Day 15: How Big is the Angle?

On Sunday morning we headed to Fort Vancouver, WA, which is just over the Columbia River from Portland. Unbeknownst to us, they were running the Vancouver marathon that morning, and all the road closures and detours turned a simple drive into a comedy of errors. After driving the wrong way down a one-way street and then riding along the course next to the runners for a few minutes (at the direction of the race organizers), we finally made it.

Fort Vancouver was a trading post for the Hudson Bay Company, and a replica of the post is open for tours. Next to the fort is an airfield and museum that describes the role this part of the country played in the early history of aviation, especially during World War 1 when Sitka spruce trees from the area were cut, milled, and then sent around the world for use in airplane construction. Among the displays was the frame of a biplane, the wing of which caught my attention:    

 The wing is framed with trusses, much as you would see on a bridge or in the roof of a building. I could go on for quite some time talking about the specifics of these trusses and the overall wing construction, but I’ll spare you (my physics students may not be so lucky). The geometry of these trusses also caught my attention. I think there’s something here that might help shape the way we do geometry in our integrated math course.

Definitions are the foundation of mathematics. They set the terms of the conversation, so to speak; by telling us exactly what something is, we can know when we can and can’t use that thing, and we can draw logical conclusions by applying the definition. I believe that students struggle with math in part because they memorize definitions instead of understanding them. Memorization without understanding leaves them without the ability to recognize when a defintion applies, or when one has been violated, which in turn makes the logical conclusions harder to draw and the proofs harder to write.

With that background, here’s my idea. Consider this picture of a display at the musuem, which contains one of these wing trusses:   

 In the second picture, I’ve marked an angle with an arrow. What if we ask students how big this angle is, and we give them protractors to measure the angle? The left side of the angle is a curve, so measurements to different points would yield different results. The conversation that follows this realization should be rich, and given the time the definition of an angle will likely come up at some point. An angle is the figure made by two rays that share a common endpoint (rays are like arrows – they have a starting point and extend infinitely in one direction). In Euclidean geometry, rays are straight, like lines. So, measuring this angle in the Euclidean sense isn’t possible because, by definition, it isn’t an angle. 

This is an incredibly powerful realization – “we don’t have the math to do the thing we’re trying to do” – that should help students understand the definition and understand the immportance of definitions in general. As an extension of this, we could then have the students write a definition for an “angle” like that shown in the picture. That would also be a fun, challenging activity with its own pay-offs. I wonder if there’s merit to this approach in general: having students learn Euclidean geoemtry by challenging them with violations of that geometry drawn from the world around them.


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