Note: this is really designed as a sample page for my fifth grade daughter to practice order of operations on. If you don’t have a fifth grader studying order of operations, you may not find this to be riveting information:
My daughter is studying order of operations right now, and I made her a page of problems in Graspable Math to solve, and thought I’d share them publicly, in case someone finds them useful someday.
If you don’t know Graspable Math, the big thing you should know is that it’s a dynamic software–it let’s you control the actions taken in an algebraic setting, but you don’t have to write them yourself. The program was written and conceived by Erik Weitnauer, Erin Ottmar, myself, and some other folks. It allows you to do actions that aren’t strictly PEMDAS, as long as the action yields the PEMDAS-approved answer, so YMMV, depending on your personal pedagogical goals. Here’s a video demonstrating the basics of this particular page:
Viviani’s theorem is this really cool proof about equilateral triangles. I’m not sure who first proved it (it’s named for Viviani, obviously). It states that if you take any point inside the triangle, and draw the shortest straight lines to the sides, the sum of the three resulting lines is equal to the height of the triangle.
I’ve been playing around with making representations and demos in Graspable Math, a research project and free teaching tool my post-doc Erik Weitnauer, Erin Ottmar, and I are making together (all beta disclaimers apply). I thought I’d make a quick proof of Viviani’s theorem, and I was pretty pleased with the result. If you want to play with it yourself, here’s a link to the canvas–but be forewarned, as of Dec 1, 2016, there’s some glitch with our saving and loading, which breaks some of the links. You’re better off deriving the proof yourself next to the proof that loads. I demo that here.
Viviani’s theorem has some funny implications. For instance, barycentric coordinate plots–the coolest way to plot three values constrained to a constant sum–couldn’t work without them.
Beauty powered by Viviani
These plots come up all the time in my work, because we often have tasks where subjects have to choose among three items. They are also the best way to think about how you pay your money to humble bundle: you gotta pay all your money, so the sum is fixed, but the values are ‘free’ to vary.
Giving money to humble bundle is a good idea, no matter where in the triangle you end up.
It’s pretty clear from counting triangles that the answer is 1/8, but I like algebra, so I made a quick solution in Graspable Math, and learned something from it, so I thought I’d share.
Here’s the basic idea:
So at first glance, this is one of those proofs that feels like drudgery–didn’t the geometric flipping thing make more sense?–But the great thing about algebra is that it gives you new insights you wouldn’t have probably had otherwise. Let me ask you this: why is it 1/8th? Why did this construction lead to that ratio? Counting provides little insight–it just is. now look at the algebraic proof, and you get a quick hint: The 8 came from three factors of 2.
Two of these 2’s appeared from the size difference between the large and small squares, and one appeared because we used the pythagorean theorem–essentially because we rotated the square while keeping it bounded in extent.
We can picture it like this: we started with the big square, then we shrunk it down, and rotated it in a particular way (keeping its total width fixed). The first of those operations took the area down by a factor of four, and the second by a factor of two.
This immediately suggests, and tells us the answer to, two related problems, both of the "find the shaded area" type:
Some people say that algebra ‘proves things rigorously’. Maybe, but the real advantage of algebra is that it gives you the opportunity to see things in new ways, that help you understand why things are the way they are. Of course there are lots of answers to the ‘why’ question, and indeed lots of ways to get to the three magical 2’s in this example. But the insights we get from algebra tend to be compelling, non-obvious, and powerful. And that’s what algebra is for (okay, that and a lot of other things too).