When Refereeing and Maths Collide

PenaltyD

Regular followers of my blog know that I like my sport and some of you are aware that I am a qualified football referee and regularly ply my (alternate) trade as the “man in black.”

With my referee’s hat on, I am an active member of a forum that discusses the finer points of the offside law and whether or not any particular offence as seen on Match of the Day warranted a red or yellow card. All part of the learning process, everyday is a school day, and all that.

Today someone posted a problem that was a little left field for the average referee discussion – what is the area of the “D” on the edge of the penalty area.

What a great question!

Lets fill you in with a few dimensions. From the goal line to the edge of the penalty area is 18 yards. The penalty spot is 12 yards from the goal line.  When a penalty is being taken all players (other than the penalty taker and the goal keeper) must be both outside the penalty area and at least 10 yards from the penalty spot. Hence the “D” on the edge of the penalty area – it’s arc marks the points that are 10 yards from the penalty spot.

Its not a simple question, but one that should be within the grasp of of a “good” GCSE student. It’s a problem involves trigonometry and Pythagoras, sectors and segments; the area of a triangle and fractions …

I got the answer to be 44.73 square yards (to two decimal places), do you?

If you’re not sure how to approach the problem you can see my back of a yellow card working below.

penaltyAreaworkingoutMidsize

 

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The Magic Washing Machine

2016 will be remembered for many things, Brexit, the election of Trump, a surfeit of celebrity deaths, but it was last week, 7th February 2017, that a true giant passed away, although you might have missed the news of his passing.

Hans Rosling was a statistician, an educator, a communicator. He used facts to explain and enlighten. Four years ago, I wrote a short blog embedding one of his videos, where he illustrated the change in family size and life expectancy over the last fifty years. He really was a remarkable man with the ability to communicate some difficult and challenging ideas with ease and clarity.

Please, take the time to watch the video above – it’ll only take ten minutes of your time and, even if you learn nothing (but you will) you will enjoy the show.

In these times of alternative facts, mis-speak and other Orwellian horrors, I hope that Hans’ legacy will be a willingness to use facts and statistics to inform and shape understanding.

Hans Rosling, 1948 – 2017. The world is a better place for your passage through it.

(A quick youTube search for Hans Rosling produces a wealth of results. If you’ve watched the video above and would like to see some more, below are a few links you may enjoy)

Where are the Syrian Refugees?  Although made in 2015, so the numbers may have changed, this short video is quite sobering and shows that we, in Europe, have probably got it wrong.

200 Countries, 200 Years    Hans examines how the wealth and health of 200 countries has changed since 1810

Channel 4 News Interview  Interviewed on Channel 4 News. Worth a watch.

Posted in Handling Data, Large Data Sets | Leave a comment

Large Data Sets

image

Large data sets – not the most inspiring of titles, but one which we teachers of A Level maths will become increasingly familiar over the next few weeks and months.

A Level maths is changing, but two plus two remains four, most of the content that is in the current A Level syllabus is in the new syllabus, to be taught from September ’17. It’s place in the syllabus may have changed – i.e. a topic that currently appears in Core 3 may now find itself in AS maths and be taught in the lower sixth/first year of A Level, but there is nothing that will be too unfamiliar to today’s teacher or today’s student.

Except for large data sets.

Candidates are to be familiar with one or more specific large data sets, to use technology to explore the data set(s) and associated contexts, to interpret real data presented in summary or graphical form, and to use data to investigate questions arising in real contexts.

… and that is new.

So its time to start thinking about large data sets, what they are, how we will teach with them, how they will impact on the exams…

The boards have, helpfully, published the large data sets that they will be using, and I’ve put copies of them here:

Further down this page you can see some sample questions from the boards that relate to the large data sets.

So what is to be expected of the student in the exam?

I must add a caveat that I am crystal ball gazing, and this is just my own view and not that of any board, but it seems that students won’t have to have whizzy excel skills to manipulate the data in the exam. They will be examined on this part of the syllabus like all others: in questions on paper in an exam hall, no computerisation of the exam.  They could succeed on these elements of the exam with no knowledge of how to use Excel (or any other package) to manipulate data. But, as I will begin to explore below, manipulating the data with Excel (or similar) in lessons or homework will help them develop a good knowledge of the structure and content of the data, and this is important.

Have a look at this sample question from OCR, in particular part ii):

OCR Large Data Set Sample Question

OCR Large Data Set Sample Question

To effectively answer this you need to know that the data set contains different regions with different geographical characteristics, and the differences between, say Unitary Authority and a Metropolitan Borough and how the provision of public transport within different areas varies.  (If you don’t believe me, have read of the mark scheme below.)

I must confess, I am a little uneasy with this – are we examining mathematical skills or geographical knowledge?

 

OCR data set sample question answer

OCR data set sample question answer

After years of staff shortages meaning Geography teachers have ended up teaching maths  it now seems that we maths teachers are going to have to do a bit of Geography teaching!

However, we must do the best for our students and so we must familirise them with the nature, structure and content of the data sets. To do this I will look at how I can incorporate them into teaching as many aspects of the syllabus as possible. For example, I will get students to calculate the Standard Deviation of all, or a sample of, the data set – and not just using the single Excel formula.  But that’s a post (or several) for another day.

This post was hopefully a brief introduction to Large Data Sets. Download the files for yourself, have a play and a think about how you might use them.  As ever, I’d be delighted to hear your suggestions. And have a look at another couple of sample questions designed to examine the students familiarity with, and ability to explore and manipulate, large data sets.

EdExcel sample Large Data Set question

EdExcel sample Large Data Set question

 

EdExcel large data set sample question answer

EdExcel large data set sample question answer

 

MEI Large Data set sample question

MEI Large Data set sample question

MEI Large Data Set sample question answer

MEI Large Data Set sample question answer

Posted in Exam tips, Handling Data, Large Data Sets | Tagged | Leave a comment

Pint, anyone?

pint

One of the casualties of the forthcoming changes to the A-Level syllabus is the Decision Maths module(s). The “standard” A (and AS) Level will consist of Pure Maths and a combined applied maths paper, made up of content from both Mechanics and Statistics.

I’m not sure how I feel about this – on the one hand, Decision 1 contains some really interesting maths: want to know how your SatNav determines your route? Decision 1 has the answer. Want to know how Excel sorts data? D1 explores the  Bubble Sort and Shuttle Sort algorithms, as well as explaining how to determine the efficiency of an algorithm. Plus I make a cameo appearance in the EdExcel D1 text book – my 15 minutes of fame!

But answering some (many) of the exam questions can become a little tedious, with the student forced to replicate the steps taken by a computer to use an algorithm to solve a problem.

But regardless of the above, I was delighted to discover a new application of the Traveling Salesman Problem that we study in the course.

The Traveling Salesman Problem (or TSP for short) is simple in principle, but soon escalates to become very calculation heavy.

A salesman needs to visit all the towns in his area, before returning home from where he started. In achieving his goal he wants to travel as short a distance as possible and the solution to the TSP is this route.

Imagine he only has 3 towns to visit – he has 3 choices for his first town, then 2 for his second, and only 1 for his third, so there are 3x2x1 = 6 possible solutions, although we can halve this as the length of the route ABC will be the same as CBA.

Add a fourth town and there are ½ of 4x3x2x1 = 12 different routes, five towns gives ½ of 5x4x3x2x1 = 60 – the numbers soon start getting very big and even a computer would begin to struggle to compute all the various different permutations to find the shortest route.

So we use an algorithm – in fact we use two. Its such a difficult problem that we don’t necessarily find the optimal solution. We use the “Nearest Neighbour” algorithm to find the upper bound – a distance we know that can contain a route, and then we use the Lower Bound algorithm to find the shortest what the optimal route could be. We now have a range of values between which the optimal solution must lie. If the range is small we may be happy to go with our upper bound route, if not we can try the tour improvement algorithm.

Rather than try to explain these algorithms, this short video offers a good insight into how they work.

“But you promised me a pint!” I hear you cry. “Whats all this got to do with the price of beer?”

I’m getting to that now.

Earlier this week, I stumbled on a superb piece of mathematics that found the shortest route that would take in all 24,727 (yes – twenty four thousand, seven hundred and twenty seven) pubs in the UK.

You can read all about it here …

… or you can just enjoy the majesty of the map that shows the route:

A walking tour of the UK’s pubs. A fine application of the Traveling Salesman Problem

A walking tour of the UK’s pubs. A fine application of the Traveling Salesman Problem

 

Cheers!

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2017 – A prime year

2017-a-prime-yearIts been a fallow time for “Prime Dates” of late – the last was way back in November 2013 (a quick moments thought should reveal why there were no prime dates in 2014, ’15 or ’16) but now the drought is over, let the deluge begin.

2017 is, itself, a prime number and the year kicks off with a prime number date:

1 1 17, or 1117, is a prime number

and, with a little poetic licence, the next day, 2nd Jan, is also prime, as long as we write it as:

2 01 17

20117 is prime.  There are other primes in January, both the 23rd and 25th (23117 and 25117) are prime and there are many, many other prime dates throughout the year.

The 3rd February, 3 2 17, is both prime in its own right (3217) but is also “made” from 3 prime numbers: 3, 2 and 17

However 2017 unfolds for you, I hope you have a great year – and keep an eye out for those prime dates.

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