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The Exploration


freshfaced333

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Hi all,

 

I had a thread about a possible exploration idea, but I scrapped that idea, as I was afraid I might venture too deeply into economics, which is something I do not want to happen since this is a math exploration.

 

I have always wanted to utilize calculus in my IA, and just find rates or something, but I am having so much trouble just looking for a real-life situation that I can use to model a function or anything. I am DESPERATE to have an IA topic by the end of next week, and I am willing to just list down all of my ideas and hopefully someone has a suggestion as to how I can expand on an idea? (I'm not sure if this is really allowed on this forum, so if I am breaking an rules, please just delete this post!)

 

An idea I have involves crowd counting, and how mathematicians actually formulate an equation to count the number of people in a crowd, instead of simply estimating. I thought that this idea has potential, as it involves quite a lot of math. The thing is, I'm afraid that it contains 'basic' math, and I need something that can be used at the HL level. I got my idea from this article: http://www.popularmechanics.com/science/the-curious-science-of-counting-a-crowd and just thought it was pretty cool.

 

Another idea I had was the mathematic in farming. I think that there is a lot to be explored in this topic, for example, how much soil a farmer needs, how much water he needs to water the crops, how far apart to plant the seeds, etc. I was thinking that optimization might be useful in that I can use it to determine max area of field, max volume of water etc? But then again, the math needs to be more advanced, HL level.

 

Finally a third idea I had was the graph theory, with the seven bridges of konigsberg. http://mathworld.wolfram.com/KoenigsbergBridgeProblem.html after looking around, I thought that this could potentially have a lot of advanced math, but I just have no idea as to what to explore!

 

As you can see, I am willing to just keep digging and trying to find an idea, and start, but I'm about to be burned out very soon. To this date, I have spend about 30 hours over 3 weeks just looking through the internet for an idea.

 

thank you so much!

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Calculus is extremely useful, you could always find something related to it whatever you are interested in, it's really not that hard to do so.

The first link does seem elementary, it's based on the fact that density=(total mass/people/etc)/(volume/area/etc) and I don't think you have anything to actually explore with it.
The second one is still quite vague, but yes you could use optimization with almost anything, you just need a situation that's more than a homework problem.
For the third one, I don't know if you've done the discrete mathematics option or not, but it is covered in said option, or rather, the concepts behind it is covered in the option. The seven bridges is just a problem of finding a Euler path for a graph, and since the graph doesn't satisfy the conditions for a Euler path it is impossible to cover all bridges once and only once.
Assuming you know something about graph theory the problem is rather just an ordinary problem with some interesting historical background. However you can try proving Euler's theorem on this topic, but if you don't know anything about the option (I'd assume so as otherwise you won't come out with the idea) it might be difficult to work on.

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Regarding the third option, we are not doing discrete math as an option, so I guess I'll scrap that idea. It is so difficult to find a situation that can be modelled into a complex situation suited for a math HL level! now that you have kinda 'deconstructed' my ideas, I feel like I need to look for more stuff now! I totally am having the 'homework problem' thing where I can find a situation and apply something I have learnt, but not being able to extend it into a full paper - I can solve the question, but I can't find other situations to model, and I find it quite difficult to formulate an aim and rationale for the exploration. my teacher stressed that aim+rationale is essential to establish before I even start doing anything.. 

 

anyone else has suggestions pretty please?  :blowkiss:

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Finding a suitable topic worth of exploration is probably the most difficult task when doing Math IA in my opinion. I think I really spent 75% of my total time just to find a good topic.

 

With your first 2 ideas, i see really no potentials in them. Even though there can be many things to explore here (which may be quite useful), but it is really not up to the advanced level. As ssy already said in the above post, optimization can be a good topic only if you find a problem that is complex enough. Notice that optimization is also part of the SL Math as well!

 

The 3rd topic seems to be very interesting. However considering that I've done the Discrete Math Option, I don't think the '7 bridges of konigsberg' alone can take up to 10 pages of exploration. I mean it's an extremely easy problem once you read a few pages of graph theory. It's also a 'limited' topic, because i don't think there's really an alternative way (other than the common way) of solving this problem, so you are left with little spaces to explore. However, you can always try to extend this topic. For example, figuring out how to apply the solving method of this problem to computer science...... But yeah, I think this topic is not worth of 10 pages exploration.

 

If you like Discrete Math, you may however want to research a bit about the 'Chinese Postman Problem'. This problem is kind a similar to the '7 bridges of konigsberg', but much much harder. Instead of finding the shortest 'Eulerian path' as in the '7 bridges' problem, you now have to find the shortest 'Hamiltonian cycle', which is way harder, even for a computer to do (if you have a huge graph). Thus it is considered to be one of the most difficult problem within graph theory; and if I am not mistaken, the Chinese Postman problem hasn't been solved to a fully satisfied extend, so here you can discuss and then evaluate different methods of approaching the problem.

 

Calculus also has a huge potential for you to write a good IA. However, don't limit yourself to just the simple problems of 'finding the rates' or 'differentiation'; you need to find something that is good for exploring. Because Calculus is a huge area of mathematics, I suggest that you look for a topic on smaller branches of calculus. For example, in calculus, there are Differential Equations (which is perfect if you are into science), Partial Differential Equations (extremely useful in all areas of life), Calculus of Variations, etc. Google each one of them, and see if there are any good problems that you can explore!!!

 

Note that all of the calculus topics mentioned above are not only up to the level of Math HL, but they are all extended outside the HL syllabus. So they have lots of potential. But there're also areas of calculus that stay within the syllabus, but not too simple, such as 'Integration by Partial Fraction, Integration by Part, Volumes of Revolution, etc. One of the topic i found interesting when i was looking for my own topic was to use calculus to model water drainage from a sink (you might notice that depending on the shape of the sink, the water is drained in different way; at least for most of the sinks that i saw, the water always follows a clockwise circular path before it is drained out of the sink). You might want to look into that :)

 

Finding a good topic is not always easy. But once you found a suitable topic that you're really passionate about, it takes only a little time to write a complete IA. I spent 4 weeks of searching for a topic, but only 1 week of writing it. The same went with my EE :P

But good luck to you! :)

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That's exactly what I did for my IA - integration for the Fourier Series. 

 

Study it a bit on your own, and if needed, talk to your teacher to try and understand the key concepts. 

This really helps with the use of Mathematics criterion -  the sophistication of mathematics 

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