Archive for March, 2010

Birthday

March 31, 2010

As of tomorrow, my age will no longer be a perfect number. A perfect number is the sum of its proper divisors (the numbers that go into it, not including itself). 28 is divisible by 1,2,4,7, and 14, and 1+2+4+7+14=28. The last time my age was a perfect number, I was too young to appreciate it really (1+2+3=6) and I doubt I’ll live to see my next perfect birthday, which would be 496.

It’s been a good few years. Last year, I was a cube (27=3*3*3) something which won’t happen again until I’m nearly retired (64=4*4*4). The year before that, I was one more than a square and one less than a cube. The year before that I was square (no nasty comments about me always being square please!) 24 was exciting because it had lots of factors, and of course 23 was the last time I was prime. Tomorrow I’ll be in my prime again!

Next year’s birthday I’m struggling to see any reason to look forward to, but the year after that will be very exciting as my age in binary will be 11111, and I will be a “teenager” for the last time in hex, being 1F years old. When we were young, we sometimes had binary candles on our birthday cakes, with candles lit or unlit to represent 1s and 0s. In two years, it’ll be the last birthday I can do with just 5 candles.

Finally, some maths – when I was a teacher, more often than not each class would have a couple of pupils in it who shared a birthday. Should we be surprised by this? If you’ve never come across it before, the Birthday Problem is an interesting bit of probability theory, with a nice graph to show the probability that there will be at least two people who share a birthday for increasing sizes of group. How many people would you need to be 100% sure that there would be at least two with the same birthday?

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Homework sucks

March 16, 2010

I usually listen to Simon Mayo’s Radio 2 Drivetime show on my way home from work. One of the features is Homework Sucks, where a listener explains a piece of homework they’re having problems with, and then other listeners contact the show to offer help.

Since the feature started in the new year, I’ve only heard a couple of maths-related questions. The more interesting of these was the following question: how many ways can you arrange three eggs into a six-egg box? The respondant is a regular poster on the TES maths forum, and you can read the thread here.

Some teachers have objected to a radio feature called Homework Sucks, but it’s a sentiment I find myself agreeing with. As a learner, homework didn’t have a great impact on me – at primary school we were expected to read every night (which for me was a joy rather than a chore), learn a few spellings, and learn our tables. Secondary school brought with it the structure of a homework timetable, and tasks which were intended to take an hour or so each evening to complete, although more often than not I finished them off on the bus or in the library at break or lunchtime. It was only really at sixth form that the workload associated with four A levels dictated that I had to spend lots of time on independent study.

As a teacher, I had to fit in with the homework policy of the schools where I taught, which sometimes meant that I had to set a homework task even if there was nothing appropriate which fitted in with what I had been teaching. I would much rather have had the freedom to set homework when I felt it would support the learning that had been going on, and the freedom to give my pupils a night off when there was nothing I wanted to set. I know that a homework timetable avoids the congestion of nineteen subjects all wanting you to spend an hour on a task on the same night, but it should not constrain teachers into feeling they have to set busy work in order to be able to tick a box on their markbook.

I could rant all day about the pro-homework arguments about instilling discipline and work ethic, preparing children for the long days and overtime they will put in as adults in corporate careers, giving children something purposeful to do with their evenings… I could rant about the unfairness of expecting a child living in a cramped untidy flat with several younger siblings to produce the same piece of work as an only child living in a mansion with a spacious study with an anglepoise lamp… but instead, I will just say that in a perfect world, I would want to encourage learners to discuss their maths with their family and friends between one lesson and another, and maybe do the odd practice-paper in the run up to their GCSEs and A levels, but otherwise, I don’t really mind what they get up to in the evenings.

Another foray into Facebook Maths

March 4, 2010

After the success of my last problem-posed-on-facebook, I decided it might be fun to post some more maths for my friends, so on Tuesday afternoon I updated my status:

Alison would love to share some maths with you this afternoon – do you want number patterns or shape?

Well the bad news is that the first couple of responses were very negative – despite the friendliness of my language, and the offer to share some maths rather than impose it on people, led them to post things such as “eeek”, and a sentiment that they wished to run away and hide, with the follow-up comment “Maths is seriously Not Fun”.  This is from people I respect greatly and who I know to be intelligent, articulate, witty individuals, and I dread to think what their experience of maths at school must have been for them to feel such distaste for my favourite subject.

Luckily for my self-esteem, another old friend came along and requested a shape problem, so I posted the following:

You’ll need a ruler, something to write with, and something to write on. Got them? Great, let’s get started!

Draw a triangle. Any triangle you like, as long as it has three straight sides I don’t really mind. Now find the middle of each side, and mark it with a dot. If you wanted to be really clever, you could find the midpoint using a straight-edge and compasses the way the Greeks did, but if you’d rather just measure with the ruler (or even fold your triangle) I won’t mind.

Once you’ve found all three mid-points, join them together: this should split your original triangle into four smaller triangles. Do you notice anything interesting about the smaller triangles?

What happens if you try joining the midpoints of each side of a quadrilateral (four-sided shape)?

And sure enough, a few people rose to the challenge, and started sharing their ideas. One commented several times, first explaining that he’d found it split the original triangle into four that weren’t necessarily the same size or shape, and then coming back to say he always ended up with four congruent triangles, all similar to the original.

Another commenter used vectors to prove the conjecture about joining the midpoints giving congruent triangles with sides half the length of the original, and went on to think about quadrilaterals. His first comment suggested that repeating the process might get you to a smaller version of the original quadrilateral, but he sent me a delighted message later on to say he’d experimented and found he always ended up with parallelograms, and that he’d proved it using his result for triangles.

Someone else saw the triangle problem in a very visual way – he could convince himself very easily what would happen with an equilateral triangle, and then just imagined all other triangles as being transformed, or viewed from a different angle. Some of my facebook friends are easily as imaginitive and happy to explore as the classes I’ve worked with in schools.

There have been plenty of comments along the lines of “Keep them coming” to the maths I’ve posted, which heartens me, but I know that my friends on facebook are not representative of the population as a whole – many are people I studied maths with, or who share an interest in technology and geekery.  I’m glad that each time I do this, some people I wouldn’t necessarily expect to respond are willing to dive in and have a go, and share their insights which are every bit as valuable as the fully and carefully constructed mathematical proofs that others send, but the real challenge will be to engage some of the doubters and maths-phobes, and convince them that actually, Maths is Seriously Fun!