This was supposedly demonstrated by Gauss, aged eight or something, when he was set a problem by a teacher desperate to get on with something more interesting to add together all the numbers from 1-100. He's supposed to have realised that if you paired up the numbers, they all had the same sum ie 101. It was then a matter of spotting the number of pairs and multiplying the two (5050).
The sum of an arithmetic series is done in a similar way. Think of a series as being:
a1 + (a1 + d) + (a1 + 2d) +......
since d is a constant...
You want to find the sum of the first n terms.
Sn = a1 + (a1 + d) + (a1 + 2d) +....
....(a1 + (n-1)d)
This is probably quite a few terms (otherwise you wouldn't bother trying to find the sum, would you?).
So the wisest option would be to pair up the terms, like Gauss did, and find the sum of each pair.
The formula is derived from writing the above series out from beginning to n, and the opposite way, and adding the terms.
However, just as Gauss's addition always gave him 101, in our case we will always get 2a + (n-1)d
Since we want the sum to n of the series, there will be n of these pairs. So the sum is n(2a + (n-1)d)
BUT -
We have just added TWO series! Because we wrote it out twice to make adding the terms easier.
Therefore, the final formula will be: n(2a + (n-1)d)/2.
Quite fiddly to prove, but easy enough to use.
Friday, 30 October 2009
Thursday, 29 October 2009
Series and Sequences 1 - nth terms
There are a few of these.
Let's just start with a definition. A sequence is a load of numbers in a list, with there being a common difference between the numbers.
A series is the same thing, but added together.
Here is an arithmetic sequence: 0,2,4,6,8,10....
Here is an arithmetic series: 0+2+4+6+8+10.....
In an arithmetic series or sequence, there is always a common difference between the numbers. In the ones above, the common difference is 2. It is always a constant (and not anything weird like n2 - that's more like geometric series).
So finding the nth term of an arithmetic series or sequence is easy enough.
You need the first term a1 and the common difference d.
an = a1 + (n-1)d
So say I wanted to find the 42nd term of the above sequence.
a42 = 0 + 41d
= 0 + 41x2
= 82
The 42nd term of the sequence 0,2,4,6,8...is 82
Because the difference is a constant with arithmetic sequences, this formula is simple enough to grasp. To find the nth term you need the first term, and then the number of terms before the nth one timesed by the common difference because there are that many lots of the common difference.
Let's just start with a definition. A sequence is a load of numbers in a list, with there being a common difference between the numbers.
A series is the same thing, but added together.
Here is an arithmetic sequence: 0,2,4,6,8,10....
Here is an arithmetic series: 0+2+4+6+8+10.....
In an arithmetic series or sequence, there is always a common difference between the numbers. In the ones above, the common difference is 2. It is always a constant (and not anything weird like n2 - that's more like geometric series).
So finding the nth term of an arithmetic series or sequence is easy enough.
You need the first term a1 and the common difference d.
an = a1 + (n-1)d
So say I wanted to find the 42nd term of the above sequence.
a42 = 0 + 41d
= 0 + 41x2
= 82
The 42nd term of the sequence 0,2,4,6,8...is 82
Because the difference is a constant with arithmetic sequences, this formula is simple enough to grasp. To find the nth term you need the first term, and then the number of terms before the nth one timesed by the common difference because there are that many lots of the common difference.
Wednesday, 28 October 2009
Slowness
Hmm, 50 posts in over a year isn't much is it? Well, I get bored easily.
You should see the other place...
I will try to post more, as I come back to maths after a few months' absence and try to tackle C3.
You should see the other place...
I will try to post more, as I come back to maths after a few months' absence and try to tackle C3.
Radians
These are little blighters used to describe the angles of circles. Like degrees, they can be used to express arc widths, or can be used in equations (especially trig equations). Basically they're just like degrees, but are based on π and the radius of the circle (hence radian). Imagine a line from the centre to any point on the circle. That's a radius. Then, in any direction, draw an arc around the surface of the circle with the same length as the radius. Stop. Draw a radius from this point back to the centre.
The angle subtended is one radian.
There are 360 degrees in a circle and one degree really isn't very large.
A radian is much bigger (around 57 degrees). Every circle has 2π radians, like every circle has 360°. 180° is therefore π radians.
Most calculations involving radians will also involve fractions. Using π enables you to be exact without lots of fiddly decimals. The same is true of fractions. In some ways then, using radians is satisfyingly precise.
It is also easy to convert one to the other.
360° = 2π radians
Therefore
1° = 2π radians/360.
and
say
12° = (2π/360) x 12
= 24π/360
= π/15
and it's usually sufficient to leave it like this for C2. At other levels you might want to do the calculation to n dps or whatever.
Similarly
2π radians = 360°
1 radian = 360/2π
and
so
3 radians = (360/2π) x 3.
Easy.
The angle subtended is one radian.
There are 360 degrees in a circle and one degree really isn't very large.
A radian is much bigger (around 57 degrees). Every circle has 2π radians, like every circle has 360°. 180° is therefore π radians.
Most calculations involving radians will also involve fractions. Using π enables you to be exact without lots of fiddly decimals. The same is true of fractions. In some ways then, using radians is satisfyingly precise.
It is also easy to convert one to the other.
360° = 2π radians
Therefore
1° = 2π radians/360.
and
say
12° = (2π/360) x 12
= 24π/360
= π/15
and it's usually sufficient to leave it like this for C2. At other levels you might want to do the calculation to n dps or whatever.
Similarly
2π radians = 360°
1 radian = 360/2π
and
so
3 radians = (360/2π) x 3.
Easy.
Thursday, 20 August 2009
Saturday, 23 May 2009
New Maths Specifications
As it's a mere five years since the last new A Level maths specifications came out the QCA are clearly in dire need of updating their syllabus. Accordingly they're now in the consultation stage for a new curriculum to begin in 2012. This means of course that uncompleted A Levels will come to the end of their shelf-life at this time. Hopefully that won't affect me. But I was interested to see some of their recommendations.
1) They want to abolish the non-calculator paper. I think C1 is great. You learn how to do a lot of manipulation of formulae and a lot of arithmetic in your head. Differentiation, integration, surds, quadratics (factorising) can all be easily done without recourse to a calculator. This, surely is good for fast, flexible thinking and for confidence. The down side is that it does mean C1 can't be that challenging - you can't really put radians into it for example, even though conceptually radians is a piece of the proverbial. The same goes for basic trig. Trig would be better introduced in C1 rather than stuffed into C2 (which is I think around 35-40% trig, all told).
2) They are trying to bring it down to four modules again. This is not a bad idea. The current A Level maths six unit system is complicated and contradictory, though it is flexible for people with different specialisms. It also throws up anomalies. For example as it currently stands maths is the only A Level where you can do 4 AS modules and two A2 ones - which I am doing, because in addition to C1,C2,C3 and C4 I took S1 for AS and am doing M1 for A2. M1 is an AS module. So I'm only doing C3 and C4 as actual A2 Levels. The situation can be reversed - Pure Maths AS is two AS modules and one Further! And you can also do AS modules in Further Maths (you could do FP1-4 and S1 if you hadn't done it). This leads to anomalies in the awarding of grades - the A* grade is only awarded on the basis of C3 and C4.
Four modules would simply take us back to where we started in the 90s with the equivalent of P1 and P2 - ie the four Core modules collapsed into two again. This is probably where this idea dovetails with the abolition of the calculator paper.
3) There are going to be no formulae to learn. This strikes me as disastrous. It makes an expecation of zero knowledge on the part of candidates - and this is an advanced qualification we are talking aobut here. The authorities do not view knowledge as important, but expect candidates to problem solve. Well you can only solve problems if you know how to approach them - which strategies you know of to use. I think the motivation for this is clear dumbing down, or as the QCA write, to "provide students with equality of opportunity and a common basis for progression.". Hmmm. An advanced qualification should be for people who are advanced in their knowledge and understanding of that subject - it should not pander to political ideas of equality, and believe me the QCA does just that. In the questionnaire I answered yesterday I was asked whether the new specifications promote gender and race diversity, or whether they were discriminatory against disabled people. For crying out loud, why is even pure knowledge infected with this leftist rot?
4) They want slightly to change the balance between pure and applied. This probably isn't a bad idea. Right now applied is 33% of AS and A2, but they would like it to be more flexible, up to 40%. With fewer papers I don't know how they will do this, except by weighting, although the QCA do suggest that they might allow certain exams to be longer - this would be an excellent idea. At the same time they think the pure content should be kept the same. So without changing weightings or lengths of exams or indeed mixing up applied content into pure modules, I'm not sure how this will work.
5)There is a lot of guff on the QCA site about being more stretching. They conceive it as using longer, less structured questions. This is clearly an excellent idea, where possible and where practicable. I don't know if they just mean in Further or in Core maths as well.
You can go and look all this up for yourself at the QCA website.
1) They want to abolish the non-calculator paper. I think C1 is great. You learn how to do a lot of manipulation of formulae and a lot of arithmetic in your head. Differentiation, integration, surds, quadratics (factorising) can all be easily done without recourse to a calculator. This, surely is good for fast, flexible thinking and for confidence. The down side is that it does mean C1 can't be that challenging - you can't really put radians into it for example, even though conceptually radians is a piece of the proverbial. The same goes for basic trig. Trig would be better introduced in C1 rather than stuffed into C2 (which is I think around 35-40% trig, all told).
2) They are trying to bring it down to four modules again. This is not a bad idea. The current A Level maths six unit system is complicated and contradictory, though it is flexible for people with different specialisms. It also throws up anomalies. For example as it currently stands maths is the only A Level where you can do 4 AS modules and two A2 ones - which I am doing, because in addition to C1,C2,C3 and C4 I took S1 for AS and am doing M1 for A2. M1 is an AS module. So I'm only doing C3 and C4 as actual A2 Levels. The situation can be reversed - Pure Maths AS is two AS modules and one Further! And you can also do AS modules in Further Maths (you could do FP1-4 and S1 if you hadn't done it). This leads to anomalies in the awarding of grades - the A* grade is only awarded on the basis of C3 and C4.
Four modules would simply take us back to where we started in the 90s with the equivalent of P1 and P2 - ie the four Core modules collapsed into two again. This is probably where this idea dovetails with the abolition of the calculator paper.
3) There are going to be no formulae to learn. This strikes me as disastrous. It makes an expecation of zero knowledge on the part of candidates - and this is an advanced qualification we are talking aobut here. The authorities do not view knowledge as important, but expect candidates to problem solve. Well you can only solve problems if you know how to approach them - which strategies you know of to use. I think the motivation for this is clear dumbing down, or as the QCA write, to "provide students with equality of opportunity and a common basis for progression.". Hmmm. An advanced qualification should be for people who are advanced in their knowledge and understanding of that subject - it should not pander to political ideas of equality, and believe me the QCA does just that. In the questionnaire I answered yesterday I was asked whether the new specifications promote gender and race diversity, or whether they were discriminatory against disabled people. For crying out loud, why is even pure knowledge infected with this leftist rot?
4) They want slightly to change the balance between pure and applied. This probably isn't a bad idea. Right now applied is 33% of AS and A2, but they would like it to be more flexible, up to 40%. With fewer papers I don't know how they will do this, except by weighting, although the QCA do suggest that they might allow certain exams to be longer - this would be an excellent idea. At the same time they think the pure content should be kept the same. So without changing weightings or lengths of exams or indeed mixing up applied content into pure modules, I'm not sure how this will work.
5)There is a lot of guff on the QCA site about being more stretching. They conceive it as using longer, less structured questions. This is clearly an excellent idea, where possible and where practicable. I don't know if they just mean in Further or in Core maths as well.
You can go and look all this up for yourself at the QCA website.
Long Time, No Blog
It's been a tricky old year so far, like a particularly knotty equation, so I haven't found the time for maths blogging, or even for maths.
So it was with some trepidation that I looked at my calendar about a month ago and saw the date for the AQA C2 exam. At that point I knew nothing of radians, logarithms I couldn't even spell, and geometric series I thought were just lots of pretty pictures (lots of geometry type stuff). So I spent a month fairly hectically doing the last four chapters of C2, missing out a few questions on the way and also, disappointingly, not really bothering to look at the proofs for the various formulae but just learning them.
C2 was yesterday. It was ok, though exams are always hard to assess until you get the marks. I answered all qs, but I think the differentiation with fractional indices one might have cost me a few %.
As before, the exam was thrilling. No I really mean this. I had bags of adrenalin, lots of excitement, masses of determination to show what I could do. I am like this with things that don't matter. Only important things have me quivering in the corner like a wobbly jelly who's just been made professor of wobbling at Oxford (ok ok that's a Blackadder joke sort of). Lining up beforehand with fifty sixth formers all pointing at me and whispering was a bit strange but it just reminded me of how utterly uninterested in it I was at their age. You do it for reasons, to get to uni, or because you happen to be good it, but you rarely do it just because you love it - you learn that later. Sometimes.
So it was with some trepidation that I looked at my calendar about a month ago and saw the date for the AQA C2 exam. At that point I knew nothing of radians, logarithms I couldn't even spell, and geometric series I thought were just lots of pretty pictures (lots of geometry type stuff). So I spent a month fairly hectically doing the last four chapters of C2, missing out a few questions on the way and also, disappointingly, not really bothering to look at the proofs for the various formulae but just learning them.
C2 was yesterday. It was ok, though exams are always hard to assess until you get the marks. I answered all qs, but I think the differentiation with fractional indices one might have cost me a few %.
As before, the exam was thrilling. No I really mean this. I had bags of adrenalin, lots of excitement, masses of determination to show what I could do. I am like this with things that don't matter. Only important things have me quivering in the corner like a wobbly jelly who's just been made professor of wobbling at Oxford (ok ok that's a Blackadder joke sort of). Lining up beforehand with fifty sixth formers all pointing at me and whispering was a bit strange but it just reminded me of how utterly uninterested in it I was at their age. You do it for reasons, to get to uni, or because you happen to be good it, but you rarely do it just because you love it - you learn that later. Sometimes.
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