For Valentine’s Day…

The excellent Maths Careers site is managed and maintained by the Institute of Mathematics and its Applications. If your students wonder where Mathematics is used they will find plenty of answers here. See for example Who employs mathematicians?  

For Valentine’s day we have an appropriate article from Maths Careers, did you know that 6 is a kissing number?!

careers-heartsAlso, see this post with instructions on how to make this wonderful pair of linked Möbius hearts.

If you wish to get creative and try this I advise watching the Numberphile video carefully; following the instructions worked as you can see from my creation here! I can verify that unless you follow the instruction to make sure the twist in each strip is in a different direction you will end up with a mess! Quite an interesting mess but certainly not two hearts!….
use-to-create-mobius-heartsNote the Desmos graphs on my strips. I created a file in Word valentine-mobius-hearts (or pdf: valentine-mobius-hearts) with Desmos images in a table. Adding dotted borders to the table gives guidelines for cutting. I began each cut by using the end of a paperclip to pierce the paper. See Valentine’s Day for the Desmos details.

To create my strips I printed the document and then printed again on the reverse. I then cut out and trimmed the strips so there was no white space at the end – the picture here has been made using strips 10 cells long.

Save your money on cards and send your loved ones (or anybody!) a math-o-gram!

Click on the image and move that slider!

Desmos have provided you with the means to send a math-o-gram to the mathematicians in your life!

Valentine’s Day seems an appropriate time to express love for Desmos!
Geeky people, you could even use the Desmos API …

NASA Valentines
For an alternative source of Valentine’s cards, we can turn to NASA! Take your pick from this post or these fabulous NASA images. (And don’t forget that NASA provides us with many Mathematics resources.)

Remaining with the loving theme you can also express your feelings for WolframAlpha!

I Love YOU

For those in school this week try some Valentine class activities. Transum Valentine Puzzle
Here’s a Valentine logic starter from Transum.

This Valentine Relay from Chris Smith is excellent as are all the other relays in this excellent set of resources. You can find more excellent resources from Chris on TES and follow him on Twitter here.


Transum Valentine Puzzle
and here’s a Valentine logic starter from Transum.

Mathematical Miscellany #18

crashMATHS has some very useful free resources for GCSE and A Level; the site is under development but you will find plenty of useful resources already. Checking for some additional resources on Completing the square for my Year 10 students I came across a GCSE worksheet from crashMaths; this has a good variety of questions including questions to stretch your students aiming at the highest grades. The worksheet solutions are noted as coming soon. For a topic such as completing the square, this is an ideal time to use technology to check our work. Any of Desmos, GeoGebra and WolframAlpha could be used. (Select for Desmos page.)
Completing the Square

crashMATHS AQA styleStaying with crashMATHS, note the A Level Practice Papers and mark schemes, this looks like a very valuable resource. There are several papers and mark schemes available with more coming. Versions are provided for AQA and Edexcel, these use the style of papers we see from the exam boards. Currently, there is nothing for OCR. For Mathematics the content is the same for all the boards and for Further Mathematics we have a prescribed core which must comprise approximately 50% of its content. This common content as we have with GCSE is very useful indeed as we can use resources from all the examination boards.

Continuing with an A Level theme we have a very interesting read, published 25th January 2018, “An evaluation of the item difficulty in AS and A level maths“. This compares the difficulty of items in sample AS and A level maths assessment materials from 2016 and 2017 with the A Level papers from 2015. The overall objective of the exercise was to compare the profile of item difficulty within the SAMs with that of the corresponding 2015 assessments, a question I believe is on the minds of Maths teachers!

SAMS & Legacy A Level comparison

Ofqual: Expected item difficulty Legacy & Reformed Specs



Clearly, we can look at the specimen materials ourselves and make our own judgement on the difficulty but this seems a robust study which used Comparative Judgement. This is a technique where each reviewer reviews many pairs of items and decides each time which item is more difficult to answer.

Items from the sample assessment materials submitted for 4 specifications,  AQA, MEI, OCR and Pearson were used.

The study shows slightly higher levels of expected difficulty for items from the sample assessments relative to the 2015 assessments but the increase in difficulty is small.  The paper states that ‘Such small differences can easily be accommodated by the setting of grade boundaries at awarding. The choice of specifications to teach should be based more on content and style as there is little appreciable difference in difficulty.’

For further reading on Comparative Judgement, look at the work of Daisy Christodoulou.


If you have not yet signed up for the new home of Edexcel’s Maths Emporium then do so! This is such a valuable resource. Latest additions include some great new GCSE maths practice papers. There is a wonderful set of practice papers by topic. Look first at GCSE Mathematics, then choose Cabinet 11 for the current specification. Under Practice Papers you will find the themed set – brilliant!
Emporium Resource

NCETM Secondary Assessment MaterialsTo finish this collection, from NCETM look at their Secondary Assessment materials which have been written to support teachers in making judgements on the degree to which pupils have mastered various components of the KS3 mathematics curriculum. This follows the primary Mastery Materials, which was published in 2015.

See further posts in the Mathematical Miscellany Series.

Integration – Desmos & WolframAlpha

With my year 12 students (UK age 16-17) we have been looking at definite Integration. Desmos and WolframAlpha are both excellent for checking work and by using the technology we have a very clear visual representation adding to our understanding.

One of the homework questions for my students involves finding the total shaded 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. I like the visual representation showing students clearly that they are dealing with areas above and below the x-axis.
Definite Integral
Scrolling down the page we see that this query also returns the indefinite integral.
Indefinite Integral
For the total shaded area, students could change the limits of the query to evaluate each section.

Or we could turn to the excellent Desmos where we can very simply change the limits.

Definite Integration Definite Integration 2

If you are unfamiliar with Integration with Desmos, turn to Learn Desmos: Integrals.
Note you can explore the graph shown in the video.

We could also show students this introduction to Integration on GeoGebra.

A version of this post for students which also includes links to some useful notes and examples can be found on Mathematics for Students.
Integration notes University of Plymouth

MEI Ritangle Competition (& Technology!)

From MEI comes Ritangle, a competition for teams of students of A level Mathematics, Scottish Highers or the International Baccalaureate. This year’s Ritangle competition  launches Monday 2 October!

For the main competition, one question will be released daily for 21 consecutive weekdays, the first question will be released on 9th November 2017. I am looking forward to seeing the problems which I am sure will be of interest to many Mathematics students even if they are studying for alternative qualifications and cannot enter the competition.

Something which caught my eye is that Technology can help with some Ritangle questions which led me to some excellent Excel resources, which you can use whether or not you are participating in the competition.

Excel is a favourite for me – I use it in both my Mathematics teaching and in my Deputy Head role. Note from Nrich, we have many uses of Excel for Mathematical Investigation. I like this very clear illustration showing fraction multiplication.

Nrich Fractions multiplication

Excel Interactive Resource – Fraction Multiplication

For teaching sequences, Interactive Number patterns will be useful.

Quadratic Sequences

Nrich – Interactive Number Patterns 2

…and one of my favourites – Happy Numbers! (For the Excel spreadsheet – Nrich investigation)
Happy Numbers

Talking of technology, the Excel fractions resources here reminded me to try fractions with Graspable Math. Having tried Graspable Math on the Interactive whiteboard recently I can confirm it works perfectly.
Add Fractions

And finally, talking of Graspable Math, this led me recently to Dave Taylor’s excellent Increasingly Difficult Questions, a wonderful collection – and with an eye on the copying budget too!
Change the subject DT

Looking at the Graspable Math Teacher Resources, I see some of the Increasing Difficult Questions have been added to a canvas. Graspable Math lets you save your work so you can come back to it later or share it with others. Saving requires a google account. To share a file, use the share menu to get a link that gives others read access to a file. I decided I liked the idea of having a canvas ready for Simplifying Expressions, starting with IDQ-Simplifying Expressions 1, I opened it on my own canvas and adapted it slightly. I do like to keep all steps of the working displayed, so I have put the exercises on the left, creating a good space on the right.
Simplify Expressions IDQ

Writing Mathematics Online

I have written various posts on the available tools online for writing Mathematics and this is a topic which remains consistently popular. Time for yet another revisit and update as new possibilities are now available – all resources / links here have been checked. I have removed resources where blogs / twitter have not been updated for some considerable time as I think this is a cause for concern regarding the longevity of the resource.

Something I use a lot. I should explain my requirements – I want tools to communicate Mathematics online, for example I may wish to provide some model solutions or answer students’ questions. Writing mathematics can be a pain (and yes I know about LaTeX). Note that there are various possibilities – sometimes just a static picture is required, sometimes you may want to display how to solve a problem in stages, or perhaps you require a collaborative space. You will also need to consider if you want the examples to be permanent or whether you just want a collaborative space for discussion. A graphics tablet is essential.

My favourite method for illustrating Mathematics online (and in fact the one I use most often) when I just need a series of static displays is to turn an interactive whiteboard flipchart (or a PowerPoint) into a pdf file; the pdf file can then be sent to students or uploaded to whatever virtual learning environment or online storage your school uses. If you do not have access to interactive whiteboard software there are alternatives, one could use Windows Paint for example; there are also various free online tools available; see some of the resources below.

flipchart to pdf example

flipchart to pdf example

For sharing resources, it is possible to upload a PowerPoint or pdf file to Slideshare. There are many examples on this blog of my SlideShare slideshows – see this for example

I should mention that I find Slideshare excellent – I use the free version which offers me everything I need – it works every time – I use it a lot!
I created the PowerPoint for the slideshow above by writing on the interactive whiteboard software using my graphics tablet and taking a picture of each page using the Windows snipping tool (it’s in Accessories) – this takes seconds – the snipping tool is something I use every day! (Alternatively I could have saved the interactive whiteboard flipchart as a pdf).

There are as always several options:



If you wish to record a screencast of the moving pen / step by step solution variety and save your work, Screencast-o-matic is an excellent option. It is very easy to use to capture the screen and your recording can then be uploaded to YouTube if you wish.

Illustrating how to simplify an algebraic fraction : simplify-algebraic-fraction

Screencast-o-matic offers everything I want in this category. It is very easy to use indeed – I can write very smoothly whilst recording.

Further resources offering various solutions for writing Mathematics online:



For a collaborative board, try twiddla which seems excellent for collaboration – voice as well if you want. Twiddla offers some very sophisticated features including the ability to use mathematical formulae and upload files and images. Use of the board with all it’s features is free but you cannot save any of your work (possible with a subscription). This would be good for working online with a group of students. offers a very easy way to have a quick scribble and then share your thoughts! See quadratic example (and note that I created that just to test years ago – a good sign that it’s still there!) Select Doodle Now, doodle, then if you want to publish you will need to sign in. You can sign in with Twitter or Facebook. I like the different pen tools available, the full screen option and the ease of publishing. Apps are available for Android and iOS.

Writing RepeaterFinally – your students may find this amusing – the Writing Repeater from ICT Games – write something and play it back – now this is a lovely tool for little ones learning to write but I’m sure we can think of some uses!