Lesson 11-5: The Midpoint Connector Theorem (Day 115)

Today I subbed in a middle school P.E. class. Normally I wouldn't do "A Day in the Life" for a P.E. class that is non-representative of what I'd like to teach this day. But in addition to eighth grade P.E., this teacher also has seventh grade health in a classroom. Thus it's worth doing "A Day in the Life" -- especially since two students (one in each grade) end up earning referrals.

8:15 -- This is the middle school that begins with homeroom and first period, which happens to be the first eighth grade P.E. class.

It seems as if I've been writing about rain a lot on the blog lately. Actually, it doesn't rain that much this week at all -- but it just so happens that the one day it rains is the one day I sub P.E. this week. In fact, the rain was mostly last night (with a little after school), but it's enough for the other P.E. teachers to cancel outdoor activities on the muddy ground. I actually spoke to the regular teacher on the phone before class. He assumes that there would be outdoor P.E. on the track today, and has no idea that the other teachers would cancel it.

So instead, all students meet in the gym. The activities are similar to the one period of P.E. that I mentioned in my January 18th post -- students can either play Dodgeball or watch a movie. I stay out to watch the Dodgeball game.

9:20 -- And once again, we know that this school has a period rotation after first period. The rotation for Thursday goes 1-5-6-2-3-4. Fifth period is the first of the seventh grade health classes.

These students have an assignment from the health text. They must read the chapter on alcoholism and answer 18 questions in complete sentences. This includes a few fill-in-the-blank problems -- so they must write out the entire sentence with the blank filled in.

10:15 -- Fifth period leaves for snack.

10:30 -- Sixth period arrives. This is the second of the seventh grade health classes.

11:25 -- Sixth period leaves and second period arrives. This is the last seventh grade health class.

12:20 -- Second period leaves for lunch.

1:05 -- Third period arrives. This is the second of the eighth grade P.E. classes.

2:00 -- Third period leaves and fourth period arrives -- that's right, this regular teacher has all six periods without a conference. This is the last of the eighth grade P.E. classes.

This time, I watch the movie instead of Dodgeball. For some reason, the movie is Elf -- and yes, I check the calendar and see that today is February 21st, not December 21st!

Actually, if you watch the whole film and count backwards from Christmas Eve, you'll notice that the part of the movie we watch today takes place on December 21st. (The day begins as Buddy the Elf wakes up in Gimbel's after decorating it for Santa, and it ends when he goes home with his long-lost father and his new family.)

I've long been fascinated with school calendars and whether movies that feature school scenes follow a logical calendar (ever since Frosty the Snowman has kids in school on Christmas Eve). We see that the following day is December 22nd, and a bell rings to mark the end of school for Buddy's newly found stepbrother. The students and I decide that this is a good time to mark the end of school for us, and so I stop the movie at this point. (In case you want to know how I determined the dates, notice that the next day is when Buddy goes out on a date with Jovie -- a Thursday night -- and the next day is Christmas Eve.)

And in case you think my dates are wrong since winter break should have started by the 22nd, recall that the film takes place in New York, where there is school on the 22nd and even the 23rd. The short winter break is made up for with a full week off for President's Day -- and that's this current week in real life. (Yes, so I'm finally talking about February, not December!)

2:55 -- The bell rings, and the students go home (a few minutes after Buddy's stepbrother).

Today's subbing indicates that in this district, middle school health is taught by P.E. teachers. I'm not fully sure how it works in this district -- presumably, one of the units in a P.E. class is health, and so students go to a classroom instead of the P.E. area. We know that seventh graders get health, but I don't know whether eighth graders get a health unit or not.

In the LAUSD, health is part of science class, not P.E. class. It's logical to assume that at my old charter school (which is chartered with LAUSD), health is also taught by science teachers. But hold on a minute -- the year I was at that school, I was the science teacher. This means that logically, I was the one who needed to teach the middle school health classes!

Of course, I had all sorts of problems teaching science -- and now we see that I could have taught health on top of science! But at no point did anyone ever ask me to teach health. The Illinois State text contains a science component, but not a health component.

But health could be considered to fall under the umbrella of life science. In the old days, life science was the seventh grade curriculum, and so I could have taught my seventh graders health. But under the NGSS, life science is integrated into all three middle school years.

The following link is to the LAUSD health curriculum:

http://www.heplausd.net/wp-content/uploads/2018/11/CA-Health-Education-Standards_MS.pdf

But these standards haven't been updated since 2008 -- well before NGSS. So the link doesn't explain how to incorporate health into NGSS science courses.

According to the link above, middle school health includes sex ed. I once subbed for a health class in another district -- it was the last class I subbed in before starting at the old charter. (I never mentioned this on the blog.) I'm not sure whether I would have felt comfortable teaching sex ed. But once again, no one at the charter ever told me that I, as the Science/Health teacher, needed to teach sex ed.

OK, let's finally get to classroom management. Here is our focus resolution for today:

2. Keep a calm voice instead of yelling at students.

I definitely feel like yelling in second period health class. And it all starts even before the tardy bell rings, when I see several students gathered around a broken lamp. It's one of those room lamps that's as tall as a human being. Apparently, someone's backpack knocks it down. But no, it's not the light part that's broken -- it's the base. Thus the lamp can no longer stand on its own.

At this point I panic -- when I'm the sub and something breaks, it looks bad on my part. And so while I believe that my voice stayed below the "yell" level, I do begin the class with a long argument about how the students shouldn't go around playing around and breaking things.

This is the class where I must write my first referral. It's not for the breaker of the lamp -- instead it's for one boy who keeps throwing pencils and other objects. Instead of hitting other students, the projectiles land near the window. I'm afraid that he might break the window after the lamp base in the classroom is already fractured. And so I write the referral and call security to escort him from the room before anything else breaks.

I've said it before and I'll say it again -- I consider any day on which I must write a referral and call security to be a failure. The ideal classroom manager is able to handle the students in the room without outside help.

Is there anything I could have done to prevent this incident? I'm wondering whether I might have provoked this boy and the other students by making a big deal about the lamp. When I first see the broken lamp, I think to myself, "This second period is the class from hell -- they can't even enter the room without breaking things. I must keep my eyes on this class!"

But in reality, this class probably isn't that much worse than the other two health classes. The lamp fell when someone brushes it with a backpack -- not because the class enters the room only to start playing with the lamp. (Indeed, there's even a slight chance that the one who brushes against the lamp to knock it down is a student leaving sixth period, not a student arriving to second period. The first student arrives to second period before the last sixth period student leaves.)

And so I argue with the students fearing that this would be the class from hell when it isn't -- and then this becomes a self-fulfilling prophecy. The students get upset that I assumed malice behind the breaking of the lamp, and so they start disrespecting me. Many students refuse to do the assignment who might have worked if only I'd begun the class with "So the lamp broke -- no big deal!" It's possible that this might even include the eventual pencil thrower!

At the start of this post, I mentioned a second referral. I'm not sure whether there's anything I could have done to prevent this referral. It's during fourth period P.E. class. Two P.E. classes enter the gym with their teachers -- yours truly and a regular teacher. Since she's a P.E. teacher, the regular teacher takes Dodgeball while I go to the side room to watch Elf. Thus students are in either the main gym or side room based on their choice of Dodgeball or Elf, not who their P.E. teacher is.

It's the other P.E. teacher who writes the referral -- so I'm not even the one who writes this kid up. But apparently he isn't in her class, but in my regular teacher's class. Thus it counts as a second referral, since my regular teacher will return tomorrow to find out that two of his students were written up.

(This is why the Elf watchers are eager to leave when the bell in the movie rings. I'm standing by the door waiting for the other teacher to dismiss her students so I know when to let mine leave. But she's likely too busy writing the referral to dismiss them!)

Fifth period is the best health class of the day. First period is the best P.E. class of the day. Both of these classes are loud, but these are much better than the referral classes.

Today on her Mathematics Calendar 2019, Theoni Pappas writes:

7/120 rpm = ____ angular velocity in degrees per minute

Since rpm = revolutions per minute and one revolution is 360 degrees, we use dimensional analysis:

(7 rev/120 min)(360 deg/1 rev) = 21 deg/min

Thus the desired angular velocity is 21 degrees per minute -- and of course, today's date is the 21st.

Some might wonder whether this should even count as a Geometry problem. Well, it does use angles and degrees, and there's no mention of negative angles or angles greater than 360 degrees. And so I decided to post it today.

Lesson 11-5 of the U of Chicago text is called "The Midpoint Connector Theorem." In the modern Third Edition of the text, the Midpoint Connector Theorem appears in Lesson 11-8.

Unlike the rest of Chapter 11, this is a lesson I covered well last year. And so this is what I wrote last year about today's topic:

Lesson 11-5 of the U of Chicago text is on the Midpoint Connector Theorem -- a result that is used to prove both the Glide Reflection Theorem and the Centroid Concurrency Theorem. Last year I only briefly mentioned the Midpoint Connector Theorem on the way to those higher theorems, and then when we actually reached Chapter 11 I only covered it up to Lesson 11-4, as I knew that I'd already incorporated 11-5 into the other lessons. But this year, I'm giving 11-5 its own worksheet.

Midpoint Connector Theorem:
The segment connecting the midpoints of two sides of a triangle is parallel to and half the length of the third side.

As I mentioned last week, our discussion of Lesson 11-5 varies greatly from the way that it's given in the U of Chicago text. The text places this in Chapter 11 -- the chapter on coordinate proof -- and so students are expected to prove Midpoint Connector using coordinates.

It also appears that one could use similar triangles to prove Midpoint Connector -- to that end, this theorem appears to be related to both the Side-Splitting Theorem and its converse. Yet we're going to prove it a third way -- using parallelograms instead. Why is that?

It's because in Dr. Hung-Hsi Wu's lessons, the Midpoint Connector Theorem is used to prove the Fundamental Theorem of Similarity and the properties of coordinates, so in order to avoid circularity, the Midpoint Connector Theorem must be proved first. In many ways, the Midpoint Connector Theorem is case of the Fundamental Theorem of Similarity when the scale factor k = 2. Induction -- just like the induction that we saw last week -- can be used to prove the case k = n for every natural number n, and then Dr. Wu uses other tricks to extend this first to rational k and ultimately to real k.

I've decided that I won't use Wu's Fundamental Theorem of Similarity this year because the proof that he gives is much too complex. Instead, we'll have an extra postulate -- either a Dilation Postulate that gives the properties of dilations, or just one of the main similarity postulates like SAS. I won't make a decision on that until the second semester.

Nonetheless, I still want to give this parallelogram-based proof of the Midpoint Connector Theorem, since this is a proof that students can understand, and we haven't taught them yet about coordinate proofs or similarity.

[2019 update: Some of this old discussion is relevant to last Tuesday's traditionalists' post regarding rigor and the Common Core.]

But even with last year's Lesson 11-5 worksheet, I must make a change. Today is Thursday, and so I must replace last year's activity page with an assignment page.