About Michael Pershan

Teaching math in NYC.

A fun problem, as long as you know the Pythagorean Theorem

I wanted to start the year off with this, until I realized (duh) that it really uses the Pythagorean Theorem.

Screen Shot 2013-08-29 at 10.51.34 AM

 

But it’s a great problem, and so rich for spinning off. What about triangles? What about pentagons? What about irregular triangles? Can you always gets a circle around the polygon, no matter where you start? (An honest-to-god context for drawing a circle through three points.)

More than ever, I’d like to really focus on problem posing this year, and this problem is very rich for jumping off from. I love it, and can’t wait to bring it in to class.

Bendy Straws and Quadrilaterals

IMG_2964

What shapes* can you make with the bendy straws?

* Do the ends have to connect? We can either require that the ends connect, or not. 

You can try to make a triangle.

IMG_2965

But can you always make a triangle out of the two straws? What requirements are there? Did you try extending the straw out all the way? Did you try scrunching it in?

IMG_2969

I see a shape there, but I notice that the ends don’t connect perfectly. Can you make them connect perfectly?  IMG_2967

IMG_2966

What categories of shapes are we able to make if the ends have to connect perfectly? (And we can use a bit of tape to make sure that the ends are sticking together.)

Are there more categories? How do we know?

What’s special about these shapes? What happens if you flip one of these shapes onto its back side? What changes? Why?

I think that this might be a nice mini-investigation that could motivate some proofy concerns.

Game: Polygon Races

Rules: Each partner gets a board that looks like this, and you win the race by crossing the edge (any edge) of the box before your opponent does.

inscribed polygon game

Inside the circle, you’re going to draw a shape. Maybe you’ll draw a triangle, or a square, or something else. The next thing that you draw is going to be the same shape, but it has to touch all the corners of your original shape.

So, if your first move looks like this:

move 1

Then your second move can look like this:

move 2

 

And your third move will be a third square, touching all four points of the second square. If you keep on drawing squares, eventually you’ll hit the edge. If you get there before your opponent does, you win.

I’d have kids play this game a couple of times and have them figure out strategies. I’d encourage them to try various shapes, and see if any of them get them to the edge faster. If a kid thinks that certain shapes are winners, I’d encourage them to explain why.

That’s one possible opener for a series of days investigating inscribed polygons.

Parallel Line and Tranversals

Standard:angle chasing

Flipped:

Make your own parallel line puzzle. Make an easy one and a hard one.

Meta:

What’s the minimum number of angles that you have to provide in a puzzle with 2 intersecting pairs of parallel lines?

There’s a worthwhile impulse here, but I don’t like the way the meta-problem turned out. But I think that there’s something really worthwhile here, the notion that after solving a problem we then study the class of problems itself.

Sequences of Inscribed Polygons

I had an incredibly rich mathematical investigation this week, and I’m trying to collect all of the math that I learned from it. I’m also attempting to record it as a sort of memoir of the week, in the hopes that I can work out how to help kids pursue these sorts of investigations. (Since, ultimately, I care more about whether kids can do this sort of noodling around than any of the particular mathematical content that they’ll see in a Geometry course.)

I started by collecting a library of hexagons, and then I discovered something interesting about regular hexagons. After generalizing to regular n-gons, I scribbled down a few questions in my notebook:

  1. Does the generalization hold for star-polygons?
  2. Is there a way to use this to find exact values for trig values i.e. cos(72)
  3. The ratio between a polygon and it’s midpoint-inscribed child approaches 1. What’s a geometric interpretation of this?
  4. How quickly does it approach 1?
  5. Does this not just work for regular polygons, but for all convex polygons? Concave polygons?

I tried 2 first, but it didn’t really get me anywhere. Then I thought about 3, and the way that if you had a ton of vertices and edges, it would basically look like a circle, and that it would still look like a circle if you connected its midpoints. I graphed my general formula for n-gons, and I saw visually how quickly it approached 1.

Then I dove into the convex/concave question. I took a concave polygon from my library (an “hourglass”) and connected its midpoints:notebook1

This was enough to yield a conjecture: that every sequence of inscribed hexagons constructed in this way ends up being entirely convex. As I struggled with my incredibly shoddy drawings, I realized that this would all be better done on a computer.

So I built a thing in Processing:

[Which is not showing up for some reason. Working on the embedding. Until then, just go here.]

I felt increasingly confident that the conjecture was true, but I still didn’t have a proof. I was able to come up with an argument that every convex polygon would yield a convex polygon, so what I needed to show was that every polygon has a convex polygon somewhere in its sequence.

In the meantime, while playing with my program I landed on a star/pentagram, and I saw that it had a poorly shaped star in the center. I figured that the poorly-shaped star was just because I wasn’t able to get precise with my vertices in my program, so I turned to Geogebra for help:

pentagram

So this is pretty clearly a counter-example to my conjecture. Once again, we have options. We could try to bar this polygon somehow. Restrict the conjecture to polygons whose edges don’t touch (except at vertices). That would be a shame, though, since the conjecture applies to many non-regular stars. We could allow everything but non-regular stars, but that seems like a downer.

As the week went on, I continued thinking about how to prove this conjecture, in any form. What I wanted to figure out was a way to measure the “concavity” of a polygon, so that I could argue that each step in this sequence reduces the concavity in some way, eventually resulting in a convex polygon. That wasn’t working particularly well, so I turned to twitter for help:

In response, Dave Radcliffe helped out:

 

The Jason Davies visualization is fantastic:

Davies visualization

And there’s even a paper that inspired the visualization, and it claims to prove something akin to my conjecture:

“If this process is repeated…then in the limit the vertices of the polygon iterates converge to an ellipse E that is centered at the origin.”

And that’s great. I haven’t read their paper carefully yet, but I’m looking forward to it. There’s one problem: we already found a counter-example to their theorem in the pentagram!

And that’s the story so far.

Some reflections on this process:

  • All of the ideas came in a big spurt on Monday. Then I spent Monday building the programs, and Tuesday and Wednesday trying to organize and clean everything up. Monday was exhilarating; Tuesday and Wednesday were tougher.
  • By Thursday I felt like I needed a break from polygons and these iterated sequences, and I couldn’t look at polygons from any other perspective. After a frustrating morning without any real progress, I decided that it was time to reflect and write everything down, and move on.
  • This is real math. Even solving posed problems isn’t real like this stuff is. I make no claims that doing real math increases your capacity to solve problems or anything, but I do think that it’s the most fun I’ve personally had with math in a long time. I’d love to share this feeling with students.
  • At the same time that I was working on all of this, I was under the sway of Proof and Refutations by Imre Lakatos. (Thanks again, Justin.) That’s some great stuff.

The next step is to figure out what all of this math means for kids. I have two competing impulses:

  1. Divide up this investigation into a series of good problems that will lead them to some of the cool things.
  2. Figure out a way to help kids have the sort of open-ended, personal experience that I had.

I’ll sort that out in writing tomorrow.