I’ve always planned to go back to school and finish my degree. I still plan to, but I do have a bit of a conundrum… I’ve already taken my first couple of years of Math classes two or three times. Unfortunately, Calculus is not the sort of thing that sticks with you unless you are using it. So like it or not, I’m pretty much looking at taking everything once again.
This time however, I am thinking to do it differently. This time I’m just going to pull the books from my own shelves and start working through them myself. The plan is, once I am able to actually get back to school, I will already have reviewed everything and still have it rattling around in my brain.
For those of you thinking I might actually be making a good plan, fear not! I have no intention of leaving off with so simple a plan. To help complicate matters, I will also take the time to learn some more tools of the trade. In order to present my work electronically, I shall endeavor to learn LaTeX and MathML. I will explore any math software I can find, including my old copy of Maple (if it will still install and run). I even found my old graphing calculators, a Casio fx-7000G and an HP 28C (with the printer!).
Now that sounds like a plan fraught with all kinds of mishaps and misadventures!
I decided to pretty much go all the way back and start with Algebra. I always remembered one of my Calculus teachers explaining to me most mistakes made in Calculus were due to mucking up the Algebra. I also figured, there would likely be less used parts of Algebra (and Geometry and Trigonometry for that matter) I would likely not remember and would be good to brush up on. Finally, I figured Algebra would provide me an easy way to start to testing out the other things I mentioned.
Who knew my re-learning would start not just with Chapter 1, but the inside cover of my Algebra book! And yet, there it was…
\[A=\frac{1}{2}h(b_{1}+b_{2})\]
The area of a Trapezoid.
Now, it was not the fact I’d forgotten this formula that caught my attention, nor was it the fact the reason this formula was correct was not instantly obvious. The interesting part was, after a moment of reflection, I realized it was just a simple derivation, but it required a small, algebraic “trick” to solve easily.
First, you had to realize the area of a trapezoid can be seen as the sum of two different shapes; a rectangle and a triangle. Imagine if you will, cutting out the rectangle in the middle, then shoving the two triangles left together to form a single triangle. The reason for doing this rather than calculating them as two separate triangles is because by merging them, we know the base of the triangle can be found by subtracting the shorter base from the longer base (the two parallel sides). Thus, the area of the trapezoid is the area of the rectangle plus the area of the triangle(s).
\[A=hb_{1}+\frac{1}{2}h(b_{2}-b_{1})\]First, we see the height (h) can be factored out.
\[A=h(b_{1}+\frac{1}{2}(b_{2}-b_{1}))\]Now initially, the next part just looks kinda icky. We can multiply the one-half through and do the math, but it just doesn’t feel right and of course, does not match the formula from the beginning. Here is where the trick comes in and is a precursor to the kind of thinking needed when we get back to Calculus. We remember our identity property and decide to multiple the shorter base by one. Or more specifically:
\[A=h(\frac{2}{2}b_{1}+\frac{1}{2}(b_{2}-b_{1}))\]Now things just got interesting as we can now factor out the one half. As an added bonus, the arithmetic left inside the parentheses now involves only whole numbers.
\[A=\frac{1}{2}h(2b_{1}+b_{2}-b_{1})\]Now it is just quick work to simplify and we arrive at the formula we desired.
\[A=\frac{1}{2}h(b_{1}+b_{2})\]
But the coolest part of all this you probably just glossed right over. Did you notice how pretty those formulas looked? Heck, they looked almost like they came right out of a textbook! That’s because the formulas were not just typed in. They were created using LaTeX and the MathJax-LaTeX plugin for WordPress. Right click on the formulas for added bonus features…
oooooohhhhhhh…..aaaaaaaahhhhhhh……shiney!
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