The Americans have their Pi Day later this month – March 14th – 3.14, as they write the month then the date.
We have our own “Approximate Pi Day” on 22nd July – 22/7 – a good approximation for Pi.
But this year, we have “Pi Month” – March 2014 – 3.14.
Enjoy every glorious day!
Picture courtesy of Adelaide Archivist http://www.flickr.com/photos/adelaide_archivist/2391956010/
License: http://creativecommons.org/licenses/by/2.0/deed.en_GB
The Six Nations Rugby – my sporting highlight of the year.
Each year, the six nations – England, Ireland, Scotland, Wales, France and Italy – play each other once in a round robin Rugby Union Championship. The rivalry, as it so often is amongst neighbours, is intense, the rugby passionate and the outcome is never a forgone conclusion.
And, like most sports, rugby lends itself to a bit of mathematical analysis.
In rugby union there are three different ways to score:
So what, if any, scores are impossible to get in rugby? Hopefully you can quickly spot that 1, 2 and 4 are the only scores a team can’t get in rugby union. (Rugby League is a different code of the game, with a different scoring system. For simplicity, from now on when I write “rugby” I will mean rugby union.)
But can you prove that you can get every possible score above 4?
Mathematical proof is often seen as ‘hard’ by pupils, but they may well be able to come up with an intuitive proof for this problem:
As soon as you can show how to make three consecutive scores, you can prove all others are possible as a penalty is worth three points and you can add a penalty on to the three consecutive scores to get the next three consecutive scores etc.
E.g. lets say P = penalty (3 points) T = try (5 points) C = converted try (7 points)
to make three consecutive scores we could have:
we could then add a penalty to each of those scores to get the next three consecutive scores:
and to get the next three consecutive scores, add another penalty etc. So we have proved that you can score every possible score in rugby, above 4 points.
So what is the most likely score?
A great question that bookies, gamblers and pundits alike all would love to answer – and a great question to try and solve (either yourself or as a challenge for your pupils.)
We know that the only possible individual scores are 3, 5 and 7, so “all” you need to do is work out how many different ways you can make a number from those adding those numbers.
There is only one way to make to make the scores up to 9, but there are two ways to score 10 points – a penalty and converted try, or two (unconverted) tries – but there is only one way – 2 penalties and a try – to make a score of 11, so we would expect a team to score 10 points more often than 11 points.
(I’m considering combinations, as the order of scoring is unimportant. e.g. Penalty, Try, Penalty is the same as Try, Penalty, Penalty. When the order is important it is a permutation.)
Its an interesting investigation to work out how many different ways there are to score up-to, say, 20 or 30 points, to maybe give you an edge when predicting the outcome of a game of rugby.
With so many different scores possible, it is notoriously hard to pick the final score of a game of rugby. If you read the press, the pundits don’t do it, instead predicting the winners, and the margin of victory, e.g. England to win by 5 points, Wales to win by 10 etc.
Which leads me to think, is there a differential that is more common than others?
Perhaps we could work that out by looking at the most likely scores, or doing some statistical analysis of previous games, but that is a problem for a different day. As half-term draws to a close, I’m looking forward to settling down in front of the TV and watching England v Ireland.
Oh, and my prediction? England to win by 3.
In December I wrote about 7 Great Resources for Maths teachers and parents and one of my top seven was the Desmos online graphical calculator, a great resource for plotting graphs and functions.
They’ve now added to their stable of offerings and I’ve recently used their new “Function Carnival” app, and I have to say, its brilliant! Pupils watch a short animation – a man being blasted from a cannon, before parachuting safely back to earth, bumper cars bumping or a Ferris wheel spinning – and then they have to plot a distance time graph of the event. They then replay the animation and this time their cannonman (or bumper car or Ferris Wheel) is superimposed over the actual cannonman (or bumper car … ) so that they can check the accuracy of their graph.
It has brought the Shell Centre for Mathematics graphs into the 21st century.
And, as a teacher, you get a great dashboard which collates your class’s answers, so you can quickly and easily see who has done what, who has done well and who needs some help.
Below are a few screenshots, but rather than me try to describe it, head over to “Function Carnival” and have a play, I think you’ll like it.
I tested it with a bright year 9 class, working with iPads, but you could just as easily use it with desktop PCs and I’m sure it would be valid for many age groups and ability levels.
Oh, I had a question about the site, so I fired of a support ticket and had an answer within 24 hours – thanks Desmos for the great support as well as the great resource. And I’ve made a mental note – the Ferris Wheel example would make a great introduction to the Sine Curve.
Some screenshots:
Teacher’s Dashboard
I stumbled across this the other day and I found it rather poetic, in a melancholic sort of way.
I can’t help but imagine the unrequited love between two parallel lines, destined never to meet. Or the brief moment of passion as two lines meet before moving further apart throughout all eternity.
It must be getting close to Valentine’s Day!
[I tried to find the original source of this gem, but, alas, I have failed to do so. If you know who first penned either the words or picture, please let me know so that I can attribute credit where its due.]
