You could generate a random quiz, using Mathster’s UKMT Mathematics Challenge Online Quiz. Choose Junior, Intermediate or Senior and one of three difficulty levels; you can also choose the number of questions, a time limit and the order the questions are presented in – random or in order of difficulty.

Nrich publish new problems every month. Why not try and get a solution published on their website? There is a menu specifically for students. You can sign up for an Nrich student newsletter if you want to be notified of new developments on the site.

Signing up toBrilliant (including an easy option for sign in for Facebook users) will allow you to join an international community and try numerous.questions at various levels.

One of my students told me recently about MadAsMathsby Dr Trifon Madas. She likes the Practice Papers, particularly the way the papers are rated according to their difficulty, see C1for example. All papers come with full solutions.

MadAsMaths question & solution

Not only do all the questions come with full solutions but most have very clear mark schemes too.

MadAsMaths mark scheme example

The papers cover the Pure Mathematics content of the UK A Level course. Note the Special Papers designed for extremely able students; ideal for students capable of the top grades. This is a really valuable collection of questions.

There are further questions and solutions available in the large collection of booklets, originally samples they are free to use. These are pdf files, if you zoom you will see a clear solution as illustrated in the image here. Some of these booklets are aimed…

For a really clear plotter showing the connection between the Cartesian graph of r=f(θ) and the graph in polar coordinates try this Polar Curves and Cartesian Graphs applet. Watch the display carefully as you move the slider; you can easily see when r is negative for example.

For more on Polar Curves – including notes and the use of Desmos, see this post.

Complete GCSE Maths Revision text from CLCnet. Don’t be put off by the 2007 date – this is still useful. The text includes numerous set of questions for each topic by grade with solutions for all the examples.

Craig Barton’s E-Book of Notes and Examples is a comprehensive set of notes with very clear worked examples; this is extremely useful for students age 11 to 16 and for any older students who need a refresher.

‘Street Fighting Mathematics‘ by Sanjoy Mahajan, with the excellent sub-title ‘The Art of Educated Guessing and Opportunistic Problem Solving’ (note the link on the left to the free Creative Commons edition under Essential Info).

Check Amazon for Kindle books, a small number of which are free or very low cost. Now you don’t have to own a Kindle to read Kindle books as the Kindle app is free to download for any device you may own: PC, phone or tablet. It is possible to search Kindle books by price, so a search on Mathematics books by price from lowest to highest will give all the free entries; there are lots of toddler books and samples there but the odd useful book is available. A very useful search for low cost books is a search on popular Mathematics books, price low to high. which returns popular Mathematics and Science books for as little as 99p.

Try Henry Ernest Dudeney’s – Amusements in Mathematicsa puzzle collection (with solutions). The first set of puzzles offers a little History, money – pre-decimal! There are several categories of puzzles available. Or try Edwin Abbott’s Flatandthe tale of a two-dimensional world occupied by geometric figures.

Project Gutenbergincludes numerous Mathematics books, including classics such as Flatland. (Mark Twain’s comment on “Classic’ – a book which people praise and don’t read.” always amuses me!)

You could even write your own book! If publishing your own Kindle eBook feels a little too much right now – you could try something simpler using Storybird – click the image to see this story on Storybird

For any Excel users this formula will determine whether or not a year is a leap year:
=IF(OR(MOD(A1,400)=0,AND(MOD(A1,4)=0,MOD(A1,100)<>0)),”Leap Year”, “NOT a Leap Year”)

Why not send your mathematical friends (or anybody else!) a Desmos Valentine? The wonderful team at Desmos have made their brilliant math-o-grams available earlier this year….and watch out for new designs coming soon.

The math-o-grams are really easy to create; why not give it a try? Just select your design, add your message and share!

Is your writing clear and well structured? Easy to read and follow?
You will find some excellent examples from Dr Kevin Lee in his Guide to Writing Mathematics.

These sites are mentioned elsewhere on this site but these all include notes and exercises on Algebra. Answers are included in these resources and of course you can check answers on WolframAlpha.

Note there are resources for older students further down the page.

The following example – note the colours shows how to expand three brackets.
You could check your answers on WolframAlpha and change the query to make up your own examples.

The Maths Teacher (age 11-17)
David Smith’s site, The Maths Teacher has an extensive collection of videos to help you study Mathematics. GCSE (age 14-16, though many of these resources would be helpful for younger students also) and A-Level (age 16-18) lessons are available. For each topic not only is a video available but also a transcript and exercises with worked solutions. This makes the site ideal for revision – you have the choice of perhaps just trying the exercises or if you feel you need more help you can watch the video – whatever is right for you.

Trinity school (age 11-16)
Trinity School have very helpful Mathematics resources – many examples for you to try (answers included) including Algebra.

Trinity School Nottingham – numerous questions and answers

For older students who have already studied many topics in Algebra, CIMT’s Step Up to A Level Maths includes notes and examples on Basic Algebra, Quadratic Functions and Equations and Inequalities. These exercises could also be used by GCSE students aiming for the highest grades. Another very useful publication is the Algebra Refresher from The Mathcentre which has many questions with the answers provided at the end of the document. Also from the Mathcentre you can view all the topics available – several different resource types such as notes and videos are available for each topic.

Sections 1 (Basic Algebra) & 3 (Equations, Inequalities and Partial Fractions) of the very helpful Helm Workbooksinclude worked examples and exercises (with answers).

If you want a little something to do then try the Math Forum Year Game. How many numbers from 1 to 100 can you generate using the digits of 2016? Unlike Countdown which uses only the four operations and brackets, additional operations are allowed – see the detailed rules.

5 – looking it up – there are several excellent resources online for you to look up definitions or find extra examples. Never rely on just one source if you are finding a topic tricky, it can be helpful to see explanations written by different authors.

6 – To learn Mathematics you need to do Mathematics. You can never do enough examples! There are plenty of questions here(with answers included). Have you looked at the problems on Brilliant? New problems are published regularly and are suitable for younger school students all the way through to university students.

10 – on writing the language of mathematics correctly – see this clear guide to writing Mathematicsfrom Dr Kevin P Leeand from John Kerl some excellent tips for mathematical handwriting, many of these tips these apply to students of all ages – do you distinguish carefully between a 1 and a 7 for example? Perhaps it is hard to tell whether you have written a 2 or a z or perhaps your 5s look a bit like a letter s?

Have a look at Peter Alfeld’s guide to Understanding Mathematics which he wrote for his students at Utah University.

Are you familiar with all the excellent (free) resources online to help your studies? Have you tried the brilliant Desmos graphing calculator for example, or used WolframAlpha to check your work? Do you have the GeoGebra app on your phone?

Having downloaded the app to my phone I can confirm it works really well and i was easily able to follow the instructions given in the links here.

I very quickly created the diagram in the screenshot from my phone below. Using a finger one can drag point D around and note the angle at the circumference. With an account you can also save your files which are standard GeoGebra files.

So, time to relax and what better way to relax than with a puzzle or two?!

On Nrich you’ll find a collection of advent calendars a Sudoku for each day perhaps? (Solutions are supplied if you want them). Or a tangram? Maybe you want to play a game?(Clearly the year doesn’t matter for an Advent Calendar!)

Plus Magazine also have an Advent Calendarwhich this year has puzzles and articles designed with your creativity in mind.

Perhaps you could make your Mathematical friends some cards! Why not put a Desmos tree on the cover?! Note this is simply a collection of lines and circles, as you can see from the syntaxit is very easy to restrict x or y values. (For more on getting creative with Desmos, see Graph Art).

Staying with the subject of cards you can practise your coding skills with this Scratch project.

Turn the sound on!

If you are feeling creative you might like these Christmas fonts.