## Saturday, 16 January 2016

### Introducing surds - cutting out squares...

The 'recent' changes to the KS3 curriculum suggest that we should be given our pupils a basic understanding of surds and surd calculation prior to GCSE. Some people may be wondering about how to introduce the idea of surds to KS3 pupils and so I thought I would share one of the things I do with pupils, and that is to look at drawing and possibly cutting out squares.

Typically this will start with some nice big squares. I actually quite like using inch-long squares; not because I have a real hankering to return to pre-decimalisation (I am too young to remember anything other than pounds and pence anyway!) but because I find centimetre squares too small and fiddly for this sort of work, and anyway I think there is something nice about reinforcing the concept of area by looking beyond the normal cm or metre squares. You can't buy inch square paper these days (or if you can I don't know where you can) but you can download square grids from the internet and stretch the image so that each square is 2.5 cm long. I need plenty of this paper as kids will need lots of attempts to try and fail; kids get to fail a lot here, so if you are looking to examine mindset as well this is a great activity to try.

I will start kids off by getting the to draw a square with 25 inch-squares inside, which most will do quite quickly. Next I will tell them to draw a square with 16 inch-squares inside of their 25 inch square so that they have this smaller square inside the larger square. The next part will be a discussion about what we might be able to say about squares with areas between 16 and 25 inch-squares, with the aim that pupils will realise that any square with an area between these two will have to have sides between 4 and 5 and therefore will be (a) drawable between the two squares we have already drawn and (b) have area made up part squares. Then comes the challenge (which at first to some pupils doesn't appear as much of a challenge): draw me the square that has an area equivalent to precisely 20 inch squares. Depending on the group this will proceed in one of two ways; either they will draw an attempt between the two squares they previously drew, bring it to me to measure (I keep a ruler that measures inches for this and other purposes), and then become frustrated when I show them that their area cannot be quite 20 inch-squares or alternatively (which I prefer) I will get them to draw a 4 x 5 rectangle and then cut their square up and see whether it can completely cover the rectangle (prompting to leave the 4x4 square intact and just cut the excess from around it and try and make it fit if necessary). This second approach is definitely nicer provided the pupils have the resilience to keep re-drawing the squares every time they make a mistake: of course sometimes because of small gaps between their pieces they will think they have completely covered the rectangle and you will have to show them (either by measuring the hole they cut or by talking about very small gaps etc) that they haven't got to exactly 20.

The purpose of the activity of course is to plant the idea in pupils head that the task may be impossible to perform in reality. I will often talk about accuracy of measuring instruments here as well, and get pupils to imagine rulers that could measure down to a millionth of an inch or more. What this allows me to do is introduce the idea of an irrational number in a way that speaks to pupils experience; they have seen first hand that the square root of 20 cannot be found as a decimal or fraction of the length of a real square. I can then talk about the fact that mathematically the number needs representing exactly rather than as a rounded value, and so the surd form is required, which leads to the need to be able to calculate with numbers in this form and so on. For me this approach is much more powerful than a simple calculator investigation and not just because it is more engaging for kids than just mindlessly punching numbers into a machine, but also because it really highlights the reality of irrational numbers: these are numbers that cannot be measured and cannot be represented in ways that have been used before.

So if you are looking for a nice concrete way to introduce the idea of surds that gives (for me) real insight into the fundamental nature of these type of numbers, then try getting your kids just drawing and cutting out squares.