INTRODUCTION TO MATHS A LEVEL 3
I am going to show you a quick example of what can come up in an exam question so that
you can see.
Generally in A level Maths, there’ll be about 10 questions in your C1 paper.
Now that doesn’t mean just one quick question, two quick questions; each question goes into
several different parts.
So question one might have a part a, part b, part c and a part d and generally the A
level syllabus is quite nice because it leads you through questions quite nicely with these
individual sections.
So for example, you may potentially get a question like this as
part a, where you get 6x plus 8 over 4.
And the main reason why this is in A level Mathematics is because it really does identify
individuals who are good mathematicians and going to survive the A level topic and those
of you who may not understand the true background behind what you are learning at GCSE.
This is where a lot you who’ve learnt things at GCSE are going to find it very challenging.
If I asked you to simply this expression, i.e. write it in a nicer way.
I can imagine that a lot of students who do not quite appreciate Mathematics that they
see, wanting to cancel out this 4 down here with the 8 up there and writing a 2.
Now the reason this does not work is because you’re telling me that that 4 and that 8
can be isolated and cancelled out with no consideration for the 6x up here.
However, that would give you 6x plus 2 which is not the equivalent of what you see on the
screen at the moment.
In order to cancel out any algebraic part or numerical part of an expression, you have
to ensure that there are common factors that are being deleted.
So the top of this equation, which is a linear expression.
It’s linear because there are no powers higher than x to the power of 1 – can be
written by factorising out the number 2.
So I can pull a 2 out of the 6x and I can pull out a 2 from the 8 over there.
So you’ll get 2 and then in brackets what you have left over is 3x plus 4 and underneath
the 4 still remains.
Now here I’ve got a far safer example to look at.
Here I could have made a silly mistake deleting the 4 and the 8 but because we spotted that
there was a common factor within the 2 at the top, we can re-write it like this and
now you are allowed to cancel that 2 and that 4.
Not this 4 and this 4, because once again this is still separated.
Here that 2 impacts everything that’s in the bracket.
It’s 2 x all of this.
So that 2 can now be cancelled with the 4 to simplify the expression.
So we can say bye bye to this and divide that 4 by 2 as well to give us a 2 remaining.
So finally for this particular question my final simplified answer would be 3x plus 4
over 2.
Now this particular example is quite a nice one.
It was a linear expression and that were asking to cancel it down and simplify.
This would only be worth possibly just one mark in your A level examination and most
of you can appreciate that because it could appear in your GCSE as well quite happily.
However, in your GCSE it would probably be worth about 3 marks because there are clearly
3 different steps you are doing.
First is to recognise that something can be factorised.
The second is to factorise out the content, which is that 2 there.
And the third would be to cancel it down and write down a simplified answer.
However, in A level Mathematics they assume what you have learnt at GCSE is a given.
So they will not award you extra marks for things that you ought to be able to do at
the end of Year 11.
Let’s have a look at something that’s slightly more complex than that.
Let’s have a look an algebraic expression which involves quadratics.
So let’s give you one like this – and this is more likely a very real exam question
you might get.
I think it’s from 2004, 2006, I can’t remember where I’ve got this question from.
There you go.
It’s x squared plus 3x minus 10 over x squared minus 25
OK, so you’ve got x squared plus 3 x minus 10 over x squared minus 25 and once again
the same question is applied here, simplify this algebraic expression as fully as possible.
Now for those of you who’ve been aiming to get an A or A* at iGCSE or GCSE should
automatically recognise that we have two quadratic equations written on the screen.
For those of you who do not spot that immediately, the best thing about this module in particular,
is that they actually do teach you how to factorise quadratics once again in C1.
So again, a lot of C1 is revision; however, if you do make the effort to revise what you’ve
learnt at GCSE you’re going to find C1 very easy, and particularly this chapter will be
wonderful, a nice little breeze through.
So let’s have a look at the top line here.
You’ve got x square plus 3 x minus 10.
That minus 10 tells you that one of your numbers has got to be negative in one of the brackets
that you’re factorising with and the other number has got to be a positive number, because
the plus in the middle of one bracket and the minus in the middle of the other bracket
will create the minus that we see here.
Then you need to consider two numbers that will give you an answer plus 3x when they
are added together.
The two numbers here are going to be 5 and 2, because 5 and 2 make up the 10 quite comfortably
when you multiply them together and they add together to make up the three.
So x here and a 5 as well.
Now the key here is to make sure that your plus and minus are in the correct place.
We want to make a positive 3x, so you want the final result when adding these two numbers
to be positive.
That means that the 5 has got to be positive and the 2 has got to be negative.
So if you were to take a little bit of time and expand this up, multiply using FOIL or
some of you might use a box method, which I’ll discuss later on in one of our sessions
if we do need to go through that.
If you multiply that up you get x squared plus 3 x minus 10, which is what we had before.
If you are checking your work whilst doing the examination, I would advise multiplying
out things like this just to ensure that you haven’t made a mistake.
Now that will all be over another quadratic here.
Now this in your GCSE is an A, A* question, x squared minus 25.
If you’re asked to factorise this they are aiming to test whether you are of an ability
of a student who is an A or an A* grade student.
The method that most people will know when seeing this is something called the difference
of two squares.
You’ve got x squared, which is a square number, take away 25, which is also a square
number – 5 squared.
So when you notice that it’s a difference of 2 squares, you should know straight away
that that is factorised into x minus 5 as one of the brackets and the other bracket
is x plus 5.
Now if I ask you to have a look at what I’ve written up on the screen now, I hope that
most of you will notice something that’s very important here, and if I was a student
it would give me a little bit of glee because I know I’ve done something right.
You can see that you’ve got an x plus 5 on the top in the numerator and you have an
x plus 5 in the denominator at the bottom.
Now if you do not see this when you factorise two quadratics the way I have, and if you
do not see any common factors at the top and the bottom of your quadratic and your expression,
this is where I’d stop and think ‘hang on a second, maybe I’ve made a mistake’,
because the idea is that you want to simplify the expression, so hopefully by factorising
both of them, you will notice that there’s something you can delete from the top and
the bottom that you can cancel from the top and the bottom.
So here I can say bye bye and I can cancel out the denominator as well as I’ve got
something far more simple than the complexity that I had earlier on.
So I’ve created two linear expressions, one over the other.
So x minus 2 on the top and underneath x minus 5.
And this is exactly the same as the complex example from above.
It’s exactly the same as writing two quadratic expressions above each other, it’s exactly
the same as that, just written in a far simpler way – x minus 2 over x minus 5 like that.
And that would be your final answer.
Now in A level Mathematics because you actually get taught how to factorise these expressions,
you’ll get awarded marks for factorising that correctly.
So you get one mark for the top, you would also get another mark for factorising the
bottom appropriately and then a third mark for cancelling out those two.
Because it’s addressed specifically in your A level modules as something that they want
to teach you, they will award you marks, whereas the question that we did a minute ago you
would not be given an extra mark for it because it isn’t directly addressed in your A level
syllabus.
Now I hope that gives you a little idea as to the start-up of your core 1 module.
There are lots of other topics within that module that you will have revised before,
that you will have met in your GCSE, for example, inequalities; you’ll meet indices, so things
like 27 over 64 to the power of two thirds.
Again that is an A* topic from a lot of GCSE syllabi but that does come up in your A level
and again they do teach you that from scratch.
A lot of the A* topics in your GCSE will be re-visited in A level C1 and C2; however,
they will be taught as if you’ve never met them before, because several GCSE Boards do
not actually cover those modules at all.
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