Extended Essay

[AmeliaC]

I need to do an extended essay and for the summer we were told to think of two prospective topics. I'm interested in doing math as my subject. do you guys have any ideas on cool topics? I've only studied through algebra 2 and some trig.

[abskebabs]

There are so many possible interesting topics you could cover in the wonderful world of mathematics. To name a few that interest me, there is geometry(both Euclidian and Non Euclidian), calculus, abstract algebra, the list goes on... I don't know what level you are at, or what you think your mathematical competency is like, but you could really do an essay on any of the major branches of mathematics. If you are at high school level though I would recommend trying to at least get the gyst of what is going on so that you can elaborate this in your essay rather than go through the entire full detail. Try and set yourself targets.

 

The reason I am giving this kind of advice is because I did a similiar kind of essay during my A levels on Qunatum theory, and this is basically a summary of some of the lessons I learned along the way when trying to do this kind of work.

 

Good luck with your essay:-)

[AmeliaC]

we have to be able to argue our topic and the essay has to be no more than 4000 words. and i'm at the high school level. i was thinking maybe geometry, but i'm not sure what to specifically investigate. any guidance is helpful

[abskebabs]

If it's geometry, then again there's a lot of choice. Basically you could pick Euclidian geometry(which is basic geometry which you will have been taught at school), or one of the Non Euclidian geometries. For your level you may find a number of the latter a considerable challenge, so that's your call. Not everything involving Non Euclidian geometry is beyond you though, if I were you I would try Taxicab geometry perhaps. Or even some basic topology. Actually some parts of fractal geometry are not too bad either, e.g. considering notions of self similiarity dimensionality etc. If you want to find out more about this then just type in google something like "middle thirds set" or "cantor dust", and you may find plenty on the subject matter. Actually self similarity dimensionality may get you quite a few too. Its your call though what you do...

[AmeliaC]

i think i need a bit of a challenge. and even if i dont pick the "harder" topics, then i would have at least learned a little bit about them.

[rigadin]

I can't answer your question because I need to know what you like in math. Personally I love graphics and geometry. I think it is really fun. But maybe you don't!!

[AmeliaC]

i really like geometry and algebra, i havent studied much else

[lakmilis]

AmeliaC, at your level , and you are fond of Geometry; perhaps you could look into Plato's perfect solids etc, read a bit about Archimedes and Euclid and you might find an interesting idea in mathematics which typically crosses into philosophy (typical trait of geomtry due to it's intuitively axiomatix proofs)

[Tom Mattson]

i really like geometry and algebra, i havent studied much else

 

Since you like those two subjects you may enjoy studying the synthesis of the two: analytic geometry. Here's an outline of what I think would be a great problem for you to work on.

 

Start with the equation of a circle in general form:

 

[math]Ax^2+Bx+Cy^2+Dy+E=0[/math]

 

Problem: Find the line tangent to the circle at some arbitrary point [math](x_1,y_1)[/math] on the circle.

 

Proceed as follows.

 

1.) Put the equation into standard form:

 

[math](x-h)^2+(y-k)^2=r^2[/math]

 

(Research "completing the square" for this).

 

2.) Find the equation of the line joining the center (which is the point [math](h,k)[/math]) and the point [math](x_1,y_1)[/math].

 

3.) Show that perpendicular lines have negative reciprocal slopes (requires trig).

 

4.) Find the equation of the line that is perpendicular to the line found in step 2 and that passes through [math](x_1,y_1)[/math].

 

You will at this point have found the line that is tangent to the circle at [math](x_1,y_1)[/math]. This problem is interesting because it is one of the few tangent line problems that can be solved without calculus.

[the tree]

This problem is interesting because it is one of the few tangent line problems that can be solved without calculus.

It is. What other ones are there?

 

Also, with step 3 isn't it o.k. to just accept that horizontal lines have negative reciprocal slopes without having to show why in this context?

[Tom Mattson]

It is. What other ones are there?

 

Off the top of my head the only one I can think of are tangent lines to a parabola. Specifically, the tangent line to a parabola at point P makes equal angles with:

 

1. the line passing through P and the focus, and

2. the axis of the parabola.

 

This is a theorem of course, so it requires a proof. But if you're learning this stuff in the sequence algebra/geometry-->trigonometry-->calculus, I suppose you could just define the tangent line to a parabola at P to be the line that satisfies those properties, and later show that it is identical to the tangent line from calculus.

 

Also, with step 3 isn't it o.k. to just accept that horizontal lines have negative reciprocal slopes without having to show why in this context?

 

You mean perpendicluar lines, right? The reason I say to prove it is because anyone who has studied mathematics through trigonometry can prove it. So why not do it?

[lakmilis]

ye but that would involve just picking up a calculus book containing it, and she could reproduce the proof. ok bladdi bla about it a little bit...but a 4000 word extended essay on it??

 

don't know

[the tree]

I just sketched the proof of the perpendicular lines thing (no calculus), and I imagine I could get 500 words out of that, probably more if I were explaining everything. And I don't think I've ever done that before, so there might be more proofs that I don't know of, so maybe there'd be even more to write about on just that.

 

Just running through the process and what each step means could be another 500 words, that's a thousand already. You could precede all of that by doing it with calculus and then say that the calculus you just used wasn't needed.

 

Go through some of the other tangent-line problems that Tom mentioned, you'd be surprised how quickly the word count fills up, some notes on the history of analytic geometry, who came up with it and what problem they were trying to solve, I don't think 4000 words would be too much to manage with this problem.