An exploration of the beauty and value of algebra in apparently geometric problems

@solvemymaths asks this question:

What fraction is shaded?

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.
screen-shot-2016-12-01-at-2-40-59-pmOneFromPythagoras

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:

and

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).

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