Why go in a straight line? Triangles.

I've read Lockhart's Lament so I've been thinking about problems to pose to students that will get them to own their definitions, theorems, etc.

Proposition 22 of Book 1 has me thinking. It is a construction proof to build a triangle given 3 line segments such that the sum of any two segments is larger than the other (third) segment.

I'm pretty sure more or less everyone feels that "the shortest path is a line" so we ask "Let's say you see a sandwhich somewhere in the room. Why do you walk in a straight line to it (assuming there is a clear path between you and said sandwhich)?" The answer will probably be because that's the quickest/shortest/whatever way to get there. Okay, let's put a small wall between you and the sandwhich.

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You   |   Sandwhich
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How do you get to the sandwhich now? Hopefully the students say walk in a straight line to the edge of the wall and then in another line to the sandwhich. Good. It might be interesting to ask if the path they took it shortest since it would bring up a lot of itteresting points about "shortest" beign subject to constraints. Now we make it interesting. Put students somewhere and a sandwhich somewhere else. Measure the distance between and then say "I want you to show me a path to take from where you are to the sandwhich that is (1) A feet long and (2) is made of two line segments like in the wall example". My guess is that they will come up with the isoceles triangle whose base contains the sandwhich and start point. Make it more interesting. If the path has to be A feet long what are all the different paths you can take? Maybe to push them in the right direction give them string and a measuring tape and split up the groups with half at the start point and the other at the sandwhich.

The interesting idea for me is that you can do all this with their intuitive notion that straight is shortest. The legs of the triangles have to add up to more because otherwise that notion would be wrong. Also, the discovery process for the harder question is similar enough to the proof construction that going through the formal proof will illustrate how in mathematics exploration and proof are complimentary pursuits.