List of resources for A-level Further Mathematics
Hi everyone !
Last year, I took up the challenge to self-learn A-level Further Mathematics and in the process I came across a lot of interesting free high-quality online resources which I wanted to share.
The resources listed below were originally part of my blog post but I decided to share it here for greater visibility. I still recommend checking the blog post for the latest updates.
I am not affiliated in any way with the resources mentioned below and the order in which the resources appear is arbitrary.
Books
| Resource | Description |
|---|---|
| Oxford Further Pure Mathematics | A free ebook provided by PapaCambridge. |
| 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. |
Questions
Practicing 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 | Description |
|---|---|
| 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)/) | Questions in these past papers are relevant to Cambridge A-level Further Mechanics. |
| 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. |
Notes
| Resource | Description |
|---|---|
| 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/Other%20Resources/9231_y17-19_sw_01.pdf) | 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 |
All of my notes have also been published here. However, these notes are not meant to help you understand a topic deeply.
University lecture 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 of the lecture notes that I used :
- 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 | Description |
|---|---|
| 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, ... |
| 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 | Description |
|---|---|
| Cambridge Examiners Report/) | If you are stuck on a question from a Cambridge paper, the examiner's report can be of great help. It points out the common mistakes students are making on each question and occasionally contains explanations. Examiners' reports have the _er tag in their names. For example 9231_w20_er.pdf/2020/9231_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
| Resource | Description |
|---|---|
| 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. |
Other resources
| Resource | Description |
|---|---|
| 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. |
Let me know if there are any dead links or mistakes in my post. Feel free to expand this list of resources in the comments.