Lesson 4-1: Reflecting Points (Day 41)

Today I subbed at a continuation school. This is my first visit to the continuation school in the new district -- I was at the continuation school in the old district around this time last year. Today's teacher has three English classes and two Film and TV elective classes. (I'd never have believed that they would have a film elective at a continuation school until today.)

Hmm, should I do "A Day in the Life" for today? I usually focus on either math classes or middle schools, and this is neither -- and besides, two of the classes have aides, so I'm the classroom manager in only three classes. But maybe I'll do "A Day in the Life" anyway, just so you can see what the bell schedule looks like at this continuation school.

8:00 -- First period is the first Film and TV class. The students have two assignments -- the first is to read a short passage and answer four questions on Chromebooks, and the second is to make a storyboard based on a modern version of Sherlock Holmes.

One student refuses to do the reading passage, because it requires him to visit the Newsela website and log in with his Google password. There is a huge concern regarding privacy and technology in the classroom, especially when it comes to tech giants like Google. (Oops, I don't wish to bite the hand that feeds me, since Google now owns Blogger as well!) Indeed, some traditionalists have cited privacy concerns as a reason to oppose the online Common Core tests such as SBAC and PARCC (and possibly even the online SAT -- tying this back to yesterday's PSAT post).

The issue of privacy and student technology deserves its own post, so I won't say any more today.

8:55 -- First period leaves and second period arrives. This is the first sophomore English class. The students have two assignments -- the first is to read two articles about how to think positively after suffering a traumatic experience. One of the articles is written by a Holocaust survivor. The second assignment is to highlight and annotate key passages in the two articles to prepare the students to write their own essay on this topic.

9:55 -- Second period leaves. It is now the teacher's conference period, which lasts through the nutrition break -- the only break of the day. During this time I'm to supervise the basketball court near the stairs, but in reality, more students hang out near the stairs to eat than at the basketball court.

11:10 -- Fourth period arrives. This is the second Film and TV class. It's one of the two classes that has an aide, and so she makes sure that the students remain on task. The aide also tells me to write the instructions on the board since these students are more visually-oriented.

12:05 -- Fourth period leaves and fifth period arrives. This is the second sophomore English class.

1:05 -- Fifth period leaves and sixth period arrives. This is the only freshman English class, and the aide returns to help out with this class. It's the smallest class of the day with only four students. (In fact six are enrolled but two are absent, including one who has been suspended.) The students are to read eleven pages from The House on Mango Street by Sandra Cisneros. Then they must complete a worksheet with questions about five vocabulary words from the story. In Part I, they are to use context clues to guess the definitions of the five vocabulary words. Then in Part II, they match the words with their real definitions.

2:00 -- Sixth period leaves. School days tend to be shorter at continuation high schools than at comprehensive high schools, and so I go home to type this blog entry.

Since there are aides in fourth and sixth periods, the worst-behaved class is fifth period. This is a continuation high school, so I expect many students not to do the work (otherwise they wouldn't be sent here). Fifth period is completely off-task when it's time to highlight the sentences -- after all, in all the classes today, the students tend to be off-task when they are given something less specific to do (to find and highlight their own sentences as opposed to "popcorn reading" the two articles).

Though I don't like doing so, I tell them which sentences they should highlight, so that they have something specific to do. My chosen sentences aren't necessarily the best ones to highlight, and the students also fail to gain the skill of doing research on their own. But as a sub, student behavior is my priority, not student learning (unfortunately). Well, at least I'm not writing the essays for them.

During the periods with the aides, the only issue in fourth is a student who asks to go to the restroom right after nutrition. I make him wait until the midpoint of the period, 11:38 -- but he continues to complain until that time (and meanwhile, a second student asks to go before the midpoint). I feel like arguing about how both guys should have gone during break and try not to miss a second of class -- but I don't, since I've learned that doing so wouldn't solve anything. As this is the last class to use the Chromebooks, I have the students return them to the cart to be charged. This time, the Chromebooks aren't numbered, and I don't waste time forcing the students to charge them, since I have the luxury of time during sixth period with the aide to make sure each is charging).

Speaking of the other class with an aide, in sixth period, some students fail to make guesses for the definitions in Part I. Instead, they peek down at Part II and just write the correct definitions for each of the words from there (except for one girl who does make proper guesses).

The worksheet is short, and so this is the only class with extra time left to spare. The four students spend their time drawing on the desks -- the top of which doubles as a dry-erase board. I've seen some math teachers tweet pictures of similar dry-erase desks in their own classrooms. It's interesting, though, that the first time I would see such desks in real life would be in an English classroom rather than a math classroom.

Meanwhile, today on her Mathematics Calendar 2018, Theoni Pappas writes:

Side AB < ?

(Here is the given info from the diagram: In Triangle ABC, AC = 7.4" and BC = 3.6".)

Actually, this is a straightforward example of the Triangle Inequality from Lesson 1-9 of the U of Chicago text. We find that AB < AC + BC = 7.4 + 3.6 = 11. Therefore, AB must be shorter than 11 inches -- and of course, today's date is the eleventh.

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'.