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Parallelogram Mazes & Introducing Proof

October 6, 2011

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.

The (very brief) lesson Introducing Parallel Line Mazes

Mazes Handout #1

Mazes Handout #2

Mazes Handout #3

Bellwork & Exit Ticket for Day 3


From → Curriculum, Math

  1. Would you mind sharing the file for this lesson? I would love to use this!

  2. Thanks! The link says maze handout #3 but the file is titled day 1. Is that what you used first?

    • Oh – whoops. Well, the way I taught these, I spent a while just using vocab correctly (are these alternate interior? consecutive? etc) then spent time finding missing angles using the angles. The first two worksheets are just using vocabulary correctly – students aren’t finding the values of any of the angles. The third worksheet came after we developed which angles are congruent and which angles are supplementary. Then we did more practice finding missing angles, etc. Anyway – that’s why it says Day 1 – it was the first in a series of ‘use parallel lines to find missing angles’. These are some of the problems we did next:

      • Jasmine permalink

        Do you have the answer keys by any chance? Thank you!

  3. Got ya! Thanks so much for the help, this fits perfectly into what I’m teaching this week. I will definitely blog about how it goes!

  4. Felicia permalink

    Thank you so much!!! This is an amazing idea!!! =D

  5. Jasmine permalink

    I’m just starting my parallel lines unit, and this is a goldmine! Thank you so much!

  6. Erin permalink

    I wish I would have found this about two weeks ago…it would have worked so much better than what I did! Definitely trying this next year!

  7. Mike permalink

    I love these worksheets. I use the parallel line puzzles from Key Curriculum Press – Discovering Geometry — although they are not the publisher anymore. This is a Michael Serra book. But I was looking for some more good puzzles — challenging like the Discovering Geometry puzzles, but that stress justification with reasons. Fantastic! Thank you!

  8. Mary @ kennedyfamfive permalink

    I love the idea of treating the process like a maze and the specific instruction that “each angle must be connected by a straight line.” That’s such a simple way to put it, but it’s something concrete for kids to use as a guide. Thanks for sharing.

  9. Aine permalink

    Hello, I was just wondering if you have the files mentioned here. I have tried downloading but it won’t let me for some reason. They look awesome. Thank you

  10. Teresa L Carter permalink

    I love this unit! It looks pretty cool and I think my students would love it, however the links will not allow me to download the documents. Would you be able to share the files with me?

  11. Hi, is there another way these documents can be shared? They are not downloading on Box…

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