In Need of Interesting and Motivating Math Problems

[xaenn]

Hello all,

I'm in the process of working on developing a course to prepare students for calculus. The emphasis is to build up students' conceptual mathematical thinking, particularly that related to the notion of the function. Recognizing that motivation is one of the biggest issues to overcome, I am looking for is interesting problems that are related to the function concept, or other pre-calculus concepts (such as exponentials, trigonometry, algebra). The students are mostly going into engineering, so problems that are intriguing from a scientific standpoint are also welcome.

Any help is greatly appreciated!


-Xaenn

 

[arildno]

Well, to take trigonometry:
This is heavily used for distance/height measurements that future engineers ought to find useful.

Remember that, in general, angles are fairly easy to measure directly, whereas numerous length measurements (say the height of a mountain) are practically impossible to do directly.

To take a typical example:

You are to measure the height of some mountain M. Now, you can't bore yourself into the middle of the mountain and then take the elevator straight up to the top and measure the length traveled thus with the elevator!

But, something you CAN do, is the following:

From one point on the (suitably horizontal) ground A, measure the angle "u" the ray from the top of the mountain makes with the ground at A. Then move some easily measured distance d along the line of sight, as projected on the ground, till you reach some point B.
Measure there the angle v made between the line of sight with the ground.

Knowing the readily measurable quantities u,v and d, we can easily get our height h! (Height then understood as elevation above the horizontal line going through A and B.)


In order to motivate your students, it can be helpful to devise problems where they see how the skill of manipulating math can SIMPLIFY the task at hand; to get out quantities that are difficult to measure directly is one such way.

 

[skeptic2]

Along this same line, the way in which the altitude of Mt. Everest was determined is a fascinating story.

http://en.wikipedia.org/wiki/Mount_Everest

 

[Tac-Tics]

The emphasis is to build up students' conceptual mathematical thinking, particularly that related to the notion of the function.

Give physical examples. A ball's motion can be represented as a map from time to position. The state of a heated metal rod can be expressed as a function from position to temperature. A waveform is expressed as a function from position and time to height.

Just as importantly, give counter examples.

Not all equations have corresponding functions. The equation x^2 + y^2 = 1 is geometrically a circle, but there is no "function for a circle."

Functions are never multi-valued. Never ever ever. If they are, they are something more general, like a relation, or a function from stuff onto subsets of a set. I'm looking at you Square Root. (Drill it into their heads that the square root of a real number is always positive).

Maybe emphasize that functions can be defined in many different ways as long as each input has a single output. They can be defined algebraically, like polynomials, geometrically, like trig functions and logarithms, through a table, etc. They can be defined piecewise. They can be defined with properties of the input, such as "f(x) = 0 for rational x and f(x) = x for irrational x". They can be defined recursively or computationally with an algorithm, such as the fibonacci function f(0) = f(1) = 1, f(n) = f(n-1) + f(n-2) for n >= 2.

You might also want to mention a little bit about the notions of injections, surjections, and bijections. Bijections are an extremely important concept in higher-level mathematics. But even at the high school level, it explains many confusing phenomenon: why division by 0 isn't allowed, why $$\sqrt{x^2} \ne x$$ in general, and why the graphs of the inverse sine and inverse cosine functions look so puny (because they are only defined between 0<=x<=1 and they get "trimmed" during the inverse, due to their non-one-to-one-ness.