Let's start with the postulate itself. Last year, I stated that the most convenient postulate for proofs is the Perpendicular to Parallels Postulate (a theorem in the U of Chicago and most other texts).
But now, with all of the changes that I'm making to the curriculum, I'm starting to wonder whether it's better just to follow Dr. Franklin Mason and adopt Playfair's Parallel Postulate as our main postulate for parallel lines.
One concern I had last year was to avoid indirect proofs. But we've opened that can of worms -- as soon as we stopped defining lines as being parallel to themselves (that is, once we've decided that parallelism is not reflexive), many of our direct proofs about parallel lines became indirect. There were also a few proofs where the Perpendicular to Parallels form could be used directly in the proof --this include the part of the Two Reflections Theorem for Translations where we show that translations slide every point the same distance, as well as the Perpendicular Bisector and Altitude Concurrency.
The sequence of theorems as given on the old worksheet were as follows:
Perpendicular to Parallels -> Transitivity of Parallels -> Playfair -> Parallel Consequences
If we let Playfair be the main parallel postulate, we can use this sequence instead:
Playfair -> Parallel Consequences -> Perpendicular to Parallels -> Transitivity of Parallels
This works because once we have the Parallel Consequences, then Perpendicular to Parallels follows trivially as a special case. In fact, any of the statements along this chain can be chosen as a postulate (including Transitivity of Parallels). Mathematicians would say that all of the statements happen to be equivalent statements.
It was a very tough decision, but I choose to keep our Fifth Postulate the way it's written. I'll post the first part of the worksheet that I posted last year. But we will save the derivation of the Parallel Consequences from Playfair until next week, since these deserve their own lesson. So the proofs that will show up on today's lesson are:
Perpendicular to Parallels -> Transitivity of Parallels -> Playfair
Notice that as the worksheet is written, the proof of Transitivity of Parallels -> Playfair is written as a direct proof, but it sounds much more natural as an indirect proof. So it has now been rewritten.
Today may be Friday the 13th -- the day that triskaidekaphobia reigns supreme. But today's lesson is all about the number five, not thirteen. I had actually tried to set this up last year such that on Day 55, I would post the Fifth Postulate -- the Perpendicular to Parallels Theorem that is equivalent to Euclid's Fifth Postulate. I'd post it when we reached Lesson 5-5 in the U of Chicago text. And it would be the fifth named postulate posted on the blog -- but that only works out if we count the Plane Separation Postulate as the fourth named postulate. (The first three are Point-Line-Plane, Angle Measure, and the Reflection Postulate.)
Last year my timing was thrown off due to substitute teaching, and I ended up posting the Fifth Postulate on Day 52 rather than 55. This year, I'm actually able to post the lesson on Day 55. But now there's not as much a connection to Lesson 5-5 of the text now that I've delayed proving any of the Parallel Consequences. But for the superstitious readers trying to fight triskaidekaphobia with pentaphilia, I point out that Perpendicular to Parallels originally appears in Lesson 3-5, so there's still a five involved somehow.
Notice that the only theorem equivalent to the Parallel Postulate in Lesson 5-5 is the Trapezoid Angle Theorem, which is essentially the Same-Side Interior Angles Consequence. It's possible to adopt this as our parallel postulate if we want -- just like Perpendicular to Parallels, Same-Side Interior Angles has a special connection to Euclid's original Fifth Postulate. After all, the ancient sage essentially stated its contrapositive -- what happens when the two same-side interior angles do not sum to two right angles (or 180 degrees, since Euclid doesn't use degrees)?
Since I cut off the second part of the worksheet, I wish to replace it with something. I notice that in the U of Chicago and many other texts, students are given a taste of what would happen if we didn't have a parallel postulate. Yes, we'd have non-Euclidean geometry.
We've spent so much time on the blog discussing what's possible in neutral geometry and what requires a parallel postulate. The U of Chicago text introduces the students to spherical geometry -- as did I over the summer on the blog. But note that spherical geometry is not neutral -- neutral geometry includes only Euclidean and hyperbolic geometry. Technically, if we want to show students what impact the parallel postulate has on geometry, we should be showing them hyperbolic geometry, not spherical geometry. But since we live on a globe, spherical geometry is far easier for students at this age to understand.
Indeed, I often point out that the Fifth Postulate and its equivalents are, believe it or not, true in spherical geometry! This is because most of these statements begin with, "if two parallel lines..." or "if two lines are parallel." But in spherical geometry there are no parallel lines, so all Parallel Consequences are vacuously true! We might as well replace every occurrence of "parallel lines" with "unicorns," and all of the statements are still vacuously true:
Perpendicular to Parallels: If a line is perpendicular to one unicorn, it's perpendicular to the other.
Transitivity of Parallels: If a line is parallel to one unicorn, it's parallel to the other.
Parallel Consequences: If two unicorns are cut by a tranversal, corresponding angles are congruent.
Playfair: Using the version of Playfair given in Lesson 13-6 of the U of Chicago text doesn't work, but it does if we use Dr. Franklin Mason's version: "Through a point not on a given line, there's at most one line parallel to the given line." The phrase at most one allows for the possibility of zero.
Euclid's Fifth Postulate: If the two same-interior angles add up to less than two right angles, the lines will intersect on that side. This is true in spherical geometry because all lines (great circles) will intersect on both sides no matter what the angles add up to. Euclid's Fifth says nothing about what happens when the angles do add up to two right angles, only when they don't.
But still, I will mention spherical geometry as an example of a non-Euclidean geometry in which parallel lines don't work the way we expect them to.
By the way, you may be wondering what's going on with Dr. Katharine Beals and her weekly Common Core problem this week. Well, there is no question this week -- recall that our traditionalist post for this week was yesterday's post on the article she and another traditionalist wrote. But still, I want to make a quick comment on something she wrote on her blog this week:
In math she [Beals's daughter -- dw] continues her alternation of Weeks &Adkins Geometry (my husband's high school text) and A Second Course in Algebra (my mother's high school text). She especially enjoys geometry proofs (even though some education gurus say she's supposed to hate them).
So Beals is teaching her daughter both Geometry and Algebra II this year, going back and forth between the two texts. So this would make her curriculum this year an integrated curriculum. But that's odd, since I thought that traditionalists like Beals hate integrated math (and she's criticized integrated math several times on her blog)!
As far as the "education gurus" who say that she's supposed to hate proofs, I believe that these gurus are being descriptive, not prescriptive. They're not saying that students are supposed to hate proofs -- they're saying that students actually do hate proofs.
Beals says that her daughter likes proofs. I say that the student I tutored last year hated proofs. And I suspect that most students actually sitting in a Geometry class are more like the student I tutored than her daughter. And so, after including two proofs on my worksheet (those of Transitivity and Playfair), I include a semi-activity -- thereby keeping students like the one I tutored in mind. Most of the questions come from Michael Serra's Discovering Geometry except for the last question, a classic.