Lesson 14-4: The Sine and Cosine Ratios (Day 144)

Today I subbed in a high school special ed class. It's in my first Orange County district -- and indeed, it's the class I've visited a lot before, most recently in my March 16th post (two days after Pi Day).

As you already know, one of the classes is Business Math, and so I will do "A Day in the Life" today.

9:00 -- Third period arrives (odd day, "first period" = zero period). It's a senior English class. This class used to have an aide, but she is no longer here.

These students are working on their -- performance tasks? Yes, in most years the district CER performance tasks are done by now -- and in most gen ed classes, they're indeed done by now. Since this is a special ed class, the teacher decides to give them until the end of this week to complete it.

There's not much opportunity for me to sing any songs in this period today. First, I don't want to disturb the students when writing the essays, and then the counselor comes in. She asks some of the students to sign some documents. (I don't fully know what the documents are for, and even if I did, it's likely to be confidential, not for blogging.)

Instead, I'll jump directly into the resolution. Today is Elevenday on the Eleven Calendar, and so it's all about communication today. Since today is the first in-person day after spring break, I ask some of the students the questions from Paige Sheehan's Desmos activity from yesterday. One guy found joy in being with his family during the break, while another found hardship in not being able to do much. And as these students are seniors, we talk about graduation -- and whether there'll be a real, in-person ceremony this year.

Two of the students have already finished their essays, but the third doesn't even start it. Once the counselor arrives, I find a way to segue from her conversation into some ideas for the essay. Since it's a district assessment, I choose not to post details about the essay or this conversation on the blog.

9:55 -- Third period leaves for snack break.

10:10 -- Fifth period arrives. This is the Business Math class. An aide returns for the last two classes.

The students continue to learn about investments. They watch videos about 401(k) plans and Roth IRA plans, and learn about their similarities and differences.

This time, I have a new song prepared for them -- a completion of the old "Ratios" tune from back at the old charter school. I'll discuss this song later in this post. After all, "Ratios" are related to today's lesson -- in a 401(k), an employer contributes 50 cents for every $1 the employer contributes.

As for communication, I tell one guy about my days at that old school, when he asks me why I have so many songs to sing. Another guy tells me about his spring break joys -- he was one of the first fans at Angel Stadium when it reopened to the public. Oh, and his hardship was that the Angels lost that night.

11:05 -- Fifth period leaves and seventh period arrives. It's a junior English class.

This class also has a performance task, though it's not on the same topic as the senior essay. The aide makes sure that the students understand what the task is all about.

The regular teacher returns with a few minutes left in class. His presence distracts me a little from singing, especially since he must remind me to fill out my sub notes for him. I have one girl choose a song for today, and she selects "Ghost of a Chance" from Square One TV. Unfortunately, there's enough time left for me to sing only part of the song.

12:05 -- Seventh period leaves, thus completing my day. There's no need for me to do academic support since the regular teacher has returned.

Here's are lyrics and Mocha code for the "Ratios" song that I perform today:

RATIOS:
Ratios are everywhere.
Ratios surround you,
Probably here and there.
For every "for every"
There's a ratio.
Divide at the colon,
And away we go.
That's all there is
To a ratio!

Bridge:
Ratios are everywhere,
But not everything's rational.
Square root of two and pi,
Are proved to be irrational.

https://www.haplessgenius.com/mocha/

10 N=8

20 FOR V=1 TO 2

30 FOR X=1 TO 102-V*28

40 READ A,T

50 SOUND 261-N*A,T

60 NEXT X

70 RESTORE

80 NEXT V

90 END

100 DATA 12,4,11,2,11,2,12,2,7,2,6,4

110 DATA 12,4,9,2,7,2,11,2,12,6

120 DATA 6,4,6,2,8,2,10,2,8,6

130 DATA 7,2,10,2,9,4,7,2,10,2,9,4

140 DATA 8,4,8,4,9,6,10,2,11,4

150 DATA 12,4,9,2,7,2,11,2,12,6

160 DATA 6,4,6,2,8,2,10,2,8,6

170 DATA 7,4,7,2,10,2,9,8

180 DATA 8,4,8,4,9,6,10,2,10,4

190 DATA 12,4,11,2,11,2,12,2,7,2,6,4

200 DATA 11,2,9,2,10,2,10,2,10,2,10,2,10,2,10,2

210 DATA 6,2,8,2,9,4,9,2,12,2,12,4

220 DATA 6,2,6,2,6,2,6,2,7,2,7,2,8,2,8,2

Don't forget to click the Sound box before you RUN the program.

I wrote this song in 12EDL, an easier scale to randomize. Once again, the rhythm isn't random, but something that matches the lyrics.

I wonder whether I should have reversed my "Ratios" and "Ghost of a Chance" songs. Then I get to sing "Ghost" in fifth period when I had more time. Then I could have compared the efforts I made in writing "Ratios" to the efforts the students should make when writing the performance task.

Then again, I was also considering making "Ratios" into a rap and fully dispensing with 12EDL. Since I promised a 12EDL tune for this song I provided it. I'll decide whether to change it to a rap later on.

Lesson 14-4 of the U of Chicago text is called "The Sine and Cosine Ratios." In the modern Third Edition, the sine and cosine ratios appear in Lesson 13-6.

This is what I wrote two years ago about today's lesson:

Lesson 14-3 of the U of Chicago text is on the tangent ratio, and Lesson 14-4 of the U of Chicago text is on the sine and cosine ratios. I have decided to combine all three trig ratios into one lesson.

David Joyce was not too thrilled to have trig in the geometry course. He wrote:

Chapter 11 [of the Prentice-Hall text -- dw] covers right-triangle trigonometry. It's hard to see how there's any time left for trigonometry in a course on geometry, but at least it should be possible to prove the basic facts of trigonometry once the theory of similar triangles is done. The section of angles of elevation and depression need not appear, and the section of vectors omitted. (By the way, who ever calls the sum of two vectors the "resultant" of the two vectors?) The one theorem of the chapter (area of triangle = 1/2 bc sin A) is given for acute triangles.

As the trig functions for obtuse angles aren't covered, and applications of trig to non-right triangles aren't mentioned, it would probably be better to remove this chapter entirely.

Yet most geometry books include trig because most state standards require it. And this most certainly includes the Common Core Standards:

Define trigonometric ratios and solve problems involving right triangles

CCSS.MATH.CONTENT.HSG.SRT.C.6
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

CCSS.MATH.CONTENT.HSG.SRT.C.7
Explain and use the relationship between the sine and cosine of complementary angles.

CCSS.MATH.CONTENT.HSG.SRT.C.8
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.*

And all three of these standards appear in this lesson.

Some people may wonder, why do we use the same name "tangent" to refer to the "tangent" of a circle and the "tangent" in trigonometry?

[2021 update: This link is dead. See James Tanton's post from yesterday's link.]

On the other hand, the reason that "sine" and "cosine" have the same name is less of a mystery. In fact, the U of Chicago tells us that "cosine" actually means complement's sine -- since the cosine of an angle is the sine of its complement. This is Common Core Standard C.7 above.

Last Friday's post was a whirlwind of ideas, and today's post continues these ideas. In the last two days, I linked to a variety of sources in search of answers to questions such as:

• Should activities be taught during the trig unit?
• Should a trig unit be taught during the Geometry class?
• Should a Geometry class be taught during high school?

We searched high and low, from traditionalists to their opponents, seeking these answers. I fear that when I post links to all these competing sources, my own opinions are obscured. The blog readers know what David Joyce and the traditionalists believe, but not what I myself believe.

Well, here's my belief -- I answer all three of those questions in the affirmative. High school should have a Geometry class, Geometry class should have a trig unit, and a trig unit should have activities -- and I posted my activity for the trig unit of a high school Geometry course yesterday.

I also think back to the activity that sparked this debate -- proofs and the courtroom. We saw how the traditionalists objected to the courtroom activity on the grounds that it is too long.

I admit that I'm fascinated with the idea of using a courtroom to highlight Geometry proofs. I took Geometry back during the 1994-95 school year -- the year of the famous OJ Simpson trial. And so I often fantasized that my Geometry class was a courtroom -- the People's Court. Actually, that TV show was off the air during that year. But it made a comeback in 1996, the first full year after the Simpson trial, as TV stations were trying to capitalize on the Simpson trial's popularity. (This was the same year that another famous courtroom show debuted -- Judge Judy.)

So I might organize a People's Court during my Geometry classes. When I would teach the lesson depends on what textbook I was using. If I had Michael Serra's text, People's Court would occur at the end of the year, around Chapter 13. With the U of Chicago text, court may occur in Chapter 3 (when the class first learns about proofs), and in many other texts, it may occur in Chapter 4 (where triangle congruence proofs appear).

One way to prevent the unit from taking too long is to assign each group a different medium-level proof -- then they present those proofs when the class actually reaches that unit! So one group may be assigned the Isosceles Triangle Theorem to put on trial a week later, while another is assigned some of the Parallelogram Theorems to put on trial a few months later. As long as all groups present before the end of the first semester, it works out in the end.

By the way, many of our students may have trouble with trigonometry, but one youngster who had no trouble with trig was -- you guessed it -- Ramanujan. When he was in the equivalent of seventh grade, an older friend lent him a trig text book, and the young genius mastered it that year!

Here's one more connection between Ramanujan and trig. The U of Chicago text tells us how to find some trig values exactly, but not others. For example, cos(60) = 1/2, but cos(20), cos(40), and cos(80) aren't as easy to find. Well, the Indian genius found an interesting formula connecting the three cosines whose values we can't find. (All values are in degrees -- "cbrt" is cube root.)

cbrt(cos(40)) + cbrt(cos(80)) - cbrt(cos(20)) = cbrt(1.5(cbrt(9) - 2))

A 20-degree angle is not constructible and so its cosine can't be written exactly using square or cube roots -- of real numbers, that is. Complex numbers are a different issue:

cos(20) = (cbrt(a) + cbrt(b))/2

where a and b are the complex cube roots of 1 -- that is, a = (1 + i sqrt(3))/2, b = (1 - i sqrt(3))/2.

Now you can see why we have high school students memorize cos(60) and not cos(20).

OK, let me post the worksheet: