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## Nottingham NQT Differentiation

On Thursday this week I went to the Nottingham University School of Education NQT session on differentiation.

I was quite concerned at the beginning when someone presented slides titled “Differentiation by Gender” and “Differentiation by Learning Style” but the subject specific discussion was your interesting. In true ITT style we had to summarise our thoughts on a big piece of paper resulting in this:

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The﻿ Venn diagram exercise I have used before with KS4+ but I’ve been inspired to try something similar with KS3 now.

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## My First @solvemymaths Problem

This morning I woke up to the problem shown below by the excellent @solvemymaths on my Twitter feed.

I’ve always been very lazy and not properly solved any of his puzzles before, partly because I have a strange irrational fear of geometry. Wanting to conquer this fear I thought I’d have a quick go at this.
It turned out to be quite straight forward really (although I did make a silly mistake and got an original answer that seemed odd). My workings with my mistake are shown below:

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## Classic Maths Books 2 – Numerical Solution of Partial Differential Equations by the Finite Element Method

This book isn’t actually that old, it was first published in 1987, the edition I have is a reprint by Dover Publications.

For my PhD I studied the application of a particular class of finite element methods for the solution of the neutron transport equation. Finite element methods are a way to numerically solve a partial differential equation – maybe I’ll write more about this in the future. It works by discretising the domain into regions known as elements and then producing an approximate solution on each element. Importantly, these approximate solutions converge to the true solution as the mesh is refined and the elements become smaller.
In the field of finite elements, it is widely regarded as a classic, and is often the first thing that someone new to the area reads, indeed it was the first published book on finite elements that I read. This is due to the relaxed and easy to read style of the book.
There are the following 13 chapters:

1. Introduction
2. Introduction to FEM for elliptic problems
3. Abstract Formulation of the finite element method for elliptic problems
4. Some finite element spaces
5. Approximation theory for FEM. Error estimates for elliptic problems
6. Some applications to elliptic problems
7. Direct methods for solving linear systems of equations
8. Minimization algorithms. Iterative methods
9. FEM for parabolic problems
10. Hyperbolic problems
11. Boundary element methods
12. Mixed finite element methods
13. Curved elements and numerical integration
14. Some non-linear problems

A really nice thing about this book is the mix of theoretical analysis and implementation details it provides. After reading this, you would be able to go away and code up a simple two dimensional finite element method. The book is well illustrated which helps with the understanding of concepts such as ‘basis functions’

As well as discussing finite elements the direct methods for solving linear systems is a fantastic introduction to this important area. The efficient solution of linear systems of equations is vital for any numerical solution of a real life problem that has been modelled by partial differential equations. This chapter serves as a great introduction to this topic (which will be considered more in future posts, and in a future classic books post). The chapters also contain a few stimulating questions, such as these from the linear systems chapter.

I think it is really nice that this book is written from a computational viewpoint (in addition to being mathematically rigorous) as after reading it, you can go away and code up the discretisation and a solver to try the methods out.

I would recommend this book to anyone who wants to learn about finite element method. The author did write an expanded book after this with some other authors, but it doesn’t read as nicely, and hasn’t generated the wave of good feeling in the finite element community as the original. In fact the sequel it is colloquially referred to as “The Black Death” (supposedly a term coined by one of the authors) as it is so bad!!

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## Towards a Silent Aircraft

A good friend of mine, Ed Hall had told me about a lecture at the University of Leicester this Tuesday (10th February 2015). This lecture was the 17th Annual Industry lecture organised by department of Engineering and was given by Professor Dame Ann Dowling. Professor Dowling is President of the Royal Academy of Engineering and a professor of engineering at the University of Cambridge where she ran the University Gas Turbine partnership with Rolls-Royce between 2001 to 2014. She is a world authority on combustion and aero acoustics and researches efficient, low emission aero turbines and low noise vehicles among other things.

The lecture hall was completely packed, I’m guessing over 200 people were in attendance. Professor Dowling started the lecture by showing how the noise from modern civilian airliners is significantly lower than that from airliners of the early jet age and thar they are also more fuel efficient.

As you can see from the graph, this decrease in noise has plateaued (is this a word!?) in recent years. Perhaps a reason for this is that environmental concerns have led to an increase in attention paid to improving efficiency, she mentioned that increasing the size of the engines has increased fuel efficiency but increased the noise level (from the large fans).

More recently, during take-off the noise of the rear jet has been balanced by the noise of the fan at the front of the engine (on approach the noise of the airframe becomes significant too, I will be doing a write up of another talk which discussed this in more detail), and so, the noise of the rear facing jet is not the only thing to be controlled.

Below is a section diagram of the Rolls-Royce Trent 1000 (the picture is taken from their excellent infographic which is available here), I have included it here as I will refer to some of the key components.

The large size of the fan – ~2.8m for the Trent 1000 – contributes to the large noise from the fan, so making this as light as possible can help, for this reason Rolls-Royce now use hollow fan blades with an internal girder like structure. Something, that to me is less obvious, mentioned by Professor Dowling is the noise generated by the wake from the fan impinging on the static stators that are downstream of the fan. This can apparently be improved by designing the stator so that the wake hits them at different points leading to the different frequency components of the noise cancelling out (this is how I understood it anyway – I am not an aero-acoustic engineer!) – this is very cool! Modern jet engines also often have chevrons on the trailing edge of the jet nozzles to reduce noise by smoothing the mixing of hot and cold air, since the source of the noise is the velocity fluctuations in the air coming out of the nozzle. However, these are generally designed empirically since running a full large eddy simulation of a jet flow through a chevron-ed nozzle takes around 4-6 months on a national supercomputer. Professor Dowling’s team developed a far quicker method that was validated experimentally.
She then presented the SAX-40 concept design for a silent aircraft:

This image is taken from some slides available here. As you can see, this looks very different from todays conventional jets!
As the engines are buried inside the airframe, instead of in nacelles hung below the wings the airframe can be used to shield some of the noise, with more space being given over to the use of acoustic liners to contain the noise. The noise from the engines would also be reflected above the aircraft reducing the noise level when compared to a conventional aircraft even further. The wings of conventional planes reflect the noise below the aircraft and towards the ground. As mentioned earlier, the airframe noise is significant when a plane is coming into land; one factor contributing to this is the use of the flaps and slats on the wings which introduce large holes into the wing surface – the air flowing around these holes creates a lot of noise. For this reason, the concept features a deployable drooping edge.
The noise due to the undercarriage of an aircraft is also significant and Professor Dowling’s researchers looked into this too. To accurately measure the source of the noise from the undercarriage they installed 108 microphones in a section of floor from an aircraft which could then be used in a windtunnel. The undercarriage of modern civilian jets are pretty similar with most having multiple wheels that are exposed. The analysis discovered that having two wheels produced the largest reduction in noise, however this obviously limits the weight that can be born by the undercarriage. So, a design with the rear set of wheels staggered inside of the front, different shaped wheels at the front and rear and a fairing were found to reduce the noise from the undercarriage significantly.

This talk was very interesting and thought provoking. However, I’m not convinced we will ever see a plane like this due to how radical it is!!

It was also nice that there was a buffet reception after the talk where I had some interesting conversations with a retired engineer – you can’t beat free food! I had also been lucky enough to arrive in Leicester early and have time to talk to Ed about upwind fluxes, Runge-Kutta time stepping methods, discontinuous Galerkin methods and the Navier-Stokes equations – all exciting topics which I don’t have much of an opportunity to talk about anymore.

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## FMSP Favourite Problems – Number 2

Here is the second post in my ‘FMSP Favourite Problems’ series looking at the problems selected for the Further Maths Support Programme’s favourite problem series of posters.
This second problem I’m not very keen on as it just seems a bit dull.

As usual I started with a small sketch, annotated with what I knew – that fact that at every bounce the ball rebounds to a height 75% of the previous bounce.

As this is repeated for every subsequent bounce you can easily obtain a relation between the height of the initial bounce and the height of the $$nth$$ bounce:

Clearly this is easily solved using logarithms

The answer above isn’t the full answer, since you don’t have decimals of a bounce, the ball must be on the 9th bounce before it rebounds to less than 10% of the original height.

Unfortunately, the typical GCSE student will not have come across logarithms – though I sort of feel that logarithms should be in the GCSE syllabus – and so couldn’t solve it this way. Unless I have missed something obvious, the only way that they could solve it would be to either calculate the powers of 0.75 until they got the correct value or to plot a graph. Not wanting to plot a graph by hand, I quickly loaded Matlab on my iPad and plotted the following:

In conclusion, as a problem I don’t find this particularly satisfying, however I think it could be used as an opening to an interesting lesson discussing exponential decay, mathematical modelling, ensuring answers you report are meaningful and interpreting graphs.

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## Symbols are meaningless without understanding

NB: this post is more opinion led than most of mine!

On Tuesday evening I attended a webinar on the ARK Mathematics Mastery programme (@MathsMastery I found this a very interesting hour or so, but the one quote that I took away from it was the following

Symbols don’t mean anything unless you understand the concept.

They showed a picture similar to the one below, and commented that the symbol for ‘5’ means nothing unless you associate it with the concept of ‘five-ness’.

I’m constantly thinking the same thing when teaching something like expansion of brackets. When faced with the following question “Expand $$3(2x -4)$$” I have had a fair number of students ask how you know if the second term is a plus or a minus – this just shows that sometimes I end up teaching the mechanics of a concept of method without ensuring there is solid understanding to build on. This is most definitely not a good thing – there is no point, in my opinion, in being able to complete the square of a quadratic for example without understanding why it is useful, and what it tells you about the quadratic. In my experience there are a significant number of undergraduate mathematics students who don’t realise that completing the square gives them important information about the turning point of a quadratic.

Having a concrete understanding of number and the number system is incredibly important. I see the overall goal of education to be “to prepare children for the world”. If by the time they leave school they don’t have a solid understanding of how numbers work and how to use them but can mindlessly expand a linear bracket because we have been told that it is on the scheme of work, have we really prepared students to deal with real life?!