Author: tombennison

I have some catching up of posts to do so today I shall be sharing 3 fairly short posts.

For the first I’m sharing a resource that I have used with both Year 12 Further Mathematicians and Year 11 Level 2 Further Mathematicians. Matrix representations of linear transformations appear in the syllabi for both of  the qualifications just listed previously.

I’m always quite reluctant to just teach this as a memory test so I use this self guided sheet that students can work through before having a whole class discussion about the transformations.

Download the file here if you want to use it.

 

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…

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 .

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!

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!

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….

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!!!

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…

If you would like to download it to use please do so here.

It’s MEI Conference time of year again! Sadly as a Maths Hub Level 3 lead I am going to be unable to attend most sessions on Friday and I can’t make Saturday due to family commitments so I need to make the most of today!

Session A: Differential Equations in A-Level FM (Sharon Tripconey)

The maths for the day (well apart from Tom Button’s mathematical interlude during the opening plenary – give the question a go here.) started with a great session on differential equations with Sharon from MEI.

I loved the start of the session activity – a classic Venn diagram activity that really made you think.

Differential equations are in content areas G6, H7 and H8 in the new A-Level specification (only first order by separation of variables).

For further maths there is a lot more differential equations content in the compulsory

content than students would often meet before – including simple harmonic motion.

In the 2017 Further Mathematics Qualifications the concept of linked differential equations will be new unless you taught MEI Differential equations.

Sharon emphasised that what we do in A-Level maths should support the further mathematicians and so it is important that we expose all A-Level students to differential equations that can’t be solved by separating variables. I wholeheartedly agree with this.

A bit of time was spent looking at the integrating factor method – In my experience lots of practice of implicit differentiation needs to be done by students before tackling this topic as being able to spot the function which after application of the integrating factor method is less likely to result in mistakes than just remembering the IF method by wrote (in my opinion anyway!)

Sharon shared a lovely Geogebra worksheet from the MEI Geogebra Institute. In this you can explore how the solutions to a second order homogenous change as the coefficients change. It is also nice to be able to see the roots of the auxiliary equation plotted on a graph.

We began to look at some practical situations which lead to differential equations; possible examples are

• Motion against a resisting force
• Oscillations
• Damped oscillations
• Forced oscillations

After a brief into we performed a very easy practical involving a bottle of water, an elastic band and a plastic bag. I’m ashamed to say I have never thought of doing this in a classroom.

Great simple to do practical involving elastic bands, a plastic bag and a bottle of water. Can then be developed with some mathematical modelling to derive the differential equation. So far @MEIConference is certainly living up to how good the previous ones have been. pic.twitter.com/DB1VwZ4aVl

— Tom Bennison (@DrBennison) June 28, 2018

Once students have had a bit of a play they can easily be led through the modelling of the physical system, discussing how to model the resistive force would be particularly interesting.

The slide below has also reminded me of an old Geogebra file that I have which demonstrates oscillations for a variety of parameters. I’ll dig this out and share it on here sometime next week.

There was only one slide that concerned me in this session — however as she spoke through it she emphasised that she meant “analytically solvable”. Overall I like this slide as I think it is vital that time is taken to emphasise correct use of mathematical terminology in this topic.

Session B: Preparation for Oxbridge and Other Elite University Applications (Rachel Sharkey)

Rachel began the session by briefly discussing her research for her EdD which looked at how teacher’s cultural capital and connections impact their students’ applications to university.

She made an interesting comment that she has heard that the different questions on the TMUA are ranked differently, but this is not official. TMUA questions are intended to be like Olympiad questions, but multiple choice. I also wasn’t aware that Warwick University no longer requires STEP.

She shared her version of a personal network diagram which she advocates as a way to understand the capital that you bring to your students as a teacher.

This was interesting and provided the source of some debate in the session… I should perhaps go and think about this for myself more.

She talked about ways to help students demonstrate an interest in mathematics:

• Structured Reading: Provide structured reading chapters from a few books on the same topic – in her case infinity – send them away to read and then discuss as a group.
• Presentations: Students research and present a topic beyond A-Level maths. This increases student’s own capital and their peers when they listen.
• Independent Reading: Provide help with the reading lists provided from the universities. They then write a book review as this provides a short summary of the book which they can call on during interview or when they are putting together a personal statement.

When preparing students for the admissions tests Rachel has started grouping questions by the skill/technique used including the following:

• Completing the square.
• It’s ok to try numbers.
• Graph transformations.

For the longer form questions her advice is very similar to my default advice. I think one of the most important things is that students need to be very proficient with the algebra, checking every step of the working, watching out for signs and to ensure that they use previous steps of a question later on in the question.

Chris Sangwin, the admissions tutor for maths at Edinburgh University emphasised how useful it is for teachers to contact the admissions tutors in universities they often send students too. They are often happy to say things in person that can’t be written down on a University website.

Session C: Mechanics in A-Level Further Mathematics (Sue de Pomerai and Tom Button)

Sue and Tom had to deliver this session instead of Avril at short notice, but you couldn’t tell!

They started with a brief summary of where each mechanics topic sits in the new linear Further Maths A-Levels (which strangely are described with modules)

Hopefully before students study elastic potential energy they will be familiar with Hookes law that states the force is proportional to the extension. Tom then discussed using a spring balance in a very simple way to help students understand elastic potential energy.

If you pull on a spring balance what do you notice? (Maybe you notice that the scale is linear and that if you exert a greater force then the extension is longer – what do these mean? )

From Hooke’s law, considering the area under the graph of force against extension it is clear that the area under the curve is the elastic potential energy \( EPE = \frac{1}{2}kx^2 \). Tom talked about this being a fantastic situation to draw out any misconceptions using the fact that Work Done = Force * Distance – in this case we do not have a constant force.

I really liked this question that Sue shared:

This would really check the understanding of students, especially if you asked them to explain their reasoning, possibly bringing any misconceptions to the foreground.

There was a very interesting discussion about attempting to move a heavy rock. You have done some work (and expended energy) but what if the rock hasn’t moved – with the usual definition we wouldn’t have done any work, yet energy has been expended. Sue emphasised that with any mechanics work we are working with ideal situations, have made assumptions and so the model does not always represent reality. I agree that emphasising these assumptions with students is very important – what seems obvious to us (we probably know the assumptions made and the impact they have) is likely not obvious to the students beginning their mechanics journey.

Top Tip: If you want to do practicals with balls that don’t bounce take a a ping-pong ball, poke a hole in it (with a compass or a drill). Now cut up a lot of thin elastic bands into little strings and stuff your ball with them. Keep filling them with elastic until the ball doesn’t bounce.

With the increased emphasis on mathematical modelling in the new A-Level it is important that we develop mathematical models for physical situations with our students. Sue led through modelling a bungee jumper. We started by finding the spring constant (plotting weight against extension and finding the gradient – how is this related to k?) Developing the model leads to a quadratic in the extension x which can be solved.

We then looked at a loop-the-loop situation.

Paul Glaister (of Reading University) got to have a bit of fun with this…

@p_glaister having fun with a loop the the loop track in @suedepom and @mathstechnology ‘s mechanics session at the @MEIConference pic.twitter.com/76ZU1DpsCm

— Tom Bennison (@DrBennison) June 28, 2018

All of these practicals are contained in the classic “Mechanics in Action” book. I may blog about this at some point in the future.

Session D: Student Tasks for Integrating Technology in A-Level Maths (Tom Button)

This session was about the student tasks developed by MEI that are designed to expose students to the benefit of technology in lessons.

The first thing Tom showed us is the very cool graph \( y = x^5-5x^3+5x \).

There are two cool things about this graph – I will let you explore this..

We then investigated the graph \(y = x^3 + ax+1\)

• When (what values of a) does the graph has maxima and minima?
• What affect does a have on the intersection of the graph with the x-axis?
• What is same / what is different?
• What is important about the point (0,1) in this graph?
• How does the gradient change at (0,1)?

The point (0,1) on this graph is a non-stationary point of inflexion – now in the second year of A-Level.

Tom shared a load of the MEI Student Tasks. Their are sheets for Desmos, Geogebra and graphing calculators.

• Desmos Tasks
• Geogebra Tasks
• Graphing calculator tasks (Casio)

As a group we discussed the task below.

The emphasis on how the technology supports the learning of this session was great – too often it can be easy to focus on the cool things tech allows you to do. I know I am guilty of this!

The structure of these sheets is also really nice.

Towards the end of the session we considered a question from the MEI SAMS Paper 1.

One interesting point made, that I hadn’t really considered before is that using technology may in itself lead to students being more comfortable with functions having more than one unknown.

Plenary (Charlie Stripp, Kevin Lord, Stella Dudzic)

The plenary today was also being live streamed as it was also serving as a launch of the Advanced Maths Support Programme. It is always nice to see positive graphs such as these:

I hadn’t seen this graph below which is at least still encouraging:

However – what will happen next year?

Kevin Lord then provided a very informative introduction to the AMSP (Advanced Mathematics Support Programme). It sounds a very supportive and exciting programme which I will be keen to be involved with. The new extra focus on the AMSP Priority areas is also welcome. These are comprised of the 12 government Opportunity areas and 20 areas that have been identified as having low Level 3 provision.

One of the key tasks for the AMSP will be to increase the percentage of institutions offering Core maths – currently only around 17% of institutions enter students for this exam.

Stella Dudzic then spent some time talking about Core Maths. I’m very excited about the Core Maths Online Teaching Platform.

Food:

Everyone knows that the most important aspect of any conference is the food and luckily the food today was as good as the sessions.