Lesson 15-3: The Inscribed Angle Theorem (Day 153)

Today I subbed in a high school math class. This is my second visit to this classroom -- the first was three months ago. I wrote about this class more in my January 17th post. It's definitely worth writing "A Day in the Life" about this class.

8:00 -- Second period (since "first period" = zero period) is the first of two Algebra II classes. These students are learning the graphs of trig functions.

8:50 -- Second period leaves and tutorial begins.

9:25 -- Third period is the first of three Geometry classes. These students are currently in Chapter 11 of the Glencoe text, which is on area. A worksheet has been scanned into Google Classroom, but the students must still work it out on paper.

Ironically, the last time I was in this classroom (which was not in January, but in March as part of a period of co-teaching), the regular teacher's lesson was on the Inscribed Angle Theorem -- and today I return to this class on the day we cover the Inscribed Angle Theorem in the U of Chicago text.

When I was in this period in January, many of the students had trouble staying focused. So that day, I started up the old "Who Am I?"/Conjectures game. I have yet to play that game in this class, and so I play it today beginning with the "guess my age" and "guess my weight" questions. As it turns out, third period has improved greatly from January -- it's the best Geometry class of the day.

10:30 -- Third period leaves for snack.

10:50 -- Fourth period arrives. This is the second of three Geometry classes.

This time, since I've already told this class my age and weight in January, I play my game but jump directly into questions from the worksheet. But unfortunately, the students have trouble with the first two questions:

1. Find the area of a rhombus with 5 as the distance from center to a vertex, and side length 12.
(This looks a lot better in a diagram as opposed to written in words.)
2. A rhombus of area 72 has a diagonal of length 18. Find the length of the other diagonal.

Since this is the class with the co-teacher, I ask him to explain the answers to these questions such that perhaps the students might understand him better.

11:40 -- Fourth period leaves. Fifth is the teacher's conference period, which extends into lunch.

1:20 -- Sixth period arrives. This is the last of three Geometry classes.

I try to avoid the problem from fourth period by starting the game with the other side of the assigned worksheet on the Chromebooks. The first two questions are on the area of a parallelogram followed by that of a triangle. But since the teacher uploaded this worksheet hastily, he didn't realize that all the questions on this side -- besides the first two -- had the answers written on the worksheet!

Thus I was forced to return to the rhombus questions in my game. This class was only slightly better than fourth period as far as these questions are concerned

2:10 -- Seventh period arrives. This is the second of two Algebra II classes.

I don't originally plan on playing the Conjectures game in this class -- it doesn't go very well with a graphing assignment. But some students see points and scores remaining on the board from the previous class, and so they convince me to play it anyway.

I only have time to ask the students for my age and weight, and then just three problems from the uploaded worksheet. The first two just ask for the amplitude and period of a trig function -- only the third requires them to prove it.

3:05 -- Seventh period ends, thus ending my day.

Dialogue 12 of Douglas Hofstadter's Godel, Escher, Bach is called "English French German Suite." It is actually Lewis Carroll's poem Jabberwocky, written in the three languages in the title. Frank L. Warren wrote the French translation and Robert Scott the German version.

I'll only post the first paragraph in each language. (The last paragraph is a repeat of the first.)

English:
'Twas brillig, and the slithy toves
Did gyre and gimble in the wabe:
All mimsy were the borogroves,
And the mome raths outgrabe.

French:
Il brilgue, les toves lubricilleux
Se gyrent en vrillant dans le guave.
Enmimes sont les gougebosqueux
Et le momerade horsgrave.

German:
Es brillig war. Die schlichten Toven
Wirrten und wimmelten in Waben;
Und aller-mumsige Burggoven
Die mohmen Rath' ausgraben.

Chapter 12 of Douglas Hofstadter's Godel, Escher, Bach is called "Minds and Thoughts." Here's how it begins:

"Now that we have hypothesized the existence of very high-level active subsystems of the brain (symbols), we may return to the matter of a possible isomorphism, or partial isomorphism, between two brains."

Unfortunately, today I have very little time for typing a summary at all today. It's a shame -- today would have been a perfect day to write more from the Chapter since the Dialogue is so short.

The Chapter is all about whether minds can be mapped onto each other -- even the minds of speakers of different languages. The author explains that this is why he opens with this Dialogue:

"I chose this example because it demonstrates, perhaps better than an example in ordinary prose, the problem of trying to find 'the same node' in two different networks which are, on some level of analysis, extremely nonisomorphic."

And he writes more about the choices Warren makes when translating Carroll's made-up words into the French language (such as "slithy" into "lubricilleux"). He tells us that the present tense sounds better in French than the past tense of English.

The chapter continues with the ASU -- a made-up version of the USA, followed by a long quote by Oxford philosopher J.R. Lucas, whose views on Godel are quite opposite to Hofstadter's. But due to lack of time, we must skip directly to the end:

"About Lucas one can say that he is nothing if not stimulating. In the following Chapters, we shall come back to many of the topics touched on so tantalizingly and fleetingly in this odd passage."

And hopefully in the following blog entries, I'll come back to many of the topics touched on so tantalizingly and fleetingly in this odd post.

EDIT: I did finally return to this Chapter in my June 5th post. Here's a link to it:

Semester 2 Review and Next Year Preview (Day 180)

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

Lesson 15-3 of the U of Chicago text is on the Inscribed Angle Theorem. I admit that I often have trouble remembering all of the circle theorems myself, but this one is the most important:

Inscribed Angle Theorem:
In a circle, the measure of an inscribed angle is one-half the measure of its intercepted arc.

"Stop reading and try to solve the problem by first considering a special case."

Well, the special case is when one of the given sides of the angle contains a diameter. This is given as Case I in the U of Chicago text:

Given: Angle ABC inscribed in Circle O
Prove: Angle ABC = 1/2 * Arc AC

Proof:
Case I: The auxiliary segment OA is required. Since Triangle AOB is isosceles [both OA and OB are radii of the circle -- dw], Angle B = Angle A. Call this measure x. By the Exterior Angle Theorem, Angle AOC = 2x. Because the measure of an arc equals the measure of its central angle, Arc AC = 2x = 2 * Angle B. Solving for Angle B, Angle B = 1/2 * Arc AC. QED Case I.

Notice that the trick here was that between the central angle (whose measure equals that of the arc) and the inscribed angle is an isosceles triangle. We saw the same thing happen in yesterday's proof of the Angle Bisector Theorem -- the angle bisector of a triangle is a side-splitter of a larger triangle, and cutting out the smaller triangle from the larger leaves an isosceles triangle behind.

"Now stop reading and extend your solution to the general case."

Let's move onto Case II. Well, the U of Chicago almost gives us a two-column proof here, so why don't we complete it into a full two-column proof. For Case II, O is in the interior of Angle ABC.

Statements                                                     Reasons
1. O interior ABC                                           1. Given
2. Draw ray BO inside ABC                            2. Definition of interior of angle
3. Angle ABC = Angle ABD + Angle DBC       3. Angle Addition Postulate
4. Angle ABC = 1/2 * Arc AD + 1/2 * Arc DC 4. Case I and Substitution
5. Angle ABC = 1/2(Arc AD + Arc DC)           5. Distributive Property
6. Angle ABC = 1/2 * Arc AC                         6. Arc Addition Property and Substitution

The proof of Case III isn't fully given, but it's hinted that we use subtraction rather than addition as we did in Case II. Once again, I bring up the Triangle Area Proof -- the case of the obtuse triangle involved subtracting the areas of two right triangles, whereas in the case where that same angle were acute, we'd be adding the areas of two right triangles.

"In proving the theorem for each of these two more general special cases, the truth of the theorem for the special case was used in the proof."

Here's the rest of what I wrote last year:

The text mentions a simple corollary of the Inscribed Angle Theorem:

Theorem:
An angle inscribed in a semicircle is a right angle.

The text motivates the study of inscribed angles by considering camera angles and lenses. According to the text, a normal camera lens has a picture angle of 46 degrees, a wide-camera lens has an angle of 118 degrees, and a telephoto lens has an angle of 18. I briefly mention this on my worksheet. But a full consideration of camera angles doesn't occur until the next section of the text, Lesson 15-4 -- but we're only really doing Lesson 15-3 today.