Friday, February 19, 2016

Lessons 14-3 and 14-4: The Sine, Cosine, and Tangent Ratios (Day 106)

Today I subbed in a seventh grade English class. But during the fourth period conference, I was assigned to an eighth grade math class. Like many eighth grade classes in this district, this class was based on the computer. I saw copies of MathLinks Student Packet 8-8 lying around the room, and it did appear that the students were working on this material on the computer:

8-8 Slope and Slope-Intercept Form of a Line
8.1 Introduction to Slope
8.2 Input-Output Investigation
8.3 Slope-Intercept Form
8.4 Skill Builders, Vocabulary, and Review

I was wondering whether I'd sub for such a class last week when I was posting my slope lessons here on the blog. Alas, I cover this class one week too late for me to post a lesson here -- and besides, it was another day when I interacted very little with the students as they were on the laptops. As it turns out, the fifth and sixth period classes did have MathLinks worksheets I could have helped them with, but by then I had to return to the seventh grade English class.

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

Lesson 13-3 of the U of Chicago text is on the tangent ratio, and Lesson 13-4 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

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.
Explain and use the relationship between the sine and cosine of complementary angles.
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? A Michigan math teacher, Mike Shelly, discusses the reasons at the following 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.

Yesterday'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.

OK, let me post the worksheet:

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