Tuesday, February 17, 2015

Section 14-1: Special Right Triangles (Day 113)

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. Last year Easter was two full months after the February holiday, so that stretch was especially tough -- even more considering that some schools ended up giving the state exam (in California, it was the practice SBAC) the week before the holiday.

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.

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 today being Mardi Gras, the last day before Lent begins (Ash Wednesday).

I spent my President's Day -- the last day of freedom before the Long March -- watching the movie Imitation Game. I seldom watch movies -- and I've never posted about movies on this blog until today -- but this upcoming weekend is Oscar night, and this Oscar-nominated movie is about Alan Turing, a 20th-century British mathematician, so this movie is appropriate for my math blog.

Here is a link to a biography on Turing.


The above website is one of my favorite websites for finding mathematicians' biographies -- the University of St. Andrews in Scotland. I've been finding math bios here for well over a decade (the site is dated 2003)!

Turing is famous for the Turing test -- the "Imitation Game" mentioned in the title. The Turing test refers to artificial intelligence -- specifically, an AI is said to pass the Turing test if it can fool humans into thinking that the AI is human as well. Brian Harvey -- whose Berkeley Logo page I linked to last week -- discusses a program called Eliza that, for a short time in the 1960's, actually did fool some people into thinking that it was human, but eventually Eliza failed the Turing test as well. Here is a link to Harvey's page, where he discusses how to write an Eliza-like doctor program in Logo:


There was a flashback scene in the movie where Turing is a teenager -- about the same age as the students we are teaching. And Alan's teacher was covering the last lesson before a holiday break -- it's not mentioned whether this was winter or spring break. Anyway, this lesson was on the proof of the irrationality of sqrt(2). Notice that I, coincidentally, mentioned this proof in my last post before watching the movie as an example of an indirect proof!

Spoiler Warning! Skip this paragraph if you don't want to be spoiled! Anyway, we find out that during these holidays, Turing's only friend, Christopher Morcum, dies of TB. According to the St. Andrews link above, Morcum died in February 1930. This implies that the break during which he died wasn't winter or spring break, but rather the half-term break. 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. END SPOILER

There is also the concept of a Turing machine. Alan mentioned a Universal Turing Machine, a computer that could solve any problem. As it turns out, a UTM is impossible -- that is, we can prove such a machine is impossible. Indeed, Turing gave an indirect proof that there can be no UTM. The following link describes Turing's proof better than I can:


Assume that a machine that can solve any problem exists. Then in particular, there is a machine that can solve the halting problem -- that is, we can feed the machine a program and its input, and it will tell us whether the program will eventually stop or run forever (called DOES-HALT at the link). We could then write a program that does the following: it will basically do the opposite of whatever its input does -- that is, it stops if its input would run forever, and run forever if its input would stop (called SELF-HALT at that link above). We then take this program and feed it to DOES-HALT, with itself as the input. To make a long story short, this program will halt if and only if it doesn't halt, which is a contradiction. Therefore, there is no UTM. QED

And so that is my takeaway from Imitation Game, a movie I greatly enjoyed. I'm hoping that some time during Oscar week, I'll be able to watch the other nominated movie about a famous scientist -- Stephen Hawking's Theory of Everything. If I get to watch it, I'll discuss it here on the blog.

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 17th -- Section 14-1: Special Right Triangles
Tomorrow, February 18th -- Section 14-2: Lengths in Right Triangles
Thursday, February 19th -- Section 14-3: The Tangent Ratio
Friday, February 20th -- Activity (includes Section 14-4: The Sine and Cosine Ratios)
Monday, February 23rd -- Section 14-5: Vectors
Tuesday, February 24th -- Section 14-6: Properties of Vectors
Wednesday, February 25th -- Section 14-7: Adding Vectors Using Trigonometry
Thursday, February 26th -- Review for Chapter 14 Test
Friday, February 27th -- Chapter 14 Test

Unlike Chapter 13, there is no plan to rearrange the chapter to match my geometry student. We are not scheduled to having tutoring this week, and even if we were, he is no where near the trig chapter of the Glencoe text.

So the plan for this chapter is straightforward. The one thing to note is how the day that Section 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 activities are based on some of the Exploration questions in the U of Chicago text, where the students are to use a calculator to discover some of the trig identities.

But that's for later this week -- how about today's lesson? Section 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 Section 14-2, and the other one is about how to simplify radicals, so we can explain in Section 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!

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