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Before we get to the rectangle, first we should look at the table you built for the square. Make sure you keep it. You'll want to refer back to it as we go along with our inventing. Great inventors, like Thomas Edison or Grace Hopper, always keep notebooks full of their scribbles. So do mathematicians. It's a good habit you may as well start here and now.
So here's my table:
| + | SP | RF | AF | LF |
|---|---|---|---|---|
| SP | SP | RF | AF | LF |
| RF | RF | AF | LF | SP |
| AF | AF | LF | SP | RF |
| LF | LF | SP | RF | AF |
I hope your table looks like this. If it does, send me an e-mail, and I'll send you a gold star. If your table doesn't look like mine, I probably didn't explain it very well, so send me an e-mail about what I should have told you that I didn't, then I'll send you a gold star too.
There are just a few things I want to point out about the table before we move on:
There's more to say about this, but first I want to experiment with a rectangle. So stop now for a few minutes and build yourself a model like this rectangle. But before you go, you need to know that we're going to being flipping the rectangle over, so when you build it, the red corner has to be red on both sides, front and back, and the same for the other three corners. Okay, here's the new model. See you back here in a few minutes.
-- So, what are we doing now with the rectangle?
Good question. This is Step 8. Collect more data. One important way that mathematicians look for new math is by what we call "relaxing the constraints." Boy, that's a mouthful of twenty-five cent words, isn't it!
What I mean is that we take something we already know, like rotating a square, and see what happens with something that's like a square, but is different somehow. The rectangle has 4 sides and 4 right angles, but it's different from a square because not all the sides are the same length. Let's see what happens.
Well, right away we have a problem. Remember the rule we made for rotations? The figure has to fit back into the same "shadow".
So RIGHT FACE is not a legal move with a rectangle. It doesn't fit into the same shadow, and it doesn't end up with the corners in the corners. Now, I may not fully understand what this means yet, but that's okay. This is not a race. No need to jump to conclusions. Right now we need more evidence. But I suspect that something different is going to happen with the rectangle.
STAY PUT is no problem. LEFT FACE will have the same problem as RIGHT FACE. What about ABOUT FACE?
No problem there. In our "shorthand" it looks like this:
If you think this looks familiar, it is exactly the same as for the square, and you may want to check back to refresh your memory.
-- Wait a minute. What good is our "shorthand" if you can't tell the difference between a square and a rectangle?
Excellent question. Now you're thinking like a mathematician! Here's what's going on. We are abstracting certain details from all the facts about these motions. Now, what does this "abstracting" mean? It means that out of all the details of the motion of the square and the rectangle, we are deliberately ignoring some things, many things, and pulling out, or abstracting, only the details of interest. It doesn't matter if you made your figures out of cardboard or construction paper, painted them or used crayons or magic markers. It doesn't matter if you use bright red or dark red. It doesn't matter if you turn your figure quickly or slowly. For our purposes, we are ABSTRACTING only the details about how the corners are rearranged by certain motions.
-- So what's so terrific about abstraction? Isn't it just a way to take something simple and make it complicated and hard to understand?
Another excellent question, and sometimes it does seem like that. But here's the most important discovery I hope you make during your visit to these pages. Abstraction is the key to the power of mathematics
-- Gee, that sounds terrific, but what does it mean?
It means that by ignoring some of the details, we mathematicians can begin to see structural similarities in things that, on the surface, look very different. Did you ever see an x-ray of an elephant's foot? Elephants have the same bones in their feet that we do. They may be bigger, and some of them are shaped a little differently, but they have the same skeletal structure.
-- So what? What's so powerful about that?
Well, for one thing, it means that if you know the bones in your own foot, you already know the bones in an elephant's foot. When it comes to foot bones, you and I have something in common with Dumbo.
-- I thought this was about math. Now you're talking about elephant skeletons?
Let me put it this way. Every time you read a page in a book, you are reading sentences that you have never seen before. But you understand them. In the same way, you when you learn some fundamental, abstract, mathematical ideas, you can use them to understand new mathematical ideas that you have never seen before.
So, abstraction is your friend. We'll see about face many times, dressed up in lots of different costumes, pretending to be special, but you'll recognize the same old about face. Once you know about face, you know it, no matter how it may be disguised. You don't have to keep learning it over again. Abstraction cuts down on the wear and tear on my brain. Time to move on.
So far, with the rectangle we have stay put and about face. Is that all there is? Maybe it's time for the Spudmeister's secret weapon. Remember Step 1? Ask a question. What if ... we flip the rectangle over face down? Try it, and see what happens? Is it a legal move - does the rectangle still fit into the same shadow? How many ways are there to flip the rectangle? Can you write down the "shorthand" notation of the motion? You take a minute to experiment with your model, and then scroll on down and we'll continue.
Ready? Well, here's what I found. I think there are 2 different ways to flip a rectangle. There's one flip that swaps the upstairs and downstairs. You pick up the top edge of the rectangle, and flip it over to the bottom, like this...
You can see a small animation of a flip if you'd like, but it's not a requirement. I just did it to see if I could. I know a lot of times the clock is ticking and you're in a hurry to finish your assignment, but it's good to stop once in a while and see if you can have some fun with what you're learning. I know it works with math, but it probably works with other subjects, except math is more fun.
... and then there's a different flip that swaps the left and right sides, like this...
-- Hey, wait a minute! How do you know it's different? Different from what?
Good questions. Excellent questions. It reminds me of another one of the mathematicians' secrets I want to give away to you. Be skeptical! Nothing in math is true just because the teacher says so. It is always your right to demand to be convinced. So the next time I just say that "so-and-so is thus-and-such" I want to hear you yelling out "Oh yeah? Prove it!" Okay?
So let's analyze what I claim are new motions. The first one, that I'm going to call an UP/DOWN FLIP, looks like this:
Red and Yellow trade places, and Green and Blue trade places.
In our "shorthand" notation, it looks like this:
If you compare this to what we have already done, you'll see that this is a new motion. It's different from anything we have done so far. That's my proof that it's different. Look at the other motions, and none of them match, so this one is different.
The other new motion is what I call the LEFT/RIGHT FLIP. It looks like this:
The description goes like this: "If I flip the rectangle over sideways, while keeping the bottom edge at the bottom, then Red and Green trade places, and Yellow and Blue trade places.
We abstract the key information from that description, and we write:
Now it's time to record our data. I'd like you to build a table for the motions of the rectangle. I'm going to use these abbreviations so I don't have to write so much in the little boxes:
You've done this before for the square, so I think you can figure this one out without too much help. I'm going to go do MY homework now.
You make your table for the rectangle, and then we can compare notes. Just click on the link to continue when you're ready.
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