdExcel exams. |
| CIE books | Google Drive folder containing complete Cambridge A-level Further Maths book and some other books on Pure mathematics. |
| Advanced Statistics by Crawshaw and Chambers | Highly-recommended A-level statistics notes and questions. |
| Other Further Maths books | A Google document containing download links for other Further Maths books. |
| OpenStax | Peer-reviewed, openly licensed and 100% free books. |
| Libretext | Online textbook platform. |
| Multivariable Calculus By James Stewart![] | Not catered for A-level syllabus but some chapters (Polar coordinates, vectors, differential equations) are worth a read. |
Practice questions
It is important, therefore, that those who prepare for an examination of this type do not fall back entirely on routine examples.
– Cambridge Examiners’ Report [9231_w03_er]
Practising similar questions from past papers over and over again leads to diminishing returns. Here are websites where you can find non-routine questions ranging in difficulty :
| Resource | Comment |
|---|---|
| Madamaths | Great range of topical questions. |
| Isaac Physics | Online learning platform created by Cambridge itself. Questions are based on past papers and are arranged by topic. Hints are offered for each question. |
| Underground Mathematics | A database of questions, some dating from the 1950s, across multiple exam boards past papers. Hints and step-by-step explanations are offered for each question. |
| Maths and Physics Tutor | Topical questions along with their solutions. |
| Dr Oliver Maths | Topical questions along with their solutions. |
| Dr Frost Maths | Online learning platform with worksheets from a range of examination boards. |
| Pre-2010 Cambridge A-level Maths (9709) M2 past papers (Paper 52) | Some questions in these past papers are relevant to Cambridge A-level Further Mechanics and are worth practicing. |
| mathedup | Topical questions along with their solutions. |
| STEP papers | Admission test past papers with quite challenging questions. The difficulty level is way above the Cambridge past papers questions but the questions are still worth a shot if you want to test your understanding of Further Maths. |
| mathdistinction | Challenging questions for A-level Maths with some questions relevant to A-level Further Maths. |
Leaning notes
| Resource | Comment |
|---|---|
| Cambridge AS & A Level Further Mathematics Full book | I used this book extensively for mechanics and statistics. It covered my Further Maths syllabus (9231) entirely. |
| Kinematics Fundamental by Sunil Kumar | Openstax book which I used to practice the projectile motion chapter in Further Mechanics. It has notes and questions different from the usual past paper questions. Solutions are available as well. |
| Cambridge A-Level Further Maths Scheme of Work | Contains links to learning resources for each chapter. |
| Interactive Linear Algebra PDF by GaTech | Highly recommend reading. You can easily skip the sub-chapters not relevant to the A-level syllabus. |
| STAT 415 Online notes by PennState | Online repository for statistics notes and some sample questions. I personally used it to learn inference testing and non-parametric tests. |
| Kinematics Fundamental by Sunil Kumar | Great notes, questions, and explanations on Kinematics. Some questions are quite challenging. |
| Paul's Online Notes | University notes some of which are relevant to A-level Further Maths. Also has a practice problems section. I personally used the website to learn about the integral test. |
| Hyperbolic notes | Notes from Calculus 1 book from OpenStax. |
| Brilliant.org | Has good notes on roots of polynomial. |
You can access all my Further Maths notes here. These notes are designed primarily for quick reference purposes rather than to serve as a comprehensive guide for understanding a topic.
Technique for finding notes
You can easily find university lecture notes of prestigious universities for most topics by searching <chapter_name> university lecture notes pdf on Google. Here are some useful lecture notes that I found with this method:
- Polar curve sketching
- Mathematical induction by the University of Sydney
- Mathematical induction by Cornell University
- Hyperbolic functions by Western Sydney University
- Vectors by the University of Oxford
Youtube videos
| Resource | Comment |
|---|---|
| GenieusHub Further Pure playlist | Topical tutorial videos for Cambridge A-level Further Pure. |
| TL maths | YouTube playlists on A-level Further Pure maths. |
| Organic chemistry tutor statistic playlist | Video explanations + walkthroughs on how to solve statistics questions. |
| 3Blue1Brown Essence of Linear Algebra | Videos to understand the geometry behind concepts like eigenvectors, matrices, etc. |
| Crash Course for statistics | Some helpful statistics videos to understand normal distributions and Chi-Squared tests. |
| blackpenredpen | Videos on integrations, polar curve sketching, differential equations, and much more. |
| Tom Rock Maths | Videos on linear algebra and differential equations. |
| MIT OpenCourseWare | Their maths and physics courses are the most relevant for FM. Example: differential equations video. Lecture notes and questions are available on their website. |
Support
Getting stuck on a question or not understanding a chapter can be frustrating especially when you are self-learning. Here are some places to find help :
| Resource | Comment |
|---|---|
| Cambridge Examiners Report | It points out the common mistakes students are making on each question and occasionally contains explanations. Examiners' reports have the `_er` suffix in their names such as `9231_w20_er.pdf`. |
| Math Exchange | Question-answer forum for maths questions. Use it for questions on Pure mathematics. |
| Physics Exchange | Question-answer forum for maths questions. Use it for questions on Mechanics. |
| Cross Validated | Question-answer forum for statistics questions. Use it for questions on statistics. |
| physicsforums | Consult the threads on maths and physics. |
| r/furthermaths, r/alevel | These subreddits are used by A-level students who might be able to help you. |
| studentroom | studentroom has some study help forums that are worth mentioning even though I've never used them. |
Some of the above forums have a strict etiquette for posting questions. To increase the chances of someone helping you, you must ask a good question in the right place. Read here and here on how to ask a good question.
Calculators
Verifying the correctness of your work is essential as answer sheets and mark schemes are prone to errors. Here are some calculators which may help you:
| Resource | Comment |
|---|---|
| Wolfram Alpha | Advanced online calculator that can be used for integration, differentiation, calculating area under curve, solving polynomial and differential equations, and much more. |
| Desmos | 2D graphing calculator. Extremely helpful for polar curve sketching and transformations in Pure. |
| Geogebra matrix transformations | Extremely helpful for transformations in Pure. |
| Geogebra 3D calculator | Can be used as a visualisation tool for vectors, planes, and matrices. |
| Integration calculator | Step-by-step explanation on how to integrate an expression. |
| Derivative calculator | Step-by-step explanation on how to differentiate an expression. |
Visualising planes in Geogebra
Progress tracking
I used Notion, a digital note-taking app, for storing my notes and tracking unsolved questions and past papers.
My past papers working schedule for the few weeks leading to my exams
My notes on polar coordinates in Notion
Other resources
| Resource | Comment |
|---|---|
| OCR Further Maths blog | Other helpful resources for self-learning FM. |
| Advice for FM | A YouTube video where Ray Amjad gives some tips for learning FM. I remember watching this video just before starting to learn FM. |