Underground Mathematics

For many challenging questions to really get you thinking, try the brilliant Underground Mathematics site. You can follow them on Facebook or Twitter too.


thinkig-about-algebraYou can search by Station, suppose you want to practise your Algebra – try the Thinking About Algebra Station for example where you will find everything from Equation Sudoku to some challenging Surd manipulations.

On the subject of surds – try Scary Sum!

If you create a (free) account you can save and categorise your favourite resources.

There are many Underground Mathematics Resource Types. Try the Review Questions for example, which in the words of the Underground Maths Team:

These are questions designed to test students’ understanding of one or more topics and to exercise their problem-solving skills. In many cases they can also be used as a classroom resource to help teach concepts and methods. They are mostly drawn from past examination questions and have been chosen as ones that are interesting in nature and require non-routine thinking. The hints and solutions are designed to explain the reasoning and highlight connections as well as giving the answer. In many cases, alternative methods or solutions are presented.


Note the various question types available; these include very challenging questions for students age 16+.


Select image for question & solution

The Oxford MAT collection includes an extensive selection of Multiple Choice Questions.

O/AO-level questions are included. These questions provide excellent challenge for sudents aspiring to the top grades for examinations taken at age 15-16 and beyond..

Can we fully factorise x4+4y4?
Starts with a Show that….
We could get very sophisticated and look at those quadratic factors too; useful for those studying the Level 2 Further Mathematics Qualification.

Can we simplify these algebraic fractions?
Review algebraic fractions

Can we simplify these simultaneous equations of degree 1 and 2?
Solve simultaneous equations. We will need to factorise a quadratic in this problem with a coefficient which is not 1 for the square term. My students and I are fans of the Box Method where a factorisation cannot easily be done by inspection.


Calculators & Tools

casio-modelsNote the updated Calculators and Tools series of pages including a page for Calculator Manuals.

All pages:

have been checked amd updated.

A new page on Collections has been added which will be updated …

The ultimate Calculator has to be WolframAlpha which as you can see has a page of its own with several slideshows to help you learn how to use it.

For drawing and exploring graphs, use the outstanding Desmos Graphing Calculator.

Mathematics – Transition Time

With the new academic year approaching are you about to start advanced level studies? Or perhaps you are heading off to university?

This post, Transition Time has been updated with new resources and has many useful questions and activities to help you prepare for your next stage of study.

Note in particular the addition of Underground Mathematics and Jonny Griffiths’ Carom Maths.
underground-mathematicsCarom Maths

Mathematics in the Movies

Mathematics, Learning and Technology

FB_IMG_1469860050102 Photograph by David Young

At the wonderful National Cinema Museum, Turin – holiday time this week!

Thinking about Maths in the movies led me to this great collection of movie clips featuring Mathematics from Harvard University.

We must of course include Abbot & Costello showing that 7×13=28!

Mathbits.com includes a section on using movie clips in the mathematics classroom. The site suggests several possible movie clips to use and has accompanying worksheets. The Abbott & Costello clip mentioned above is included, the MathsBits worksheet is here.

Donald in MathMagic LandI loved Donald in Mathmagic Land as a child.

For a series of excellent articles see Plus Magazine’s Mathematics in Films.

For a very comprehensive database of mathematics mentioned anywhere is a movie try MMDB – The Mathematical Movie Database from Burkard Polster and Marty Ross.

From Numberphile, Math and Movies (Animation at Pixar) we learn how 3D aminated characters…

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A perfect day…

28th June is a Perfect Day to enjoy some Mathematics!

perfect numberPerfect because 28 and 6 are both perfect numbers. You can easily check for properties of any number with Tanya Khovanova’s Number Gossip where we learn that 28 does not only have the rare property of being perfect, it is also composite, even, happy, odious, practical, triangular and Ulam! You can browse all the properties here.

28 is also happy! Happy Numbers – a favourite investigation, Dr Who knows about happy primes!

It’s also National Tau Day! Pi is wrong…..
This video provides a short version of the Tau Manifesto (14 minutes)

Spreading the Desmos Love!

Spread the Love

Have a look at this Desmos page (or select the image).

Before you take a peek in the Love folder, can you identify the functions used  for the various letters?

Love folder

Love imageThis came about because of a Twitter conversation. @WillWhittow, Admissions Tutor & Senior Lecturer at Loughborough University shared this image.  I replied that we should do some dancing too!

Desmos Dancing



Dancing Desmos 2





And Desmos thanked us both for spreading the love!

Spread the Love 2






For more on Desmos – see this series of pages.

Mathematics Challenges

UK Maths Challenges
You can practice for the UK Maths Challenges with these past papers. Questions and full solutions are provided.

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.

Note Geoff Smith’s advice for young mathematicans and as he says you will find thousands of questions are available at all levels on the Art of Problem Solving site.

Nrich have a series of short problems based on the UK Junior and Intermediate Challenges.Nrich Short Problems

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

If you have not tried Nrich problems before you may find their recommended starter problems good to try. You can search for problems by topic if you wish.

If you have not tried Nrich problems before you may find their recommended starter problems good to try. You can search for problems by topic if you wish

Brilliant problem

Signing up to Brilliant (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.


A great source of questions and full solutions for the Pure Maths content of A Level exams.

Mathematics, Learning and Technology

MadAsMaths MadAsMaths

One of my students told me recently about MadAsMaths by Dr Trifon Madas. She likes the Practice Papers, particularly the way the papers are rated according to their difficulty, see C1 for example. All papers come with full solutions.

MadAsMaths question & solution 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 MadAsMaths mark scheme example

MenuThe 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…

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Polar Curve Plotter

Polar & Cartesian PlotterFor 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.

Free Mathematics Books

Some books that might be useful for you:

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.

The Maths E-Book of Notes and ExamplesCraig 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.



AQA Further Maths Text

For Further Mathematics A Level (age 16-18) AQA have provided a set of three textbooks.
(Scroll down to Resources for students).

street-fighting-mathematicsStreet 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 Mathematics a 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 Flatand the tale of a two-dimensional world occupied by geometric figures.

Project Gutenberg  includes 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 StorybirdStorybird Sequences

..and finally, I must mention John & Betty’s Journey Into Complex Numbers!
John and Betty