Teaching Triangle Angle Sum

The Triangle Angle Sum Theorem is one of my favorite topics in Geometry class. It’s a foundational fact about triangles, and in geometry, every problem is a problem about triangles.

I love the standard proof of the theorem, where a line is constructed through a vertex that is parallel to the opposing side. It highlights the crucial role that parallel lines play in our conception of geometry, and it points to the assumptions we make about them as well. With a little nudge, this standard proof is eminently discoverable, and makes for a great classroom activity.

But I also love showing students some non-standard proofs of the theorem. Here’s a demonstration I built in Geogebra meant to mimic a paper folding activity that shows how the angles of a triangle form a straight line.

You should do it with actual paper, too! Here’s a short video. Stick around for the bonus tearing at the end!

Apart from being fun and surprising, what I like about these demonstrations is how they illuminate something important and essential about result: It’s the straight line, not the number 180, that’s important. Plus, the tearing activity works with more than just triangles! Unfortunately it’s not so adaptable to spherical geometry, but that’s another lesson.

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Beyond My Control

Teachers have great power to impact their students, yet so much depends on factors beyond their control. This is one of the many tensions of teaching.

In my first year as a teacher I noticed some students didn’t bring pencils to class. I was dumbfounded. “How are you going to do math without something to write with?” was my naive reaction. Later I realized the more pressing question was “How am I going to teach math if I can’t rely on kids bringing pencils?”

As a public school teacher you become acutely aware of what you rely on. Even the best schools I’ve worked at would run out of paper, or chalk, or chairs. Working hard only to have your plans derailed by something beyond your control really stings.

All of this has shaped my approach to teaching with technology. In many ways I’m a very technology-positive teacher: I was an early adopter of tools like Desmos, Geogebra, and Scratch. But I’ve been reluctant to grow too dependent on technology in my teaching. I’ve had Smartboards for years, but never prepared slides; I’ve had laptop carts, but designed lessons that required internet access sparingly. It’s a very real possibility that I’ll show up to school and the Smartboard or wifi just won’t work. With so much beyond my control, it’s often easier to just avoid the risk.

One of my frustrations in the current remote/hybrid landscape is that I can no longer avoid that risk. Every single moment of my teaching now depends on multiple technologies functioning properly. And teaching well requires not only that they function, but that they and I function together smoothly. Now I find myself depending on a Smartboard and Google Classroom and Zoom and so much more. And I have to learn them all while trying to figure out how to turn a video conference into math class. It’s a bit overwhelming on the best of days. And then my laptop speakers decide to stop working.

There’s a minimalism to teaching and learning math that I’ve always loved. With just a pencil and paper I can become a mathematician. With just one good question I can launch a math class. But now there’s a lot more I have to rely on, and plan for. And it’s all beyond my control.

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MfA Workshop — Computational Thinking

Tonight I’ll be running a workshop for teachers titled “Building Bridges Through Computational Thinking.”

In the workshop we’ll explore the mathematical and pedagogical benefits in taking a computational approach to mathematics. Through a variety of computational thinking tasks spanning different branches of math, we’ll see how these tasks offer alternate pathways into mathematical ideas, genuine engagement in applied mathematics and mathematical modeling, and opportunities for rich pedagogical variety.

This work is a natural continuation of the work I’ve been doing at the intersection of mathematics and computer science education for the past several years. As always, I’m grateful to be supported by Math for America and MfA’s teacher community in developing and trying out new ideas for students and teachers.

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TCM 2019

I’m excited to be heading back to the North Carolina School of Science and Mathematics (NCSSM) for their annual Teaching Contemporary Mathematics (TCM) conference.

I’ll be presenting Building Bridges through Computing, in which I’ll talk about how mathematical computing projects in Python and Scratch can build bridges between theory and practice, the procedural and the abstract, and the simple and the impossible! My talk will focus on Pre Calculus and Calculus topics, and include projects like solving systems of equations, estimating roots, and elementary numerical methods.

Estimating Intersections in Python

I presented at TCM in 2016 on Mathematical Simulations in Scratch and really enjoyed my visit. The conference is focused on big ideas and brings in lots of inspired teachers, and NCSSM is a unique school with incredible programs and exceptional teachers.

TCM 2019 runs January 24-26. You can find out more information here.

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Visualizing Cantor’s Zig Zag

A famous and intriguing result in mathematics is that there are just as many points on a line as there are in a plane. This seems counterintuitive at first: planes contain infinitely many lines, so not only should a plane have infinitely many more points than a line, it should have infinitely times as many points as a line! But this is one of the many curious consequences of the mathematics of infinity.

Here, we’ll restrict ourselves to points in the plane with non-negative integer coordinates. Think about points of the form ( c), where c is a non-negative integer. Since there are infinitely many integers, this set of points is infinite, and the points all lie on the line y = 0. The set of points of the form ( c, ) is also infinite, and these points all lie on the line = 1. Notice that, since these two lines are parallel, every point on one matches up perfectly with a point on the other: (0,0) with (0,1); (1,0) with (1,1); (2,0) with (2,1), and so on.

This matching offers a reasonable argument that the two sets have the same number of points: Every point in each set has a unique partner in the other, so counting the points in one is equivalent to counting the points in the other. In this case, we say that the two sets are in one-to-one correspondence. And if anything, this only seems to bolster the argument that there are more points in the plane than on a line: There are infinitely many lines of the form y = k in the plane, and each one contains as many points as the line = 0. So the plane should contain infinitely times as many points as the line! But the mathematics of infinity is tricky business.

Even though it seems like there are far more points in the plane than on the line, it’s possible to match the two sets up in a one-to-one correspondence. It’s not obvious how to do that, but thanks to Georg Cantor and his famous zig zag, we know it can be done. Here’s a visualization I created in Desmos to demonstrate this matching.

This animation shows how each point in the quarter-plane can be paired up with exactly one point on the half-line, and vice versa. The zig-zag pattern enumerates the points in the plane, showing that they could be rightly imagined as though all lying in order on a straight line. This one-to-one correspondence shows that the sets are the same size. And while this demonstration is limited to only part of the plane, the argument can be extended: for example, skipping every other point on the line y = 0 would create space to accommodate the points with negative y-coordinates.

The animation above is an extension of an earlier version shared on Twitter.

Thanks to Chris Long (@octonion) for inspiring this journey into the infinite by linking to this great paper on Cantor packing polynomials, which I used to create the above Desmos demonstrations. And Kelsey Houston-Edwards also recently shared a fun and related problem. I guess infinity is in the air!

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