Patterns

Three Hundred Eighty Fourth Post: Patterns

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Often young students are confused by math. In early grades in school a lot of the work involves learning vocabulary words and remembering facts. Math has these same skills but to a lesser extent. In math reading comprehension, imagination, observation, and problem solving skills are stressed. However it can still be confusing to young students about the importance of learning math and the reasons why the math work is supported with proofs.

Why must we learn from proofs. The simple answer is that we are trying to learn math the same way that the mathematician created the “math work.” The mathematician left these steps for a reason and that reason is so we can recreate his or her work by following these steps. That proof is in the universal language of math. It would be difficult to record thoughts and all the ideas considered while creating the math theory, but with this proof it translates those ideas into one’s one ideas. One can take their own meaning and understanding from the proof. That is why math is such a powerful language.

Something happens when we group the proofs and observations together. This helps us see relationships and leads to new discoveries. We are looking for new patterns. Of course this is easier with the Internet. Traditionally it involved finding some good math books in the library. Wikipedia.org is an invaluable source of mathematical knowledge. But it is once we learn from the knowledge that came before, that we can create something new.

Now I suppose you want an example.

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I had a simple observation about parabolas that is so simple it is easily missed. And it would not be of use if I hadn’t worked on other properties of parabolas. That is specifically how conic sections such as a circle and parabola are related and how their functions (such as circular functions (sine and cosine)) also relate.

So here is a rough theory. It could be wrong since I really haven’t put it to any test. “It’s just a hunch.”

I see no reason why cutting and joining a parabola with its mirror (inverse) couldn’t always create an ellipse. It would have different lengths of x and y axis’s causing the focus’s to change accordingly ( or stay the same if it is part of the same parabola.)

Once the axis are known (or focus’s known) the relative coordinates of the parabola could be found. Which is valuable since we have knowledge of the “Parabola Key.” It also (though more difficult to find) may lead to an alternative to the quadratic equation.

That is why this work with parabolas, though simple, is so interesting to work with. Learning knew properties leads to knew ways of describing the conic sections’ equations.