Somewhere between trigonometry and quadratic equations, the typical teenager comes up with a question that makes a teacher's heart sink. "Sir, why do I need to know this?" Maybe he or she has a point. The sad truth is that there are large chunks of GCSE and certainly of A-level maths which, when the exams are over, most pupils will never need again. Maths may provide an excellent grounding in logical thinking, it may even be an art form, but for a teenager distracted by topics with more instant appeal, its relevance is all too often lost.
In the past 20 years, disaffected teenagers have become maths-phobic adults, and those adults have become the parents of the generation we are trying to enthuse about the subject. No wonder there is a constant fight to prevent the decline in maths in preference to other more accessible subjects.
One solution is to meet the "so what?" question head on. Instead of tackling maths the conventional way (here's pi, isn't it fun), why not ask questions that are of everyday interest and use those as the hook for exploring the maths?
Take the National Lottery. It may have had some questionable effects on the country's gambling habits, but it has provided a perfect tool for introducing probability theory and permutations. Its organisers rightly say "it could be you". What they don't say is that the person struck by lightning could also be you. Fourteen million to one against are extremely long odds for a lottery win, and it's entertaining to look at what other events are as unlikely as this (the chance of being killed by lightning in a year is, in fact, very close to 14 million to 1).
Why should you buy your lottery ticket after Thursday? Answer: because if you buy it on Thursday, your chance of being knocked over by a car before the Saturday draw is higher than the chance of winning the jackpot.
You could devote a whole lesson to the subject "how to avoid being ripped off". The fact that the lottery takes away 50 per cent of your stake (for every pound;1 paid in, 50p goes in prize money - the rest goes in tax, good causes and administration) makes it a worse "investment" than most bets on the market, but it is important for any potential gambler to realise that in any bet, the only guaranteed winners are the bookmakers.
The aim of the gambler should really be to minimise losses. Bookmakers' odds always add up to more than 1.0 (so they can make a profit). Pupils may have fun adding up the odds for all the teams in the World Cup, say, to see what the total is. It will probably be about 1.2, allowing a 20 per cent profit for the bookmakers, the "rip-off" factor. The challenge is to find an incompetent bookmaker whose odds add to less than one.
Simple questions can often be the trigger for discovering the beauty of maths. For example, why are there so few four-leafed clovers? And why do so many flowers have five rather than four or six? An investigation in the garden shows the surprising frequency of the numbers 3, 5, 8 and even 13 and 21 in plants. These numbers belong to the Fibonacci series, in which each term is the sum of the previous two.
There are elegant geometrical reasons why Fibonacci numbers are likely to appear in the natural world, and these can be the basis of fascinating investigations. You can even bring art into this topic. Leonardo da Vinci was one of many Renaissance men fascinated by the "golden rectangle", whose long and short sides have the ratio (C5 + 1)2. Not only is this ratio claimed to be aesthetically pleasing, it also has profound links with Fibonacci. If you take any two successive terms in the Fibonacci series, their ratio is very close to the golden ratio. This is part of what Douglas Adams would call the fundamental interconnectedness of all things. And hey, it's cool!
Murphy's Law ("if it can go wrong it will") also makes for some intriguing mathematical study. For example, why does the place you are hunting for in a road atlas so often seem to be on the edge of the page or in the crack down the middle? The answer is that probability theory says it will. A page in an A to Z typically has dimensions of 12cm by 20cm, making an area of 240 square cm. The street you are hunting is equally likely to appear at any point on this page. The 2cm strip around the edge which poses all the problems when map-reading may look small, but it represents an area of 112 square cm, or 47 per cent of the page. It means that on average almost half the times you look up a street in an A to Z it will be within 2cm of the page edge.
Coincidences are another area of everyday fascination. Maths can be used to explain how coincidences aren't as surprising as they may seem. The surprising "birthday coincidence" game is one way to illustrate it.
Ask a class of 30 pupils to write down secretly the birthday of somebody they know, ignoring the year (for example, July 17). Then ask them to say what odds they would put on two of the secret birthdays being the same.
For those not familiar with the answer, odds of 10 to 1 against seem reasonable. I've even heard someone say 1000 to 1 against! Ask them to call out the birthdays. Three times in four that you try this experiment with a class of 30, there will be at least two birthdays the same.
To explain why this is so, you need to show what the chance is of every birthday being different. The chance of two people having different birthdays is 364365. The chance of three being different is 364365 x 363365. The sum expands as the number of people increases, and by the time you have 23 in the class, the chance is just 0.49 of every birthday being different (which means that the chance of at least two birthdays being the same is 10.49, or 0.51). If you want to play safe, you can ask the class to write down the birthdays of two people each. The chance of having no coincidence is now minuscule.
Incidentally, there is a fun variant on the birthday game. If you have a class of 20, get each pupil to write down a number between 1 and 100. The challenge is for them to pick a number that nobody else in the class will pick. You can then demonstrate your "psychic" powers by predicting that at least two in the class will pick the same number. Even if pupils pick their numbers completely at random, the odds of a coincidence are massively in your favour, and rather perversely - the harder they try to second guess their pals, the more the odds will favour you.
There are plenty of topics that will interest maths-phobic adults and teenagers, including sports rankings, dating and code-breaking. There is mathematics in Baywatch's life-saving, kicking a rugby ball, sex surveys and conjuring. These subjects provide friendly entry points to geometry, statistics, inverse functions and much more.
Tackling maths in this way has taught me a few things. One is that if you present maths in the form of everyday questions, there are many aspects of it that can appeal to just about anybody. Another is that there really are some areas of maths that are more important to understand than others. After basic arithmetic and the presentation of statistics, I would put an understanding of probability theory at the top of the subjects vital to thriving in the adult world.
I don't want to trivialise the subject. There is a danger of having too much fun at the expense of serious learning. However, to teach disaffected teenagers you first have to grab their attention. If it takes questions like "how can I win the lottery?" or even "why do buses come in threes?" to do it, then I'm all in favour.
These questions and more are covered in 'Why do buses come in threes? The Hidden Mathematics of Everyday Life' by Rob Eastaway and Jeremy Wyndham, Robson Books, pound;12.95