Just before I was due to teach Period 1 this morning my computer chimed, announcing the arrival of (yet another) email.

Thinking that I might be able to action and delete it before the lesson began, I clicked to open it. I’m glad I did, its the best email I’ve received all week:

Good morning

Knowing you find beauty in football and maths alike, check out the top ten in league division two. Perhaps you have already!

Have a good day,

Well, Year 11 could wait – I scuttled over to the BBC Sport webpage and checked the standings at the top of League Two:

Now, I’ve no idea if this has happened before, or how likely it is to happen (and I’ve no idea how one would even begin to work that probability out) but it made me smile, perhaps it will make you smile too.

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Regrets, I’ve had a few, but then again, too few to mention.

Mistakes, however, I’ve made (and continue to make) many of them.

I made one this morning. I was working through a difficult “ladders” question with an A level Mechanics class, a problem that I had set as a homework and most couldn’t complete.

And so, with the power of OneNote and Teams, the class were able to watch and listen to me as I demonstrated how to do the question, interjecting as appropriate for clarification or help as required.

On completing the question I asked the class if they now understood, if they were happy with what I had shown them. They all were.

Until a few minutes later, when one student suggested I had made a mistake with one part of my answer.

I asked him to talk me through his thinking, and he was right, I was wrong. (We were calculating a range of values needed for a force to stop a ladder slipping, and I had got my upper bound wrong.)

Whilst I will confess to being a little frustrated with myself for making the error, that is more than compensated with the sense of satisfaction and pride I vicariously took from my student; the apprentice has surpassed his master, and isn’t that what we all really want as teachers? For our students to leave us having learned all that they can from us?

Mistakes – we should celebrate them for what they are, whoever makes them, a chance to learn. After all, if no-one ever made any mistakes we’d never have the opportunity to improve.

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It may be nine days until Christmas, but today is *the* day to celebrate this December.

It’s Pythagoras Day: 16/12/20

Why?

Well 16^{2} + 12^{2} = 20^{2}

and

A Pythagorean triple is a set of 3 integers (whole numbers) a, b & c such that:

a

^{2}+ b^{2}= c^{2}

And its nearly five years until the next Pythagoras Day: 24th July 2025, or 24 7 25

Happy Pythagoras Day!

]]>“We live in unprecedented times” is, perhaps, a somewhat overused phrase of the last six months, but yesterday it was apt as the unprecedented happened.

In their match against Manchester United, Brighton and Hove Albion hit the woodwork* 5 times in the game. This (one team hitting the woodwork 5 times) has never happened before in a Premier League fixture.

(* “hitting the woodwork” in football is when you shoot, and the ball rebounds off either the posts or the cross bar. For the neutral spectator it is an exciting and dramatic moment as the ball cannons back – with an audible thud – onto the field of play. For the attacking side it is, of course disappointing as they hope, instead, to see the net bulge, whilst the the defending team breath a collective sigh of relief as a goal is thwarted.)

So how likely is this? Hitting the woodwork five times in a ninety minute game? Should we be surprised that it happened, or perhaps surprised that it hasn’t happened before?

On of my regularly readers, Steve from Cheltenham, got in touch with me to walk me through some maths he had done to calculate the probability of this happening:

Assume that every shot is always within a rectangle that extends to 1 metre around the outside of the goal (of course, this isn’t always the case, but we need to make some assumptions if we are to create a working model.)

Assume that there is no aiming and hence every point within that rectangle is equally likely to be hit (another assumption to simplify our model)

The total area of the aiming zone is 34.03 m^{2}

The area of the posts and bars are not what you might first think …

In our initial work, we just found the area of the posts and bars, but this would assume we were modelling the ball as a point. But actually, the centre of the ball can miss the post, but the ball still strike it, so the effective width of each post is the 12cm of the post, plus x 2 diameters of the ball (the diameter of the ball is 22cm) Hopefully the sketch picture below adds some clarity:

This make the “effective” area of the woodwork 6.83 m^{2}

Therefore the probability of hitting the woodwork is 6.83/34.03 = 0.201 (to 3 decimal places)

Brighton had 18 shots in the match, so we can use a binomial distribution with n=18 and p = 0.201 to find the probability of hitting the woodwork in 5 out of those 18 shots. Using the standard notation:

When B ~ (18.0.201), P(X=5) = 0.152

or, the probability of hitting the woodwork 5 times in yesterdays game was about 15%, so unprecedented, yes (because it hasn’t happened before), unlikely? yes, but perhaps not too surprising and maybe we should be wondering why we haven’t seen it happen before …

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Here’s the tree at the junction of South Ealing Road and Little Ealing Lane getting blown over this afternoon. Amazing car cam footage by Joanna Wolman.

(Some understandable swearing!) pic.twitter.com/zlFcDFuXWX

— Jon Ball (@JonBall) July 5, 2020

Whenever I see a clip like this my first reaction – like those in the car (and sorry for the NSFW language) – is how lucky the survivors are.

But then I begin to ponder.

Are they actually unlucky? What are the chances of being in the vicinity of a falling tree? Pretty low, I suspect.

So whilst you might count the pedestrians as being lucky to avoid being crushed by the tree, they are also pretty unlucky to be so close to a tree as it gets uprooted by the wind.

Alas, I lack the mathematical skills to determine an answer to this question so (for me at least) it will remain a philosophical question.

Lucky or unlucky – what do you think?

]]>Some good news.

For the first time since mid-March, excess deaths are below the five year average. Regular readers will know that I have been tracking “excess deaths” (based on data provided by Office for National Statistics ) as it strips out any debate as to whether a death was due to Covid 19 or not. Instead, it compares the number of deaths in a week to the five year average for the equivalent week.

The graph above does not mean we are out of the woods yet (as I write, Leicester is being placed in to local lock-down to combat a regional spike) but it does give cause for hope.

(note: week 25 is the week ending 19th June 2020. Week 11 – the last time excess deaths were negative, was the week ending 13 March 2020)

]]>I continue to crunch the numbers, and explore different ways to display the data.

In the graphs above, I have taken the total number of weekly deaths in England and Wales (as reported by Office for National Statistics) and subtracted from that the average number of deaths for that week.

Before the outbreak of Covid 19, 2018 had been a “bad year” for deaths, with the weekly death rate often being above the 5 year average (see graph of cumulative deaths, below) so I plotted that – on the same scale – to make an easy comparison with the tragedy of this year.

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Data can tell us many things, but we need to understand what data we are looking at, and what the data is showing us. As the Covid crisis has continued, there has been debate and discussion about how many deaths are due to the disease.

The government, daily, discloses the number who have died in ** hospital **from Covid-19, but this only tells part of the story as many are dying from the virus in care homes and, possibly, at home.

How do we know the true impact the virus is having?

One way is to look at total deaths per week, and compare them to equivalent weeks in years gone by.

The data is all available at the Office for National Statistics and I’ve used the data to generate the graphs above.

Its quite clear that, whilst there is variation from year to year, the trend for each year is similar.

Until you reach week 13 of 2020 (week ending 27 March) when the line takes a sinister upwards turn, and does not stop its climb.

By the week ending 17 April 2020 (the last week data is currently available) more than 22 thousand people in England and Wales died in that week, circa 12 thousand more than the average for that week.

12,000 excess deaths in one week alone. A sobering reminder of the deadly effect of the corona-virus.

]]>The unfolding tragedy that is Covid 19 is being fought on many fronts, and data, statistics and mathematics are playing a strong supporting role, by helping to inform what is happening and allow the scientists and politicians make decisions and review the outcome of the policies that have enacted.

Much data is being made publically available, and I have been experimenting with a new tool I have found to display that data. Whilst not up there with some of the excellent graphs and infographics being produced by many sources, I’m quite pleased with what I have achieved. Later this term, I will be teaching (remotely) some of my students how to access the data and use various tools to visualise and interpret said data.

I needed somewhere to host my first couple of forays in this field, so here they are.

Data sources:

]]>This cartoon is good. Very Good.

(FWIW, I definitely identify as a Pythagorean rather than a Trigonometric)

It came from “The Saturday Paper”, an online Australian newspaper. Here is a link to the original cartoon, and the paper.

And now we contrast it with something that is both bad and ugly. It is perhaps no surprise that this monstrosity comes from Fox News. Just look at that y-axis (if you dare) and weep.

Now go back up to the top of this post and look again at the brilliance of that cartoon. Hopefully it will erase the memory of that shockingly bad graph,

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