Geometric Constructions

Some of my fondest memories of being at my grandparents' house are of pencils, t-squares, the scent of WD-40. I loved being out in the garage with my grandpa, a former drafting and technical drawing teacher, as he worked...it was fascinating to me- it seemed like art.  And above all, I knew it was something that he loved. When he was working, you could see his whole demeanor relax. He was focused on something entirely outside himself, almost a form of therapy for him. Since I first learned geometric constructions as a teenager,I always associated the act of geometric construction with my time spent in my grandpa's garage. 

My first experience teaching geometric constructions was anything but therapeutic. I was teaching a group of students who were repeating geometry and had no interest in an overenthusiastic 21 year old trying to teach them a skill they felt they'd never use. I will never forget them moment a girl stood up and said, "Look at this b*tch up here preaching" as I over-enthusiastically tried to get them invested in bisecting a line. My next experience was with a group of highly accelerated 8th graders who would gobble up anything I asked them to attempt. It was cool because it was math and because I asked them to do it. 

Next year I will be in a less extreme geometric situation, so I'm trying to start from the beginning and figure out a way to make this as relevant as possible for my kiddos. Here's where I'm starting:

Instinct #1: Math Nerd Prospective

I had a professor in college who always said "the history of math is the history of the world." Major events in mathematics and science innovation are so closely tied with major events in history and geometry has a fascinating one. How can you not love a good story about Euclid and Poincare and all the other amazing geometers who have battled over things like the parallel postulate??? Unless you're 16. And focussed on your football game Friday night. Or the boy sitting in front of you in class. Got it....might need a different approach.

Instinct #2: Applications to Drafting and Architecture

With my Grandpa in mind, I contacted our drafting teacher to get her perspective. She said to think about building and construction.  

"You could look at building roof trusses with balsa wood or bass wood.  With the demise of pencil drawing and going almost all AutoCAD I don't teach my kids how to do this anymore.  The computer does it for them. "

So while it's a foundational skill for a lot of careers, it's not something that's obviously useful with all the new technology. Our drafting teacher followed up with a comment that I loved...

"I, of course, still like to draw with tools though seldom do.  The focus required is kind of meditative.  I did actually demo a complete drawing using tools and the kids were amazed I did it faster than they draw on the computer.  Some times it is fun and instructive to show off." 

This is something I want to make sure I hit in class. There are some kids who will identify with the hands on, fascinating process of creating something that is exact. There will also be kids who are more into the technological aspects to help them see the why. 

Preliminary Thoughts: 
A lot of the idea behind the history of math is creating restrictions (axiomatic system) and seeing the results that follow from it. It's why I love this Geometric Constructions Game that I found through Crazy Math Teacher Lady.  Kids are put under restrictions and need to try to create the objects in the fewest moves possible. I love how open ended this game is and for an honors class I can see this becoming a huge competition. However, for my standard kids I'm trying to develop some entry points to help them connect what they're doing with the mathematics. I spent most of this week researching and talking to peers. I'll be designing some entry level activities  and hopefully some PBL scenarios for next week!

Mean Value Theroem

One of the pitfalls of math education is when conceptual understanding is sacrificed for rote skills. I really do believe as a teacher that if kids get the concept, they can apply it much more readily than if they just know how to do some specific skill or memorize steps. And I’ll be honest, I killed it year one with my students' understanding of Mean Value Theorem conceptually.

My kids knew under what criteria it applied, knew that the slope of the secant had to equal the slope of the tangent, drew beautiful pictures and wrote beautiful descriptions of the concept. And then they kept getting MVT questions wrong on my quizzes. And tests. And reviews. It was the same mistake every time- they knew they did it, I knew they did it- and each time they’d say that they’d fix it for next time. They consistently answered with the slope of the secant line, not the value of c.

So while we discussed the nature of their mistake each time, I don't think I ever quite gave them enough practice with finding the actual value of c. I'm hoping this activity can help address that.

What Worked?

Introduction:

I introduced MVT using the idea of being given a speeding ticket when an officer never saw you speed. We debated whether it was fair or not as a class (which of course it isn't to a room of teenage drivers) and then talked about under what conditions it would be fair. What kind of proof do they think would hold up in court? 

Image from http://image.slidesharecdn.com/myserminarslide-150523230351-
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This stems from an idea I read about at Division by Zero. My kids pretty much came up with this exact idea. If they got to some "checkpoint" faster than they should have, it would definitely mean they would have broken the speed limit to get there. In other words, the average rate of change has to equal the instantaneous rate at some point along their drive.

Turns out these devices exist and there are places in the UK where people will actually get speeding tickets for their average speed The image at right demonstrates how these sensors measure speed. Cue the 17 year old's discussion about how "the man" is always watching and they can't do anything without getting in trouble.

Discovery Activity:

Last year I used this MVT Exploration from Key Curriculum Press. I liked that it made the kids examine the criteria for MVT to apply and it led to some great conversations on how this applied back to our speeding situation. It's not an activity that's been optimized to its greatest potential yet, but it provided a good segue from the real world into the theoretical. I'd like to write something that ties more directly to the speeding ticket investigation and transitions more naturally, but we will save that for later in the summer.

What's Changing?

So again, my kids nailed the concept. Really, they got it. But, y'all, they couldn't remember to find "c" to save their lives. When I've used a clothesline activity before (mostly with logarithms, fractions, or function evaluation) it seemed to help my kiddos see that things actually have values associated with them. I'm hoping this might do that same. 

Mean Value Theorem Clothesline Activity

I’m thinking this could be a great extra practice or review activity. My class this year was a yearlong 90 minute block so i don’t know that I’ll have time for it in a traditional AP class next year. However, it would be a good station activity or targeted intervention strategy. They really just needed practice and in a way that might require some collaboration and conversation with peers.

Not a post filled with rocket science, but a nice easy transition into summer blogging. Happy last week of school, everyone!