Yet Another Cycling Forum
General Category => The Knowledge => OT Knowledge => Topic started by: Martin on 15 September, 2009, 10:04:49 pm
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I've just spent many hours failing to find a solution either online or via MyMaths to my 13yo's maths homework, as someone who managed a grade A GCE 30 years ago, has maths changed that much?
WTF are Original Sequences; First and Second differences? >:(
more importantly; what possible bloody use is any of this in the National Curriculum other than to keep maths teachers off the streets?
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WRT to your last question: I don't know; never heard of this terminology in the context of quadratic equations. I must be getting old, or becoming to specialised in solutions of PDEs!
What was your son's mathematical problem?
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Dunno - we never used those terms.
Has anyone ever used a quadratic equation in Real Life (TM)? ISTR they tell you where a quadratic curve cuts the axes, which must be really useful when the final breakdown of society comes and you've lost both the matches and the tin opener.
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What was your son's mathematical problem?
"write down an nth term formula for the original sequence
12 26 46 72 104 142" ???
a Freddered gold 1 point star to the first (non maths teacher) who can come up with the correct answer :thumbsup:
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I use quadratic polynomials to reconstruct numerical solutions; e.g. in the case of a thin film model in fluid mechanics (imagine an oil film) I know the velocity at the wall, the mid-depth velocity from a depth average solution and the shear at the film surface from which I must reconstruct a velocity profile to obtain the velocity at the surface. In this case a cubic interpolation would be better but more difficult to implement. This is just one example of a mathematical solution to an engineering problem I work on right now (for an aeroengine application, so very practical :)).
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I don't think designing aero engines features in year 9 :-\
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I don't think designing aero engines features in year 9 :-\
I'm not a designer... ;)
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What was your son's mathematical problem?
"write down an nth term formula for the original sequence
12 26 46 72 104 142" ???
Gee, I'm not even sure that I understand the question. [I did study mathematics, I promise]
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3n^2 - 7n + 6 ...?
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The first differences are the differences between the numbers, so 14, 20, 26, 32 etc. The second differences are the differences between these, which is a constant 6.
So from this you should be able to write a formula for the nth term, but I can't at this time of night ???
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3n^2 - 7n + 6 ...?
nope
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I noticed that the differences between the consecutive numbers increased by 6 each time. Can I have a C please?
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12 + 11n + 3n2
I think
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3n^2 - 7n + 6 ...?
nope
No, I've been running that one through in my head, and it doesn't fit, but I can't remember how to derive the function.
Anyway, we're not really talking quadratics; this is Series work.
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12 + 11n + 3n2
I think
Not that either, but it's close. Should there be a minus sign?
*too late at night*
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12 + 11n + 3n2
I think
how can 12+11n+ 3n[sp]2=12? (unless n=0)
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12 + 11n + 3n2
I think
Not that either, but it's close. Should there be a minus sign?
*too late at night*
It works if you start at n=0
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12 + 11n + 3n2
I think
And n is the rank of each number in the sequence, n = [1 ... n]? Doesn't seem to work.
How is this quadratic function or polynomials math?
[Edit: Beaten by Clarion]
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How is this quadratic function or polynomials math?
[Edit: Beaten by Clarion]
Cos a quadratic is just an eqn with x2 in, innit ;)
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How is this quadratic function or polynomials math?
[Edit: Beaten by Clarion]
Cos a quadratic is just an eqn with x2 in, innit ;)
Just a shape function in this case... <shrug shoulders>
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Doesn't seem to work.
It does.
slightly edited:
y= 12 + 11x + 3x2 is the equation of a quadratic curve, and if you insert the discrete points n = 0, 1, 2 etc you get the series that Martin gave us.
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Doesn't seem to work.
It does.
It's the equation of a quadratic curve, and if you insert the discrete points n = 0, 1, 2 etc you get the series that Martin gave us.
Aha! I started at n = 1...
[Edit: at least I'll be ready for Baby G when she reaches that age! ::-)]
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the problem I have with all this is that it it just maths for maths sake; nobody is ever ever going to need this apart from maths teachers
my work involves similar levels of anality but it does at least relate to something that exists in the real world and may one day affect people which this doesn't.
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the problem I have with all this is that it it just maths for maths sake; nobody is ever ever going to need this apart from maths teachers
or anyone who does a Physics degree or a PhD....
Academic subjects are always like that though, eh ?
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Yes, but, mathematics is a toolbox or a vehicle to help one formulate solutions to science and engineering problems. When I work with people who can't express themselves or read it, it is, in my field, quite difficult to be creative.
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Doesn't seem to work.
It does.
slightly edited:
y= 12 + 11x + 3x2 is the equation of a quadratic curve, and if you insert the discrete points n = 0, 1, 2 etc you get the series that Martin gave us.
yebbut if you were to enter that as the solution in MyMaths you would get a big fat 0
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3n^2 - 7n + 6 ...?
nope
This was the source of my 'efforts'... ;) perhaps it is wrong though...
id:A027599 - OEIS Search Results (http://research.att.com/~njas/sequences/A027599)
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the problem I have with all this is that it it just maths for maths sake; nobody is ever ever going to need this apart from maths teachers
or anyone who does a Physics degree or a PhD....
My PhD had some practical applications; most of the PhD students I am involved with will eventually compute solutions to say engine or oil industry or sportswear ;) problems, possibly idealised at first but quite applicable next. The math and physics are just a way to formulate a solution to a "complex" problem.
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the problem I have with all this is that it it just maths for maths sake; nobody is ever ever going to need this apart from maths teachers
I love maths, so I find quadratic equations fun for their own sake. And I'm sure that the mental exercise is beneficial in the same way that being forced to do ten mile cross country runs in the freezing rain was beneficial for the physical health, even if they didn't have any practical real-world applications.
However, if you want to know just what is the point of quadratic equations, read these fascinating articles:
101 uses of a quadratic equation (http://plus.maths.org/issue29/features/quadratic/index-gifd.html)
101 uses of a quadratic equation: Part II (http://plus.maths.org/issue30/features/quadratic/index-gifd.html)
d.
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Doesn't seem to work.
It does.
slightly edited:
y= 12 + 11x + 3x2 is the equation of a quadratic curve, and if you insert the discrete points n = 0, 1, 2 etc you get the series that Martin gave us.
yebbut if you were to enter that as the solution in MyMaths you would get a big fat 0
Why ? What form do they need it in ?
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3n2 + 5n + 4
Seems to work; it gets 12 for n=1, 26 for n=2 etc.
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Nice link Smutchin! Didn't know this mag'.
That (below) what I do for a living... :D (except that the NS equations describe other fluids than air and that these eqns have very few analytical solutions, esp not in the case below! A bit approximative for a mathematical mag' :-\)
The team instead turned to the Navier-Stokes equations, a set of complex differential equations which describe air flow. In most real-life situations, as in this one, it's impossible to solve these equations analytically. But with the help of powerful computers, you can employ some educated guesswork to come up with approximate solutions that tell you just how the air will behave. The branch of maths that develops the computer algorithms for this task is called computational fluid dynamics. It requires massive amounts of computing power, but comes up with effective results.
Source: Career interview: The fastest mathematician on Earth (http://plus.maths.org/issue52/interview/index.html) (latest issue, 52, of +plus)
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Doesn't seem to work.
It does.
slightly edited:
y= 12 + 11x + 3x2 is the equation of a quadratic curve, and if you insert the discrete points n = 0, 1, 2 etc you get the series that Martin gave us.
yebbut if you were to enter that as the solution in MyMaths you would get a big fat 0
Why ? What form do they need it in ?
n = [1 ... n] I think
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Aha! I started at n = 1
In which case you'd need to transform it from n to n-1. That'd give you 3n2 +5n +4
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Aha! I started at n = 1
In which case you'd need to transform it from n to n-1. That'd give you 3n2 +5n +4
Yep. Viz Rower40 above as well.
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Why ? What form do they need it in ?
n = [1 ... n] I think
But they want the formula for the nth term. That's what we've derived.
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Why ? What form do they need it in ?
n = [1 ... n] I think
But they want the formula for the nth term. That's what we've derived.
Je ne parle pas l'anglais. ::-)
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I had a look at Smutchin's link as well. My problem is that the moment letters are used to represent numbers, I have to turn them back into example numbers again and do the arithmetic before I can work out what is going on. By the second example the frustration starts to well up and I do something else instead.
So my WFT (White Flag Threshold) is a good deal lower than most people's.
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My problem is that the moment letters are used to represent numbers, I have to turn them back into example numbers again and do the arithmetic before I can work out what is going on. By the second example the frustration starts to well up and I do something else instead.
You're algebraphobic ;)
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Wow, I thought you enjoyed chess? Surely you can link letters to a strategy (e4 x...) and solve a problem (you do iwith chess, don't you?)!
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Wow, I thought you enjoyed chess? Surely you can link letters to a strategy (e4 x...) and solve a problem (you do iwith chess, don't you?)!
Chess isn't analytical, it is pattern recognition. That is why you learn all the standard moves..
..d
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I see no link whatever between quadratic equations (or any other maths for that matter) and chess problems. Although many of this country's best chess players have been mathematically very gifted (the "English Chess Explosion" of the 1980s was largely based upon some of the top Maths brains from Cambridge) there are plenty of examples of brilliant mathematicians who have been duffers at chess. Einstein himself, apparently, was a very weak player. Alan Turing also, although as I mentioned on one of the recent threads about him, Bletchley was crammed with top chess players.
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3n2 + 5n+4
(http://www.astro.washington.edu/images/gold-star.png)
(no I didn't deduce this it was a maths teacher)
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Why ? What form do they need it in ?
n = [1 ... n] I think
But they want the formula for the nth term. That's what we've derived.
Oh come on you lot.
It must surely be n+1 :thumbsup:
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Wow, I thought you enjoyed chess? Surely you can link letters to a strategy (e4 x...) and solve a problem (you do iwith chess, don't you?)!
Chess isn't analytical, it is pattern recognition. That is why you learn all the standard moves..
..d
Rote-learning the moves won't help you. You have to understand the underlying ideas. Matthew Sadler, who I think was World no. 7 at his peak, used to sit down after every game and explain it all to his mum (a Frenchwoman! :thumbsup:) who so far as I know never played competitively. I watched him on a few occasions and it was amazing how much he saw, and in fact how good he was at explaining it.
Edit: whether we are talking about different things I don't know, but chess certainly is analytical. We talk about the "tree of analysis". Alexander Kotov, in his 1972 masterpiece "Think like a Grandmaster" goes into a lot of detail about this.
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3n2 + 5n + 4
That looks good to me.
How did you arrive at that? Was it by trial and error? That's how I was trying to do it, and I might have got there eventually, but not for a while yet. Seems like a pretty tough task for this level of maths unless there's a specific technique that has been taught as a means of making it more straightforward.
My son is doing very similar number sequence stuff at the moment, but on sequences with much simpler rules (he's 11, ie year 7).
d.
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What was your son's mathematical problem?
"write down an nth term formula for the original sequence
12 26 46 72 104 142" ???
a Freddered gold 1 point star to the first (non maths teacher) who can come up with the correct answer :thumbsup:
At 13yo your child is probably only just being introduced to quadratics.
The first and second differences are precisely that:
The first difference being the difference between each consecutive number in the series.
The second difference being the differences between each of the first differences.
series: 12 26 46 72 104 142
1st diff: 14 20 26 32 38
2nd diff: 6 6 6 6
The second diff being constant implies function including x^2
your son is therefore familiar with series and y=ax+b I expect this is to then build on this to reach y=ax^2+bx+c
For instance if you write the series x^2
1 4 9 16 25 36, the first differences are 3, 5, 7, 9, 11 ... the second differences are 2, 2, 2, 2, 2
therefore A in the term Ax^2 is the second differential /2
then write out the series 3x^2 and subtract from the original series to leave 9 14 19 24 29 34. This new series is of the form 5x +b as the first difference is 5, this is 5x+4. Therefore the final function is 3n^2 + 5n + 4
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My son is doing very similar number sequence stuff at the moment, but on sequences with much simpler rules (he's 11, ie year 7).
best stop wasting your time going out on that bike and boning up for the next 2 years then ;)
BTW can anybody remember how we solved all this utter crap back in the '70s? it certainly didn't involve drawing chain link fences on pieces of paper.
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Oh, I'm late to this party.
In fact I'm late to this party because I've been trying to help the Beloved Stoker recover the formula used by the regulator to model a proposed incentive scheme, where the rate of the incentive payment and the value you're incentivised to hit are both dependent on the same thing.
So we've been doing exactly this---second differencing (approximate) data to try to get a plausible quadratic formula out.
It's a real-world problem!
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The first difference being the difference between each consecutive number in the series.
The second difference being the differences between each of the first differences.
series: 12 26 46 72 104 142
1st diff: 14 20 26 32 38
2nd diff: 6 6 6 6
The second diff being constant implies function including x^2
your son is therefore familiar with series and y=ax+b I expect this is to then build on this to reach y=ax^2+bx+c
That was exactly how I did it.
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Back to school you lot...
Bletchley was crammed with top chess players.
A lot of people at Bletchley were top crossword experts too with a fair smattering of polyglots.
For a quick maths lesson:-
This is the 'original sequence': 12 26 46 72 104 142
The 'first differences' are, funnily enough, the differences between successive terms in the 'original sequence': 14,20,26,32,38
The 'second differences' are, funnily enough, the differences between the terms of the 'first differences': 6,6,6,6
To build a general formula we observe:
x1=12
x2=12+8+6
x3=12+(8+6)+(8+6+6)
x4=12+(8+6)+(8+6+6)+(8+6+6+6)
So, for xn we have 12 plus 8*(n-1) plus a number of 6's that correspond to the 'triangular numbers' (1,1+2,1+2+3,1+2+3+4,1+2+3+4+5,... == 1,3,6,10,15,...) which we all know the formula for that. Think of the triangle like this:-
6
66
666
6666
To calculate the number of 6's we imagine a similar triangle spliced on to it to make a rectangle:-
6xxxx
66xxx
666xx
6666x
This makes a rectangle of size 4 * 5, of which the 6's make up only half. So a triangle of size has 4*(4+1)/2 elements or, in the general case: n*(n+1)/2
In this case the input is n-1, (with n=1 we have no 6's) so the formula for the number of 6's in xn is (n-1)*(n-1+1)/2 = n(n-1)/2
So: xn = 12 + 8*(n-1) + 6*n*(n-1)/2
= 12 + 8*(n-1) + 3*n*(n-1)
= 12 + 8n - 8 + 3n2 - 3n
= 3n2 + 5n + 4
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3n2 + 5n + 4
That looks good to me.
How did you arrive at that?
Just looking at it you know that for n=1, an equation of an2+bn +c , a+b+c will equal 12.
for n=2 that becomes 4a+2b+c=26
and n=3 9a+3b+c =46
This then becomes three simultaneous linear equations.
9a+ 3b -46 =c
4a+2b +9a +3b -46 =26 so 13a+5b =72
a+b+4a+2b-26=12
so 5a +3b=38
Two simultaneous linear equations.
13a=72-5b
a=(72-5b)/13
5/13x(72-5b)+3b=38
360-25b+39b=494
14b=134
sub into the other equations and out pops your answer.
Probably got a booboo or two along the way.. but the system seems to give some answers. Maybe it needs a fourth term?
Or maybe I am barking up the wrong tree after too many glasses of wine tonight?
..d
Edit: just seen the more elegant solutions above.
Oh well. We never did that kind of analysis at school - just calculus..
..d
Was it by trial and error? That's how I was trying to do it, and I might have got there eventually, but not for a while yet. Seems like a pretty tough task for this level of maths unless there's a specific technique that has been taught as a means of making it more straightforward.
My son is doing very similar number sequence stuff at the moment, but on sequences with much simpler rules (he's 11, ie year 7).
d.
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rats.. it is standard calculus isn't it (facepalm).
dx^2/dx of ax^2 = ax/2
And the next level up would be 2ax^2/3?
Doh!
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Coming late to this, I got there without Greenbank's triangles (and wrote at the same time as David, which looks pretty similar).
12 = a.1^2 + b.1 + c = a + b + c
26 = a.2^2 + b.2 + c = 4a + 2b + c
so, difference tells us 3a + b = 14
46 = 9a + 3b + c
difference tells us 5a + b = 20
difference of the differences says therefore 2a = 6, a = 3
therefore b = 5 (as 3.3 + b = 14)
therefore c = 4 (as 12 = 3+5+c)
Re rote learning / pattern matching - I'd say maths is a lot of pattern matching, but you need to know the ideas to know which patterns to apply and how to make them go. Its one of the reasons I'm really rusty at maths - I only use bits of it occasionally, so the patterns take time to find in my head.
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Thanks everyone :thumbsup: keep it coming
much like the GPS this will probably end up as a more useful resource than reading the manual...
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Anyway, it beats my current assignment which is on Number Theory (due in on Thursday so has to be posted tomorrow)...
M381 TMA04 Question 3 (Number Theory) - 10 marks (out of 100).
i) Show that the continued fraction of sqrt(18) is [4, <4,8>].
ii) Find the two positive solutions of the Diophantine equation x2-18y2=1
Rather messy answer to be posted when I find a camera...
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Has anyone ever used a quadratic equation in Real Life (TM)?
Quadratic equations describe the kinetic energy of moving objects, say your head. They allow you to relate the energy at a given speed, say 12mph, to the energy at some other speed, say your normal cycling speed. Is that useful?
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Has anyone ever used a quadratic equation in Real Life (TM)?
Quadratic equations describe the kinetic energy of moving objects, say your head. They allow you to relate the energy at a given speed, say 12mph, to the energy at some other speed, say your normal cycling speed. Is that useful?
not if it's going to become another h****t thread
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Coming late to this, I got there without Greenbank's triangles (and wrote at the same time as David, which looks pretty similar).
Same idea, much tidier than mine though.
..d
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I've been teaching my niece maths and we covered reciprocals last week. Can someone explain the point of giving 'to the power of minus one' a name? Who's ever used it?
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I've been teaching my niece maths and we covered reciprocals last week. Can someone explain the point of giving 'to the power of minus one' a name? Who's ever used it?
Ah, you want really truly bonkers stuff like:-
e-i*pi + 1 = 0
Rather messy answer to be posted when I find a camera...
http://www.greenbank.org/misc/IMG_0392.JPG
http://www.greenbank.org/misc/IMG_0393.JPG
and the final 3 questions on Mathematical Logic *hack spit* that I should really do tonight...
http://www.greenbank.org/misc/IMG_0394.JPG
It's all greek to me...
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Ah, you want really truly bonkers stuff like:-
e-i*pi + 1 = 0
I've used that, as part of a running check on a program I wrote to calculate Bessel functions of complex numbers, itself part of an analytical model of a very large helical waveguide.
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I've been teaching my niece maths and we covered reciprocals last week. Can someone explain the point of giving 'to the power of minus one' a name? Who's ever used it?
The rev counter of many cars is labelled "min-1".
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Thanks everyone :thumbsup: keep it coming
This won't be of any help to understanding the how and the why of this kind of problem, but wolfram alpha will often give you a solution for maths problems.
Input your sequence and it returns this:
- Wolfram|Alpha (http://www.wolframalpha.com/input/?i=12+26+46+72+104+142)
which shows the answer you were looking for and the differences. Having the answer can often help you work backwards and provide insight into what's going on, so can be useful.
I've been doing some maths revision lately and I've been astonished how useful a resouce youtube is for this. There are heaps of very good, bitesized lectures/tutorials on all manner of maths subjects.
Lots of subjects are covered by more than one uploader so if you don't get it by watching a particular explanation you can try with another "teacher" to see alternative methods etc.
A youtube search will generally offer several videos to watch, or you could try these, who cover lots of subjects with good explanations:
YouTube - ExamSolutions's Channel (http://www.youtube.com/user/ExamSolutions)
YouTube - patrickJMT's Channel (http://www.youtube.com/profile?user=patrickJMT&view=videos)
YouTube - densmath's Channel (http://www.youtube.com/user/densmath#grid/uploads)
There's also this guy, Sal Khan (a clever clogs MIT graduate), who also has a whole heap of videos up, but additionally links them to his free "adaptive maths program" to help teach maths:
Khan Academy (http://www.khanacademy.org/index.html)
You or child can watch the videos on their own, but if you sign up to the adaptive maths wotsit they become linked to a logical progression of subjects accompanied by practice problems. So, learner starts with straightforward subjects, watches a video or two then is presented with some questions for practice. Once those are sucessfully dealt with the next subjects are opened up, with more tutorial videos and practice problems, and the whole thing progresses in that format.
The web is stuffed with maths resources, which is very fortunate for me, because I need all the help I can get with my engineering maths.
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I see no link whatever between quadratic equations (or any other maths for that matter) and chess problems.
No. You were not talking about quadratic eqns. You mentioned thinking of a problem with/using letters!!
Although many of this country's best chess players have been mathematically very gifted (the "English Chess Explosion" of the 1980s was largely based upon some of the top Maths brains from Cambridge) there are plenty of examples of brilliant mathematicians who have been duffers at chess.
Indeed. Not in Britain only.
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Wow, I thought you enjoyed chess? Surely you can link letters to a strategy (e4 x...) and solve a problem (you do iwith chess, don't you?)!
Chess isn't analytical, it is pattern recognition. That is why you learn all the standard moves..
These are quite mathematical I'm afraid. And as Wow noted chess is analytical. Not a shot in the dark on my part!
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Conversely, there have been a lot of good chess players whose maths is weak. Nigel Short is probably the best recent example - poor O levels / (no?) A levels and no university education. Whereas most of the world's top players are very good at lots of things, Short has only chess with no academia whatever to back him up. I think Fischer was like this too.
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I've been teaching my niece maths and we covered reciprocals last week. Can someone explain the point of giving 'to the power of minus one' a name? Who's ever used it?
You did. ;D reciprocal is so much shorter than to the power of minus one.
Just jargon really, like so many other fields.
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Conversely, there have been a lot of good chess players whose maths is weak. Nigel Short is probably the best recent example - poor O levels / (no?) A levels and no university education. Whereas most of the world's top players are very good at lots of things, Short has only chess with no academia whatever to back him up. I think Fischer was like this too.
I think you are right. They were also, Fisher?, socially unskilled etc. Would that link to some sort of health/mental syndrom?
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Having said that, Frenchie, my problem with anything mathematical over and above basic arithmetic is specifically substituting letters for numbers. You are not actually doing this in the thinking process in chess but you do when you record the moves you have played. We actually call it algebraic notation (1 e4 e5 2 Nf3 Nc6 etc). There is also the old-fashioned "descriptive notation" (1 P-K4 P-K4 2 N-KB3 N-QB3) which represent exactly the same moves.
Edit: the word "reciprocal" is also part of the chess vocabulary. There are certain positions, I think exclusively in K & P endgames, which are said to be "reciprocal zugzwang". In such positions, the player with the move is at a disadvantage.
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Having said that, Frenchie, my problem with anything mathematical over and above basic arithmetic is specifically substituting letters for numbers. You are not actually doing this in the thinking process in chess but you do when you record the moves you have played. We actually call it algebraic notation (1 e4 e5 2 Nf3 Nc6 etc). There is also the old-fashioned "descriptive notation" (1 P-K4 P-K4 2 N-KB3 N-QB3) which represent exactly the same moves.
So you don't think or talk about a sequence you might play using the letter symbols, right? Is this what you mean? You see the pattern in your mind?
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Just noticed this thread. A slightly different, and possibly simpler way of solving the original problem -
Original series -
n= 1 2 3 4 5 6
Original series 12 26 46 72 104 142
First differences 14 20 26 32 38
Second Differences 6 6 6 6
This can be extended to the left to get the 0th member of the series -
n= 0 1 2 3 4 5 6
Original series 4 12 26 46 72 104 142
First differences 8 14 20 26 32 38
Second Differences 6 6 6 6 6
and the value of n and n0 substituted into our polynomial -
0n2 + 0n + c = 4
or
c = 4
We can now consider the instance where n=1 -
an2 + bn + c = 12
ax12 + bx1 + c = 12
or
a + b + 4 = 12
which simplifies to
a + b = 8 or b = 8 - a and a = 8 - b
and the instance where n = 2
an2 + bn + c = 25
or
4xa + 2xb + 4 = 26
which simplifies to -
2a + b = 11 or b = 11 - 2a and a = (11 - b) / 2
From the above we have b = 8 - a = 1 - 2a and a = 8 - b = (11 - b) / 2
and solving for a gives a = 3 and for b gives b = 5
Therefore the equation we are looking for is -
3n2 + 5n + 4
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Having said that, Frenchie, my problem with anything mathematical over and above basic arithmetic is specifically substituting letters for numbers. You are not actually doing this in the thinking process in chess but you do when you record the moves you have played. We actually call it algebraic notation (1 e4 e5 2 Nf3 Nc6 etc). There is also the old-fashioned "descriptive notation" (1 P-K4 P-K4 2 N-KB3 N-QB3) which represent exactly the same moves.
So you don't think or talk about a sequence you might play using the letter symbols, right? Is this what you mean? You see the pattern in your mind?
I think most of it is visual but when I think ahead it's "I do that, he does that, I do that...". Also, in a tricky situation where maybe you have the choice of two or more possible plans, it helps to verbalise your problem - actually express it inwardly in sentences.
A few years ago the head teacher with whom I run a junior chess club one evening made me play "blindfold" against all the kids. They were huddled around a demonstration board and I had my back to it. After about 8 moves I won a piece (it was simply a matter of counting the number of my pieces attacking a certain square and theirs defending it. I had one more attacker than they had defender, so I won a piece). After an hour's play and about 20 moves in, I was still a piece up but was really struggling to keep the position in my head. 7.30 came and we ran out of time. I was declared the winner because I was still a piece up, but when I looked at the board I was really surprised because the position didn't look anything like the one I had in my head!
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...and the value of n and n0 substituted into our polynomial...
The thing is, regardless of whether you use this method or any of the others that have been suggested, the key is this:
your son is therefore familiar with series and y=ax+b I expect this is to then build on this to reach y=ax^2+bx+c
I suspect that for whatever reason, Martin's son is not as familiar with these concepts as the exercise assumes. Not surprising. It's hard to take it all in when you're that age, and doing homework at the end of a tiring day, you might not readily grasp that you're supposed to use the information you were given in class. I see it all the time with my own son.
And the fact that several very clever people here didn't readily get the correct answer shows how hard this stuff can be if you're unfamiliar/rusty with the relevant concepts, and how straightforward it can be if you know the right technique.
d.
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I think I will have to re-learn English when I do homework with G. I didn't quite understand the question I'm afraid. :-[ :-[ :-[
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I didn't quite understand the question I'm afraid. :-[ :-[ :-[
Nor did several native English speakers!
d.
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You did. ;D reciprocal is so much shorter than to the power of minus one.
Just jargon really, like so many other fields.
But add to that the time taken to explain what the reciprocal is and it's far longer!
Besides, everyone I know has always called it the inverse.
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You did. ;D reciprocal is so much shorter than to the power of minus one.
Just jargon really, like so many other fields.
But add to that the time taken to explain what the reciprocal is and it's far longer!
Besides, everyone I know has always called it the inverse.
I'm pretty sure we were taught 'reciprocal' (but at some point I think I switched to 'inverse' - possibly on my degree?). I'm a couple of school generations older than Gonzo.
I would guess that reciprocal is a more familiar everyday word than inverse to 'normal' people (i.e. before they study any maths!)
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p.s. I've never heard of this 2nd difference stuff. I could solve the problem, and I know about arithmetic and geometric series etc (Proof by recursion anyone? A simple idea that took me ages to get my head round).
I think these things are a way of introducing quadratics without the world of graphs, equations of a straight line, crossing the x-axis etc. I think.
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I used to love series. I remember in one of my maths O level papers, I couldn't remember whether the differential of cos was sin or -sin (or the other way round), so I derived the series and differentiated that ;D
I have not the first idea now how I would go about deriving the series for cosine, let alone in exam conditions and as a piece of rough working out.
I got an A, in case you're worrying ;)
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I think I will have to re-learn English when I do homework with G. I didn't quite understand the question I'm afraid. :-[ :-[ :-[
You were not the only one. It was some of the unfamiliar jargon that was used that would have confused me.
..d
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You did. ;D reciprocal is so much shorter than to the power of minus one.
Just jargon really, like so many other fields.
But add to that the time taken to explain what the reciprocal is and it's far longer!
Besides, everyone I know has always called it the inverse.
But inverse is a broader term than reciprocal. cf. average and mean.
What is the inverse of a matrix? is it a reciprocal?
As to 'the time taken to explain' that only occurs if the person you are talking to hasn't already labelled the concept with that particular piece of jargon.
..d
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I used to love series. I remember in one of my maths O level papers, I couldn't remember whether the differential of cos was sin or -sin (or the other way round), so I derived the series and differentiated that ;D
I have not the first idea now how I would go about deriving the series for cosine, let alone in exam conditions and as a piece of rough working out.
I got an A, in case you're worrying ;)
I got an A for GCSE maths and I don't think I ever learned what cos and sin were. I knew you had to use them for certain algebraic functions, but only got as far as learning when to press the appropriate button on the calculator. I think I missed a large chunk of the maths syllabus because I moved schools during the second year (what they call year 8 these days) and my new school was somewhat further ahead in maths. Cos and sin were probably covered in the bit I missed.
d.
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Should old Harry
Catch a herring
Trawling off America?
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I used to love series. I remember in one of my maths O level papers, I couldn't remember whether the differential of cos was sin or -sin (or the other way round), so I derived the series and differentiated that ;D
You could have worked out which to use by just drawing the 2 curves. Much quicker! ;)
(I quite like maths when different concepts overlap like that, and when there is a visual/geometric answer for something difficult. )
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Never heard that, but I remembered anyway.
ftr, I think I was in the last year not to be allowed calculators in maths A levels (1983). When I went to University to study Engineering, we had to submit our calculators for individual approval. Mine caused a stir because it was neither an approved nor disapproved model, being bought in Japan. :-\ It was OKd eventually, but I was too crap to pass the exams anyway :-[
Calculators are used too readily. Kids don't tend to understand what numbers mean. And, because it can't be keyed into a calculator easily, they tend to struggle with algebra, geometry, set theory and matrices (I confess, these last two I found tricky), let alone calculus, which, once you have the key, you unlock a whole new world. </soapbox>
Edit: mattc - perhaps I could, but I don't tend to work visually, and it was much more satisfying doing it my way ;)
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I got an A for GCSE maths and I don't think I ever learned what cos and sin were
They're the real and imaginary parts of eiθ
If you know that the nth term of the Taylor Series for ex is xn/n! you can derive the power series for sin(θ) and cos(θ) from that. If Clarion did that at O-level, he deserved an A :)
edit: that may seem supremely obscure, but it's something I use regularly now. It's a fundamental tool in both RF/radio propagation and AC circuit theory.
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ftr, I think I was in the last year not to be allowed calculators in maths A levels (1983).
We were allowed them from the start of the O-level syllabus onwards. That would have been about 1976 or 77.
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ftr, I think I was in the last year not to be allowed calculators in maths A levels (1983).
We were allowed them from the start of the O-level syllabus onwards. That would have been about 1976 or 77.
I took 'O' level maths in 1980. We had a choice of whether to use a calculator or log tables / slide rule. I chose the later.
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ftr, I think I was in the last year not to be allowed calculators in maths A levels (1983).
We were allowed them from the start of the O-level syllabus onwards. That would have been about 1976 or 77.
You must have had an indulgent exam board. I did AEB. We used calculators (sparingly) in class, but there was no chance of having them in exams.
And, er..
If you know that the nth term of the Taylor Series for ex is xn/n! you can derive the power series for sin(θ) and cos(θ) from that. If Clarion did that at O-level, he deserved an A :)
I did, and thanks :-[
I went on to get three maths A levels and another in Physics, which is broadly the same thing. ;D
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I went on to get three maths A levels and another in Physics, which is broadly the same thing. ;D
(http://imgs.xkcd.com/comics/purity.png) (http://xkcd.com/435/)
(Click for original source...)
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Where do logicians fit in that scheme?
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Engineers? (Applied physics and maths)
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I went on to get three maths A levels and another in Physics, which is broadly the same thing. ;D
I'm not sure about that. I've always been reasonably good at physics, but my maths is not great. Not surprisingly I found a PhD a lot easier than a first degree...
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Funnily enough, given the cartoon, I found that (at A level at least), there's an awful lot of crossover between applied maths & physics. Questions are slightly different, but the concepts are much the same ;D
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Engineers? (Applied phyics and maths)
There's not much that's "applied" about formal logic - it's pretty darn abstract.
Although I suppose it relates more closely to the "real world" than pure maths.
d.
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Just noticed this thread. A slightly different, and possibly simpler way of solving the original problem -
Original series -
n= 1 2 3 4 5 6
Original series 12 26 46 72 104 142
First differences 14 20 26 32 38
Second Differences 6 6 6 6
This can be extended to the left to get the 0th member of the series -
n= 0 1 2 3 4 5 6
Original series 4 12 26 46 72 104 142
First differences 8 14 20 26 32 38
Second Differences 6 6 6 6 6
and the value of n and n0 substituted into our polynomial -
0n2 + 0n + c = 4
or
c = 4
We can now consider the instance where n=1 -
an2 + bn + c = 12
ax12 + bx1 + c = 12
or
a + b + 4 = 12
which simplifies to
a + b = 8 or b = 8 - a and a = 8 - b
and the instance where n = 2
an2 + bn + c = 25
or
4xa + 2xb + 4 = 26
which simplifies to -
2a + b = 11 or b = 11 - 2a and a = (11 - b) / 2
From the above we have b = 8 - a = 1 - 2a and a = 8 - b = (11 - b) / 2
and solving for a gives a = 3 and for b gives b = 5
Therefore the equation we are looking for is -
3n2 + 5n + 4
I would have solved it in a similar manner. Despite having had my early twenties taken away by the French education system to get maths and physics stuffed in my head, I had never heard of Greenbank's solution! I find it interesting to see the different approaches to a similar problem!
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I would have solved it in a similar manner. Despite having had my early twenties taken away by the French education system to get maths and physics stuffed in my head, I had never heard of Greenbank's solution! I find it interesting to see the different approaches to a similar problem!
Indeed. My solution was designed to highlight a couple of extra things along the way, i.e. the composition of the terms, and working from their composition towards a general formula, and the use of triangular numbers representing accumulation of the second differences. I just wanted to be a bit different...
Without trying to explain things, and just looking for the answer, my solution would have been similar to yours (calculate the term for n=0 which gives c, then use the next two terms to generate two simultaneous equations with two unknowns (a and b) and then solve.)
There is no wrong way of working it out as long as the working is shown and the logic at each step is correct and obvious or described if not. As with all questions like this, the bulk of the marks are for the working/logic with only 1 mark for getting the final answer right.
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I got an A for GCSE maths and I don't think I ever learned what cos and sin were
They're the real and imaginary parts of eiθ
Yebbut you aren't told that until A-level ;).
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Yebbut you aren't told that until A-level ;).
Indeed
If Clarion did that at O-level, he deserved an A
:)
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There is no wrong way of working it out as long as the working is shown and the logic at each step is correct and obvious or described if not. As with all questions like this, the bulk of the marks are for the working/logic with only 1 mark for getting the final answer right.
The joy of maths! Just like with routefinding for cycling, there are lots of different ways of reaching the result, with different things to see & experience along the way. Like chess, too, in fact, Wow ;)
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If Clarion did that at O-level, he deserved an A
:)
Quite right too.
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Yebbut you aren't told that until A-level ;).
You might not have been, but I was told that during GCSEs, but then I was a swot and I had various inspirational maths teachers during my education...
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This thread makes my brain hurt.
The answer is x.
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I burned my maths books (and many others) when I got my "A" level pass.
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This thread makes my brain hurt.
The answer is x.
Wrong all the world knows that the answer is 42 now what is the question?
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This thread makes my brain hurt.
The answer is x.
The skill comes not in knowing that the answer is x but knowing y
Is that the first time I have punned in algebra?
..d
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Yebbut you aren't told that until A-level ;).
Indeed
If Clarion did that at O-level, he deserved an A
:)
If GCSEs have been getting gradually easier, surely we can therefore work out how old Clarion is? If the change of ease of GCSEs has been variable, we could solve the data by writing a quadratic equation too!
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The flaw with that, Gonz, is that I'm a smidge older than he, and I didn't do that stuff 'til A level...
Different exam Boards...
edit: oh yeah, and we're not talking GCSE anyway ;D
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If MV was with JMB, he will have done more 'modern' maths, like sets & matrices, which we only skimmed through, and I really wish we'd done more of... :-\
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Spot on. Right Board, right syllabus. Loads of probability theory too, which has been useful.
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I was one of the "guinea pigs" subjected to the Cambridge University's "School Mathematics Project" when I was 11 in 1965. We did modern maths, with sets and matrices etc. With the latter I always had premonitions of men in shades and lots of green numbers flashing before my eyes.
My teacher was a certain Dave "three hundredweight" Hanneman, so-called because it was rumoured that that he could lift this amount above head height.
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;D We used to say 'SMP' stood for 'Simple Maths Please!' ;)
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As a maths duffer I didn't find it simple. I scraped a grade 6 in my O level.
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Now you say it, we did SMP, not JMB. It was fun. Possibly why I found A level so much of a shock :)
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Ouch! Yes, SMP to A Level would have been a big step. SMP was good on concepts but not too good at the mathematical method (AIUI: IANAM*. My brother was at a different school & did SMP)
* I Am Not A Mathematician ;D
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The flaw with that, Gonz, is that I'm a smidge older than he, and I didn't do that stuff 'til A level...
Different exam Boards...
edit: oh yeah, and we're not talking GCSE anyway ;D
I think I fall between the 2 of you age-wise and I didn't do that until A-level. I think it was Cambridge exam board for O, AO (remember them?) and A.
I do have a vague memory of SMB but that was probably at middle school.
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I think I fall between the 2 of you age-wise and I didn't do that until A-level. I think it was Cambridge exam board for O, AO (remember them?) and A.
I just scraped into O levels ... and we got to do maths a year early and then do AO. That seemed much harder, but it might have been that we swapped to a teacher that was much less good. Decent Physics teachers and a friend who went on to be a teacher got me through maths after that. I forget which board they all were - we did enough different boards at different times that we got a load of certificates - so hiding the odd duff result could be quite easy :)
I still have my maths O level revision book (me, obsessive hoarder? no...), and I had cause to compare the contents page to a new GCSE syllabus earlier this year, which made me do this: :o
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Loads of probability theory too, which has been useful.
Ouch. I am a determinist.
More numerical analysis please...
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Ouch. I am a determinist.
I'm a physicist. Probability is the name of the game :)
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I'm an Acoustics Engineer. If in doubt I just add 3dB ;D
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I am a Structural Engineer. If in doubt I go up one size.
Always better to make sure that lawyers don't get involve in design.