Wednesday, April 26, 2017

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

This is what Theoni Pappas writes on page 116 of her Magic of Mathematics:

"Take the year you were born. To this add the year of an important event in your life. To this sum add the age you will be at the end of 1994. Finally, add to this sum the number of years ago that the important event took place."

Here Pappas is showing us some "Number Magic." Notice that 1994 is the year in which she first wrote this book. The following is the example that she gives:

1953 -- Year born
1980 -- Year Pappas (claimed to have) traveled into outer space. (Really, Pappas?)
    41 -- Age as of 1994
    14 -- Years elapsed since 1994 of important event
3988 -- Total

According to Pappas, the answer is always 3988. Of course, this isn't really "magic" at all -- notice that there are two pairs that add up to 1994. The year of her birth and her age add up to 1994, as do the important event and the number of years since that event. So it's not surprising that the numbers always add up to 1994 * 2 = 3988.

Let me try another example. This time, I'll change it to the current year, 2017, so my four numbers should add up to 2017 * 2 = 4034. The year of my birth is 1980 -- hey, that's the same year that Pappas (supposedly) traveled into space. And as for the year of my important event -- I just think I'll choose the year 1994. After all, it's the year I took my favorite math class, Geometry. Oh, and a lady with the initials TP wrote one of my favorite books in that year.

1980 -- Year born
1994 -- Year of important event
    37 -- Age at the end of this year
    23 -- Years elapsed since important event
4034 -- Total

Unlike Pappas, here's something that really did travel into space -- Cassini. There was a special Google Doodle marking the space probe's arrival at the rings of Saturn. Again, I'm a science teacher, so I like to point out important scientific events. In this case, researchers are excited over the possibility that there is life on at least one of the ringed planet's moons. Cassini will orbit Saturn 22 times to gather data from the Cronian moons.

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 26th -- Lesson 14-1: Special Right Triangles
Tomorrow, April 27th -- Lesson 14-2: Lengths in Right Triangles
Friday, April 28th -- Activity (includes Lesson 14-3: The Tangent Ratio)
Monday, May 1st -- Lesson 14-4: The Sine and Cosine Ratios
Tuesday, May 2nd -- Lesson 14-5: Vectors
Wednesday, May 3rd -- Lesson 14-6: Properties of Vectors
Thursday, May 4th -- Lesson 14-7: Adding Vectors Using Trigonometry
Friday, May 5th -- ActivityMonday, May 8th -- Review for Chapter 14 Test
Tuesday, May 9th -- 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.)

[2017 Update: The activity day is now on Friday and the trig ratios are on Monday. Thus the other teacher's original intent has been restored.]
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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