Lesson 1-7 of the U of Chicago text introduces postulates. In the last lesson, the undefined terms -- the primitive notions -- point, line, and plane were introduced. Since these are undefined, we don't really know what they are unless we have postulates -- also known as axioms -- to describe them.
I reproduce the main postulate of this lesson, the Point-Line-Plane Postulate:
Point-Line-Plane Postulate:
(a) Unique line assumption: Through any two points, there is exactly one line.
(b) Dimension assumption: Given a line in a plane, there exists a point in the plane not on the line. Given a plane in space, there exists a point in space not on the plane.
(c) Number line assumption: Every line is a set of points that can be put into a one-to-one correspondence with the real numbers, with any point on it corresponding to 0 and any other point corresponding to 1.
(d) Distance assumption: On a number line, there is a unique distance between two points. If the points have coordinates x and y, we define this distance to be |x - y|.
Let's look at each of these four assumptions -- since postulates really are assumptions, or statements that are obviously true -- in detail. The first assumption, that two points determine a line, goes all the way back to Euclid's First Postulate. In Hilbert's formulation of Euclidean geometry, this is Axioms I.1 and I.2.
The second assumption, about dimensions, are often different in other texts. Some texts, for example, emphasize that three noncollinear points determine a plane -- and give the example of a tripod standing on its three legs, the ends of which are the three points determining the plane of the floor. Hilbert's Axioms I.3 through I.8 roughly correspond to this assumption.
Assumptions (c) and (d) often appear in geometry textbooks as the "Ruler Postulate." The Ruler Postulate was first formulated by the American mathematician George David Birkhoff, about eighty years ago. The Ruler Postulate basically states that rulers work -- that is, we can measure line segments.
The lesson continues with its first theorem, the Line Intersection Theorem:
Line Intersection Theorem:
Two different lines intersect in at most one point.
And then we have the definition of parallel lines:
Definition
Two coplanar lines are parallel lines if and only if they have no points in common, or they are identical.
Now, as I pointed out in last year's post, the last four words of this definition are controversial: "or they are identical." Last year, I thought that using this inclusive definition of "parallel" -- one that allows a line to be parallel to itself -- would make proofs simpler. Under the exclusive (or strict) definition of parallel, we can prove two lines to be parallel by assuming that the two lines intersect and showing that the assumption leads to a contradiction -- that is, an indirect proof. But under the inclusive definition, we can use a direct proof instead -- to prove two lines parallel, we show that if they have at least one point in common, then they must have every point in common.
Many postulates and theorems can be stated more easily if we are allowed to use the inclusive definition of "parallel." For example, Playfair's Parallel Postulate:
Through a point not on a line, there is exactly one line parallel to the given line.
can be simplified to:
Through a point, there is exactly one line parallel to the given line.
And the following Common Core Standard:
A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
can be simplified to:
A dilation takes a line to a parallel line, and leaves a line passing through the center unchanged.
(Notice what effect the inclusive definition would have in spherical geometry -- parallel lines would now exist, except that parallelism would be equivalent to identity. We've already seen that in spherical geometry, similarity is equivalent to congruence.)
But since last year, I've changed my mind about the U of Chicago's definition of "parallel." The problem is that there exists a problem on the PARCC Practice Exam which requires the strict definition of "parallel" -- using the inclusive definition could lead to a student giving the wrong answer. In particular, Question 13 asks the students to find the dilation image of a pair of intersecting lines, with the center of the dilation on one of the lines but not the other. Choice A claims that the dilation maps both lines to respectively parallel lines, while Choice B claims that the dilation maps the line not containing the center to a parallel line and the line containing the center to itself. Under the strict definition, only (B) -- the intended answer -- is correct, but under the inclusive definition, both (A) and (B) are correct. So we see that using the U of Chicago definition of parallel may lead to an incorrect answer on the PARCC.
Here are some links to discussions regarding the parallel line debate. Mathematics Stack Exchange is a well-known question and answer site:
http://math.stackexchange.com/questions/45506/is-a-line-parallel-with-itself
At the next link, someone asked on Quora whether parallel lines can intersect. One of the posters who responded was Dr. David Joyce -- the very same Joyce I often quote here on the blog! Here he wrote that to Euclid, no line is parallel to itself, but one can consider the real projective plane, where it's sometimes convenient to define a line as parallel to itself. (In particular, parallel lines intersect at the same "point at infinity," and so every line has the same "point at infinity" as itself.)
http://www.quora.com/Can-two-parallel-lines-intersect
This is a Common Core blog. Therefore, what appears on the PARCC and SBAC exams matter more than trying to avoid indirect proofs or shorten the statements of theorems by a few words. The Common Core, PARCC, and SBAC take priority over all other considerations. And so if the U of Chicago definition could lead to a student choosing an incorrect answer on the PARCC, then that definition should not be used. This is especially true in light of the fact that it is the U of Chicago that is the outlier -- most other texts don't allow a line to be parallel to itself.
And so we must use the standard definition of parallel -- the definition originally intended by Euclid:
Definition
Two coplanar lines are parallel lines if and only if they have no points in common.
Using this definition will, as I said earlier, force the upcoming Parallel Test proofs to be indirect. I'll decide what to do about those proofs when we reach them. As it turns out, I don't have to change today's worksheet at all. Last year, I wrote that on the following images, I'll just leave a space for "parallel" in the vocabulary section and leave it up to individual teachers whether or not to include those four extra words in the definition. (Notice that in my exercises derived from the U of Chicago text, I preserved the true or false question "a line is parallel to itself." Of course, the answer will depend on which definition the teacher decides to use.)
The lesson concludes with some Postulates from Arithmetic and Algebra. As I mentioned yesterday, I want to avoid mentioning "algebra" -- the subject that causes many students to hate math -- yet these are important properties that show up in proofs (for example, the Reflexive Property of Equality). And so I'll just call them "Properties from Arithmetic" and just leave "algebra" out of it.
As an aside, let me point out that I was once a teenager studying geometry. Like those of many students at that age, my thoughts turned toward girlfriends and boyfriends -- except that I was the geometry nerd who wasn't a part of all of that. Like most nerds, I could make the following assumptions:
David Walker's First Postulate:
I am physically attracted to no other person.
David Walker's Second Postulate:
I am physically attractive to no other person.
These are statements that were obviously true -- in other words, they're postulates.
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