Day 1
The Koch Curve is a fractal, generated by repeatedly following a simple rule: everywhere you see a line segment, erase the middle third of the segment, then draw the other two sides of an equilateral triangle whose base is the missing segment. It starts with a single line segment, then becomes 4 line segments, with one triangle. In the 2nd step, it becomes 16 line segments, with 5 little triangles growing off of it.
- Draw several more iterations of the fractal. Measure how long the line segments are each time. Count how many segments there are. Count the triangles. Figure out the total length of all the line segments. Make a table showing all of thse values.
- Build the table in Desmos and look at the resulting patterns. Can you find an equation that fits any of the patterns? What kinds of equations should you try? Why is it challenging to find an equation?
- Can you at least find a way to calculate/predict how many line segments there will be in the next step? Like if you know that there are 16 segments at step 2, and you know how to follow the rules to construct step 3, can you use that to predict how many segments there should be at step 3?
Day 2
The Koch curve question was pretty challenging, wasn't it? We don't really have a satisfactory answer yet. But just looking at the graphs we made, it seems pretty clear that none of them are straight lines. Why not? Because the number of new line segments doesn't increase by the same amount every step. Nor does the length of those lines change by the same amount every time. With the equations we've been doing so far, in the form $y = \mbox{something} x + \mbox{something}$, the number next to the x cause the line to increase or decrease by the same amount every time you increase the x. That's what it means to be straight.
So what do we do with this question that we don't know how to answer? Is it unsolvable, just because we haven't figured it out yet? Let's think about another example: you're all familiar with the Pythagorean theorem, that for a right triangle, the lengths of the sides of the triangle always have a simple relationship $a^2 + b^2 = c^2$. You've also seen that there are certain sets of three numbers that fit that equation perfectly, like 3, 4, and 5 (because $3^2 + 4^2 = 9 + 16 = 25 = 5^2$). Other sets are 6,8, and 10; 5, 12, and 13; and many more. Apparently many of you even memorized some of those sets in your geometry class.
Well there was a very old question in math, coming from the 16th century -- does that equation ever work for integers when the exponent is 3? As in, is there an extension of the Pythagorean theorem that looks like $a^3 + b^3 = c^3$ that works for special sets of integers? It's a really old question, and no one has ever been able to find any sets of numbers that worked. No one has ever found sets that work when the exponent is 4 either -- like $a^4 + b^4 = c^4$, or for 5, or for 6, or ... for any other exponent than 2. When mathematicians notice that a particular problem is unusually difficult, and surprising, they begin to wonder whether there is some deeper reason for it. Maybe it's because they are used to being clever and figuring things out quickly -- when it's more challenging, they start to wonder "Maybe it's impossible!" So a mathematician named Fermat suggested that there aren't any other exponents for which the Pythagorean theorem works. That it really is impossible, and he claimed to have a way to prove it -- meaning that he believed he understood why it couldn't work.
But he never wrote it down.
So generations of mathematicians tried to figure out the problem, believing that Fermat was probably right, but not being able to figure out why it didn't work, or to prove that it couldn't. Every important mathematician from the 16th century into the 20th tried to solve that question. And none of them succeeded.
So they began to wonder if this was a question that just couldn't be solved -- couldn't be proven one way or another. They started to use computers to search for enormous numbers that might fit, or to look for patterns that might explain why they couldn't find any. Most of them gave up on the problem. Decided that it was a dumb question, or that they weren't smart enough to do it, or that it was just too hard for humans.
So in the 1980's a mathematician from England named Andrew Wiles started working on it, but by this point it was considered kind of a crackpot kind of question to work on, so he didn't tell anyone (which is pretty unusual in mathematics -- it is a very collaborative field). He spent years working on it. Making a lot of progress in figuring out why it couldn't work. Close enough that he thought he might have the solution, and then he got afraid that he might have made a mistake. Working together is really important because talking to other people helps to point out mistakes. Helps to avoid blind spots. For the same reason you should have someone else proofread really important essays or letter.
So he asked some colleagues at Princeton, and they organized a class for graduate students, and Wiles started presenting the proof as though it were a class, but without telling the students what he was really doing. And it was such a hard class that all of the students dropped out,leaving just other professors sitting in, helping Wiles check his work.
When he finished, he organized a talk at Cambridge, and presented his proof over the the course of several days. By the time it was finished, the mathematical world had gotten wind of what was going on. The last presentation was to a packed audience, and he got a standing ovation when he finished -- and some champagne. And a lot of awards. And a knighthood.
What's the point? Not the trite point that "you too can become great if you work hard" (even though it is true), but the more important point that Some problems really are hard. They take more than 2 minutes to solve. And sometimes we just don't know enough yet to do it. Persistence matters.
So our exploration of the Koch Curve -- we're going to have to table it for a while. We don't know enough yet, but we will. But we do understand that equations of lines won't work. We need something more sophisticated than that, and when we start to build up those skills, we can return to this question and get deeper into it.
- Return quizzes, how to do retakes
- Create a linear model that describes Mark walking around the room at a constant speed. graph it.
Day 3
Construct graphs using desmos (an online app at desmos.com) that show the following:
- A person walking at one step per second. The y-axis should show how far they get, the x-axis how many seconds they walk. Make a table of values and the equation that matches it using the app.
- A person walking at two steps per second.
- Make one up for a different speed.
- Make one that shows someone walking backward
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