When I am approached by a new student who is working towards their GCSE in Maths, the first thing I ask them to do is to try a “skills check” sheet of questions. The purpose of this exercise is for me to establish how good a grasp they have of the basic skills that they should have covered before the end of Key Stage 3, so that I know what level we are starting from and can plan accordingly.
I don’t want them to see it as a test and feel under pressure. Calculators aren’t allowed, but I don’t mind if they need to look a few things up to remind themselves of how to do them; I’m more interested in their understanding of the underlying concepts, and the way they apply this understanding. Maths should make sense; not only is it a lot more enjoyable if you understand it, but if the basics don’t make sense to the learner then what hope do they have of getting to grips with the harder material needed for the exam?
Skills that I ask them to demonstrate are:
Some may question when they would ever use these skills in the real world, and sometimes that’s hard to answer. However, every one of these skills is still examined at GCSE, and some of the techniques – such as arithmetic with fractions – will be needed if the student goes on to study Maths at a higher level. It always surprises me how many students starting A-level Maths can’t remember how to divide by a fraction!
At the student’s initial consultation we’ll run through the exercise – which ideally they’ll already have worked through beforehand – and identify the gaps in their knowledge, addressing some of them there and then. The rest – as well as revisiting those newly addressed – can be worked into the first few lessons.
We’ll also discuss any areas of the GCSE course that they feel need special attention; perhaps they have struggled with the material in a particular topic or maybe they have missed out on teaching due to illness or a change of teaching set. We will of course aim to cover all the topics in the relevant exam specification (provided that there is enough time left before the exams), but this initial discussion allows me to plan for that particular student’s immediate needs.
Think of a number, multiply it by 2 and take away 6.
Now halve your answer and add 3. You should end up back at the number you first thought of.
Can you use algebra to explain why this happens?
Hint: start by using a letter for the number that you first thought of.
Try it before you read on...
Okay, so let's call the number that you started with "n".
We're going to start by "dressing it up" to make an algebraic expression.
"Think of a number, multiply it by 2 and take away 6."
If you multiply n by 2 then you get 2n.
Now take away 6 and you have 2n - 6.
Now we're going to "undress" it again to get the n on its own... but by a slightly different route!
"Now halve your answer and add 3."
How can we halve it? Well, 2n - 6 is the same as 2 lots of n and one lot of -6...
... and -6 is the same as 2 lots of -3
... so if you halve 2n - 6 then you get half of 2n and half of -6, giving n - 3.
The last step is easy: we now have n - 3, i.e. 3 less than n. If we add 3 to that then the -3 and the +3 cancel out
and we're left with just n, the number we started with!
How about this one?
Think of a number, add 4 and treble it (i.e. multiply it by 3). Take away 12. Tell me what you got and I'll tell you what number you started with. How do I do it?
Think it through and see if you can work it out yourself first.
Of course, I could do the whole calculation backwards (add 12, divide the answer by 3 and then take away 4) but the numbers are carefully chosen to make it much easier than that. In fact, all I have to do is to divide by 3. It's a bit like going through a labyrinth of side streets and eventually finding yourself just a few steps away from where you started!
Here's why: If you start with n and add 4 then you have n + 4. Treble it then you get 3 lots of n + 4, i.e. 3(n + 4).
This is the same as 3 lots of n and 3 lots of 4, which we can write as 3n + 12 (this is called multiplying out the bracket).
When you take away the 12 you're left with just 3n...
... so to get back to n I just have to divide by 3.
What we're doing here - in a mathematical sense - is building up equations and then solving them.
In the first example, you found the value of 2n - 6 and then "undressed the n" to get back to its original value.
In the second one you found the value of 3(n + 4), multiplied out the bracket and again "undressed the n" to find its value.
Here's a slightly harder one for you to mull over:
Think of a number. Double it, add 2 and multiply your answer by 3. Take away 6 from your answer. Now take away the number you first thought of.
If you were to tell me the number you'd ended up with, how could I work out the number you started with?
Can you make up some more of your own?
Actually, "first-time blogger" isn't really accurate; in fact, quite a while before blogging was a "thing", I published what today would be described as a blog. When I lived in Japan, 1999-2001, I kept an online diary of my time there. It was all done in basic HTML - if such things as blogging platforms existed back then then I wasn't aware of them - and can still be found at www.tanuki.org.uk, although not all of the photos are still there.
Anyway, life has moved on since the last update on that site (published in 2009). Still happily married to Craig but we now have Xander (7) as well as Freya. Still living in Hall Green but now in a different house. Still teaching Maths but now striking out on my own rather than being employed by a school or college. It will be nice to be able to focus on the teaching and not worry about crowd control! I'm carrying on a family tradition really; my mum did maths tuition at home for many years. I remember, as a small child, frequently being urged to be quiet because Mummy was "chootering"!
I'm not expecting to be a frequent blogger but I'll post occasional updates, mostly Maths-related.