Tuesday, April 10, 2018

Lesson 14-1: Special Right Triangles (Day 141)

Lesson 14-1 of the U of Chicago text is called "Special Right Triangles." In the modern Third Edition of the text, special right triangles appear in Lesson 8-7 (as I explained in yesterday's post).

Meanwhile, I subbed today in a seventh grade P.E. class. As it turns out, today -- Day 134 in this district -- was a CPR training day for all P.E. teachers. Of the four P.E. teachers at this school, three were at the training, but the fourth -- a long-term sub -- is not required to attend the training.

Last week, I wrote that I want to writing more "Day in the Life" posts in order to focus on classroom management especially in middle schools, even in classes other than math. But in this situation, it's tricky -- all the P.E. teachers meet on a field, and the long-term sub does most of the management. (I can't really say "classroom" management, since it's not a classroom.)

With so many regulars out, the students didn't dress at all today. Instead, they were to walk three times around the field, then have free play.

The main management issue was breaks for water. As it turns out, today is the day that sixth graders from the elementary schools make their visit -- the counterpart to Pi Day, when the eighth graders had their high school orientation. So the P.E. teachers implement a strict water rule -- only one student at a time is allowed to go to the water fountain. The fear is that the seventh graders might remember their friends from last year and disturb them on the tour.

After the sixth graders left around halfway through the day, two students at a time were allowed to use the fountains. This came in handy because the temperature was well above average today. (And yes, I'm aware that there are parts of the country where the temperature is still below average for this time of year.) But even the students in fifth period -- the first class of the day according to today's rotation -- complained that the weather was too hot (at 72 degrees). I suspect that if it had been one degree cooler, they would have complained that it was too cold. I observed this often at my old school last year -- some students always complain about the weather at P.E. time. The temperature is always too hot or too cold -- it's never just right or absolutely perfect.

Meanwhile, the restrooms are supposed to be locked -- but they're difficult to lock. I keep leaving the restroom thinking that it's locked, only to find out that another student got in.

After lunch, this school has SSR time. This is often awkward for students who rotate into P.E. during SSR time, but today many students would rather read than walk in the hot sun.

The three Harry Potter teacher archetypes -- Flitwick, McGonagall, Snape -- are relevant here. I must make sure to avoid the two extremes and not swing from one to the other. As I wrote earlier, I've been thinking about management over the spring break. For example, we think back to the incident I mentioned in my post from a week ago, when a student in sixth period is talking loudly. If I ignore the student, then I'm being Flitwick -- but if I write down the name and start arguing and yelling at the student, then I've just turned into Snape.

Many students today might have believed that making them stand in the hot sun to wait for water is Snape-like, but as long as they understand the rules and I avoid arguing, then I avoid Snape. During each class, I walk away from the pathway leading to the water and allow one of the other subs to control the line. This way, the students realize that the water rule really is a rule, rather than something that I personally came up with just because I'm a mean, Snape-like teacher. Again, I make sure that I don't give in to the complaints and turn back into Flitwick.

That's a makeshift "Day in the Life" rather than a true DITL post. But I'll continue to focus on my classroom management. I will still mark this post with the "subbing" label anyway.

This is what I wrote last year about today's lesson:

Chapter 14 of the U of Chicago text is on Trigonometry and Vectors. Here's the plan:

Today, April 10th -- Lesson 14-1: Special Right Triangles
Tomorrow, April 11th -- Lesson 14-2: Lengths in Right Triangles
Thursday, April 12th -- Lesson 14-3: The Tangent Ratio
Friday, April 13th -- Activity (includes Lesson 14-4: The Sine and Cosine Ratios)
Monday, April 16th -- Lesson 14-5: Vectors
Tuesday, April 17th -- Lesson 14-6: Properties of Vectors
Wednesday, April 18th -- Lesson 14-7: Adding Vectors Using Trigonometry
Thursday, April 19th -- Review for Chapter 14 Test
Friday, April 20th -- Chapter 14 Test
Monday, April 23rd -- Activity (as explained in yesterday's/last Friday's posts)
So the plan for this chapter is straightforward. The one thing to note is how the day that Lesson 14-4 would have occurred, there is a planned activity day. I've noticed how many texts, including the U of Chicago, discuss the tangent ratio in a separate lesson from sine and cosine. I suppose that in many ways, sine and cosine are alike in a way that tangent isn't. The sine or cosine of any real number is between -1 and 1, while the tangent can be any real number. Therefore the graphs of sine and cosine resemble each other. The tangent ratio involves two legs, while the sine and cosine ratios involve one leg and the hypotenuse. Even the name "cosine" includes the word "sine," while the name "tangent" doesn't include "sine."

Yet I will end up covering sine, cosine, and tangent all on the same day. In the past, I've seen many teachers simply teach SOH-CAH-TOA all in the same lesson, and then when they come to me for tutoring, they look at each triangle in the homework to determine whether sine, cosine, or tangent is needed to solve the problem. But as it turns out, all of the questions require tangent because the student is actually reading the tangent lesson in the text! If the student is going through all of that, then we might as well have all three trig ratios in the same lesson.

And so this is exactly what I'll do. This will then free a day for an activity. My planned activity will actually be one that I found off of another teacher's website. (Actually, I'm still debating whether to do the activity on Friday or on Thursday, since this teacher presents this activity before teaching the students about sine, cosine, and tangent.)

But that's for later this week -- how about today's lesson? Lesson 14-1 of the U of Chicago text is on Special Right Triangles -- that is, the 45-45-90 and 30-60-90 triangles. The text emphasizes how these triangles are related to the regular polygons. In particular, the 45-45-90 and 30-60-90 triangles are half of the square and the equilateral triangle, respectively. We can obtain these regular polygons, in true Common Core fashion, by reflecting each right triangle over one of its legs. The regular hexagon is also closely related to the 30-60-90 triangle.

The questions that I selected from the text refers to these regular polygons and using the triangles to measure lengths related to the regular polygons. I mentioned today how I like to watch baseball over summer break -- well, a baseball "diamond" (really a square) appears on the worksheet. Also, a honeycomb, with its hexagonal bee cells, also appears.

The review questions that I selected are also preview questions. Two of the questions involve similar right triangles in preparation for geometric means in Lesson 14-2, and the other one is about how to simplify radicals, so we can explain in Lesson 14-4 why the sine and cosine of 45 degrees are usually written as sqrt(2)/2.

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