Friday, July 25, 2014

Source 2: Franklin Mason

My next source is Franklin Mason (also known as Dr. M), a high school math teacher in Indiana:

Unlike Joyce, whose webpage was last updated before Common Core, Dr. M actually has the Common Core Standards in mind. His website contains a complete course in Common Core Geometry, complete with Powerpoints and worksheets, along with an actual textbook that is still in the works.

Dr. M focuses on rigor and proofs. In fact, some of Joyce's desires for a rigorous proof-based geometry course are fulfilled in Dr. M. For example, Dr. M states that the intersection of two lines being at most one point is a consequence of an earlier postulate (not a postulate itself). Under Chapter 2 (of Prentice-Hall), Joyce writes:

"In a return to coordinate geometry it is implicitly assumed that a linear equation is the equation of a straight line. A proof would depend on the theory of similar triangles in chapter 10."

This is actually omitted from most traditional texts -- after all, it's so "obvious" that a linear equation is a line that neither algebra nor geometry texts prove this. But Dr. M now actually uses similar triangles to prove this in his final chapter, Chapter 13, on Analytic Geometry.

Regarding circles, Joyce writes:
"The tenth theorem in the chapter [5 of Prentice-Hall] claims the circumference of a circle is pi times the diameter. It is apparent (but not explicit) that pi is defined in this theorem as the ratio of circumference of a circle to its diameter. So the content of the theorem is that all circles have the same ratio of circumference to diameter. This theorem is not proven. It would require the basic geometry that won't come for a couple of chapters yet, and it would require a definition of length of a curve and limiting processes. It would be nice if a statement were included that the proof the the theorem is beyond the scope of the course."

Dr. M actually has a section on least upper bounds in his Chapter 9. These are the "limiting processes" to which Joyce refers. Thus, Dr. M is able to give even a rigorous definition of pi! And of course, he uses translations, rotations, and reflections to derive SAS and the others.

Since Dr. M's website is so thorough, one may wonder what I can provide on this blog that would improve upon it. Well, regarding parallel lines, Joyce writes:

"Chapter 7 is on the theory of parallel lines. It begins by postulating that corresponding angles made by a transversal cutting two parallel lines are equal. One postulate is enough, but for some reason two others are also given: the converse to the first postulate, and Euclid's parallel postulate (actually Playfair's postulate). A proliferation of unnecessary postulates is not a good thing. One postulate should be selected, and the others made into theorems."

Dr. M improves on Prentice-Hall by giving two postulates for parallel lines. One of them is Playfair's Postulate as mentioned by Joyce. The other is called "Exterior Angle Postulate." As it turns out, this is Euclid's Proposition 16:

Euclid, Proposition 16:
"In any triangle, if one of the sides is produced, then the exterior angle is greater than either of the interior and opposite angles."

Dr. M uses the Exterior Angle Postulate to prove that if two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel (as Euclid in Proposition 27). But Euclid proves Proposition 16 itself. In fact, Dr. M gives the proof right on his blog -- as of today, it's the third post from the top (dated August 3rd of last year). This proof derives from SAS. And so we wonder, why does Dr. M call it a postulate rather than prove it as a theorem?

I think that earlier this year, Dr. M actually did include the Exterior Angle Inequality as a theorem, but he recently changed it to a postulate. Notice that this proof is what I call a low-level proof. It's used in proofs of other results (including both the Alternate Interior Angles Theorem and the Triangle Inequality), but is not truly an interesting result in itself -- after all, the Exterior Angle Equality tells that the exterior angle actually equals the sum of the remote interior angles, not merely that it's greater than either one. So, as many authors do with low-level proofs, Dr. M just makes the Inequality a postulate rather than cause students to worry about its proof.

But there are other ways to prove the Alternate Interior Angles Theorem besides the Exterior Angle Inequality. I will discuss them in my next post.

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