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I'm struggling to motivate a 14yr old son to learn his maths. In the case of trig I can assure him that I still find it useful - even in the last week doing some woodwork. However in the case of quadrilaterals, algebra etc I can offer no such. In retrospect perso I find it interesting - but its hard to convey that - offers anyone ?
Post edited at 18:22
In reply to LeeWood:
A quadrilateral is a four-sided polygon. Is that really what you're referring to? If so, what's the exact context?
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In reply to LeeWood:
Do you mean quadratic equations?
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In reply to LeeWood: if you mean quadratic equations, then yes, I use them a lot , or for similar purposes, use crossplots of data, usually a geologic property versus depth
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In reply to LeeWood:
Assuming you’re talking about the algebraic solution of a quadratic polynomial then I’m with your son.
I often find myself solving far more difficult equations, and I almost invariably do so by telling my computer to solve them for me.
There are lots of things in maths that help me build the mental and paper models I then ask my computer to solve, but the formula for solving quadratic equations isn’t one of them. I can’t see any reason to learn it - if one needs it a few seconds on google brings it up.
The only motivation I can see is to get a good GCSE result.
Take such applied frippery out and put a bucket load more linear maths in...
Post edited at 18:58
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In reply to LeeWood:
They're made up of two triangles struck together and you've already established that trig is cool......
And just look around - there are quadrilaterals all over the place.
Post edited at 19:07
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In reply to wintertree:
> ...........the formula for solving quadratic equations. I can’t see any reason to learn it - if one needs it a few seconds on google brings it up.
Or just look on the formula sheet in the exam! Obviously one needs to know how to use it and understand why there are two answers and how to interpret them etc.
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In reply to Robert Durran:
> They're made up of two triangles struck together and you've already established that trig is cool......
Wouldn't that look like a partially eaten Toblerone?
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In reply to LeeWood:
20 years in engineering and I don't recall ever using the quadratic equation solution outside an exam.
I did actually use it a couple of weeks ago to help one of our lads work out an altitude profile for his drone to fly in a vertical semi-circle over a fixed point.
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In reply to FactorXXX:No
> Wouldn't that look like a partially eaten Toblerone?
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In reply to LeeWood:
When I was young I didn't really get the value of mathematics. As I (rather slowly) wised up, I got to realise the extraordinary value and power of mathematics. Now, I would say to any youngsters, get as smart as you possibly can with mathematics.
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In reply to John Stainforth:
> When I was young I didn't really get the value of mathematics. As I (rather slowly) wised up, I got to realise the extraordinary value and power of mathematics. Now, I would say to any youngsters, get as smart as you possibly can with mathematics.
Absolutely.
On the other hand, I've just been trying to look up a cartoon, which, as a Maths teacher, I find painfully sharp (couldn't find it). Anyway, pupil puts up hand: "Please Mr, when are we ever going to use any of this in real life". Teacher: "You won't, but some of the bright kids might". Tragically probably pretty true a lot of the time. I've come to the conclusion that children should, in an ideal world, stop learning any new maths a year before they drop the subject and just concentrate on doing stuff that applies what they already know and understand. Our current exam driven system largely uses most pupils' final year studying maths to drive them to the limit and weed them out from progressing to the next stage. It's a destructive process.
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In reply to wintertree: how do you know the computer is giving you the correct answer? And that's not a facetious answer as if I get the computer to extract trends from my data it will get it very wrong as context is lost.
Perhaps the value of quadratics lies in an understanding of intercept and gradient and what they mean
In reply to Robert Durran:
It's here: https://www.reddit.com/r/funny/comments/5cqpm4/why_i_cant_be_a_math_teacher...
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In reply to Thugitty Jugitty:Thanks, brilliant!
In reply to wbo:
> Perhaps the value of quadratics lies in an understanding of intercept and gradient and what they mean
Why?
In reply to wbo:
> how do you know the computer is giving you the correct answer?
When solving an equation (as in my previous post), I plug the answer back in to the equation and look at the residual error. A good solver will be extensively tested, and if using a numerical one it will also supply confidence or accuracy measures in the values it solves for. So manually plugging it back in and checking the residuals is never necessary but it does flag up user error in calling the solver etc.
> And that's not a facetious answer as if I get the computer to extract trends from my data it will get it very wrong as context is lost.
Looking for trends in data is not the same as solving an equation though. Some would have you believe that’s what the emperor’s new clothes, erm sorry AI, is here to solve.
Post edited at 20:11
In reply to Robert Durran:
> Or just look on the formula sheet in the exam!
You don't get given this at GCSE level any more.
For those questioning why you need to learn this: a lot of Maths at school is about learning to how to manipulate the structure. Quadratics are the simplest form of a conic equation, they crop up in thousands of modelling scenarios, e.g. optimising sale prices or simple problems involving acceleration. You couldn't study more advanced ones without studying these first.
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In reply to Wil Treasure:
> For those questioning why you need to learn this: a lot of Maths at school is about learning to how to manipulate the structure. Quadratics are the simplest form of a conic equation, they crop up in thousands of modelling scenarios, e.g. optimising sale prices or simple problems involving acceleration. You couldn't study more advanced ones without studying these first.
Ultimately this is the truth, but for a small and specialist group of mathematicians/engineers. As a teenager I more or less took this on board and scraped through - but now looking back on a working life in which quadratics have only just resurfaced *only because they're taught in school* I lack energy to insist on their value.
As for 'being a bright kid' - question of nature or nurture, can we really expect to change the direction for a child of average intelligence just by 'challenging him to be bright' ?! Aptitude for maths is not always present and other successful career paths do exist.
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In reply to LeeWood:
> Ultimately this is the truth, but for a small and specialist group of mathematicians/engineers. As a teenager I more or less took this on board and scraped through - but now looking back on a working life in which quadratics have only just resurfaced *only because they're taught in school* I lack energy to insist on their value.
There is very little utilitarian value in the solution of quadratic equations per se. So if that's what your original question is asking - give up.
There's obviously intellectual value and interest in the study of mathematics in general - but we should accept that it's not for everybody.
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In reply to wintertree: I'd agree that looking at residuals is interesting and important , but I'd disagree that computers get always get it right - the answer can be mathematically sound, match the data and still wrong
That would, as you say, be picked up by AI - I spend a lot of time looking at crossplots coloured, flagged by a third variable looking for predictive combinations.
In those datasets it's useful too know the value and physical meaning particularly of the gradient
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In reply to LeeWood:
Personally I find maths fascinating, but I hated it at school. A few years ago I read the below link which I always find to be a slightly depressing comment on how maths is taught at school.
https://www.maa.org/external_archive/devlin/LockhartsLament.pdf
In reply to Wil Treasure:
> You don't get given this at GCSE level any more.
You get it for National 5 in Scotland where I am!
> For those questioning why you need to learn this: a lot of Maths at school is about learning to how to manipulate the structure. Quadratics are the simplest form of a conic equation, they crop up in thousands of modelling scenarios, e.g. optimising sale prices or simple problems involving acceleration. You couldn't study more advanced ones without studying these first.
Precisely. The example I use is, given 20m of fencing, what's the maximum area that can be enclosed by using it to make three sides of a rectangular enclosure with an existing wall forming the fourth. Symmetry, zeros, optimisation - lots of ideas that obviously transfer to more complicated situations and types of function. Those who say that as an Engineer, or whatever, they never solve quadratic equations are, I think forgetting that they are probably using ideas every day that are most readily introduced by studying quadratics.
Not to mention that parabolas crop up all over the place!
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In reply to wintertree:
"I often find myself solving far more difficult equations, and I almost invariably do so by telling my computer to solve them for me.", that's fine and dandy until we give up producing anyone who can program the computer...
Quadrilaterals come up rarely for me unless I've got an unusual geometry to work with but are widely used in computer solvers which I use frequently and like to be able to understand.
Quadratics I use ~1/month - by the time something like this emerges from my work it's usually pretty complicated, and lamenting my poor maths ability, I resort to a symbolic solver or a numerical root finding algorithm.
Complex numbers and sqrt(-1) I use every working day.
Should you son keep using this stuff - yes - this is about building a toolbox and it is a tool - if you have an extensive range of tools you can do a lot more and you have many more options. Depending on the directions you go in you may find some of the tools never get used but it is hard to tell which you'll actually want in 3, 5, 10, 20 years time. Give up now if you want to give up these possibilities. Until last week (in a discussion about cyclic quadrilaterals) I hadn't used the word quadrilateral for over 20 years - it doesn't mean it was a waste knowing what one is.
Post edited at 22:23
In reply to gravy:
> that's fine and dandy until we give up producing anyone who can program the computer...
Indeed, but most people who do GCSE maths will never in their life write a numerical solver for a computer, and of the few that do, most will never do so again after their undergrad degree.
I’d love to split GCSE Maths into a “maths for life” course and an alternative “maths for people who like maths” course. I doubt quadri-anything would go in to the former.
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In reply to Robert Durran:
> Precisely. The example I use is, given 20m of fencing, what's the maximum area that can be enclosed by using it to make three sides of a rectangular enclosure with an existing wall forming the fourth.
But that's a stupid and 'theoretical' problem - straight out of a high-school textbook - which no practising tradesman would ever pose.
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In reply to Rob Parsons:
> But that's a stupid and 'theoretical' problem - straight out of a high-school textbook - which no practising tradesman would ever pose.
Quite. A tradesman would build the half-circle that maximises area and be done with it. A lazy tradesman would note that a half-square approximates a half-circle and go with that.
You also don’t need the quadratic roots equation to answer this problem if you do it with maths. Assuming one writes an equation for the area in terms of an unknown side length ‘x’ and finds the root of dArea/dx, the equation will be trivial enough that I suspect one just has to divide 4 in to 20 m to get the answer...
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In reply to mattrm:
> Personally I find maths fascinating, but I hated it at school. A few years ago I read the below link which I always find to be a slightly depressing comment on how maths is taught at school.
> https://www.maa.org/external_archive/devlin/LockhartsLament.pdf
What an absolutely brilliant article - one of the best but equally uplifting and demoralising thing I've ever read about maths education.
In reply to Rob Parsons:
> But that's a stupid and 'theoretical' problem - straight out of a high-school textbook - which no practising tradesman would ever pose.
It's not a stupid problem. It's a decluttered problem which introduces several important ideas without unnecessary distractions which transfer to numerous situations. Given to pupils as an open investigation it can work brilliantly to illuminate these ideas. I'm not suggesting it as a standard problem to be solved by use of a standard recipe.
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In reply to wintertree:
Oh dear...........
> Quite. A tradesman would build the half-circle that maximises area and be done with it. A lazy tradesman would note that a half-square approximates a half-circle and go with that.
See my last post. An inquisitive pupil, having discovered that the answer is a rectangle twice as long as it is broad, might wonder if a greater area might be possible with other shapes. They might eventually hypothesise that a semicircle maximises the area. And then wonder why.......... It is a brilliant question that leads naturally to other questions - proper education.
> You also don’t need the quadratic roots equation to answer this problem if you do it with maths. Assuming one writes an equation for the area in terms of an unknown side length ‘x’ and finds the root of dArea/dx, the equation will be trivial enough that I suspect one just has to divide 4 in to 20 m to get the answer........
Yes, of course you would. But you're not a fourteen year old who has never heard of quadratic functions, let alone differentiation.
I suggest you read the article linked earlier.
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In reply to LeeWood:
Define “in real life”. I have used them on a *lot* of occasions in game development, does that count? :-P
Usually solving basic mechanics equations.
In reply to Robert Durran:
> Oh dear...........
> See my last post. An inquisitive pupil, having discovered that the answer is a rectangle twice as long as it is broad, might wonder if a greater area might be possible with other shapes. They might eventually hypothesise that a semicircle maximises the area. And then wonder why.......... It is a brilliant question that leads naturally to other questions - proper education.
I’m not at all convinced. This question still feels massively contrived.
It’s much more natural to think about “whole” shapes when it comes to the relationship between perimeter, area and shape.
What this question has done is added a lot of complexity and words on top of that.
You can set the same concept with rectangles without a pre-existing wall for one side, and compare to a circle. For more keen students this version is easier to generalise to n-sided polygons including differentiating the area with respect to side count. They can then derive a proof for a square with regards aspect ratio and a circle with regards side count. That’s a long way towards proving the circle as the most efficient shape.
The brick wall aspect changes the formulation a bit and wordifies the question a lot. It also gives an excellent opportunity for schools to teach pupils to do paper working paying no attention to physical units...
> Yes, of course you would. But you're not a fourteen year old who has never heard of quadratic functions, let alone differentiation.
I might not have been being entirely serious - but the point arrived at by going through the maths was that the example you gave does not require the equation for the roots of a quadratic to be solved. It is best solved by re-arranging of the equation, something which can be taught quite independently.
My point would be that many 14 year olds will never find the root of a quadratic or differentiate again in their lives. Other 14 year olds would really benefit from having learnt to differentiate already. Maths is far to “one size fits all” currently.
> I suggest you read the article linked earlier.
The 25-page PDF? Looked a bit excessive, like...
Post edited at 07:07
In reply to LeeWood:
Most maths at school is not needed for most people doing most ordinary jobs.
Basic, simply arethmetic will get most people through life quite adequately.
For anything else we have graduates in careers who pursue subjects which need more challenging maths - such as quadratic equations.
We have computers now.
No wonder your 14 yr old isn't interested. Once I found out I was never going to need logarithms etc., I gave up. I needed education - not something I might remotely need 20 years later - by which time I'd have forgotten anyway.
Post edited at 07:34
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In reply to wintertree:
> I’d love to split GCSE Maths into a “maths for life” course and an alternative “maths for people who like maths” course. I doubt quadri-anything would go in to the former.
Definitely. We could do the same for English, as soon as someone can read and knows all their sounds they can get by. No need to bother worrying about writing, metaphor, poetry, different styles or spelling or anything like that.
And science. Who needs that, really? Only kids who like it should have to learn it.
As for languages, just let the kids who go to foreign places learn them.
PE? Noone in this country exercises after 16 anyway so scrap it.
In fact, why bother learning anything? Get the kids down the pits and up the chimneys again. Save a fortune on education and we can spend all that cash on other stuff we're actually going to use.
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In reply to XXXX:
Nice rant but it overlooks some things and seems to wildly miss my point.
Basic maths is an important life skill for everyone - budgeting, estimating, understanding compound interest etc. Without it people are in a worse place for life.
A key reason I’d split maths is so that far more children learn more maths that will be useful to them and their futures uncluttered by things they will never use. There’s still plenty of elegant maths to teach and generate interest with in a refocused course.
Science is split (or used to be) in to double and tripple. That’s not controversial is it?
Post edited at 08:06
In reply to Wil Treasure:
> You don't get given this at GCSE level any more.
Depends which board you follow; some still give the quadratic solution and others don't. Of course, a relatively common question is 'solve this by factorising' or solve this by completing the square' so the formula isn't helpful anyway.
I've ben asked this question, with varying levels of apathy and anger behind it, plenty of times. I've generally used one of three answers - for the brightest who wil clearly go on to do matematical a-levels, you can talk about it being the gatway to higher polynomials and the simplest conic section etc. For middling pupils, you can esily describe questions that have two solutions (throw a ball in the air - at what time is it 10m above your head?) and point out that linear equatisn can't solve those.
But, actually, quite a lot of the time, I'd simply point out that they are studying for a peice of paper that hey can show to prospective employers that demonstrates that they are able to do maths, and this is one of the ways they are tested. They have to be able to solve quadratics, not to actually slove quadratics, but to prove that they are capable of learning techniques such as solving quadratics! It's usually surprisingly effective.
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In reply to XXXX:
Exactly, let's decide 20 years in advance that our kids are only fit for mundane trivial labour and that computers will do all the work anyway and just not bother with school. Why bother to learn to drive, add up, say anything other than "like" and "innit" respect! No point in learning anything unless you've got a written* guarantee that in later life it will definitely be useful.
* don't worry, your robot will read it for you.
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In reply to gravy:
Nice rant, but as with Irk’s it rather misses my point in my view.
> Why bother to learn to drive, add up, say anything other than "like" and "innit" respect!
And yet some people currently leave school like that.
What was achieved by trying to teach them a formula for finding the roots of a quadratic equation?
How has knowing how to use that formula enhanced their future prospects? Perhaps if they’d had a “maths for life” class instead it would have been more likely to give them tools of more relevance.
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In reply to wintertree:
So you would pre-determine a child's future at 14 would you? Wouldn't want to clutter up their tiny minds? Wow, how awful that would be, to confuse their little heads, bless.
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In reply to XXXX:
> So you would pre-determine a child's future at 14 would you? Wouldn't want to clutter up their tiny minds? Wow, how awful that would be, to confuse their little heads, bless.
Where did I say that I would pre-determine a child’s future?
Do you honestly thinking a refocusing of maths towards real-life situations, for those who struggle the most with algebra, is going to pre-determine their future?
I agree with Jamie’s 3rd option he explains to many of his pupils - “do it to get this important piece of paper”. Why not make the stuff they do as useful to as many of them as possible?
By denying them useful maths in favour of a formula probably 95% of them will never use again in their life, I suggest you are advocating more strongly than me for controlling their future, by wasting their time and sending them out in to the world unprepared.
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In reply to LeeWood:
> However in the case of quadrilaterals, algebra etc I can offer no such. In retrospect perso I find it interesting - but its hard to convey that - offers anyone ? <
I've found algebra useful for DIY, personal finance and hospital lab work. hospital lab especially simply balancing equations to find 'x' be that length, weight, area whatever.
Also geometry and trigonometry for DIY and helping me to wrestle with elementary engineering problems.
At least today many students seem to have more basic understanding of maths and may realize they can use it when and if a need arises. At least I know quadratic equations are available as a tool and I could ask my son to apply them for me if need be.
In the 50s and 60s we spent so much time on arithmetic, for example long division of pounds, shillings and pence; furlongs chains etc, which has been rendered largely redundant by metrication and electronic calculation. I used logarithms purely as an arithmetical tool and had no real understanding of them.
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In reply to LeeWood: I'm a physicist and work with a lot of engineers, yes I use that numeracy and mental process from school all the time. It formed the basis of my undergraduate, post graduate and professional career. Quadratic equations are everywhere but that's not really the point, they are a fairly simple example of a language which describes our world and which is one of the foundations of our civilisation.
Obviously you won't find yourself repeating those classroom exercises in your career, you should be much more fluent by then, but that applies to all disciplines.
This basic algebra has to be introduced to reasonably able pupils, to not do so is to deny those pupils the chance of of a scientific/technical career.
This "do it for the piece of paper" attitude really stinks, and is quite depressing, but of this time. The discipline to apply yourself to something you aren't gifted at, the mental agility you WILL develop by applying yourself, the glimpse into another way of seeing the world.... all invaluable whatever you end up doing in life.
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In reply to LeeWood:
I was taught to use geometry to measure the width of a river when you only have access to one bank. It worked but was a faff and difficult at night. The easier way was to tie a length of string to an adjustable spanner and throw that over and pull it back and measure the string. At night you waited for the splash/change in resistance and added a bit on. Easy.
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In reply to nniff:
> I was taught to use geometry to measure the width of a river when you only have access to one bank.
Of course the cynics on here are going to moan that it's a contrived, unrealistic problem while failing to acknowledge that it is just a simple introduction to the techniques use to make maps or even measure the distance to a star.........
> It worked but was a faff and difficult at night. The easier way was to tie a length of string to an adjustable spanner and throw that over and pull it back and measure the string. At night you waited for the splash/change in resistance and added a bit on. Easy.
Do you know the joke about the mathematician, engineer, physicist and economist who were all given a barometer and told to use it to measure the height of a church spire?
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In reply to wintertree:
> And yet some people currently leave school like that.
> What was achieved by trying to teach them a formula for finding the roots of a quadratic equation?
No that outcome is not due to quadratic equations. Decades of lowering of academic standards and slack working discipline most likely was enough.
CB
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In reply to wintertree: Would you bother teaching any algebra? Quadratics are a pretty straightforward example of solving an equation, and the results are fairly obvious when plotted on a graph. While you might think they are not relevant to anything useful they are a fairly comprehensible demonstration of a function
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In reply to Mike Stretford:
> The discipline to apply yourself to something you aren't gifted at, the mental agility you WILL develop by applying yourself, the glimpse into another way of seeing the world.... all invaluable whatever you end up doing in life.
Yep - makes perfect sense to me - as an adult- but conveying that to an adolescent (who may not be math-smart) remains problematic. I can at least remind him of my own life experience: I quit french at 13 believing I would never need it - and had a big catch-up struggle at 40 having decided to move out to France.
Otherwise - as others have commented - it would help a significant proportion of kids if maths was more vocation oriented - and life-applications featured in all cases.
It is re-assuring to know that many of you have used algebra etc - would anyone hazard a guess what % this represents ? (my guess would be < 5%) - help us keep this discussion in proportion !
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In reply to Robert Durran:
> Do you know the joke about the mathematician, engineer, physicist and economist who were all given a barometer and told to use it to measure the height of a church spire?
No - you gonna tell us ? sounds good !
In reply to wintertree:
> I’m not at all convinced. This question still feels massively contrived.
> It’s much more natural to think about “whole” shapes when it comes to the relationship between perimeter, area and shape.
I think you are completely missing the point. Of course it's a bit contrived, but the whole point is to give an accessible problem which introduces some mathyematical ideas about quadratic functions and it does it very well.
> You can set the same concept with rectangles without a pre-existing wall for one side.
The reason that I prefer the version with the wall is that the symmetry of the areas/graph of area against width is not at all obvious (whereas it is in the situation without a wall where swapping width and breadth will obviously give the same area) and should come as a surprise leading to the discovery that the important property that there is symmetry in all quadratic functions.
> For more keen students this version is easier to generalise to n-sided polygons.
I agree if the main aim is to find the maximum area of many shape with a given perimeter, but we were talking about the aim of introducing quadratic functions.
> It also gives an excellent opportunity for schools to teach pupils to do paper working paying no attention to physical units...
Why?
Post edited at 12:33
In reply to LeeWood:
> Yep - makes perfect sense to me - as an adult- but conveying that to an adolescent (who may not be math-smart) remains problematic. I can at least remind him of my own life experience: I quit french at 13 believing I would never need it - and had a big catch-up struggle at 40 having decided to move out to France.
Thumb up!
> Otherwise - as others have commented - it would help a significant proportion of kids if maths was more vocation oriented - and life-applications featured in all cases.
This comes down to how flexible and effective the schools streaming policy is, it worked well at my school in the late 80s. The 2nd and 3rd steams did work like that. It's a tough call though sometimes, which pupils should be challenged and pushed in which subjects? Either way there'll be complaints.
> It is re-assuring to know that many of you have used algebra etc - would anyone hazard a guess what % this represents ? (my guess would be < 5%) - help us keep this discussion in proportion !
I'd estimate nearer 10% or more.... but then I have a broad definition of 'used'.
In reply to mattrm:
Great link. Thanks very much.
In reply to Robert Durran:
> I think you are completely missing the point. Of course it's a bit contrived, but the whole point is to give an accessible problem which introduces some mathyematical ideas about quadratic functions and it does it very well.
My point was that a case without the wall is less contrived, puts more focus on the maths, still covers quadratics, and has a larger number of more accessible and intuitive routes to understanding the deeper geometry.
> The reason that I prefer the version with the wall is that the symmetry of the areas/graph of area against width is not at all obvious (whereas it is in the situation without a wall where swapping width and breadth will obviously give the same area) and should come as a surprise leading to the discovery that the important property that there is symmetry in all quadratic functions.
I take that point. Still very contrived and doesn’t really get at the origin of the symmetry in the function. Plotting some quadratics and relating the parameters to the plots feels a lot less contrived to me.
> > For more keen students this version is easier to generalise to n-sided polygons.
> I agree if the main aim is to find the maximum area of many shape with a given perimeter, but we were talking about the aim of introducing quadratic functions.
Well the original task was finding an application for which one might use them, but we digressed.
> > It also gives an excellent opportunity for schools to teach pupils to do paper working paying no attention to physical units...
> Why?
Because I’ll bet you even money the source document just uses the number “20” willy nilly without a unit postfix. Plenty of past papers (through which I have trawled) encourage this by going so far as to print the unit next to the result box, encouraging the student to not bother writing it or considering it.
In reply to wintertree:
> What was achieved by trying to teach them a formula for finding the roots of a quadratic equation?
Of course, the quadratic formula is also a thing of great beauty. Would you similarly rubbish studying a Leonardo painting, listening to a Beethoven symphony or seeing a Shakespeare play. OK, the quadratic frormula may not be the best example, but there ought to be an argument for studying some maths for it's intrinsic aesthetics alone (quadratic functions in general I would argue are a good example).
In reply to wintertree:
> My point was that a case without the wall is less contrived, puts more focus on the maths, still covers quadratics, and has a larger number of more accessible and intuitive routes to understanding the deeper geometry.
I've no idea why you say any of that.
> I take that point. Still very contrived and doesn’t really get at the origin of the symmetry in the function. Plotting some quadratics and relating the parameters to the plots feels a lot less contrived to me.
How would you suggest getting to the origin of it? The fence thing is only meant as an introductory exercise leading to further questions.
> Because I’ll bet you even money the source document just uses the number “20” willy nilly without a unit postfix.
You are simply assuming that! And why would the version without the wall be any different?
In reply to Robert Durran:
> Of course, the quadratic formula is also a thing of great beauty.
Blimey. It's not exactly exp (i x pi) = -1, is it ...
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In reply to Rob Parsons:
> Blimey. It's not exactly exp (i x pi) = -1, is it ...+
An nor is every great painting a Leonardo, or every great bit of music a Beethoven symphony.
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In reply to Robert Durran:
> I've no idea why you say any of that.
Because
(1) adding a fence adds more words not related to the maths, and then adding a wall adds more words not related to the maths. Simple elegant maths has become word soup. There is a skill in translating word soup to maths, but muddling up the teaching of words-to-maths formulations and an understanding of quadratics makes it harder to do either well. You don’t neee the wall to get to the underlying quadratic, just the fence, so you’re throwing in unneeded word soup.
(2) Because there are several different, easily accessible ways of tackling and maximising the area/perimeter ratio for a rectangle of (x-a)(x+a) without needing the quantity 20 m or the concept of a wall, and without either they are simpler and make it easier to understand.
> How would you suggest getting to the origin of it?
Like I said - by plotting some quadratics and identifying features on them and how they correspond to the constants in the quadratics.
> The fence thing is only meant as an introductory exercise leading to further questions.
Where as I would take the graphical approach with the maths, and then look at applying it to free fencing and wall attatched fencing showing the translation of the real world quantities into the maths they’ve gone through.
Then I’d ask them “so why isn’t the ideal shape where you have a wall a square?”...
> You are simply assuming that!
As I made clear by giving a confidence in my guess. Based on reading lots of past papers.
> And why would the version without the wall be any different?
To be fair it wouldn’t - I was jumping another step to a purely symbolic solution.
In reply to Rob Parsons:
> Blimey. It's not exactly exp (i x pi) = -1, is it ...
Quite. One encapsulates an awful lot, and the other the third most basic polynomial form (counting zero order). Is a higher order polynomial more beautiful?
I recall being presented with a lot of quadratics of the form y = ax^2 + bx + c. Not very elegant and hides the main parameters of the quadratic form away from view.
Presented as y = k(x-x0)^2 + y0 you have an equation where the steepness, centre and top/bottom of the curve are all directly represented in the equation, and where solving for the roots doesn’t require recall or use of a formula sheet.
I’d probably start with teaching the second form along with examples of how it arises, and then teach mapping between the two forms.
Post edited at 15:11
1
In reply to wintertree:
> Because there are several different, easily accessible ways of tackling and maximising the area/perimeter ratio for a rectangle of (x-a)(x+a) without needing the quantity 20 m or the concept of a wall, and without either they are simpler and make it easier to understand.
The algebra to get an expression for the area with perimeter 20 is actually simpler in the case of the wall, and I am not at this stage generalising the perimeter to P or whatever. I just want a nice simple example (and really the wall is not exactly complicated!) to plug some numbers in, plot a graph, note that we do get a maximum and that the graph has symmetry. Further investigations of quadratics might follow.
In reply to wintertree:
> Quite. One encapsulates an awful lot, and the other the third most basic polynomial form (counting zero order). Is a higher order polynomial more beautiful?
Well, the fundamental theorem of algebra is certainly extremely beautiful, but realistically it makes sense to start with linears and quadratics.
> I recall being presented with a lot of quadratics of the form y = ax^2 + bx + c. Not very elegant and hides the main parameters of the quadratic form away from view.
> Presented as y = k(x-x0)^2 + y0 you have an equation where the steepness, centre and top/bottom of the curve are all directly represented in the equation, and where solving for the roots doesn’t require recall or use of a formula sheet.
> I’d probably start with teaching the second form along with examples of how it arises, and then teach mapping between the two forms.
I'd be interested to know whether there is any school in the world where the completed square form is introduced before the standard form. You could certainly make a mathematical case for it but as a sensible approach for most pupils, I would suggest it is lacking.
Are you a maths teacher?
1
In reply to Robert Durran:
> I'd be interested to know whether there is any school in the world where the completed square form is introduced before the standard form.
“Standard form” - what is standard about it? It’s not an ISO or a BS. It’s no more or less a standard than the other form.
Schools curricula seem to be an odd mix of doing things the way they have always been done because that’s how they’re done, and doing them as the political whim of the month dictates.
> You could certainly make a mathematical case for it but as a sensible approach for most pupils, I would suggest it is lacking.
Why do you find it lacking? I find it relates much more directly the algebraic symbols and the functional form, that it makes the location of the minima or maximum trivially obvious, and that it puts the common form of all quadratics into stark clarity. I can’t say any of these about the other form.
> Are you a maths teacher?
I teach a maths heavy science as part of one of my jobs, and do regular small group tutorials looking in part to relate understanding through the maths. I see people after their A-levels. I’ve plenty of experience of choice of functional form leading to head scratching or lightbulb moments.
Post edited at 15:35
In reply to Robert Durran:
> The algebra to get an expression for the area with perimeter 20 is actually simpler in the case of the wall
Wall, specified perimeter: Area = x * (20 m - 2 x)
No wall, no specified perimiter: Area = (x-k)(x+k).
The later expands and simplifies to area = x^2 - k^2. Bingo - biggest area when the difference from a square (k) is zero.
1
In reply to wintertree:
> I find it relates much more directly the algebraic symbols and the functional form, that it makes the location of the minima or maximum trivially obvious.
If you think it is trivially obvious to the average or even quite bright 15 year old then I can guarantee you are mistaken!
1
In reply to wintertree:
> > The algebra to get an expression for the area with perimeter 20 is actually simpler in the case of the wall
> Wall, specified perimeter: Area = x * (20 m - 2 x)
Which quite a few of my 15 year olds will come up with independently. And mixing up letters standing for units and letters standing for numbers/quantities really is asking for trouble!
> No wall, no specified perimiter: Area = (x-k)(x+k).
To which, quite rightly, most fifteen year olds will ask where the perimeter has gone! It really is far more subtle. At this stage, I'm trying to keep things accessible.
In reply to Robert Durran:
> If you think it is trivially obvious to the average or even quite bright 15 year old then I can guarantee you are mistaken!
It is however a lot more obvious and a lot easier to explain.
In reply to Robert Durran:
> And mixing up letters standing for units and letters standing for numbers/quantities really is asking for trouble!
You are right. I should have italicised my variables. UKC needs inline LaTeX... I also should have used the multiply symbol where appropriate but again typesetting for maths isn’t great on a climbing forum...
Or - even better - set the work without an arbitrary number and its unit in there in the first place...
> To which, quite rightly, most fifteen year olds will ask where the perimeter has gone! It really is far more subtle. At this stage, I'm trying to keep things accessible.
So replace x with p/4. Or tell them that x is 1/4th the perimiter. If the relationship between the perimiter and length of a side is not accessible to your students, there’s really no point in them doing quadratics...
Post edited at 16:07
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