pythagoras – Welcome to infinitefreetime dot com

I have removed the Second Skin from my new tattoo, and the itching is absolutely maddening, so I’m going to distract myself with math. Because that’s why you come here, right? As a reminder, this is the original image, and the question is the ratio of the inner square to the outer square:

The first thing we’re going to do is draw the two diagonals of the inner square. These are, by definition, perpendicular to each other, and they are also equal to the circumference of the circle. Let us define the radius of the circle as x:

What we have now is four right triangles inscribed inside the circle. Pythagoras tells us that the sum of the squares of the two legs are equal to the square of the hypotenuse, which is the line on the left of the square there. Therefore, defining the hypotenuse as Y, we get:

x² + x² = y²
2x² = y²

Take the square roots of each side, and we get:

√(2x²) = √(y²)

And therefore:

√(2x²) = y

Which means that all four sides of the inner circle are equal to √(2x²), thusly:

To get the area of the inner square, all we have to do is multiply √(2x²) by √(2x²), which, conveniently, just gets rid of the square root symbols. The area of the inner circle is 2x².

Now, we need to realize that since the radius of the circle is x, the diameter of the circle is 2x, and that the diameter of the circle also equals the width and the height of the outside square. So that outer square is 2x high and 2x wide:

Therefore, all we have to do to get the area of the outside square is multiply 2x by 2x, which gives us 4x². Which, conveniently, is exactly twice the area of the inner square, which was 2x².

The outside square is therefore twice the size of the inner square, and the ratio of the inner square to the outer square is 1:2.

Or, y’know, you could just rotate the fuckin’ inside square, which makes it visually obvious.

I did something today that I’ve never done in twenty years of teaching– I would estimate, without a shred of exaggeration, that 2/3 of the teaching I did during my fourth hour was in Spanish. It was time to sit down with my newcomers and see where they were at, and the only way to do that was to communicate with them in their own language. To wit, I generated this for them:

And then I banished about half of the class from the room, sending them with my co-teacher to her classroom, mostly to cut down on the number of other kids who might want to talk to me and also to prevent a certain student from getting Valentine’s Day-related harassment, and sat down with the kids and went through a bunch of problems with them. I’m hoping that document is translated well from what I typed; based on my meager Spanish it looked okay, and the kids didn’t have questions. The boy read through it, smiled at me, and proceeded to get nearly a perfect score on his assignment with only a small number of questions, all of which, I’m proud to say, I understood; the girls are a little bit behind for 8th grade but not enough that I’m terribly concerned about it. I have English-speaking kids who, based on this one assignment, have bigger problems than they do. One of them does seem to rely kind of heavily on the other, who did most of the talking and also appeared to do the lion’s share of the work, but we’ll see how that shakes out in a couple of weeks.

You may notice, even if you don’t read Spanish, that the actual Pythagorean Theorem doesn’t appear anywhere in that document. That’s entirely intentional; I generally deemphasize the formula itself in favor of the process of figuring out a missing leg or a missing hypotenuse. They know the formula, but I treat this as mostly calculator work, and I drill the phrases “square-square-add-square root” and “square-square-subtract-square root” into their heads until they’re repeating them in their sleep. Since I didn’t have any real idea where these kids might have been in terms of their math skills I decided I’d leave it out entirely for now.

We are taking it easy tomorrow, across the board. I kinda feel like I’ve worked the kids (all of them, not just the new ones) like dogs this week, and between talking a lot more than usual and the added stress of teaching in a foreign language today, I’m ready for a day where I can wave them vaguely in the direction of a Quizizz or something else that has a chance of being fun rather than being at the board or hunched over someone’s shoulders all day. They’re picking this up pretty well so far so I think if I have a calm Thursday before a four-day weekend God will forgive me.