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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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## Matlab in the Classroom

One of the things I have missed since teaching in a school is the fact that I don’t have access to Matlab….

Matlab is a product from The Mathworks, designed originally for numerical computation (indeed the name derives from MATrix LABoratory as the software was originally focussed towards operations on matrices) but which now is capable of much more. More information about Matlab and The Mathworks is available here.

I had got used to using Matlab in teaching an demonstrating, not least because of the ease with which you can use it to check matrix computations, plot graphs of functions and demonstrate numerical methods.

Last term I set my Year 12 Further Mathematicians some homework which involved them researching the Jacobi Method and performing 2 iterations of this method on a $$2 \times 2$$ matrix system. I wanted to be able to demonstrate this straightforward numerical method to them, without having to work out how to use the Python IDE on the Windows computers at school, so I decided to give Matlab Mobile a try.

Matlab Mobile was originally released in 2010 (see here for an early blog post discussing Matlab Mobile) but I hadn’t had much reason to use it before. Originally the app needed to connect to your computer with a running copy of Matlab, but since it’s release it has become a lot better. Th app is currently on version 4.1.1 which is a universal app suitable for both the iPad and the iPhone. With this version (as long as you have a valid Matlab Licence for desktop) you can connect to a copy of Matlab running in the cloud to execute commands. You can also upload your files to the cloud so that they can be run from the app.

On opening the app you are presented with a screen that looks like this:

The left hand portion of the screen displays commands that you have typed and on the right hand side you can toggle between Figures and viewing your command history.

Using a keyboard (I recommend using a bluetooth keyboard here as the on screen keyboard takes up so much space) it is easy to type short commands that accomplish complex things. For example, in the screen shot below, I create a matrix and then find its eigenvalues using the  eigs  command. As you can see this command can also return the corresponding eigenvectors of a matrix simply by explicitly giving two output arguments to the function.

One of the great uses of the Matlab desktop product is to prototype functions before coding in another language. Unfortunately, the mobile app does not allow you to edit or create  .m  script files or function files. This is a real shame, especially as it extends to the inability to make small edits to files you have already stored in the cloud. Luckily though, you can at least view the test of functions that you have written by using the command  type  in the command window, as shown below.

The code shown above is a naive implementation of the iterative Jacobi scheme for solving systems of linear equations – I stress it is not the most efficient way to code this algorithm, but it is clear for teaching purposes. Using Matlab Mobile I was able to discuss this short function with my class, and relate it to the algorithm that they had to research and then express in pseudocode.
If you are going to use Matlab Mobile for a similar purpose I would recommend avoiding writing script files and instead use function files with enough arguments to allow you to vary everything you will want to demonstrate the affect of. I would also say it is probably useful to print more of the internal workings of the function to the command line than you would normally.

Another great piece of functionality in Matlab is the ability to produce professional looking graphs with very little effort. You can still generate figures such as the one below (a visualisation of the Barnsley fern after 10000 iterations – I may write a blog post about this in the future) in the Cloud, but a lot of the data exploration tools in the figure window of the desktop product are not usable. You can however, save the figure to your camera roll or email it to yourself.

As a final example of the graphical abilities of Matlab Mobile the 3D surface plot below was created using 4 lines of code.

One important thing to note is that I couldn’t connect to the Matlab Cloud through my schools wifi (I suspect because of the firewall) and so had to resort to using my phone as a 4G hotspot.

All in all, if you have a Matlab licence it is well worth familiarising yourself with the Mobile app, and I managed to do all that I wanted to do in the classroom with it. I just needed to do a bit more forward planning than I would have done if I had had access to the fully fledged Matlab product! Hopefully as the app develops they will remove some of the limitations that are currently present.

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## A Starter for all Groups

Sometimes I like to do a quick starter with my groups where they are just playing with numbers and which isn’t related to the work of a particular lesson.
One day last week I decided I wanted a starter that I could use across all abilities and year groups. I remembered the following problem that I had recently read on Dan Meyer’s (@ddmeyer) blog here:

• Choose three consecutive whole numbers and add them. Write your sum on the board. What do you notice?

I chose to use this problem as it seems simple but can revel lots about the nature of numbers. The only modification I made was to restrict the numbers that could be chosen to be less than or equal to 50 – I did this to make the arithmetic manageable.

I was very happy with the response I got to this problem from all of my groups, they were all engaged with thinking about what they could say about the results, especially once they had come and written them on the whiteboard.
I found this task good for the lower attaining KS3 as a way of getting them to practice mental / written addition methods and to have a look at how to tell if a number is divisible by 3 or not. With the higher attaining students it led on to a discussion of knowing something and how to rigorously show it using algebra. We also discussed how it doesn’t matter how the consecutive numbers are labelled, we will always get a sum that is divisible by 3.

Overall I think this activity benefited my students and that it was a valuable thing to do, which I will try to repeat in future. Of course, having to plan one activity that can be used with all classes is a nice bonus in terms of planning time!

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## Classic Maths Books 1 – Bostock and Chandler textbooks

This week I posted the following picture on Twitter:

A few people retweeted, favourited and commented how good they were for questions to stretch A Level students so I thought I would write a bit more here.
They were first published in 1982 and are still available from Amazon here and here.
I think I got my copies when I was doing my A Levels and asked my teacher if there were any other books he would recommend that weren’t the Edexcel ones.
I think they are still fantastic books, vastly superior to the Pearson published textbooks, though I think some students may find them a bit dry. They certainly assume more knowledge than the current books and the later questions are generally more involved (or it is less clear on how to start them). There are some really nice examples in the book, such as:

• Find the complex roots of the equation $$2x^2 + 3x + 5 = 0$$. If these roots are $$\alpha$$ and $$\beta$$, confirm the relationships $$\alpha + \beta = – \frac{b}{a}$$ and $$\alpha \beta = \frac{c}{a}$$

All these examples are well explained, and it is easy to follow them line by line.
One other aspect of them that I appreciate are the multiple choice questions at the end of each chapter; these are great for a quick test of understanding and I am using some of them in some QuickKey quizzes that I am trialling this half term.
I may make a blog post about a classic mathematics book a fairly regular thing… gives me an excuse to get some of the older books off my shelf again 🙂

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## #Nurture1415

A lot of people seem to be doing a #Nurture1415 post – Sue Cowley (@Sue_Cowley on Twitter) has compiled a long list here.

So here are my, hopefully short, 5 positives and 5 negatives.

5 Positives from 2014

1. I saw the final completion of my PhD in applied computational mathematics by graduating from The University of Nottingham
2. I joined Twitter, properly. I had a while a go signed up to Twitter, but never really used it. This year I decided to set up a professional Twitter account (@DrBennison). I was a bit dubious at first, but I now use it all the time. Everyone on Twitter is very welcoming, happy to get involved in discussions and I have learnt a lot from everyone I have interacted with. It was particularly good to meet people I had spoken to on Twitter at Mathsconf2014.
3. I started working at a great school, with a great department that is very supportive of each other.
4. I also started a blog in the summer, which I have enjoyed updating!
5. Doing maths by hand. It has been nice to practice integrations etc by hand again. I had become very reliant on Mathematica over the last few years.
6. First wedding anniversary. It has been a great first year married to my beautiful wife and I feel very lucky to always have her support.

5 Wishes for 2015

1. Develop my website properly. At the moment my website just hosts my blog, a few files and a Javascript game – I need to spend time designing a proper homepage.
2. Do some coding. After 5 years of programming everyday I have been missing doing some programming. I have two projects I would like to work on: 1) an iFEM finite element iOS app; 2) a useful online computer based assessment and grading system. Hopefully if I am disciplined in school holidays I can make a start on these.
3. Blog more frequently. I’ve learnt loads in the last six months reading other people’s blogs and updating mine, however my posts have been infrequent. My goal for 2015 is to post at least one decent post every week – fingers crossed I can manage this.
4. Continue learning from colleagues at work, people on twitter and in any other way so that I can be the best teacher that I can be.
5. Manage a good work-life balance

You got one bonus positive.. sorry for not sticking to the rules.

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## Merry Christmas

Merry Christmas to everyone who has taken the time to read my posts since I started this blog. I know they have been infrequent recently and I intend to post more often this year.
People always mock me for my inability to colour very well, so here is a Christmas Calculated Colouring I did this morning for you all 😉

Colouring sheet courtesy of 10ticks.

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## Why are there so few derivations in A Level Mathematics?

A thing I have noticed with the A Level textbooks (I use the Edexcel books published by Pearson) is that there are very few derivations of results, with them preferring to just present a formula for the student to use.
I don’t understand why this is the case… I get that there is a lot of content to be covered in a relatively short time (maybe this will improve with going back to terminal exams) but aren’t students missing out on some really important concepts by not including derivations.
I’m sure many teachers do show derivations, but why are they not in the text books we expect students to use to help their studying?
I recently asked my Year 12 FP1 students to derive an expression for the inverse of a general $$2 \times 2$$ matrix, in terms of the entries of the matrix that you are trying to invert. Of course, a straight forward way to do this is to use the definition of the matrix inverse to derive 4 equations with 4 unknowns which can be solved in two pairs to obtain the entires of the inverse matrix in terms of the original matrix.
I was surprised to find that no one managed to do it. Perhaps the amount of letters floating around put them off, but I think this kind of exercise is important for building confidence and developing fluency with methods used in other modules.
I typed up a solution for them, if it is of interest to anyone it is available on my website here. Of course there are likely to be typos…..
I’d be interested to hear other peoples views / experiences of derivations with A Level maths.

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## The Perils of Grading Work

Last night, just as I was going to bed I was notified through Twitter of Manan’s (@shahlock) latest blog post where he looks at the grading (by another person) of some work that had been sent to him.

As he rightly points out commenting on just the written grading is fraught with problems as you do not have any idea of what has been said in class in relation to the work. I agree completely with everything Manan has said about this particular piece of work.

The assessor has been very thorough; looking at every line and not just skipping to the final solution. This is of course a good thing and should be applauded, as should the detailed feedback given. However, my initial thought on looking at this assessment and its grading was that it had been marked by someone slavishly (is that a word?!) following a mark scheme with less knowledge of mathematics than the person taking the assessment. On reflection, this is perhaps a bit harsh, maybe the focus of this assessment is exposition and explaining mathematical reasoning – this would explain some of the comments given for problem number 7.
I don’t feel that this explanation really holds for the other marks lost though. The “order of operations” one probably bothers me the most. Working out $$3(-7+9) – 5$$ by using the distributive property gives the same answer as calculating $$3(2) – 5$$. Frankly who cares how this is worked out as long as the correct answer is given; it is a short calculation that doesn’t really require working out what is inside the brackets first! Similarly the marks lost on question 6 just don’t make sense…

I passionately believe that mathematics should be seen as a creative discipline (as well as a scientific one). This sort of grading completely goes against this belief and is only going to serve to diminish confidence and turn pupils off mathematics. We should be celebrating different ways of working things out and I think you can encourage and strive for correct written communication without being pedantic!

Please let me (and Manan) know what you think….