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Proofs and Geometry

Woman teaching geometry, from Euclid's Elements.

Woman teaching geometry, from Euclid’s Elements. (Photo credit: Wikipedia)

In all my years of tutoring, I’ve noticed that most students prefer algebra over geometry or the other way around.  Even when I was in high school, I preferred algebra.  Geometry required more effort on my part, especially when it came time to learn proofs.  I earned my first (and last) “F” on a geometry math test that had geometric proofs.  Of course I worked diligently to pull my grade up by the end of the year.  But I will always remember that test, the teacher (Mrs. Yarbrough), and the feeling I had when I got my test back.

This followed me for a long time.  Throughout the years, I was asked to tutor students in geometry.  I always had an internal conversation that went something like this:

“Why me, can they get someone else?  There has to be someone who loves geometry.  Now I have to tutor proofs.  Proofs made me fail my geometry test in high school.  Maybe they already learned proofs.  Okay, I can do this.”

Then I would learn that the students needed help with proofs.  Go figure!!!  Eventually, this happened a few times and through tutoring my students, I finally learned, understood, and appreciated proofs.

When I was in high school I only had access to the teacher, my textbook, and the books at the library.  With today’s technology it’s easier to get help with almost any subject.  I wish the internet were as accessible then as it is now.

Here are a few internet sites and videos to help you with proofs:

Sparknotes         Extensive notes about geometric proofs

For Dummies     Tips for “Mastering the Formal Geometry Proof”

WyzAnt                Steps for writing two column proofs

Your Teacher    A video of an example of a two column proof

Khan Academy  A video explaining how to solve geometric proofs

So, why do we teach proofs in geometry classes?  Who thought geometric proofs were important enough to confound even the best math students?  Joshua N. Cooper of the University of South Carolina made a compelling argument about the importance of proofs in math classes.  Read his article before you tackle your first proof!  It may help you accept proofs.

I hope this is helpful.  If you have other ideas or resources to help with geometric proofs, please share them in the comments section!

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Natural Geometry, Hex, and Sacred Geometry

Math Munch

Welcome to this week’s Math Munch!

People can be skeptical when some mathematicians and scientists talk about mathematics as the “mysterious code” that “underpins the world.” I mean, the natural world is so chaotic! But then you run across this:

bees on honeycellsHoneycombs are remarkably symmetrical. Each little cell is a perfect hexagon – and all bees build this way. Why? Because of mathematics.

NPR’s Robert Krulwich wrote about this in a recent post on his excellent science blog, Krulwich Wonders. I think the explanation is an amazing example of how the natural world often follows mathematical rules perfectly. Thousands of years ago, an ancient Roman scholar named Marcus Terrentius Varro conjectured that the hexagon is the shape that most efficiently breaks flat space up into little units – making honeycombs that hold the most amount of honey while using the least amount of wax. He couldn’t prove his idea, though…

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