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## 12 Days of Christmas: Day 5 – Christmas Market Simultaneous Equations

On Boxing Day, I went with my family to the Nottingham Winter Wonderland.

While in the Alpine style bar I bought 2 mulled wines, a hot spiced apple juice, 1 fully loaded raclette fries (shown in the picture above) and one ham and mushroom pizza. The total cost of these was £27.50.

In a slightly contrived situation the tables around me bought the following with the associated total cost:

Table in front: 1 ham and mushroom pizza, 4 mulled wines, 3 hot spiced apple juices and 2 fully loaded fries. £50.50

Table behind: 1 hot spiced apple juice, 1 fully loaded fries. £10.50

Table to the left: 3 hot spiced apple juices, 2 fully loaded fries. £23.50.

The question is: “How much did each item cost?” Of course I would like to know the cost of each item (my memory and keeping of receipts is poor!) but I am really more interested in the method used to get the answer…

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## 12 Days of Christmas: Day 4 – Integration by Substitution

Integration is a tricky topic to teach at A-Level (and beyond really) as so much comes down to experience. You only get good at Integration by doing lots of integrals and building up experience of  different techniques.

Integration by substitution is very powerful but deciding on a substitution can be tricky and is often something students find hard. Because of this I wrote a small card sort where students first have to match an integral with an appropriate substitution and then do the integration.

If you want to use it the file is available here .

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## 12 Days of Christmas: Day 3 – Fortran 2018

Those of you who know me well know that I am quite a fan of programming and due to my background the languages I am most expert in are probably Fortran and Matlab.

Excitingly a new standard – Fortran 2018 – was approved and published on the 28th November. If you so wish you can buy it from here.

Helpfully John Reid as usual has produced a document detailing the differences from the last standard – which can be found from the Fortran Working group page here.

Fortran has certainly moved on from the original restrictions on line length (with certain columns reserved for describing the type of information on the card).

I’m particularly interested in the new parallel features, though given my job now I’m unlikely to have much of a chance to play with them.

The paid for Intel compiler already seems to have quite good support for this new variant of Fortran. If you are interested in Fortran it is certainly worth following @DoctorFortranon twitter – this is the Twitter handle of Steve Lionel who used to work for Intel and whose comments on forums I found very useful!

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## 12 Days of Christmas: Day 2 – Mary Somerville

I first heard of Mary Somerville When I visited my sister who was studying at Somerville College in Oxford – the formerly girls college named after Mary Somerville.

Mary Somerville was a Scottish mathematician and communicator of science born on the 26th December 1780 – 238 years ago! Like most females of the time she didn’t have access to the education that was available to her brothers and complained that she “was annoyed that my turn for reading was so much disapproved of, and thought it unjust that women should have been given a desire for knowledge if it were wrong to acquire it”.

Following the death of her first husband she was able to pursue mathematical studies, corresponding with John Playfair and William Wallace. Her second husband, a cousin, Dr William Somerville was elected to the Royal Society and because of that she mingled with high society. She became the mathematics tutor to Ada Lovelace and also visited Charles Babbage frequently during the design of his analytical engines. Her first scientific paper, “The magnetic properties of the violet rays of the solar spectrum”, was published in 1826 in the Proceedings of the Royal Society. She then published a book, “The Mechanism of the Heavens”  in 1831 which became a standard textbook. For a book of that period it is surprisingly readable and can be viewed online here. The following screenshots show her discussion of a simple pendulum.

I am particularly intrigued by the notation used for the inverse function for cos(x).

Interestingly it is also a much cleaner result than if you simply ask Wolfram Alpha / Mathematica to do that integral for you….

I encourage you to spend some time having a look at this book online, it is lovely!

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## 12 Days of Christmas: Day 1

Merry Christmas to everyone reading this! I hope you have had a fantastic day whatever you have been doing.

In an effort to kick-start myself into blogging more I am aiming to post a blog post for each of the 12 days of Christmas. These posts will probably be quite short, some may be fairly random, some may contain teaching resources etc.

As it is Christmas day I thought I would write briefly about a Christmassy-themed theorem involving Pascal’s triangle. The so called, Christmas Stocking Theorem. I only came across this theorem a week ago when David Bedford (@DavidB52s) mentioned it…

This theorem states that a diagonal sum of $$k$$ entires from Pascal’s triangle is equal to the entry below and to the left of the last diagonal entry in the sum

Mathematically this is equivalent to saying:

$$\large \sum_{i=o}^{k-1} \begin{pmatrix} j+i \\ i \end{pmatrix} = \begin{pmatrix} j+k \\ k-1 \end{pmatrix}$$.

In the above, $$j$$ tells you the row at which you begin, and $$k$$ is how many entries you will sum. For example for the first picture above $$j = 2$$ and $$k=4$$, whereas for the second $$j=1$$ and $$k=6$$.

In future posts I may prove this in some fashion but I’ll leave you to experience the wonder for now….

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## A-Level Christmas Calculated Colouring 2018

I’m late posting it here (sorry) but I know I have already shared it with some people directly.

The Calculated Colouring for 2018 is now here – maybe you could do it on Christmas morning?

Same format as before – all questions should be suitable for current Year 12 students, some sample questions are below so you can get a flavour for them:

Thanks to Adam (@robotmaths) and my class for pointing out a couple of mistakes in my original version.

This year’s picture is a snowman – I need to come up with something exciting for next year.

The file is available here.

Enjoy!!!

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## An Integration a day Advent

I really should have shared this before the beginning of Advent! You have probably realised that I have been struggling to post on here – a New Years resolution for next year is to post more.

For my Year 13s I have made an “integration a day Advent Calendar”. Nothing fancy, just 24 integrals to find as students only get good at Integration by doing lots of them…