Categories

## A Bit of Nostalgia

Yesterday’s post was a bit mammoth, so today’s post is much shorter! I’m sure you are all releived by this.

Back in April I posted on my Mathematical Journey and in that post I mentioned how I felt lucky to have a granny who was a maths teacher.

Ever since I can remember she would talk about maths with me, set me problems to do and send me coded messages in the post. As she taught at the Royal College for the blind this included teaching me Braille which sadly I can no longer remember. Whilst sorting through some stuff last night I found some old letters etc so thought I would share a photo here

My handwriting hasn’t really improved much since then to be honest.

Categories

## Math(s) Teachers at Play 88

So, here is issue 88 of the Math(s) Teachers at Play blog carnival. This acts as a round up of some cool blog posts that have been published since issue 87 over at cavmaths. As usual people have submitted entries, which I will supplement with some posts that I have really enjoyed reading in the last few weeks.

It is tradition for the post to start with something about the issue number so here goes!

Firstly, here are a few calculations that result in 88 (taken from Zoo of Numbers

\begin{align} 88 &= 23^2 – 21^2 \\ &= 13^2 – 9^2 \\ &= 4^2+6^2+6^2 \\ &= 2^3+2^3+2^3+4^3 \\ &= 3+4+5+6+7+8+9+10+11+12+13 \\ &= 17+ 19 + 23+29 \end{align}

88 is also an Erdos-Woods number. An Erdos-Woods number, $$k$$ is defined to be the length of a chain of numbers $$n,(n+1),(n+2), \cdots + (n+k)$$ such that each integer $$n+i$$ for $$0<i<k$$ shares at least one prime factor with either $$n$$ or $$n+k$$.16 is an Erdos-Woods number because for each number in the chain

$$2184,2185, \cdots, 2199,2200$$

shares at least one factor with either $$2184$$ or $$2200$$. TheErdos-Woods numbers are sequence A059756 in the Online Encyclopedia of Integer Sequences, the first few are as follows

$$16,22,34,36,46,56,64,66,70,76,78,86,88,92,94,96,100$$

I am now wondering if there is a nice way to compute Erdos-Woods numbers, yet another thing to look at when I get the chance…

Right, I had some really interesting submissions this month.

Jo Morgan (@mathsjem) shared this great post about Teaching Foundation GCSE (Grades 9 and 10 in the States) classes. This is one of her most popular posts of all time, and it contains some great tips for engaging those students who really struggle at maths – I know I will be returning to it when I am teaching Foundation GCSE.

Lisa (I can’t find her on Twitter unfortunately) shared an article she has written for the Huffington Post titled “Reconsidering Math as an Art Form”. It was inspired by Paul Lockhart’s essay a “Mathematician’s Lament”. I agree with Lisa and think this should be essential reading for mathematics educators. I particularly liked his comparison of the teaching of Mathematics to the teaching of music – I fundamentally believe mathematics is as creative as any of the creative arts and it is a massive shame that a lot of school children don’t see this side of maths.

Stephen Cavadino (@srcav) shared this amusing post linking a “Quirk of Probability” to a mobile game. I really enjoyed reading this before I received the submissions, and it was nice to be reminded of it.

April Freeman (who blogs here) has a post about using bundles of popsicle (or ice lolly sticks) to help with the understanding of place value for her daughter. I think this could be a useful technique to help with the visualisation of place value.

This nice review of a year of running a maths club for 4th and 5th graders by Benjamin Leis is well worth a read, there are some great tips that are applicable to general teachers as well as those running extra-curricular clubs. The Yearly Topics Map on his blog also contains links to some really nice resources.

Mrs E (@MrsETeachesMath) has shared this fairly short post on introducing proof. Proof is always a topic that is relatively challenging to teach. The ordering a story idea is a nice analogy to emphasise the importance of logical thinking when completing a mathematical proof.

Christy (@housefulofchaos) is a homeschooler based in Ontario and she submitted this on “Minecraft Multiplication Practice”. I confess I’ve never played Minecraft, but I am now very intrigued by this detailed post on a way to make multiplication more interesting. I’m also very curious about the online course she is running on Minecraft Maths, I wonder if I would be allowed to join?!?

Pea has shared a post on using concrete manipulative to teach the addition of fractions. Too often addition of fractions seems to be based on rote learning and I like how the understanding is at the fore-front of this approach, and I like the fraction discs.

Denise Gaskins (@letsplaymath), the organiser of this carnival has shared this fantastic post on introducing infinite series to children  with a puzzle courtesy of Don Cohen and an additional puzzle for older children from James Tanton. The infinite series approach to the solution of James Tanton’s puzzle is amazing, and I certainly don’t think I would have come up with it – I used similar triangles! I hadn’t seen Don Cohen’s “Map of Calculus for Young People” before, or looked at Don Cohen’s site; I’ll be spending more time here!

The final submission comes from Manan (@shahlock) who has shared his comic for Pi approximation day. I love Manan’s comics, make sure you check out the others.

I came across this site examining the maths of banknote patterns, it is a really interesting read. I had n’t really considered the mathematics behind these patterns and I will certainly be exploring them more in the future. I came across this article thanks to Colin Beveridge (@icecolbeveridge) who tweeted it alongside this Desmos sheet . Colin has also recently shared his Mathematical Journey (in response to this post of mine), it is a really interesting read, and contains some useful advice for any budding mathematician.

Danny Brown (@dannytybrown) has, in the last couple of days shared this long post about mathematical teaching styles. I promise it is worth spending time to read to the end!!! Get involved in the conversation about it on Twitter to.

Because of my background I am very interested in computer programming and the benefits this can bring to mathematics education. I have been looking at some early programs designed to be used in the maths classroom for the BBC Micro and have discovered an emulator by Matt Godbolt (@mattgodbolt), he has a video describing the coding of this available here.

Finally a few of us our taking part in a challenge to write a blog post every day of the summer holiday, the index is available here. At the moment it is me, Christine Norledge (@MissNorledge), Kim Thomas-Lee (@kimThomasLee), Mark Wilson (@mwimaths), Archbishop Sancroft High School maths department (@ASHS_Maths), @funASDteacher and Jennifer Stice (@mathchick5) taking part. We would love you to take a look t our posts and maybe join us in the challenge?

The next edition of this blog carnival will be at Mrs E Teaches Math.

Categories

## The Magic of the Micro: Part 1

At the beginning of July Stuart Price (@sxpmaths) blogged this post looking at A Level textbooks through the ages.  I really enjoyed reading this, and was particularly drawn to a picture of a book that described computer programs written in Basic for use in the maths classroom. I was so keen to look at this that I straight away ordered a copy from Amazon and here it is:

This book is fantastic, and it’s a shame that I don’t know of anything similar today (does anyone reading this?) – cue for me to write one, yet another project….

The BBC Micro holds a special place in my heart as it is the first computer that I can remember using at Grove Primary School. I remember sitting at one in the BBC room and doing various things, including my first bit of programming in BASIC (I later had a fairly long hiatus before coming back to coding).

Having become used to modern computers you forget the limitations of older computers such as the BBC Micro. For example, at the beginning of the book some of these limitations are mentioned, such as the fact that the BBC Micro’s filing system can only cope with 31 filenames. I particularly like this quote from the introduction to the book

“The lack of ‘idiot-proofing’ means that they [the programs] may sometimes ‘crash’, but in these circumstances a consideration of why it went wrong may itself be very illuminating, This will usually be due to inaccuracies in computer floating-point arithmetic, the non-existence of solutions, or the singularities of functions, all of which children should be aware of.

I don’t know of many ‘children’ who have an understanding of floating-point arithmetic or the failure of certain numerical methods in the presence of singularities – it is even possible to get through some undergraduate numerics courses without really tackling these issues!

I am going to write a few posts over the summer looking at some of the programs in this book, with the odd modification (I’ve quite enjoyed doing a bit of Basic for the first time in over 20 years).

Of course, to run some BBC Basic programs I either need access to  BBC Micro (If anyone has a working one that they don’t want anymore please let me know!!) or a decent emulator. As a mac user there aren’t many BBC emulators available, and those that are don’t seem to have been maintained since 2012. But then I found this excellent Javascript emulator by Matt Godbolt (@mattgodbolt). On his blog there is a lot more detail about the implementation of the emulator, as well as some other cool posts – I urge you to check them out if you are interested in coding! The video about the BBC emulator is definitely worth a watch. It’s sad that the emulator doesn’t apper to work on an iPad though.

I first tried Program 1 from the book, which is a simple program to compute the first $$n$$ triangke numbers, where $$n$$ is an input from the user. This is shown on the screen shot below:

I then thought I would try the highest common factor program:

I then thought I would look at something a bit more complicated, and try plotting the graph of $$y = x^2$$. One thing that you don’t have to do with things like scaling the output explicitly to fit the graph onto the display when using Matlab or NumPy. Of course the BBC Micro and BBC Basic is not as sophisticated as this, and we are explicitly giving coordinates on the screen of where to colour a pixel – because of this osome scaling needs to be done. This explains the divide by a 400 on line 60 of the below code from the book.

This code generates the following output which is recognisably the curve of $$y=x^2$$

I’m going to play a bit more with the graphics commands and will post next week with some further programs from this book, and maybe one that I have written myself too.

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## The First #mathsjournalclub Article and Discussion Date

Firstly, put the evening of Monday 24th August into your calendars!!

Thank you to everyone who voted for our first journal article. Around 50 people made a selection and the overwhelming winner was the article “A Glimpse into Secondary Student’s Understanding of Functions” from the journal International Journal for Mathematics Teaching and Learning published online by the Center for Innovation in Mathematics Teaching of the University of Plymouth.

We will be discussing this at 8pm on the 24th August 2015 – you can follow the conversation using the hashtag #mathsjournalclub. If you can’t make the actual conversation but have read the paper it would be great if you could still contribute some thoughts under the hashtag #mathsjournalclub and they can be used to guide the discussion.

To remind you of the topic of the paper I have included a shot of the first page below. Please click on the link to download the full paper. I will publish some discussion themes about a week before the 24th.

I’m really looking forward to discussing this with you all on the 24th.

Categories

## Maths in the Media

Math’s seems to be quite popular in the media at the moment, with three great articles printed in the last week or so! I thought I would share them here and provide a short commentary.

The Singular Mind of Terry Tao

The New York Times has published this great (long) article on the life of Field’s Medalist Terence Tao by Gareth Cook (GarethIdeas) The New York Times often seems to publish these kind of long articles, and luckily they have a policy of letting you read 10 articles a month from their website for free!

I think this article nicely describes the life of a research mathematician, and I particularly like this quote

“The steady state of mathematical research is to be completely stuck. It is a process that Charles Fefferman of Princeton, himself a onetime math prodigy turned Fields medalist, likens to ‘‘playing chess with the devil.’’…”

The article is a very readable account of what it is to be a modern research mathematician, and in it Steven Strogatz sums up maths as a “… conversation with each other over the millennia”, emphasising the connections with mathematical discoveries across the generations.

The New York Times also has a puzzle from Terence Tao here.

John Horton Conway: The Worlds Most Charismatic Mathematician

This article by Siobhan Roberts appeared in the Guardian on Thursday 23rd July is an interesting portrait of the mathematician that Sir Michael Attiyah describes as the “most magical mathematician in the world”. John Conway is famous for his work in the 1970s on symmetry groups. I didn’t realise his commitment to maths education and how much time he spends at maths camps in the university holidays – not many academics would be willing to give up this precious research time. Conway is of course famou for inventing The Game of Life – I will write about this in the future I think, once I have coded it in Python.

This article is fascinating and well worth a read – I will definitely be buying the full book hen it comes out in September.

20 Mathematicians Who Changed the World

Walter Hickey has written this article for Business Insider. It is considerably shorter than the previous two and provides brief (a few lines) information on twenty famous mathematicians. I don’t completely agree with his choices – for instance what about George Boole?!? Who do you think is missing from this list? I’d be interested to know…

Categories

## AM-GM Inequality

The Arithmetic Mean – Geometric Mean (AM-GM) inequality is possibly one of my favourite inequalities (another is the Cauchy-Schwarz inequality, see a later #summerblogchallenge post for more details on this).

Earlier this week I was honoured to have received an envelope of goodies from Ed (@solvemymaths). Alongside the very cool mathematical Mr Men stickers, there was a note from Ed with a lovely geometric illustration of the AM-GM for two lengths $$a,b$$.

In words, the AM-GM inequality states that the arithmetic mean of a set of (non-negative) numbers is always greater than or equal to the geometric mean of the set of numbers. More formally, the AM-GM inequality can be stated as follows: For a set of non-negative real numbers $$a_1,a_2,a_3,…,a_n$$ the following inequality holds

$$\frac{a_{1}+a_{2}+a_{3}+ \cdots + a_{n}}{n} \geq \sqrt[n]{a_{1}a_{2}a_{3} \cdots a_{n}}$$

Example: For the set $$1,6,9,15$$ the arithmetic mean is $$\frac{31}{4} = 7.75$$ and the geometric mean is $$\sqrt[4]{810} \approx 5.3348 [\latex]. Proof: There are many different proofs of the AM-GM inequality. The excellent Art of Problem Solving website lists a few here . I particularly like the one using the rearrangement inequality as it is so concise. The AM-GM inequality is often useful for questions on STEP and Maths Challenge papers, here is a typical question and a solution. Question: Given that [latex]a,b,c$$ are non-negative integers, show that

$$(a+b)(b+c)(a+c) \geq 8abc$$

Solution: Given that $$a,b$$ then applying the AM-GM inequality we have that

\begin{align} \frac{a+b}{2} &\geq \sqrt{ab} \\ a+b &\geq 2\sqrt{ab} \end{align}

Performing similar calculations with the other pairings we can obtain

\begin{align} b + c &\geq 2\sqrt{bc} \\ a+c &\geq \sqrt{ac} \end{align}

So, since the numbers involved are all positive we can multiply the inequalities to obtain

\begin{align} (a+b)(b+c)(a+c) &\geq 2\sqrt{ab}2\sqrt{bc}2\sqrt{ac} \\ &= 8\sqrt{abbcac} \\ &= 8\sqrt{a^2b^2c^2} \\ &=8abc \end{align}

as desired.

Can you come up with a nicer solution? Can you come up with similar questions?

The AM-GM inequality comes up in higher level maths too, a friend, Edward Hall of the University of Leicester has reminded me that it is used in a modified form when proving the continuous stability of the solutions to parabolic PDEs. He also mentioned that he has recently seen it used in a paper concerning Discontinuous Galerkin (more on these in the future) methods for quasilinear PDEs.

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As I’m sure some of you know I love going out for food and am happy to spend ages reading menus as I walk down a street.

Ever since I was a child I have enjoyed looking at the price of things on menus, working out various things.

For example on the above menu (from Carluccio’s, which is one of my favourite Italian restaurants) you could work out the most expensive and the cheapest three course combination. You could also compare it to the set menu and work out the most you could save – it’s always good to maximise savings! There’s also a multitude of the “I’ve bought 4 main courses, 3 desserts and 2 starters and it costs this much, what did I buy type?” problems. In a similar vein I have used menu prices when creating simultaneous equations that are harder than the typical GCSE integer valued questions. By creating some margin percentages for each meal, and some numbers representing the number of covers of each sold in a night a matrix multiplication problem can be created to find the profit made by a restaurant from one service. This could also lead on to discussions about the conditions necessary for the matrix multiplication ABc to be the same as the matrix multiplication BAc (where c is a cold vector containing the meal prices)…
I’d recently been led to think about menu pricing when a picture was posted on Twitter. It reminded me of a study that I had read a couple of years ago on the effect of price formatting on the sales in a restaurant. This study “Effects of Menu-price Formats on Restaurant Checks” by Yang, S, Kimes, S and Sessarego was published in the Cornell Hospitality Report in 2009. In this study diners were given menus with three different price formats and the effect of these formats on the total bills paid. As you would expect the party size had the most impact, dining duration and guests willingness to pay also had a large effect (the p value was less than 0.0001 for these three variables). The overall regression analysis combined the three different price formats into one variable, which didn’t seem to affect the total price paid significantly. However, the authors then performed a more detailed analysis of the three different price formats using a linear contrast analysis and they found that parties with a numeral only format (i.e. no $sign) spent$3.70 more than the average party and those who had been given the menus with the prices written in text spent \$1.85 less. This increase translated to a 8.15 percent increase in spend per person.  Perhaps this explains why more menus such as the Carluccio’s one above don’t include a £ sign when they list prices.

Colin Wright (@ColinTheMathmo) recently posted a picture of a menu similar to the one below

I’m not sure how I feel about the use of 9.5 instead of 9.50… Though any discomfort I have with it is, I guess, only because we are used to having prices with two decimal points. Realistically there is no reason to have the trailing zeros. What do you think?

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## Rational Functions

So here is the first of my 47 (if I can count….) blog posts as part of #summerblogchallenge – not quite as many as @MissNorledge!

I thought I would briefly write about rational functions which often come up in A Level and STEP papers. A rational function is any function $$y = f(x) = \frac{P(x)}{Q(x)}$$ with $$P(x), Q(x)$$ polynomials. Quite often you are asked to sketch such a function. To do this, the normal steps are to find the roots, i.e. where the function crosses the $$x-$$axis which is easily done by solving $$P(x) = 0$$. Then we can think about asymototes: For the horizontal asymptote consider the ratio of the leading coefficients of $$P(x)$$ and $$Q(x)$$, for the vertical asymptotes try to factorise $$Q(x)$$ and find what values make $$Q(x)$$ zero. It is the normal to mark on any local maxima and minima. Of course the usual way is to use calculus and differentiate, setting to zero and solving. However, sometimes we just need to know the approximate location of the maxima and minima  and the associated y values. This is due to the fact that we can often sketch the graph using this knowledge and examining what happens near the asymptotes.

I was shown this by another teacher who recently retired and had never considered it before, always using calculus. But with practise, this could be done mentally, pretty quickly I think.

The picture below shows the method for an example. Essentially you imagine a fixed horizontal line across the graph of your rational function, so $$y$$ takes on a fixed value. We can then form a quadratic whose coefficients depend on $$y$$. A maximum (or minimum) of our rational function will occur when the discriminant of our resulting quadratic is zero. Using this fact we can find the local maximum and minimum values of the rational function.

Categories

Categories

## #summerblogchallenge

Last week I made the (what I thought) throw-away comment that I was going to try and write a blog post every day of the holiday. I said this mainly because I want to try and get more regular in my posting, and this seems the perfect incentive!

Christine (@MissNorledge) seemed keen to take part and has worked out that she has 51 posts to write!! This is a definite incentive for me to manage the whole summer as she has a much bigger summer of important events planned than I do!

A few others are joining in too, please feel free to join in!

Tomorrow afternoon I will create a post where I will continually link to all the posts everyone writes, so please remember to tweet your posts with the hashtag #summerblogchallenge so that me and Christine can link them 😉

I break up tomorrow so I will be officially starting my challenge on Friday…….