Engaging Students in Parallel Proofs

I am really starting to like, even love, teaching Geometry.  There, I said it.  Hopefully I won’t lose my math card.

I am finding that Geometry problems are just one big puzzle.  And I love puzzles!  I am also inspired to see some students who were maybe not great Algebra students have success with Geometry.  I have identified those students, and I plan to work with them one on one this year to strengthen their Algebra skills so they can keep their newfound engagement with mathematics going strong!  It is so exciting to see these student’s perception of their own mathematics ability improving, and their attitudes about math changing!

Parallel proofs were a blast on Friday thanks to the amazing Bowman and his Global Math Whiteboarding talk Tuesday night.  If you missed it, you can watch it here.  You can even take a look at the terrific whiteboard brainstorming sheet that we all worked on together during the meeting.  I have a set of giant whiteboards, but I often forget about them.  I have been wanting to use them, and this was an amazing opportunity.

I assigned two students per giant whiteboard with two different colored markers and one eraser.  The two different colored markers were a fabulous idea.  The students can color code things (so helpful in Geometry), and I could make sure both partners were equally participating.  I’m not sure who gave this tip at Global Math but THANK YOU.  As a bonus, the colored markers I found in my cabinet were scented.  I didn’t realize how much this would thrill high school students.  Some things that I think they will love they could care less about, but break out scented Expo markers and WATCH OUT because you will have some really happy teenagers.  Who knew??

Back to parallel line proofs.  I gave them the proof, but told them not to prove it yet.  I told them to just draw the picture and label the GIVENS only.  Then, I had them discuss with their partner how they knew the “prove” was true.  For example, given that m<1 is 100 degrees, how did they know that m<8 was 80 degrees?  Once they talked about it, I had them write down their thinking (steps) on their whiteboard.  Then, they shared their giant whiteboards with their steps with the rest of the class.

They discovered:

  • There are many ways to do a single proof
  • Some ways can take many more steps
  • Some ways are equally as efficient as others, there is not one most efficient way
  • How to carefully explain their steps to the class so everyone would understand what they were doing to “prove” they knew it.
  • They do not like it when other people talk while they are presenting. This is great training for when I am talking!  lol

After they shared, I modeled how to write down the proof steps, explaining that they have to SHOW why m<2 = 80 degrees, if m<1 is 100 degrees using substitution and angle subtraction.

After that, I gave them another proof, but let them do the steps and work on the proof before sharing with each other.  The sharing is crucial, as it helped students see the difference ways each group solved the proof.  My favorite part was how mad a student became when she realized she had taken TWO extra steps than another group to prove her statement.  Now that is engagement!  In fact, this activity produced some of the highest levels of engagement I have seen all year!  Thanks again Bowman!!

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