Friday, February 23, 2018

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

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.

Today I subbed in a high school ROP multimedia class. (For those who aren't familiar, ROP, or Regional Occupational Program, is a California career technical education program.)

Since it's not a math class, there is no "Day in the Life" today. But I do wish to point out that depending on the class, students are directed to create their own graphics and videos -- including a short one-minute trailer of a hypothetical movie. Footage from actual films as well as one new scene of their own is used to create the trailer.

Last year, the coding teacher wanted to do a similar project with our eighth graders. It was more successful at this school -- partly because these are high school students not eighth graders. But another factor is student behavior. Even though last year's eighth graders generally behaved better for other teachers (including the coding teacher) than for me, they still misbehaved enough to discourage him from proceeding with the project. I suspect that if I had been a stronger classroom manager last year, their improved behavior would have carried into coding -- and then they would have been allowed to do a video project.

One student I meet in today's class is from Puerto Rico. He tells me that his home has been damaged by last year's hurricanes, but he is attending school here in California, funded by a "scholarship" in his favorite sport, volleyball. I ask him about his family back on the island, and he informs me that they are doing well, even though they're still without electricity. It's one thing to read about the problems Puerto Ricans are facing -- it's another to meet a displaced victim face to face.

Unlike the rest of Chapter 11, this is a lesson I covered better two years ago than three years ago. And so this is what I wrote two years ago 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.

Today is an activity day. The old Lesson 11-5 worksheet from two years ago wasn't officially an activity page, but I included an extra questions which serve to make this an interesting activity:

First, notice that the first thing we see in the U of Chicago text is a picture of Sierpinksi's Triangle -- one of the fractals that we learned about two months ago when reading Benoit Mandelbrot's book. It's included here since it's closely related to midpoints. Then there is a problem from the text about Midpoint Quadrilaterals. Wu's proof is added as the last bonus question.

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