Efficiency of a calculation method

The proposed new curriculum for primary mathematics stresses the importance of children learning what they refer to as the 'efficient' method of doing a particular kind of calculation. That's an intriguing concept. The 'efficiency' of a device is defined as the work achieved divided by the effort put in to achieve it.

On this basis, doing a division like 784 ÷ 18 on a calculator must surely be the most 'efficient' method: very little effort put in and the best chance of the output being the correct answer = high efficiency. For some reason, they seem convinced, though, that long division must be the most efficient method of doing a calculation like this: which involves a great deal of effort put in (so lower efficiency), and a significant chance of getting the wrong answer (which would be zero efficiency!)

OK, so they don't want children to do calculations like this on a calculator; I have no idea why, but they don't. But why would long division be more efficient than, for example, this approach:

10 × 18 is 180, so 20 × 18 is 360, so 40 × 18 is 720 (so far only multiplying by 10 and doubling, which is easy; not much effort expended!)

I now need to get from 720 to 784.

Adding 18 at a time, I get 720, 738, 756, 784.  I needed a further three 18s. The answer is 43.

I recognise real mathematical and creative thinking when 10–11-year-olds share different ways of doing multiplications and divisions, How sad if the new curriculum suppresses this and sends out the incorrect message that there is only one proper way of doing a multiplication or a division.

So, here's a challenge for any reader. Can you come up with 12 ways of calculating 75 × 12?

I'll give you the two most efficient to get you started:

1) You just happen to know your 75 times table because you are an avid watcher of Countdown, so you just write down the answer, 900.

2) Use a calculator.

OK, now find 10 more ways of doing it ...