You are going to make mistakes in you’re A-level maths exam. If your teacher sat the exam, if Stephen Hawking sat the exam, they would make mistakes in their working. However, before you turn in your exam script, you must check your answers. Carefully.
Following are some ways you can make sure your answers are right – or at least in the right ball-park:
Put your answer back in. The first thing you should do if you’re not sure about your answer is check: does it answer the question? For example:
If you’ve solved an equation, put your answer back in as x or y or whatever and make sure the equation works.
If you’ve worked through a proof, take a random line in the middle and check that the left-hand side and right-hand side give you the same value if you substitute in random variables.
If you’ve done something geometric, see whether the shape you’ve ended up with ‘works’ another way; if you’ve solved a triangle with the cosine rule, for example, make sure the sine rule also holds for it.
Check your answer is reasonable. When you get an answer, especially a measurement, it often pays to ask, ‘Is this a reasonable answer?’
Suppose you’re calculating something such as the distance from Earth to Mars. To check whether your answer is reasonable, you might ask yourself the following questions:
What’s the smallest my answer could be? With Mars, you probably know that satellites are a few hundred kilometres above the Earth, so that’s a very low lower bound on your answer.
What’s the biggest my answer could be? You might know, for example, that the sun is 150 million kilometres away from Earth – so something more than a few hundred times that, say, would be far too big.
What’s a good rough estimate? If you didn’t know anything about the mathematical process you had to go through, what would your best guess be?
Go through the ‘Idiot List’. Get hold of a little notebook (available in all good stationers, bookstores and supermarkets and online) dedicated to recording the mistakes you keep making. Every time you come upon a mistake you’ve made more than once, put it on your list. Whenever you’re stuck or have the wrong answer, go through the list and see whether anything jumps out at you!
Oh, and one more thing: don’t call it the ‘Idiot List’. That’s really negative. Go for something like ‘Mistakes I’m correcting’. Be nice to yourself, eh?
Look at every line. Time on your hands? Well, you could count the tiles on the ceiling, or you could take a line-by-line approach to checking your work. Take a ruler and put it under the first line you’ve written. Check that everything you’ve written down follows clearly from the line above it. When you’re happy, move on to the next line. And the next. If you’re very careful about this and you have an error, it’ll show up – even if the process is really tedious.
Sniff your work. Related to the line-by-line method – but much quicker and less intensive – is to read swiftly over your work and see what ‘smells funny’. You’ve been doing fairly serious maths for several years now, and you presumably have a decent sense, when you’re looking at your work, for what is definitely right and what’s . . . well, a bit fishy.
Are there places your handwriting looks a bit crammed up? Are there big gaps in the reasoning? Is there anything that isn’t immediately obvious? Is something negative where it ought to be positive? Where, if your teacher or a classmate were to read through your work, would he or she say, ‘Hang on a minute . . .’? Those are the things you want to pay attention to. Even if those elements are right, making them clearer will make the air a bit less smelly.
Work with wolf fences. Back in the early days of computer programming, a wolf fence was a form of debugging your code by putting in commands every so often to say ‘Wolf fence at line 20’ (and so on) so that you could eventually see where your code was crashing. The idea was that you could catch your ‘wolf’ by building fences gradually closer and closer together. (There are better debugging tools and languages now, thankfully – at least for computers.)
For maths done on paper, though, the wolf fence is really useful if you’re trying to catch an error – especially in a proof. The usual problem is that you get as far as you can go and find that the two sides don’t match up. Nightmare! Now you have to go back and find your error. You can do it like this (assuming you’ve just worked on the left-hand side of the proof):
Pick a line midway through your proof and work out the value of your left-hand-side expression for a randomly chosen value of your variable.
Do the same thing for your original right-hand-side expression with the same randomly chosen value.
If the expressions match up, the mistake is almost certainly after the line you picked; if not, the mistake is almost certainly before it.
Repeat the process in the half of the proof you know the mistake is in, and keep going until you narrow down your options to two lines with different values. Your mistake is in going from one line to the next.
Ask an imaginary friend for help. When you’re stuck, explain your problem in great detail to an imaginary friend – out loud, if you can and want to, or in your head if you prefer. You’ll be surprised how often the solution comes to you.
Read the question again. Sometimes, you just get stuck. You’ve followed your thread as far as it will go, and you can’t see where to go next. So you could throw down your pen in disgust . . . or you can scour the question for hints:
See if there’s any information the examiners gave you that you haven’t used yet. Does it help you?
Check that the information you have used is correct – and that you’ve copied any formulas you need correctly. (This is another argument for knowing your formulas: if you know them by heart, then it’s much easier to notice when you’ve written them down wrong.)
See if there’s an earlier part of the problem that you might be able to use. Many exam questions have a ‘follow on’ structure, in which part (c) uses something you worked out in part (a) or (b).
Find another way to do it. There’s often more than one way to do any question. If you’re stuck, lost or going wrong, maybe it’s worth looking for one of them. If you’ve done something on the calculator, try doing it without. If you’ve used one equation for a straight line, try using another. If you’ve tried the quotient rule, try the problem again with chain and product rules.
Do the whole thing again. This is a real ‘in the case of emergency, break glass’ tip: if something’s gone wrong and you can’t find where, cover up your work and do it from scratch again. Yes, it’s a pain in the unmentionables. Yes, there’s every chance you’ll make the same mistake again. Yes, the whole thing might turn out to be a colossal waste of time. But, on the other hand, you might get the right answer and save yourself a few points. It’s swings and roundabouts.