Choose any 4 digit number except for 1111, 2222, 3333, 4444. I’ll choose 1502.

Rearrange the digits to give you the biggest and the smallest numbers you can. 5210 and 0125.

Find the difference of these two numbers. 5210 – 0125 = 4995.

Repeat for the new number you get (using zeros to supplement any missing digits if necessary; you must always have 4). 9954 – 4599 = 5335.

Keep repeating until you have a good reason to stop.

Example

5210 – 0125 = 4995

5533 – 3355 = 2178

8712 – 1278 = 7434

7443-3447 = 3996

9963 – 3699 = 6264

6642 – 2466 = 4176

7641 – 1467 = 6174.

I’m not going to repeat any more because 6174 => 7641 – 1467 which is the calculation I just did!

Discussion

• So every number we’ve tested came to 6174. Do you think all numbers will come to 6174? Why? In maths we can’t say we’re sure about something unless we have a good reason, so unless you want to go through every number and check it, we can’t say that we know that every number sequence we choose will go to 6174! (This is how mathematics differs from science, we don’t just make hypotheses and wait until the next time they break, we find ways to be certain!)
• Would we have to check every different number to be certain or are there shortcuts we can take? (should we check both 1234 and 1243 separately?)
• My rules stipulated that you couldn’t choose all the same number: 1111 or 2222. What would happen if you did use those numbers?
• What about two digit numbers or three digit numbers?
• Could we frame the problem as (1000a + 100b + 10c + d) – (1000d + 100c + 10b + a)?
• What about five or six digit numbers (or more…)? You may need a computer to help!

Thoughts

I just came across this interesting bit of mathematics via Twitter: http://plus.maths.org/content/os/issue38/features/nishiyama/index. It offers the opportunity for a beautiful open-ended task for secondary maths classes of all abilities. The initial task is easy and produces a startling result that feels like a trick; that hook can lead to discussion and further work about probability, permutations, algebra and programming.

On May 6th the British People voted for a hung parliament.

There is a tradition in journalism to seek to reduce complex issues down to the simple; to form a narrative which describes in terms that a casual reader/viewer/listener can consume with the minimum of effort. So far as the reduction accurately describes the situation they attempt to convey, the journalist succeeds.

Facts are simple, but their analysis is complex. Let’s start with some facts: excluding the one constituency that has not yet returned at the time of writing, 29,653,638 people voted in 649 constituencies for dozens of parliamentary candidates, the majority of whom were representatives of political parties. Within each constituency, candidates who received the most votes won their seat and became an MP. 306 of those were Conservative, 258 were Labour, 57 were Liberal Democrat. Other parties got 28 seats between them.

The first-past-the-post system means that the votes for a losing party in a constituency count for nothing. If we disregard these votes, then actually 13,863,530 people voted for their MP, which is 47% of the overall turnout. 7,279,220 voted for a Conservative MP which is 25% of all the people who voted. The turnout was around 65%, so as a percentage of people who could have voted, the people actually responsible for returning the 306 Conservative MPs constitutes 16.25% of the electorate.
(Data Source The Guardian DataBlog).

As I have suggested, analysis of these results is complex, so I will attempt to be minimalistic in my interpretation. What is clear, I think is that the signal that is produced by this (perhaps any?) election is chaotic, and complex. However the analysis that the the British People ‘spoke with one voice’ is absurd. Patently they spoke with many, many voices. How then can the journalistic trick of reducing the signal of the election down to the pithy simplification that “the British people voted for a hung parliament”? It confuses intention with consequence in the most pathetic manner. To restate it correctly, the British people voted, the result of which is a hung parliament.

So What?

The electoral system manipulates the votes of the electorate to produce a result. If you infer from the consequences of the vote to what the voters intended then you are bound to ascribe to the voters their support for the electoral system. That is to say, because the result was a hung parliament, voters voted for a hung parliament. Or, going further: Voters got exactly what they wanted. Concluding by going further still on this grotesque line of reasoning, therefore Voters rejected electoral reform; they got what they voted for.

Wait, what? If you remove the result from that, doesn’t that come down to arguing that because voters voted under a particular electoral system, they voted for that electoral system? That line of reasoning is preposterous. Surely no serious journalist would use such a slight-of-logic?

Kay Burley, Sky News

This interview exchange between Kay Burley of Sky News and David Babbs of 38 Degrees may sound familiar…

• Burley The public have voted for a hung parliament, we got exactly what we voted for
• Babbs What the people voted for [we believe] is they voted with hope for something better…
• Burley 65% of the public who were eligible to vote, voted for a hung parliament
• Babbs Yes, and people who… [gets cut off, the first of many]

Kay Burley’s analysis of the election is exactly the sort of lazy analysis I suggested above. She does herself no favours by the interview disintegrating into a series of ugly attacks on Babbs that is frankly painful to watch, but the crux of the interview is that she has swallowed the ‘voted for a hung parliament’ narrative lock-stock and barrel, and refuses to discuss electoral reform outside that narrative.

Relevance

The real argument about electoral reform is about enfranchisement. This goes back to my earlier numbers; it could be argued that if the Conservative party were to govern as a minority, then only 16.25% of people in the country are responsible for that government, or 23.5% if you include the people who voted for Conservatives who weren’t elected, or 36.1% if you further disregard the people who did not vote. Whichever percentage you choose, they achieved 47.1% of the seats at the election. There is the source of the inequality that Babbs is arguing against.

The discussion about electoral reform should be: can people live with an electoral system that has that statistical reality? The only relevance that the current election should have in this discussion is that the result brings about the sorts of power brokerage necessary for a discussion on electoral reform to happen at high levels, because the Liberal Democrats could act as king-makers.

This is not particularly relevant to this blog, but I have no other forum on which to express these views. Ordinarily I try not to get involved in such discussions. However, there is much to discuss here from a philosophical and political perspective (as well as a statistical one). If the general election is not being discussed in classrooms, then it should be. I can’t believe that it is not.

If you are interested in doing your own statistical analysis of the election (or previous elections), then the Guardian offers data for you to do so:

http://www.guardian.co.uk/news/datablog/2010/may/07/uk-election-results-data-candidates-seats

Report on Finland’s Education System

I think there are a few things that they miss out from the report, like a very homogenous population, a smaller wealth gap between rich and poor, and a very low density of population. Still, it does paint the picture of an idyllic scenario of an educational system free from interference from politics. I wish a party in the UK would put that in their manifesto!

http://vimeo.com/1908224?hd=1

As the comments underneath the video allude to, there is some utility here to attempt to consider the size of magnification that occurs during a portion of the video, and how to express magnifications of such a magnitude.

Flags By Colours

It would be a fun and worthwhile activity to use this site to exercise KS3 students’ abilities to mentally construct and destruct pie charts in a game of ‘guess the flag’

Cell Size And Scale

From a scientific perspective it is interesting to reflect upon the relative sizes of elements, from a Coffee Bean to a Carbon Atom (through various things such as a human Ovum, Sperm, various viruses, compounds and so on).

From a mathematical perspective it is also interesting for the way in which the relative scales are measured in the top-left. Exploring the different notations for small sizes would be a useful exercise in place-value for all levels of Key Stage 3 and 4.

This article is a little heavy on the politics for an average maths classroom, but is perhaps useful for A-level students, and is definitely useful for any teachers teaching the IB, as it has excellent cross-over with theory of knowledge. Worth a look