INTRODUCTION TO MATHS A LEVEL 4
One final thing from Decision Mathematics.
Now I have to be very honest, Decision Mathematics is probably my favourite, only because I’ve
had to actually learn it as an adult, I didn’t touch it when I was at school, had no idea
what it was and it was really refreshing to meet something new as a teacher and I’ve
really thoroughly enjoyed teaching the subject as well.
It’s one of those that, being very honest, if you find the essence of Mathematics that
we meet at GCSE – so if you find Algebra quite tough and it’s a bit too meaty sometimes
and you have to work really hard to get it, if you’re not naturally intrinsic at learning
mathematic concepts – I would recommend that using Decision D1 and D2 and one of your
two applied models, would be the wisest thing you can do.
The best thing about Decision Mathematics is that if you are a hard worker, if you follow
the rules carefully, if you put in the hours, it can be one of the most rewarding subjects
and it’s very easy to get full marks in if you work hard.
A lot of people misunderstand when they get introduced to D1 and D2, they think it’s
quite easy, because it really isn’t.
Anything too deep in terms of thought, they think that there is no necessary need to actually
sit down and practice it over and over again.
However, if you do put the hours in, if you do take the time to lay out your work clearly,
you can really pull up your final A level grade by working very hard in these subjects
and doing exceptionally well at them.
So I’m going to give you an example of something that you can meet in Decision Maths.
This is one of the modules in D1.
OK, I’m going to give you a list of numbers: 7, 2, 4, 9, 3, 5.
An exam question will simply ask you to use something called the ‘bubble sort’ to
put the numbers above into ascending order.
So put the numbers above from lowest up to highest into ascending order.
Now this is one of the most challenging things I’m going to ask you to do is that you can’t
just look at those numbers and go ‘oh – 2, 3, 5….’ and work out the order by looking
at it.
The idea with Decision Mathematics is that you’re meant to remove all mental human
intelligence from the situation you’re in and think about what a computer will have
to do to ensure that the right answer is achieved.
So if I asked a computer to put those numbers into ascending order they’re not going to
be able to look at them and think ‘I know that 2 is the smallest number’ just by glancing
over them, going ‘2’s the smallest number I’m going to pop it first’.
What they will have to do is go through a systematic set of steps to ensure that the
final answer they achieve is the one that you’re looking for and one method of doing
this is something called the bubble sort.
The bubble sort asks you to highlight two numbers at once and consider their order individually.
So the very first thing you do is that you circle the first two numbers and you look
at them and you say ‘are they in the correct order?’
So you can ask a computer a direct question ‘is a) the first number smaller than b)
the second number?’
If a) is smaller than b) then do not touch these numbers and move on.
However, in this particular situation, 7 is definitely not smaller than 2, so we’re
going to ask the computer to switch them over.
So first thing you do is write down 2, 7 as they are and then write down the other numbers.
Now this is again, a very long process; however, if you follow rules, if you’re systematic
– there’s plenty of time in the exam to do this – you can never really go wrong.
So now we have our new set of numbers.
Now we’ve considered the first two so now we’re going to move onto the second and
the third, so you bubble them up.
Again, the same question is asked; ‘is the first number smaller than the second number?’
and in this particular one, the answer again is ‘no’.
So the 7 and the 4 once again get switched over.
So it’s a 2, 4, 7, 9, 3, 5 OK, and they’ve gone onto the second step and once again you
ask the same question.
Now let’s look at the third number and the fourth number.
So you’ve got to bubble them and again I ask you the same question, ‘is the 7 smaller
than the 9?
Is the first number smaller than the second?’
and here thank God I don’t have to do too much – the 7 is smaller so I’m going to
leave it the way it is.
Now in D1, and this is where it splits up between those that listen and those that do
not, you can’t just leave it the way it is, you have to re-write all those numbers
once again, because a computer will just check and re-write, check and re-write, check and
re-write, and you’ve got to follow rules as if you’re a computer, so no short cuts.
So even though I’m not going to change anything here, I am definitely going to write those
numbers out again so I can show the Examiner that I understand that I have checked the
third and the fourth number and they do not have to change.
And now we look at the fourth and the fifth number, so the 9 and the 3, and here you’ll
notice that you do have to switch over, so it’s 2, 4, 7, 3, 9 and 5.
And finally you look at the last two numbers here, the 9 and the 5, and once again we do
have to switch them over because the 9 is bigger than the 5.
So finally you’ve got a 4 next a 7, 3, 5 and finally a 9.
Now if we were to look at this, not much has happened.
I’ve gone through the whole list, I’ve bubbled each pair consecutively and finally
finished up, but what you will notice if you look carefully, is what have I achieved that’s
brought me one step closer to putting those numbers into ascending order?
What you should notice – yes there is a 2 at the beginning which is the lowest, but
I think that wasn’t really what the intention was of this question, that’s just by chance
that that’s already at the bottom – the intention, the first step of this, what it’s
done is made that 9 rise up, rise up to the top of the pile.
So in effect like a bubble, rising up to the top of the water.
So that 9 is now done.
The 9 is now completed.
Now if I was to do this again, so start again and bubble the first two numbers, so the 2
and the 4, then bubble the next two numbers, the second and third, then the third and the
fourth, the fourth and the fifth and the fifth and the sixth.
What I should notice is that 7 now rises up to the top and will be in its correct place
just below the 9 and that will be my second pass in this algorithm.
What I’ve got written on the board already is the first pass, where it’s brought my
9 up to the top.
In the second pass, the 7 will come up to the top.
In the third pass the next highest number will rise up and be next to the 7, which is
the 5.
In the fourth pass the 4 will be there and then you’ll notice that the 2 and the 3
are in the correct positions.
Now again what I just said, you’ll notice that the 2 and the 3 in the correct positions
is not good enough because a computer will not notice this.
What you will have to do is do the bubble sort once more all the way through and the
moment you do the bubble sort and there are no changes at all, you have then proven that
the order of the numbers is sufficient, that the numbers are in ascending order and then
you are allowed to stop.
So an Examiner will make you lose a mark if you do not do it all the way through once
more to show that there are no changes.
And in a D1 exam you’ve got to actually write down ‘bubble support no changes, therefore
numbers are in ascending order’.
You’ve got to explain why you have stopped at that moment in order to get full marks
in a question like this.
Now here, this question I’ve only used five numbers, now in an actual A level examination
there will probably be nine numbers there on the board, nine.
And what they’ll ask you to do is to do the bubble sort like I have above once.
You’d have to do that once all the way through thoroughly showing each step.
But then after this you can write down the second, the third, the fourth, the fifth,
however many passes there are after that.
You can just write down the answers, you can work them out mentally and just write down
the final answer.
So they don’t expect you to use lots and lots and lots of paper, they expect you to
show the method once and then after that they’ve given you a mark for working out because they
can appreciate that, and then they’ll allow you to just drawn the remaining passes without
doing the bubble sort especially.
So I hope that’s given you a nice introduction to it and put a little bit of a smile on your
face – it’s put a smile on mine.
I do love Maths, it is great.
OK.
If there are any questions I’d be willing to take some later on.
I hope that’s been a good introduction for you.
I haven’t tried to introduce you to anything too complicated but these are both exam questions
that will come up.
If you do do the papers you’ll recognise a couple of them later on when you do start
revising through these modules.
OK, thank you for listening and I do apologise for my handwriting, I will practice and see
you next time, when hopefully it won’t be so bad!
Thank you.
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