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Second grade logic and rulers February 23, 2012

Posted by mareserinitatis in education, gifted, math, teaching.
Tags: angles, education, geometry, kids, rulers, sir cumference
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Today, I went back to work with the second graders.  We’ve been spending a lot of time talking about circles and degrees and Babylonian units and π.

My plan has changed from the original one of trying to teach the kids a bunch of applied stuff.  I’ve pretty much given in to teaching them historical discoveries in math simply because there’s a lot of stuff you can do that doesn’t require multiplication and division.  It’s been a lot of fun, but I decided to try something different but related: I wanted to teach them out to make a formal mathematical proof.  Okay, not terribly formal.  What I want them to learn is how to use logic to make a proof.  I suspect some of them know already (based on some of the arguments I’ve had with my son who has rock solid 7-year-old logic).  However, I’d like them to use their brains for good instead of getting out of (or into) trouble.

The thing about geometrical proofs is that they really aren’t that hard.  At least, I never found them to be.  I remember sitting in 10th grade geometry and being given T-charts.  I would race through them and ace them all.  I was horribly surprised to see that my classmates had difficulty with them as well as complaining the teacher was too abstract.  I threw the idea of proofs out to Mike, and he said that pretty much the only tool you need is a brain, so it’s probably a good idea.  (I’d have to disagree…you need a brain…but you also need a pencil, paper, ruler, and protractor.  But otherwise, I think he’s right.)

Today, we started with the concept a line and measuring its angle.  I know my former math instructor wouldn’t approve, but I’m teaching them to use degrees (aka Babylonian Units) because that’s what’s on the protractor.  Also, I’m not sure how versed they are in fractions, so we’re not going to get into fractional parts of π.  (Actually, if anyone has ever seen a protractor with units of π rather than degrees, please let me know as I’d love to buy it.)

Once we had a line, then I told them to draw a point on the line with another line coming out of it, so that it would look like this (without the measurements):

Each of them drew the line coming out an a different angle.  They all measured their angles and found that they all summed to 180°.  A couple of the kids seemed surprised that they all ended up with the same number. Incidentally, those that didn’t seem surprised were very absorbed with the flexible rulers I had brought to use.  (Note to self: second-graders are easily distracted by anything novel.) We then talked about how any two angles, if they formed a straight line, would add up to 180° and how this was known as the supplementary angle theorem.

Once we had that down, we used it to prove the vertical angle theorem.  It took them a bit to realize that the line created by adding supplementary angles doesn’t have to be horizontal (like in the picture above).

That’s all we got through today, but I plan on using this to show them that the interior angles of a triangle always add up to 180°.  It might take us a couple weeks to get there, especially since next week I’m supposed to read them a couple more of the Sir Cumference books.