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…

View original post 3 more words

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


Valentines day is coming

Desmos Valentine

Click on the image and then on the slider!

Desmos Valentine instructions

Why not send your mathematical friends (or anybody else!) a Desmos ValentineThe 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!

Mathematical Handwriting


John Kerl – Tips for mathematical handwriting

Is your writing legible? Is that a 5 or an s, a 2 or a z? Included on the Study Tips page see John Kerl’s Tips for mathematical handwriting.

Dr Kevin Lee - A Guide to Writing Mathematics

Dr Kevin Lee – A Guide to Writing Mathematics

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.

See also Ten Simple Rules for Mathematical Writing from Dimitri Bertsekas.

Algebra Examples & Exercises

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.

For younger students (age 11-16) try the Interactive Tutorials from The Centre for Innovation in Mathematics Teaching, (CIMT).
Fundamental Skills   Function Machines   Linear Equations   Basic Factorising
Expansion – Single Brackets  Linear Equations with Brackets
Factorising Expressions 
Substitution into Formulae 1     Substitution into Formulae 2
Linear Equations 1   Linear Equations 2
Coordinates   Plotting Straight Lines   Graphs and Gradients
Laws of Indices   Negative Indices   Fractional Indices
Straight Line Graphs   Linear Equations   Parallel and Perpendicular Lines
Simultaneous Equations   Equations in Context
Equations, Formulae and Identities   Simplifying Expressions   Factorising  Using Formulae
Quadratic Equations: Factorisation  
and by Completing the Square

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.
Expand Brackets Colleen Young

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

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 Workbooks include worked examples and exercises (with answers).

Plymouth University Algebraic Fractions

Plymouth University Support Materials

Plymouth University have some very clear workbooks, try Basic Algebra – Brackets for example or Algebraic Fractions.

For videos try the following sites:

Exam Solutions GCSE Maths     

Exam Solutions A Level Maths (select your exam board from A Level Maths tab)

Hegarty Maths GCSE      A Level  and KS3 for younger students (11-14)

and note The Mathcentre, already mentioned above.