Methods of Last Resort 1 - Percentages

Following on from my session in Kettering at #mathsconf8 I will be writing a series of blogs about the areas of maths I find or figure out that might be better looked at separate to any problems that might be solved using a standard approach or a 'method of last resort'. The first area I want to look at is percentages.

Because of the multiplicative nature of percentages there are lots of questions that can be solved without having to resort to approaches such as "Find 10% first..." or "What multiplier calculates...." or other standard approaches. The point I made at mathsconf is that I would want pupils to understand why these questions can be solved quickly and straightforwardly, and that actually by exploring the special nature of some of these calculations we can deepen pupils understanding of the topics - in this case percentages.

Find 32% of 75

This is the example I used at mathsconf. There are still plenty of teachers that don't realise that 32% of 75 is the same as 75% of 32, but once they see it they understand why. What I like is that in explaining why this is true really does get at the heart of percentages and how they are calculated and so it is a perfect little 'explain why' to stretch pupils as well as then serving as reinforcement of concepts for others.

Find 32% of 100

Try it; you will be surprised how many pupils don's immediately link the % with the 100 or are unsure when they want to say 'isn't that just 32?' Again this sort of question gets at the heart of percentages as parts of 100.

Find 32% of 50

If you have built up to it these are actually now becoming quite straightforward, but encouraging pupils to talk and explain why is still powerful.

Find 32% of 200, 300, 400 etc

I probably don't need to say much more at this point.

As well as calculating percentages, equally there are similar questions for writing one value as a percentage of another. Again there are standard approaches for this (writing and converting fractions or similar) but there are questions that anyone with a real understanding of percentages would look at and solve. This set of questions comes from a well known worksheet provider; see if you can spot the ones that could be done without requiring the use of a standard approach or 'method of last resort'.

Even if you don't really know your fractions, questions 3, 5, 6, 7, 11 and possibly 12 and/or 17 can be solved using some relatively straightforward multiplication and division. Do we always teach pupils though that if they can see an obvious way to write it as 'a percentage of 100' that this will be much quicker than a standard approach, and more importantly to support them in understanding why this works which would lead to a deeper understanding of percentages as a whole.