Tag Archives: undefined terms

Geometry: Starting Off

The first day or two of geometry is always point line plane. We never really use it again. Geometry has mostly been subordinated to algebra in high school, as I’ve written before, and my geometry class is best thought of as algebra applications with geometry. Or is it the other way round? Purists see geometry as the medium for introducing proofs, logic, and construction. To which I say pish tosh. Most of them are never going to see those subjects again. “But if they don’t learn rigorous logic in geometry, they won’t be able to learn advanced math!” Yeah, that’s moronic nonsense. What is “solve for x”, if not a proof?

But I love history, so I always start by telling them to put their pencils down and just listen as I explain the significance of Euclid’s Elements and the wonder of a book written 2300 years ago. Three hundred years ago is older than our country. Euclid wrote Elements 300 years before the birth of Christ, so Christ’s contemporaries (the educated ones) thought of Euclid much the way we think of Alexander Hamilton or George Washington. Take seven additional chunks of people looking back 300 years and here we are. A book written that long ago was “in print” over 1000 years before “print” existed, and since then, is second only to the Bible in published editions—not just in the English language, which had to wait another 100 years after the Latin version was published, but in all languages.

As to writing another book on geometry [to replace Euclid] the middle ages would have as soon thought of composing another New Testament.–Augustus de Morgan

Why? Because he* nailed it. For over 2000 years, his model met the world’s requirements, and when the world finally found limits to his model, it wasn’t because he was wrong.

Euclid was nagged by his “fifth postulate”, which is easier to sketch than describe:

That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

If you’re not a mathematician—and I am not—you’re like, um, duh? What else is going to happen? The lines will meet up. But Euclid and other early mathematicians knew the fifth postulate wasn’t the same as the other four, and that’s almost certainly why he established the first 28 theorems without reference to it. For the next couple millennia, mathematicians tried to prove the fifth postulate using the other four, and failed. This collected history of effort around that single postulate ultimately led to the realization that there other, non-Euclidean geometries, many of which (if I understand this properly) begin with the negation of the fifth postulate. This discovery rocked the world, robbed it of a truth previously assumed absolute, and ultimately contributed a bit to Einstein’s theory of relativity.

And 2300 years ago, Euclid needed the fifth postulate to complete his model, furrowed his brow and said, “Yeah, hmm. Something’s not right about that one.”

I tell my students that I’m not a mathematician, and that they don’t need to be, either, in order to realize what a stunning achievement Elements is, and to realize the significance of math in our world that thousands of years ago, mathematicians knew enough to be bothered by a postulate that seemed obvious but was yet somehow different from the others needed for the model. That they, my students, are studying an incredibly old math, one that holds up for our ordinary requirements to this day, but also created the foundation for deeper, more complex models. That if they don’t like math, don’t like geometry, to at least appreciate it from a historical standpoint.

I am probably fooling myself a little bit, but the kids always seem interested. Which is all I’m looking for. Just to show I’m not making this up, here are my board notes:

(Yes, my board work sucks. It’s something I build as I go through a lecture most of the time, a document in progress. I’ve started taking pictures of my boardwork to get a better sense of what I said, what I emphasized, and what I could do to improve boardwork next time.)

Then I go onto undefined terms—not just the terms for geometry, but the meaning of undefined terms. Here, again, Euclid nails the building blocks for his model. (Geometry books give point, line, and plane as the three undefined terms, but I also spend time on “congruence” and “between”.)

Then I show how the building blocks of the undefined terms allow us to define everything else in the Eucliden model. I usually use ray, segment, and angle just to give the the idea.

This year, I decided I wanted to do more with 3-dimensional graphing (xyz) and introduced it as part of this lecture. First, the students learned to represent three dimensional planes without a coordinate system, and see for themselves what happened when two planes intersected. The kids had fun with that; here’s one of the best:

Then we went into more formal xyz graphing. I’m including more 3-d graphing this year to help prepare students for 3 variable systems next year, and also to give the students more variety in visualizing images. Click on the board work below to see that I draw in the rectangular prism, which helps students grasp the difference between 2-axis graphing, in which any two points are a diagonal in a rectangle, and 3-axis graphing, in which any two points are the vertices of a rectangular prism. I heard a lot of “ahas” as I went through this. Not sure what the next 3-d graphing activity will be, but I think I’ve started with a good foundation.

So that was the first day, really. Then I went into the meat of unit one: angle types, angle pairs, perimeter and area formulas, and as always, using these relationships to set up equations and solve with the ever loved algebra.

Here’s the first test. I think I caught all the glitches after I captured this. But I’m sure I missed something; I have a pathological tolerance for typos.

*I’m assuming it was just Euclid. More fun that way.


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