Tuesday, February 16, 2016

Lesson 14-1: Special Right Triangles (Day 103)

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

Consider the three major breaks of the school year: winter break, spring break, and summer break. I suppose that now I should actually say four major vacations, since Thanksgiving break is now the fourth major break of the year. Of these four breaks, which one do you look the most forward to?

Some people might say summer break -- after all, it's the longest break. Others might say winter break, since the holidays are a fun time of the year. I enjoy going to the beach and watching baseball in the summer and opening Christmas presents in the winter as much as the rest of you, but as for the break I look forward to the most, ever since I was a student myself, that honor goes to spring break.

To me, the toughest stretch of the school year is that from President's Day to spring break. Think about it -- the first school holiday of the year is often Veteran's Day. Then, starting with Vets Day, there is another holiday every 2-4 weeks. A few weeks after Vets Day is Thanksgiving. A few weeks after Turkey Day is winter break. A few weeks after coming back from the holidays is Martin Luther King Day. A few weeks after King Day are the holidays for Lincoln and Washington. But once we reach President's Day, there are no more holidays until spring break, which may be a month or two after Prez Day, often depending on the Easter date.

Notice that the stretch from President's Day to Easter is not the longest with no holidays. When I was growing up, that honor usually went to the beginning of the year, with no holidays from Labor Day until Veteran's Day. The school district whose calendar we're following on the blog has an off day for Columbus Day, so that breaks up the Labor to Vets stretch. But even without the Columbus holiday, I still found the Prez to Easter stretch to be tougher. Think about it -- the entire first quarter fits between Labor Day and Veteran's Day, but the first quarter is an easy quarter. The work in many classes was just review, especially math. In other classes, major projects are not assigned yet. On the other hand, President's Day to Easter is mostly the third quarter. Major projects are assigned and due during the third quarter, and we've reached the harder chapters in most math texts. (But with Easter being so early this year, the longest stretch without a holiday on the blog calendar is actually the stretch after spring break, from Easter to Memorial Day.)

Therefore, I consider the current stretch, from President's Day to Easter, to be the toughest stretch of the year. I've named this difficult period of time the Long March -- this name evokes the military where soldiers often have to travel long distances on foot, and it also refers to the month of March, the month that constitutes the majority of this period. And so the break I look the most forward to is the one that marks the end of the Long March -- spring break. Notice that for those who are more religiously minded, the Long March often corresponds exactly to Lent, with last week being Mardi Gras and Ash Wednesday.

Last year, I said how the British academic calendar differs from ours. In Great Britain, the school year is divided into three terms, much like our trimesters, except that the holidays of Christmas and Easter divide the three terms. Midway through each term, there is a one-week half-term break. The February half-term break actually occurs around now -- the American President's Day. Therefore there is no concept of a Long March in the UK -- three- or four-day weekends are already rare. The major breaks are Christmas, Easter, and the half-term weeks in between the holidays.

(With all of these extra breaks in the British calendar, the school year stretches into mid-July. This explains why JK Rowling releases all of her Harry Potter books -- including the recently announced "eighth book," in July, though it doesn't hurt that the 31st is the birthday of both the author herself as well as her main character. That's right -- much of what I know about the British school system I learned through the Harry Potter books.)

Fortunately, there are some bright spots during the Long March -- most notably Pi Day, the biggest day of the year in our geometry class. But I must begin the Long March as so many of my own teachers once did, and that's with a very difficult chapter. Chapter 14 of the U of Chicago text is on Trigonometry and Vectors. Here's the plan:

Today, February 16th -- Lesson 14-1: Special Right Triangles
Tomorrow, February 17th -- Lesson 14-2: Lengths in Right Triangles
Thursday, February 18th -- Lesson 14-3: The Tangent Ratio
Friday, February 19th -- Activity (includes Lesson 14-4: The Sine and Cosine Ratios)
Monday, February 22nd -- Lesson 14-5: Vectors
Tuesday, February 23rd -- Lesson 14-6: Properties of Vectors
Wednesday, February 24th -- Lesson 14-7: Adding Vectors Using Trigonometry
Thursday, February 25th -- Review for Chapter 14 Test
Friday, February 26th -- Chapter 14 Test

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.

Thus ends the first day of the Long March. There's still more than a month to go!