Parallelogram Mazes & Introducing Proof
I’ve been playing around with how I present parallel lines and introducing my students to logical thinking, justifying their arguments, and the foundations of proof. As someone who enjoyed geometry as a high school student and who enjoys it today, I always found the proof aspect to be the most fun – all high school geometry proofs are essentially one big maze. You’re told where to start and where to end – you just gotta connect all the dots and make the jumps in between. So, I experimented with presenting angle relationships with parallel lines that way as well.
We had already discussed linear pair and vertical angles – or, the angles that are formed when 2 lines intersect. Then we jumped to 3 lines and the idea of parallel lines – which meant corresponding angles, consecutive angles, alternate interior, and alternate exterior. We drilled a bit on terminology and identification – some low-level Blooms verbs. Then I gave them this:
(Note: I realize I’ve forgotten to mark these lines parallel. I promise it is something I will fix when I revise this lesson).
Anyway – my students think about it, a few have an opinion, and one or two try to justify their answers. However, they can’t – we don’t have a name for the relationship between those two angles. Then I give them this:
Now they’ve got some extra legs along the way. At this point, the maze analogy works really well – you need to jump from one angle to the other and each angle must be connected by a straight line. Doing some modeling, I show them that you can get from 1 to 3 using consecutive angles, 3 to 6 using alternate interior, then 6 to 2 using consecutive angles. Then we try a few more, then they come up with some on their own. What’s even better: the requirement that they justify their reasoning (ie: provide an informal proof) comes naturally.
Discussions this generates: Can I jump from 1 to 4? What about from 1 to 7? From 1 to 6? These are all questions I want my students to ponder, but I’ve now tricked the class into asking them rather than me asking them. Now they’ve got motivation for really fine-tuning their understanding of these angles relationships.
Where this leads: I love the fact that there are multiple answers – almost any student can generate their own path and justify their answers. You can also ask some interesting questions – what is the shortest path. Find a path that uses every angle exactly one. Find a path that doesn’t use the same rule twice in a row (All of these are doable, by the way – I know because my honors kids came up with the questions and then found the answers themselves). These are interesting, non-trivial questions. And they’re related to one of my favorite and underutilized branches of mathematics: graph theory.
After you set the stage, you can give them something like this:
And after that, throw in some algebra:
Getting Fancy: Have students create their own maze with their own challenge, then make a class book of mazes for others to solve. You can also give some interesting challenges in the creation of mazes themselves – for example, create a maze where it is impossible to use every angle exactly once (in other words, create a maze that does not have a Hamiltonian Path).
Anyway – this is something I experimented with and the discussions it generated have been great, the productivity and motivation of my students has increased, and I like the room for differentiation – the faster students seem to get hooked by the different challenges, while the slower students get lots and lots of practice identifying these angles and using the congruence/supplementary relationship. So, there’s one lesson I can stock away and use next year.
Update: Here are some of the files from the semi-revised version this year. I got sick partway through teaching this unit, which is why some worksheets make references to me being out sick. This also means I didn’t get to do as much as I would have liked with the path-type questions (“What is the shortest path? longest path? using only alternate angles?”) But, the real gem to these (in my opinion) are the diagrams – with those, you can construct any kind of lesson or interesting series of questions you’d like.