Don’t get me wrong: I love tutoring. As long as a student is putting in a genuine effort, I’m happy to forgive the odd “I don’t know” or “We haven’t been taught that.” And my students do put in the effort. I thank them for it.

But there are some times… some times I have to clench my teeth a little bit. It’s not that they’re doing it wrong, exactly, they’re just going about it the long way.

That’s perfectly ok, by the way. Whatever gets you to the answer is fine by me, and fine by the examiners. It’s just, they’re making extra work for themselves. Here are three of the things that are especially bad for my enamel.

1. Multiplying and dividing by five

Bus stop method, fine. Grid method, fine.

Or alternatively, to divide a number by 5, you can double it and divide by 10, both of which are easy, and you can do them in whichever order is easier – you’re multiplying by $\frac{2}{10}$, which is the same as dividing by 5.

To multiply by 5, you can multiply by 10 and then halve the answer, or vice versa – that’s effectively multiplying by $\frac{10}{2}$, which is 5. Again, both of those are quicker (and safer, for many students) than multiplying out.

If you’ve got the methods down quickly and can do them reliably, great! Keep on keeping on, and if you don’t, it’s good to know them. But if you struggle with them, an obvious time-saving shortcut might be a good trick to have up your sleeve.

2. Taking away from 90º

The number of times I’ve seen students look at a right-angled triangle with one angle of, say, 53º, and say “They add up to 180, so $90 + 53 = 143$, and $180-143=37$” … ooh, my dentist isn’t going to be pleased.

If you have a right angle, the other two angles have to add up to 90. $90-53=37$ too, you know. That’s one, two-digit subtraction rather than an add (almost always with a carry) and a three-digit subtraction.

Yes, a good GCSE candidate shouldn’t have a problem with that level of arithmetic. But why bother doing harder sums than you need to?

3. Working with fractions

Ah, probability. Multiplying a quarter by two thirds? I’ll bet you half a bar of chocolate a student’s first thought is “0.25 times 0.6”, and the other half that they get their wrong calculation wrong. ((I hope to lose, my teeth are taking enough of a bashing as it is.)) It’s bad enough getting the right calculation wrong, but two-thirds isn’t 0.6, it’s $0.\dot 6$, and good luck multiplying that by a quarter.

Multiplying fractions is the easiest thing you can do with them. $\frac{1}{4} \times \frac{2}{3} = \frac{2}{12}$, or a sixth – you simply multiply the tops and multiply the bottoms. It’s genuinely perplexing that students think decimals are an easier route here. Unless you’re adding things up, decimals are usually harder to deal with than fractions.

What’s your experience? If you’re a teacher, what student habits make you grind your teeth? If you’re a student, what do you wish your teachers would stop doing?