Lesson 4-1: Reflecting Points (Day 41)

Today I subbed at a middle school special ed class. This is a self-contained class with aides, and so there is no "A Day in the Life" today. I also write about this class in my December 4th post -- the seventh graders from that class are now eighth graders in this class, along with a new batch of seventh graders.

This is the school that starts with homeroom/first period with all other classes rotating. But the schedule is mixed up due to a double-assembly schedule. (I described the double-assembly schedule back in my September 24th post, albeit from a different school.) Even though the class is for the most part self-contained, some students leave for a period or two of electives, and so the bell schedule must be followed.

Normally the day would start with homeroom, but due to the double-assembly schedule, the day starts with first period P.E., followed by the doubled fifth period for the assembly. This throws everything off. Each aide has a scheduled thirty-minute break that is aligned with the bell schedule and even takes rotation and day of the week into account -- everything except the assembly. (One aide points out that perhaps the double-assembly schedule should be only on Wednesdays, since these are already shortened days anyway.)

By the way, the topic of the assembly is "Rachel's Challenge," where "Rachel" actually refers to a survivor of the Columbine massacre twenty years ago (when I was a high school senior). We actually don't take the special ed students to the assembly -- instead we show them a video about lockdown drills (which have become more common since Columbine).

I do help a student or two with math. It's a worksheet that involves adding two dollar amounts (on a calculator) and then "dollar up" (rounding up to the next dollar -- that is, the dollar bills that one might hand an actual cashier).

I usually don't sing songs in classes like this -- and today was no exception. Instead, I do like to pass out candy and pencil gifts to these classes (since they are so small). In gen ed classes, I usually wait until it's the week of Halloween (or whatever holiday), but for this special ed class, I give the students some Halloween Kit-Kats today, October 10th.

As for the pencils, in addition to Halloween pencils, I actually found some fall pencils. Some of the pencils read "Fall in love with learning." And so I give these pencils away.

Recall that back at the old charter, I made the mistake of not having a pencil gift during the long period of time between the first day of school (mid-August) and Halloween -- and in old posts, I suggest that either Back-to-School Night or the fall equinox might be a good time to give away pencils between the first week and Halloween week. These fall pencils would have been perfect for the equinox.

Other than that, there are no major problems today for me. The aides naturally take care of any classroom management.

Today on her Mathematics Calendar 2019, Theoni Pappas writes:

0.01 km = _____ m

There are 1000 meters in a kilometer, so by dimensional analysis 0.01 km is 10 m. Therefore the desired length is ten meters -- and of course, today's date is the tenth. This isn't exactly a Geometry problem, but it is National Metric Week, so this problem is perfect for this week.

Lesson 4-1 of the U of Chicago text is called "Reflecting Points." This is what I wrote last year about today's lesson:

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 September is not much different from a traditional course. But I had to give all that preliminary material first -- after all, the Common Core Standards demand it:

CCSS.MATH.CONTENT.HSG.CO.A.4
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.

Lesson 4-1 of the U of Chicago text deals with reflections. As I mentioned last year, 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 on reflections. 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 onm.
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'.