Units 12-1 – 13.16 from AJ Hobson’s ‘Just the Maths’offers very comprehensive coverage starting with the basics and going well beyond school Mathematics
(These sources of Notes can all be found on the Notes page.)
You can use Desmos and WolframAlpha to check your work and see some excellent visual representations.
As an example, we can find the total area bounded by f(x) = x4−3x3−4x2+12x, the x-axis, the line x=−1 and the line x=3.
We could use WolframAlpha for a quick check. The visual representation shows clearly that we are dealing with areas above and below the x-axis.
Scrolling down the page we see that this query also returns the indefinite integral.
For the total shaded area, we can change the limits of the query to evaluate each section.
See Calculus & Analysis for more examples of WolframAlpha queries.
Or we could turn to the excellent Desmos where we can very simply change the limits.
For some great puzzles by Jonny Griffiths, check Digitisers. Clear instructions explaining the puzzles and notation used are given at the beginning of the document, we then have all the tasks by topic. Full solutions are provided.
— AMSP – Advanced Mathematics Support Programme (@Advanced_Maths) September 14, 2021
From AMSP have a look at this great new magazine which is aimed at high-attaining Year 12 (UK age 16-17) students. You can read more about the magazine and freely access the first issue here.
You will find out more about where Maths can take you in the future, learn more about the applications process and what it is like to study maths at university.
Of course check the Do some maths! section. There is some great advice on problem solving and puzzles to try.
MEI’s Introduction to Data Science Short Course is now freely available from MEI; this is a self-study course aimed at A level Maths students and uses the large data sets for AQA, Edexcel, MEI and OCR.
For a list of recommended books for young people interested in Mathematics, try this list from Nrich which is grouped into three different categories: History of Mathematics, Recreational and Thinking Mathematically.
From Cambridge University this list of interesting mathematics books and internet sites is mainly intended for sixth-formers planning to take a degree in mathematics. The list includes some items which are suitable for less experienced readers so may well appeal to a wider audience. The list was last updated in September 2020. I see it includes Kevin Houston’s “How to Think Like a Mathematician, see also from Kevin Houston his page on the book which includes some solutions to problems in the book, also available are sample chapters on writing mathematics.
From Imperial College, their STEM book list has many recommendations in several categories including Mathematics. Imperial College says that the list is aimed at A Level students but is suitable for anyone looking to develop their STEM knowledge and have a great read.
See the Reference page also which includes a useful list of reference materials.
If you have younger brothers and sisters, for Primary age children, these are all available as pdf files from White Rose Maths. The books are also available on Kindle.
If you like puzzles try Henry Ernest Dudeney’s – Amusements in Mathematics puzzle collection (with solutions). The first set of puzzles will offer a trip down memory lane for some of your parents who remember money – pre-decimal! There are several categories of puzzles available.
For notes and examples on polynomials, see the following resources: From the mathcentre which has an extensive collection of very clear notes and other resources, the Algebra collection includes Polynomial Division.
David Smith’s site ‘The Maths Teacher’ includes Algebraic Multiplication and Division under Algebra AS level.
In Paul’s Online Notes, Section 1-4 is on Polynomials; these start with a clear definition and include examples of polynomials and also expressions which are not polynomials .
It is really useful to have a look at the graphical representation as well as the algebraic solution. For example: Show that x = −1 is a root of x3+11x2+31x+21 = 0 and locate the other roots algebraically. The graph on Desmos is here. You can check your algebraic solution, by looking at the graph.
Cubic example, solve f(x)=0
One of the pages in the series on Calculatorsis on Polynomials where you will find the following resources.
As we come to the end of an academic year and look to a new one, it will be a time of change for many students. Perhaps you have completed GCSEs or equivalent qualifications (UK age 15-16) and are about to start on your A Levels or perhaps you have completed those and are about to start studying Mathematics at university.
To be in a position to begin your new courses well you should be thoroughly familiar with the essentials of the work you have studied to date. At whichever level you are studying your Algebra should be at a standard where you can manipulate expressions with ease.
Some resources to help you prepare and will be useful reference material for you during your course…
Dr Frost Maths is used by millions of students round the world. Amongst the extensive library of resources are videos for students age 8-18 explaining topics from scratch. There are longer Exam-topic videos, average length 10 minutes and shorter Key Skill videos which are about 2-4 minutes in length.
Hegarty Maths
Free lessons from Colin Hegarty recorded on YouTube are available to help GCSE students prepare for A Level Maths.
These Transition Takeaways from Mohammed Ladak have been specifically chosen to help with A Level Maths preparation.
You could also look at Step Up to A Level Maths from The Centre of Innovation in Mathematics Teaching which helpfully lists skills you should be confident with and provides resources to support your study of these skills.
AMSP – Transition To A Level Mathematics – Essential Skills
As you study your A level (16-18) course you may find some of the material in the section below useful.
For many challenging questions to really get you thinking, try the brilliant Underground Mathematics site.
Make sure you have some useful apps on your phone if you don’t have them already. Mathscard app from Loughborough university is free and a handy reference guide of mathematical facts and formulae. Every student should have the Desmosapp (free) and you could also get the WolframAlpha app (low cost).
If you are preparing for university, then make sure your 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. The Mathcentre has an extensive collection of helpful resources for students of Mathematics.
For a collection of forty mathematics activities bridging between A Level and University, try Carom Mathsfrom Jonny Griffiths.
Check the List of Activities, how much do you know about Inequalities for example? For a complete PowerPoint with information and questions on Inequalities, choose Carom 1-2: Inequalities.
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. 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.
The Open University has several helpful publications for students of Mathematics. Many of these resources would be helpful for students still at school.
Note the exercise at the end so you can practice, fully worked answers are shown.
WolframAlpha can of course handle polynomials.
For further notes & examples on operations with polynomials, see the following resources: From the mathcentrewhich has an extensive collection of very clear notes and other resources the Algebra collection includes Polynomial Division.
Mathematics for Students 