Monday, September 8, 2014

Section 4-1: Reflecting Points (Day 22)

At last, we have reached what makes Common Core Geometry different from traditional geometry -- transformations, including reflections, rotations, and translations. So far, what I posted in August is not much different from a traditional course -- only the previews that I posted in July give a glimpse as to how this will not be the usual geometry. But I had to give all that preliminary material first -- after all, the Common Core Standards demand it:

Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

And that's exactly what had to do in the first three chapters of the U of Chicago text -- define angles, circles, perpendicular lines, parallel lines, and line segments. Only now after defining those basic terms can we actually define rotations, reflections, and translations so that we can finally do Common Core Geometry.

Section 4-1 of the U of Chicago text deals with reflections. As I mentioned in the preview back in July, we do reflections first because the text defines rotations and translations in terms of reflections!

The definition of reflection is so important that I repeat it here. (Remember that I use a strikethrough to represent the segment symbol, since I can't reproduce the vinculum here.)

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 PQ.
For a point P on m, the reflection image of P over line m is P itself.

This definition is highly intuitive -- after all, suppose I gave a student a line m and two points P and Q such that the reflection image of P over m is the Q. Now suppose that I drew in segment PQ, and asked the student to tell, just by looking, how many degrees is the angle formed between m and PQ. Chances are that the student will say that it is 90 degrees. Then suppose that I asked the student to tell me which of P and Q is closer to m. Chances are that the student will say that the distances are the same.

(This is a trick that I often do with students -- whenever I ask a student whether AB or CD is longer, I'm almost always trying to get the student to notice that the lengths are equal!)

For this section, I'll repeat my first worksheet that I posted back in July as a preview -- but now I'm posting it again for real. Then I follow it with some exercises. Keep in mind that the method I suggested to generate reflection images is folding -- and it may be hard to fold when there is writing on both sides. As much as I want to save paper and not tie up the Xerox machine, this lesson, and the ones that follow, are very intensive on drawing and folding.

Still, there are a few more things that I want to include here. As I mentioned earlier, one way to generate reflection images is folding. Another method suggested in the U of Chicago text is utilizing a protractor. And once again, this is an important lesson, so let me restate the method for those of you who don't have the U of Chicago text:

1. Place your protractor so that its 90-degree mark and the center of the protractor are on m.
2. Slide the protractor along m so that the edge line (the line through the 0- and 180-degree marks) goes through P.
3. Measure the distance d from P to m along the edge line. You may wish to draw the line lightly.
4. Locate P' on the other side of m along the edge, the same distance from P.

We can see why this works. The first two steps take care of the "perpendicular" part of the definition -- as m is on the 90-degree mark and PP' is on 0 and 180 degrees, so m is perpendicular to PP'. Then the last two steps give us the "bisector" part -- with P and P' both d units from m, so that m bisects PP'.

This text gives a third method -- constructing using a straightedge and compass. But I wish to save constructions for a little later.

Notice that I prefer just to use the prime symbol for the image of P, giving us P'. The U of Chicago text also gives function notation, r(P). I admit that this is more precise -- especially if we, just as the book does, add a subscript to indicate the line of reflection. But once again, my priority is to avoid confusing students, and I hope that students will find P' less confusing than r(P).

The text also includes reflections on the coordinate plane over one of the axes. I decided to include this, since we'll have to use a coordinate plane at some point in this course. But I still want to avoid writing linear equations until we get to the second semester (at which point the Common Core Standards have us do something brand new with equations). Since reflecting individual points on the plane has nothing to do with equations, I have no problem including this here.

I had to throw out most of the review questions for this lesson, since most of them pertain to the second half of Chapter 3 that I've skipped for now. Only Question 22 can be included -- and even then, only the (a) part fits here. Even though the question is labeled (Lessons 3-3, 3-2), the (b) part requires parallel lines and corresponding (actually same-side interior) angles, so it's actually part of Section 3-4.

The final Bonus Question 24 is actually a preview of the next lesson, Section 4-2.

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