# Posts from the ‘KS5 (VI Form)’ Category

## Task

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.

In the BBC Website Magazine today is an article about proportions and magnitudes. It made me reflect that we often spend time teaching students how to express numbers in different forms, but rarely attempt to give students an understanding of how the numerical forms differ, and what they represent.

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

Even BBC Radio 4 journalists are unable to recognise the distinction between the following sentences:

- I do
*not*want Megrahi to die in prison; - I want Megrahi
*not*to die in prison.

There ought to be a clear distinction between the intention of the speaker in the two cases: the first does not necessarily convey any intention, while the second takes a clear intentional stance.

On BBC Radio 4 this morning, a five minute interview went frustratingly round in circles because neither the Foreign Secretary nor the interviewer could satisfactorily explain this distinction.

We often use the first form of the sentence when we mean the second, and this linguistic ambiguity was siezed upon in a piece of journalistic opportunisim. Bill Rammel was asked a question about whether the UK government ‘wanted Megrahi to die in prison’. He responded that they did not. The question asked about whether an intention existed; he replied that it did not. He was not asked, nor responded to whether there was the converse intention; he was not asked “Does the UK government want Megrahi to be released from prison before he dies?”, but it is now widely reported that he confirmed exactly that.

Increasingly, it seems that journalists exploit these linguistic ambiguities in order to create a story. No wonder politicians (of every persuasion – I am ambivalent with respect to the different parties) are so careful when asked ‘clear yes and no questions’ and sometimes simply repeat a well-rehearsed phrase. When they are misrepresented so wholly as in this case, can you really blame them?

When they occur, these stories are good opportunities to highlight the ambiguity of language and the care with which language needs to be used to sixth-form philosophy students. It is perhaps the most important practical application of learning philosophy that its students can be forewarned against the pitfalls of such exploitative misrepresentation.

This is a delightful way to revise Venn Diagrams with older students: A Venn Diagram of mythical creatures.