Lesson 8-8: Arc Measure and Arc Length (Day 88)

Today is the second day of my four-day assignment in the high school math class. Wednesdays are even periods, and Cohort A meets in person. As it turns out, the regular teacher's second, fourth, and sixth periods are all Geometry.

I'll continue to provide the class size stats for the classes I see today. The largest class is sixth period, with seven in-person students. (Ten students have signed up for hybrid out of fifteen Cohort A students). The smallest class is fourth period, with only four in-person learners. (Eight students have signed up for hybrid in this period, out of fifteen Cohort A students). Also, there are more freshmen (SteveH track) as part of the in-person groups today -- yesterday, all five in-person students were sophomores.

And as I promised in yesterday's post, today I take advantage of tutorial, which is considered to be the last 20 minutes of each block. So when it's time for tutorial, I show the students my own completed worksheet for Section 6.4. And some students do check their answers and appreciate the help I'm giving them. But some others don't bother to start the assignment -- I hide my key from them so they can't merely copy it.

But still, I don't sing any songs during these last 20 minutes. Once again, I'll be more inclined to perform when it's a genuine substitute teaching assignment, as opposed to just watching the students as they log in to see their regular teacher on Zoom.

Lecture 2 of Prof. Arthur Benjamin's The Mathematics of Games and Puzzles: From Cards to Sudoku is called "Games of Chance and Winning Wagers." Here is a summary of the lecture:

• Suppose you find yourself in Las Vegas with $60 in your pocket. There's a concert in town that you really want to see, but the tickets cost $100. So you decide to wager $1 at a time until you either get $100 or go broke. Which game should you play, and how likely are you to win?
• Roulette is the easiest game to analyze. The wheel has 18 red, 18 black, and 2 green slots. The expected value is the weighted average of how much you can win or lose. When you bet red, you win $1 with probability 18/38 or lose $1 with probability 20/38, so expected value is -5.3 cents.
• If you bet on red twice, you either win twice with probability 22.4%, once with probability 49.9%, or lose both with probability 27.7%. These add up to 100%. The expected value of the two bets is exactly twice that of one bet.
• In a charity raffle, you bet $1 and can win half the pot. Therefore the expected value of each ticket is -50 cents, which makes sense since half the money goes to charity.
• In sic bo, you bet $1 and choose a number from 1 to 6. You then roll three dice and win $1 if your number comes up once, $2 if it comes up twice, and $3 if it comes up thrice.
• The probability of winning $3 is less than 1%, $2 is 7%, $1 is 35%, and losing $1 is 58%. The expected value of this game is -8 cents. In other words, this game is very favorable -- for the casino, that is.
• The most popular dice game in casinos is craps. You bet $1 and roll two dice. You win with either 7 or 11 and lose on 2, 3, 12. Any other roll is called a point -- then you keep rolling until you either make your point and win or roll a 7 and lose.
• If your point is a 4, you have only a 1/3 chance of making your point, since there are only three ways to roll a 4 but six ways to roll a 7. If your point is a 5, you have a slightly better chance of making your point -- 2/5 or 40%.
• According to the Law of Total Probability, the overall winning chance is a weighted average of your chances from each opening roll. The weighted average of winning craps is 244/495, which is about 49.3%. So the expected value = 0.493 - 0.507 = -0.014 or -1.4 cents. An anagram for "slot machines" is "cash lost in 'em" -- enough said.
• But even with only a 1.4% house advantage, you'll probably lose $60 much more often than you'll make $40. This is called Gambler's Ruin.
• If you play a game, the probability of getting from $60 to $100 is exactly 60%. This is called a "symmetric random walk." It is the only internally consistent formula -- if we lose $1 then our probability drops to 59%, and if we win $1 then it rises to 61%
• But if a game isn't fair -- if the win probability is p and the lose probability is q -- then the probability to reach $n by starting at $i becomes a(i) = (1-(q/p)^i) / (1-(q/p)^n).
• For craps where p = 0.493, the probability of reaching $100 is 28%. If p = 0.49, the probability drops to 18%, and it becomes 1.4% for Roulette at p = 0.473. For a slightly favorable game at p = 0.51, the probability rises to 92.6%. It's much better to make a single big bet.
• In the martingale strategy, you double your bet each time. At first it seems that you're guaranteed to win eventually, but in reality there's a maximum bet.
• If you're playing a favorable game, you should bet your edge -- if you have a 2% edge, then you should bet 2% of your money. If you bet twice your edge, you might lose everything.
• In the Monty Hall problem, there are three doors with one car and two goats. After you choose a door, Monty reveals a door with a goat and asks whether you want to switch. You actually have a 2/3 chance of winning by switching.

We've already discussed Monty Hall before, and I don't wish to tie up this post with another discussion of it, so let me just provide another link:

https://betterexplained.com/articles/understanding-the-monty-hall-problem/

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

Lesson 8-8 of the U of Chicago text is called "Arc Measure and Arc Length." This and the next section are the same in the old Second and modern Third Editions -- except that arc measure appears much earlier in the new version (in Chapter 3).

[2021 update: There is some overlap between what the Geometry students are learning in class this week and today's lesson in the U of Chicago text. Thus I'm making several changes to what I wrote about this lesson based on what I see in the classroom today. Recall that what happens in the actual classroom with real, live students always takes priority over the preplanned lessons that I post on the blog -- especially when it's a Geometry classroom. Also, I'm adding the "subbing" label since I describe subbing extensively, even if I don't do "A Day in the Life."]

This is, of course, the lesson when students learn about the number pi. Two of my favorite lessons to teach each year are the Pythagorean Theorem and pi. In the first few years of this blog, I rearranged the lessons so that Pythagoras appears near the Distance Formula and pi is taught on Pi Day. But since we're following the order of the text this year, Pythagoras and pi are taught here in the same chapter!

Indeed, since following the digit pattern means that our pi lesson isn't on Pi Day, you might ask, what lesson will I post on Pi Day instead? According to the calendar, March 14th works out to be a Sunday -- which, unfortunately, isn't a school day. Luckily, I have two months to figure out how I'm going to celebrate Pi Day on the blog this year.

The seventh grade U of Chicago text, called Transition Mathematics, is much more convenient for setting up the pi lesson near Pi Day. Today's lesson on the circumference of a circle is Lesson 12-4, and Lesson 12-8 is on spheres -- whose surface area and volume formulas definitely use pi (so at least the Chapter 12 Test that I'd give near Pi Day is about pi). Keep in mind that I'm referring to my old Second Edition, not the new Third Edition -- the Third Edition of Transition Math teaches pi in Chapter 7 and stats in Chapter 12.

Much of my chapter rearrangement in past years was driven by my desire to celebrate Pi Day by teaching the famous constant. Thus I began the second semester with Chapter 12, so that we would be in Chapter 8 on measurement. The chapters following 12 are also related to similarity (such as trig) while the chapters following 8 are also related to measurement (such as volume), and so the net result was that we covered Chapters 12 through 14, and then back to Chapter 8. 

In fact, in the early days of this blog, after Lessons 8-8 and 8-9, I went directly to Chapter 15. Thus I would have a sort of "circle unit," combining the two circle measurement lessons with the circle chapter, namely Chapter 15. Not many texts include circle measurement with the other circle theorems, and as we see, the U of Chicago text isn't one of them. But I know of one curriculum that does place these lessons together in a circle unit -- APEX.

Yes, as I can see in the classroom this week, Unit 6 in the APEX Geometry course indeed combines circle measurement (circumference, area) with circle theorems (chords, secants). In fact, while much of this week's lessons corresponds to Lesson 15-1 of the U of Chicago text, one topic taught this week -- arc measure -- occurs in today's Lesson 8-8. It's been a long time since I taught the same Geometry lesson in class and on the blog at the same time, and it finally happens again today.

And so I should -- and will -- take advantage of this by replacing the worksheets I posted for 8-8 last year with what the students are working on today. Indeed, this is my big activity week, and some readers might notice that even the activities that I posted yesterday involve a worksheet and cutting, which might not be appropriate in the pandemic era. You might wonder whether there's a way to convert that Pythagorean Theorem activity to an online format such as Desmos.

Well, this week's regular teacher is continuing to assign worksheets. (Then again, she doesn't let me log in to her Zoom classes -- as far as I know, she has the students play on Desmos during the lessons.) And I see that these worksheets tend to be from Kuta. One advantage of Kuta worksheets is that each one is a double-sided page, so the teacher can make and leave copies for the in-person students. The alternative would be to print and copy APEX study guides instead of Kuta worksheets -- and APEX study guides tend to be many pages. Neither the teacher nor the students really want to print and copy all of those APEX pages. Therefore this class relies on Kuta for printed work and APEX mainly for the online quizzes.

I try to avoid posting copyrighted work on the blog, and these Kuta worksheets definitely have the copyright symbol near the bottom. But I believe that anyone can generate a Kuta worksheet, so I'm hurting no one -- and besides, I've posted Kuta on the blog before. If the company was going to come after me for posting their worksheets on the blog, they would have done so years ago.

Here are the three worksheets that I'm posting tonight:

Section 6.3: If an angle is given, name the arc it makes. If an arc is given, name its central angle. (This one should be easy.)

Section 6.4: Find the measure of the arc or central angle indicated. (One thing that makes this tricky for my students today is that in Question #8 -- as well as some questions on the other side that I don't post today -- x can be negative. The actual arc and angle measures are always positive.)

Section 6.2 (back side only): Find the circumference of each circle. (While circumference does appear in APEX Unit 6 as I wrote earlier, it's not until later in the chapter, so officially, the students in my class haven't reached pi yet. But it does match Lesson 8-8 of U of Chicago, so I post it today. But the rest of APEX Section 6.2 isn't taught until Lesson 15-1 of U of Chicago, so I don't post it. I cross out the 15-1 questions as to leave only the pi questions.)

When I show these questions to my students today, I tell them to find the answers by making as few calculations as possible, since each additional step is an extra chance to make an error. For example, in Question #4 on today's 6.4 worksheet, I have the students find Arc VXU by adding the known Arc UY of 126 degrees to the semicircle VXY to obtain a total of 306 degrees. When the students try to find Arcs VW or XY, they either make a silly calculation error or assume that two arcs are congruent when they aren't.

I'll keep one worksheet from last year -- the vocabulary worksheet. This reminds me of something I did last summer -- the Shapelore project, where I came up with new, simpler terms for Geometry terms, avoiding Latin and Greek in favor of words from Old English (Anglish). I wanted to convert the last three chapters of the text of Shapelore, but I never made it to Chapter 15 (and indeed, I didn't quite make it out of Chapter 14).

So, in honor of my in-person students learning about circles this week, let me convert a few words from both Lesson 8-8 and 15-1 into Shapelore:

circle: I kept going back and forth between "ring" and "wheel." I think that "wheel" is better since "ring" has another meaning in math. We may also choose to keep circle instead -- we're allowed to keep easier words that the kids know, even if they come from Greek. There's also an Anglish website, and it says that "umbeling" may also work for circle:

https://anglish.fandom.com/wiki/Shapelore

arc: The Anglish Moot above says that "bow" is a good word for arc.

central angle: While the Anglish Moot gives "nook," I like keeping the Latin angle, because it is a simpler word. But central can become "middle."

minor arc/major arc: "smaller bow" and "bigger bow."

semicircles: "half-wheels"/"half-rings"/"half-umbelings"/"half-circles." Whatever word we choose for circle, just say "half" of it.

degrees: Just keep degrees. I think "bowlengthworth" is too off-putting for the kids. I'm also likely to keep measure, although we could use "meting" instead.

chord: The Anglish Moot says "string," and I like this one.

circumference: Both this and perimeter can become "outsidelength" or "outlength."

pi: I know that pi is Greek, but I'll keep it anyway. If we truly want it to be Anglish, we should make it a Germanic rune such as "perth" (and "thorn" instead of "theta")m but I'd rather keep it as "pi."

intercepted arc: The Anglish website gives "forset," "waylay," and other words for intercept. So we would say "forset bow" or "waylaid bow."

Let's think about four years ago, 2017. Yesterday I wrote about how I should taught Pythagoras to my eighth graders that year, and so today I'll do the same regarding pi and my seventh graders that year.

Actually, I never reached the lesson on pi that year. That's because I was waiting, as usual, for Pi Day to teach the lesson, but I was out of the classroom before March 14th. In fact, I wrote in a post dated later that month what had actually happened on Pi Day. I decided to give my students one last surprise by delivering a pizza to my old classroom. But the bell schedule was mixed up that day, with school out early the entire week for second trimester Parent Conferences. It turned out that sixth grade was in the classroom at the time I delivered the pizza. Thus the sixth graders got to celebrate Pi Day with a pizza, even though seventh grade is the year that pi appears in the Common Core.

So had I made it to Pi Day, how would I have taught the lesson? Pi Day fell on a Tuesday that year, and at the time, Tuesdays were for projects. I assume that the Illinois State text had some sort of project where students had to measure the diameters and circumferences of various round objects -- in other words, an activity not much different from the one I'm posting today.

On the other hand, I posted that I should have made Tuesdays the traditional lesson day. Still, I see no problem with a brief measurement activity before the traditional lesson -- just as I'd given the eighth graders a brief Pythagorean Theorem activity before the traditional lesson two months earlier.

I had no control over Parent Conferences or the bell schedule. Again, I don't know when seventh grade had class that day -- only that sixth grade was the last class. If I were teaching, I wouldn't have been able to get the pizza -- but I could have sent my support staff aide to purchase it instead. After all, she'd bought a pizza for our eighth grade class four months earlier. As a bonus, I could have had her get an extra pizza to share with my fellow teachers as they waited for Parent Conferences to begin.

So that seventh grade isn't left out of the party, I could bring some other round foods -- such as cookies -- for the students to measure. They only get to eat what they measure, so this is an incentive to do the activity correctly. Meanwhile, sixth grade gets a party but isn't learning about pi. Actually, I remember that there was a pi activity near the end of the Illinois State sixth grade STEM text page as a preview of seventh grade. The Pi Day pizza party would have been a great excuse to do this -- provided, of course, that I was given more than a day's notice as to what the bell schedule would be that day (which, as you may recall, wasn't always guaranteed on shortened days).

Two years ago at the old charter school, I found out that there was indeed a Pi Day party. It's possible that the pizza I'd brought the year I left encouraged the administrators to make Pi Day a regular celebration at the school. Unfortunately, it was soon after Pi Day when the renewal petition for our school was denied.

For the second straight day, we're avoiding the elephant in the room -- classroom management. The problem is when I am the regular teacher and there's no one for me to leave names for. And my support aide -- the only adult my students respect in my classroom -- might not be present if she's out buying the pizza!

Yes, I know that many of these recent posts have turned into spilled milk and discussion of my class from four years ago again. But it's important to reflect on my past failures in order to set myself up for future success if I ever return as a regular teacher. The sky is the limit!