http://math.berkeley.edu/~wu/CCSS-Geometry.pdf

Like Dr. M, Wu is also developing a Common Core course, although he is also working on the eighth grade geometry portion as much as the high school class -- after all, the peculiarities of Common Core Geometry begin in 8th grade.

Wu begins by giving some basic postulates, then moves on to transformations. He covers rotations first and gives some of their properties. Then Wu gives his first theorem, Theorem 1 (under high school, not 8th grade):

Wu:

"Theorem 1. Let L be a line and O be a point not lying on L. Let R be the 180-degree

rotation around O. Then R maps L to a line parallel to L itself"

We are considering rotations about a point O, and we want to know what happens when we rotate lines about that point O 180 degrees (since rotations require a center and a degree). The answer is that it depends on whether the line passes through O or not. A line passing through O ends up being rotated to itself -- this is essentially because a line is a straight angle of 180 degrees. Less obvious is what happens if the line doesn't pass through O. If L is the line that we are trying to rotate and O doesn't lie on L, where is the image of L? We begin by choosing any point on line L, and Wu labels this point Q. Now we want to rotate Q 180 degrees about O, and Wu refers to this new point as P. So where is P -- can it lie on line L? We consider the line that passes through points Q and O, and Wu labels this line with a lowercase l. Like all lines passing through O, l is rotated to itself, and so P, the rotation image of Q, lies on l. This proves that P can't lie on L, for if it did, there would be two lines passing through P and Q (namely L and l) when there is only supposed to be one line through those points. (And L and l can't be the same line, since O lies on l yet not on L.) But point Q is arbitrary -- we've shown that every point Q on the line L is rotated to a point P that is not on L. Therefore L and its rotation image can't have any points in common -- and lines that have no points in common are called "parallel." QED. (Technically, parallel lines are lines in a plane that have no common points -- but "plane" is assumed here because the rotations, reflections, and translations that we're performing are transformations of the plane.)

Now this Theorem 1 can be used to prove other statements -- for example, if two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel. This is Theorem 12 in Wu. To prove this, we observe that the two lines and the transversal form a "Z" shape, and rotating this "Z" 180 degrees gives the same "Z" shape. (Wu does this more formally.) This rotation maps the transversal to itself and one of the lines to the other -- and since lines are parallel to their rotation images by Theorem 1, the original lines are parallel.

So we have two ways to prove the Alternate Interior Angles Theorem -- using the Exterior Angle Inequality following Dr. M (and Euclid), and using rotations following Wu. As a math tutor, I decided to present both the Exterior Angle Postulate and Wu's Theorem 1 to a geometry student who is reading a traditional text (not any of the texts mentioned on this blog so far) that defined transformations in its final chapter. The student is taking a fast-paced summer course in which all the assessments are multiple-choice (as there is no time to grade a free-response test), and so there are no proofs on the tests. Yet when I asked him to prove the Alternate Interior Angles Theorem, he saw the required 180-degree rotation for Wu right away. Perhaps this was because not doing many proofs in class, he was especially unfamiliar with the indirect proof that was required in the derivation from the Exterior Angle Postulate. At any rate, if an actual student found the rotation proof easier to understand, then this is how I should teach it. And besides -- Common Core Geometry is all about rotations (and other transformations), and so I should use rotations as much as possible in a Common Core class (as opposed to the derivation from the Exterior Angle Postulate, which doesn't use transformations at all).

What I will do on this blog is present the material from my U of Chicago text, which is already very similar to Common Core Geometry, and fill in the gaps using information from Wu. The first seven chapters in my U of Chicago text are as follows:

1. Points and Lines

2. Definitions and If-Then Statements

3. Angles and Lines

4. Reflections

5. Polygons

6. Transformations and Congruence

7. Triangle Congruence

Notice that while reflections are given in Chapter 4, the other transformations aren't given until two chapters later. I understand that there is a newer version of the U of Chicago text -- mine is dated 1991 -- in which all the transformations are covered in Chapter 4. But the 1991 text is the one in my hands, and so I will work from this one.

As for the other chapters, the first two chapters are similar to a traditional text. Notice that parallel lines are covered very poorly in this text -- corresponding angles are mentioned in Chapter 3, alternate interior angles in Chapter 5, and same-side interior angles not at all.

Here's how I'll rearrange the chapters so that they fit Common Core:

1. Points and Lines

(I'll focus on sections 1-6 onward.)

2. Definitions and If-Then Statements

(I'll move angles up to this chapter. Chapter 3 is awfully late to first present angles -- most traditional texts cover angles in Chapter 1!)

3. Reflections

(I might as well get to reflections right away. Since reflections are defined using perpendicular lines, some of the information on perpendicular lines fit here.)

4. Rotations

(Here's where I'll include information from Wu's page, including his Theorems 1 and 12.)

5. Polygons

(This chapter is mostly intact.)

6. Transformations and Congruence

(This chapter is mostly intact, except that rotations have been moved up.)

7. Triangle Congruence

Notice that U of Chicago presents reflections first, while Wu gives rotations first. The definitions of reflection and rotation in U of Chicago requires that reflections have priority:

U of Chicago:

"For a point P not on a line m, the reflection image of P over line m is the point Q if and only if m is the perpendicular bisector of segment PQ. For a point P on m, the reflection image of P over line m is P itself."

"A rotation is the composite of two reflections over intersecting lines."

Wu uses a similar definition for reflections, but his definition of rotations is a more traditional one in terms of the center and angles and so is independent of reflections. The problem is that for Wu, reflections are not well-defined unless we prove that there is exactly one point Q such that the reflecting line is the perpendicular bisector of segment PQ -- and his proofs ultimately derive from Theorem 1 and rotations. This isn't a problem for U of Chicago because it merely assumes, via the Reflection Postulate, that reflections are well-defined:

Reflection Postulate (U of Chicago):

Under a reflection:

a. There is a 1-1 correspondence between points and their images. This means that each preimage has exactly one image and each image comes from exactly one preimage.

There are more parts, but part (a) is the only part that matters now. Part (a) is what guarantees that reflections are well-defined.

To conclude this post, let me state that Chapters 1-7 constitute what I would like to be the first semester of the geometry course. Other concepts such as area, volume, and similarity (and thus dilations -- the transformations on which similarity is based) can wait until second semester. I'm thinking of a student who just barely passes Algebra I (earning the lowest grade that the school allows for entry into Geometry). The last thing that such a student wants to see is more algebra, yet concepts like measurement and similarity are obviously algebra-heavy. I'd like to see the first semester of Geometry be algebra-light -- the most calculation required in Chapters 1-7 is to determine whether angles add up to 90, 180, or 360 (the Polygon Angle-Sum Theorem) -- so that the barely-passing Algebra I student has a chance to earn a good grade in at least one semester of the geometry course.

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