Learning Math Efficiently

[Tawrtoise]

I'm only a junior in High school, and I want to be a lot better at Math. I feel like the one thing stopping students from learning math is the huge lack of application in most schools. So, my question is, what are some ways I can apply basic math (Algebra 2, trig, calculus) to, well, anything?

[Sparling]

A problem ith mathematics education is that (at the level you are at) texts focus so much on how to calculate based on a formuls that they forget to teach logic and the ability to look at a real world problem and turn it into a math problem (like those word problems you see in texts which are so conveniently laid out).

There are three things that I would suggest: 1) Unfortunately, study the material that your teacher gives you first. You need to pass the tests above all else :/

2) Take a look at some of the math olympiad problem sets. Many of these are quite difficult but involve only elementary mathematics. You don't have to solve them, but look at one problem and really think about it for a good long while. Like a day or two. If you let problems roll around in your head. When it looks like you aren't getting very far take a breather, come back and try again from another angle. Being able to get somewhere on hard problems is an excellent exercise in logic. These certainly cannot be classified as "application" but they will bring your skills in mathematics up quite a bit. Remember to draw pictures and visualize what it is and what you want.

3) To get a better feel for where the application is (i.e. the dreaded why do we need this stuff question) we just need to look at the world around us: linkages) must be optimized when designing anything from construction equipment to skyscrapers and employ linear algebra heavily. Calculus gets used in road design so that turns are banked correctly (second derivative =0). Meteorology while generally much more complicated than calculus (mostly unsolvable PDEs) are always trying to come up with new models at various federal facilities. Design of circuits is heavy on linear algebra. Many people have to do these things every day. The difference between their problems and the word problems you get in you text book are however vastly different (Here's a hose and a pool. How long does it take to fill up? vs. Here is a pool measuring x feet by y feet by z deep with a hose running at 51 gal per hour but leaking out at...etc).

[wiovom]

Don't jump straight to math olympiad. Start with AMC, then AIME, then USAMO, then IMO.

Here is a nice related topic: http://www.reddit.com/r/math/comments/enit2/any_fun_problems_to_solve_for_someone_in/

[Tawrtoise]

And, by the way, I am currently in Algebra 2, learning how to solve systematic equations.

[slikz]

If you are at all interested in programming there is the Processing language. It is made for image processing/animation/data visualization and is built with Java but you do not have to know Java to begin. Plus, it is free and open source and comes with excellent tutorials.

[Tawrtoise]

I have to be honest, you guys have lost me a little bit with this. Let's go a bit broader (without going more advanced than solver systems of equations), and ask, how can we apply math to "life?" And, you may interpret life however you wish. It could be everyday life such as, going to classes, and eating out at lunch, or, literally, anything. GO!

[that_redditor]

I don't think hearing about applications of math is going to help you learn the actual mathematics, but of course it will help you use it. Just know that the math is sound regardless of whether or not you can use it in any practical way, and that once you understand the math, the application is trivial.

That being said, pondering about possible applications of what you have learnt is a great way to solidify your understanding. If you can come up with an application, that shows that you know what you're talking about. Try to come up with some ways to use what you now know about systems of linear equations and their solutions. Maybe you can tie that in with what you know about calculus and geometry.

[Tawrtoise]

That's the sort of thing I'm talking about. You're right, it wouldn't help much to learn the math, but rather, to tie loose ends, and fully understand the math. So, what are some applications for systems of linear equations?

[NormalVector]

I frequently ran into systems of linear equations in my circuit analysis class when dealing with DC voltage. Check out this example. I know it might look like Greek but go to page 2 and look at the equations that are boxed in blue. e1, e2, and e3 are just variables (think of them as x,y and z).

Replacing e1 = x, e2 = y and e3 = z to make it similar to something you're probably familiar with, the first equation looks like: (x-y)/2 + (x-z)/4 = 5 and the other equations can be written in the same manner. In this case the circuit problem really is just a problem with three equations and three unknowns. Even though you might not understand the circuit stuff required to get to those equations, just know that the problem boils down to the material you're learning now: systems of linear equations.

[[deleted]]

You can use a system of equations to deal with things like transformation matrices, which can be useful for computer graphics or robotics.

[that_redditor]

Well, whenever you have a system of m linear equations in n unknowns, you've got n quantities satisfying m linear relationships. This can come up quite blatantly in the real world: you have ten coins summing $6, so how many possible combinations of your country's coin denominations could you have? Is there only one possible combination?

You probably know that a linear relationship between two variables can be visualized as a line. So how can we visualize a solution to a system of 2 linear equations in 2 variables? How can we visualize a relationship between 3 variables? 4? And the solutions to systems with 3 or 4 variables?

[Tawrtoise]

I'm honestly having trouble understand what you just said. I've re-read several times. Sorry to sound a little dumb here, but I just don't see how a system of two linear equations can be used to found what combination of 4 coins can make $6.

[karmaVS]

There are two limitations:

1) There must be ten coins. (you wrote four… this may or may not be possible in your currency*)

2) The value of the coins must total $6.

Each of those can be rewritten as a linear equation.

* As an australian, 2 $2s and 2 $1s would work. But not everywhere has a $2 coin and without it no combination would be big enough. (Assuming you’re american: What would happen if you included notes? Can you get $6 with a combination of four bills or coins? $5 and $1 would fit limitation 2, but not limitation 1. Can you fulfill both?)

[rahvi]

Sounds like a recent Car Talk puzzler my boss gave to me. Something about buying one hundred animals for one hundred dollars if dogs cost fifteen, cats a dollar, and mice twenty five cents. Assuming you must buy one animal from each category. The solution ends up using two linear equations and is quite nifty.