Sine Curve Graphs!

My first year teaching (which seems like a much longer time ago than it was), the administration asked us to observe at Providence High School. Providence is by all measures a “good” school, and I did get at least one thing out of the visit. It was just hard for me to see a connection between Providence, a predominantly white, middle-class school in a nice neighborhood, and my school, a majority-minority, high-poverty school.

Providence, a large, polished public school, bears no resemblance to HGS either. But the one thing I did get out of the visit was a project that one of the classes was doing, a project that I’ve had a lot of success doing here. Some things do transcend – creating sine curves out of colorful straws is apparently one of them.

The unit circle is an important but initially mystifying document, whereas trigonometry is strictly visual. We started with just setting up trig ratios with right triangles they could see. No problems there. Then I introduced them to the unit circle, posing the question: where can you see right triangles?

unit circle

This was more of a challenge, but an interesting one. Once they were curious, it became a lot easier to explain the connection from trig to the circle.

The project works like this: first, they wrap a string around the circumference of the circle, marking where each of the angles hits. Then that string becomes the template for their x-axis. They use straws to measure the sine, the distance from the x-axis to the angle, or the “opposite” side.

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Creating these graphs in such a physical, hands-on way helps them to grasp onto some of the more abstract stuff to come… and it does get pretty abstract.

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Euclidean and Non-Euclidean Geometries

Euclidean Geometry seems like a heavy topic for ninth graders, but it just means Geometry on flat surfaces. There are only 4 students in my IM1 right now, so I had each of them illustrate the basic axioms:

  1. There exist at least 3 distinct points
  2. Not all points lie on the same line
  3. For each 2 distinct points, there exists a unique line on both of them
  4. For every line there exists at least 2 distinct points on it

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The illustration for axiom 2 is slightly off, although I completely see where they’re coming from. They drew each point having a unique line, not one unique line though both of the points. I think that one is on Euclid/ whoever rephrased it in my textbook, although I do wish I had noticed this before it went up.

We talked about why Euclid might have thought it was convenient to create a Geometry for just flat surfaces, and the fact that the world isn’t actually made out of just flat surfaces. That’s where the non-Euclidean part comes in. I’ve been pretty excited about the idea of hyperbolic planes ever since one of my colleagues showed me this TED talk about how coral is a hyperbolic plane, and how the best way to represent those is through crochet. You should watch it!

I crocheted several of these hyperbolic planes over the summer (cool sentence, right?), so we did this investigation about parallel lines on 3 types of surfaces: planes (blue), spheres (green), and hyperbolic planes (pink):

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I had the students figure out the parallel postulate in each of these geometries: “on a given point not on a given line, exactly how many lines can be drawn parallel to the line?”

[SPOILERS] It was quickly clear that on a flat surface, there is only one parallel line through each point. We’ll be making the connection from this fact to the coordinate plane next week. It was almost immediately obvious that making parallel lines on a sphere is impossible. Creating any type of line on a hyperbolic plane was unfortunately where this investigation broke down. Most of them threaded their string through the middle of the hyperbolic plane, even though they had a vague sense that it didn’t count.

On Monday, before we bring it back down to just the Euclidean stuff, I’ll show them the TED talk because she has stitched lines onto her example. This might be a time that it’s better to not be hands-on. We’ll see.