Lesson 12-8: The SSS Similarity Theorem (Day 128)

Lesson 12-8 of the U of Chicago text is called "The SSS Similarity Theorem." In the modern Third Edition of the U of Chicago text, the SSS Similarity Theorem appears in Lesson 12-6.

Today is Pi Day -- the biggest day of the year for math teachers. Last year, I didn't post on Pi Day, with the chaos of leaving my old school going on at the time. If you recall, I was able to buy a pizza for my sixth graders that day anyway -- a rather bittersweet Pi Day. This year, I'm ready to make today's Pi Day post on time.

I often report on the deaths of famous mathematicians and scientists. Last night, I learned of the passing of Stephen Hawking. Two years ago, I wrote about the British physicist on the blog, around the time I watched the movie Theory of Everything. He died last night in my time zone, but in Greenwich it was already March 14th. Hence he died on Pi Day, a day already associated with the birth of another famous physicist -- Albert Einstein. Some people have noticed that Hawking was born on the anniversary of the death of yet another physicist, Galileo.

There is also a special Google Doodle for today -- the second such Pi Day Doodle. According to Google, this is the 30th anniversary of the math holiday, which was first celebrated by physicist Larry Shaw at the Exploratorium Museum right here in California (albeit San Francisco, not here in Southern California). Larry Shaw, unfortunately, has also died -- he passed about seven months ago, on August 19th. (As far as I know, no famous physicists were born on August 19th.)

Today I subbed at another middle school. The class is Digital Filmmaking. Actually, the teacher is taking the class on a major field trip -- all the way to Nashville, Tennessee for the annual Student Television Network Convention -- a meeting for students who produce those morning announcement videos that are common to many middle and high schools now. (The Tennessee capital is in the Central Time Zone, so it's a two-hour time difference for the travelers. On the Eleven Clock that I wrote about two days ago, only Memphis is in Central Time, while the rest of the state is in the new Eastern Time.)

Ordinarily I don't do "A Day in the Life" for non-math classes. But I will do so today (Day 120 in this district) only because it's the first of four straight weekdays in the classroom while the teacher is out of town. Classroom management is vital on multi-day assignments -- if I don't manage well on the first day, I'll have lost the students by the fourth day. So I return to the first resolution:

1. Implement classroom management based on how students actually think.

7:15 -- This teacher has a zero period, of course -- the morning announcements class that's currently in Tennessee. At high schools, zero period is actually called "first period." Well here, at least in the computers, this class is labeled "ninth period" -- go figure.

The three eighth graders whose parents chose not to pay for the trip remain in this class -- a guy and two girls. They already know that today is Pi Day -- when I arrive, one girl is actually telling the others some Pi Day jokes!

Of course, it's mostly just a free study day for the trio. Two of the students start doing their math homework. They're in Algebra I, but they still have to study the Geometry portions of the eighth grade SBAC. (I've written about this problem in past traditionalist posts.) I notice that yesterday's lesson was actually on circles and pi, but today they are working on angles and transversals.

I decide there's no harm in playing some Pi Day videos for them. And if I'd known this morning that I would have a class with only three students, I could have purchased a pie for them. I tell the students that I'll buy them some pie tomorrow. The girls want apple pie while the long guy actually prefers a chocolate pie.

8:15 -- The rotation starts with sixth period today. (This obviously isn't the school that always starts with first period.) This is the first of four Digital Filmmaking classes. The students watch a video on the history of filmmaking -- starting with the Roman philosopher Ptolemy (who first noticed that a series of still pictures can resemble motion) and ending with the digital media age.

9:05 -- Sixth period ends. As it turns out, first and second periods are both conference periods. For the teacher, this is probably to make up for having a zero -- er, ninth -- period. I spend my nearly 2 1/2 hours of break by typing up this post and watching more Pi Day videos.

10:00 -- In case you haven't heard, today is the day of a special walkout to protest gun violence. Today has been chosen -- not because it's Pi Day, but because it marks one full month since the Valentine's Day massacre in Parkland, Florida. Yesterday, prosecutors announced that they are seeking the death penalty for the defendant in this case.

I try to avoid politics on this blog, except for politics that is directly related to education. Indeed, I've only mentioned the shooting twice on the blog -- once the day after the incident, and the other two days ago (in pointing out that Sheila Danzig, a major opponent of Daylight Saving Time, lives about 25 miles away from Parkland).

But then again, this truly is directly related to education. The shooting, after all, took place at a high school. And some of the proposed solutions involve teachers -- for example, should teachers be allowed to carry concealed weapons?

When I first heard about the proposed walkout, I was subbing at a high school. It was announced that the walkout would have the full support of ASB. Some teachers were concerned about losing instructional time -- especially since many of the students might just leave class just to talk to their friends or play on their cellphones.

I noticed that this school had a special bell schedule, called "Extended Snack," that was often used during football and basketball seasons for special pep rallies. As it turned out, on this schedule, the extended snack began at exactly 10:00. Of course, I don't sub at this high school today, but it's a no-brainer to use this schedule today. Then no single period loses instructional time to the walkout.

Anyway, today's middle school doesn't have an Extended Snack schedule. But the schedule has nonetheless been rearranged to accommodate the walkout. Indeed, this school normally has homeroom at the start of the day. (This is why the early morning class is called "ninth period," because in the computer, zero period is homeroom.) Instead, homeroom is cancelled, and both sixth and first period are moved up so that the classes end at 10:00.

I don't like subbing on the day of a fire or earthquake drill, when I must be in charge of getting the students lined up. So I'm glad that the walkout occurs right in the middle of my conference period. I decide to show solidarity by leaving my classroom anyway a few minutes after 10:00, by which most students have arrived on the field. Leaders from ASB say a few things about the shooting, and then the students walk one lap around the entire school. Those concerned teachers from a few weeks ago will be happy to note that most of the kids were respectful during the walkout -- if one student talks loud, another tells him to be quiet. And I catch only one student, a girl, trying to talk to someone on the phone.

Once again, I don't know what to say about the shooting and proposed solutions myself. I know that Einstein, whose birthday we celebrate today, was a pacifist who opposed violence. But school shootings were rare during his time, so I don't know what he'd say would be an possible solution. And Hawking, whom we mourn today, was outspoken on many social issues -- but not school shootings, which are rare in his native England. So I'm not sure what two of the brightest minds of the 20th century would say about 21st century school violence.

Instead, I'll quote MTBoS blogger Sarah Lanahan, a Texas Algebra I teacher. She agrees that gun laws need to change, but writes that walkouts aren't necessarily the right thing to do:

Since Weds, Feb 14th I have observed a plethora of responses to the massacre at Marjory Stoneman Douglas High School. My emotions have run a literal marathon. While many of our youth and educators are finally speaking up and having their voices heard I cannot sit back and not listen to some of the ridiculous things I've heard and seen.

I know I'm preaching to the choir here, but the answer to this situation is NOT to arm all the teachers! 

Putting metal detectors in all schools is not the answer.

Planning walkouts for months away is not the answer; especially on state testing days. They message will be lost there.

Blaming the lack of mental health care is not the answer. 

Pointing fingers and blaming is not the answer.

Sending thoughts and prayers is not the answer.

But more than anything sitting around and not doing anything is also not the answer.

Standing up, speaking out and voting is the answer. Empowering teachers to get to the polls and to have their voice heard by lawmakers IS part of the answer.

Teaching students how to manage their emotions and how to cope with stress IS part of the answer.

Right now too many people are focused on finding the ONE right answer, but there's a problem with that approach.

There is no ONE right answer.

Our country is in need of a new approach to many of the issues we face. We need a new approach to gun regulations and purchases. We need to all incorporate a new approach to teach the WHOLE child and that requires additional training of everyone on a school campus. We need to connect with lawmakers and ensure they really hear from us.

I hope that everyone is doing well and staying strong after the difficult past few days. I also hope everyone is ready to go out and be heard while educating the whole student.

10:17-- The walkout ends (17 minutes, one minute from each victim) and snack proper begins, followed by my second conference period.

11:30 -- Third period begins. The second Digital Film cohort arrives.

12:25 -- Third period leaves for lunch.

1:10 -- Fourth period begins. The third Digital Film cohort arrives.

1:59 -- March 14th at 1:59 is the famous Pi moment as first envisioned by Larry Shaw. It just happens to be one minute before the end of fourth period. I decide to stop the filmmaking video and quickly play a Pi Day video to celebrate the moment.

2:00 -- Fourth period leaves. As fifth period arrives, this is a good time to look at the first resolution to improve my classroom management.

As expected, many students want to talk and joke around during the video. Some distractions such as cellphones aren't an issue in this class, since the teacher requires the students to leave their entire backpack on the shelf near the door, except for needed materials (a pen or pencil). But of course, they still have their voices with them.

I set up my usual "good" and "bad" lists of names for the regular teacher. I decide that I won't leave any names for the "bad" list, but I do warn them that I have no problem with leaving their names on the three remaining days.

I also threaten to have the students stay after school if they don't finish the video -- after I must stop the video many times when they're talking. One guy says that he must catch the bus after school -- to which I reply that therefore he (one of the more talkative kids) should stop talking. In the end, the students do have to stay about a minute after school -- we finish the video, but they can't leave until they put up the chairs (as often happens in elementary schools).

My real management test is whether the class degenerates into chaos by the fourth day -- especially this troublesome fifth period class. The students will be working on a multi-day assignment based on the video they watched. There could be students who finish quickly and have nothing to do on the last day, while others might waste time on the first day.

2:55 -- Fifth period leaves, thus completing my day.

All in all, it's a good Pi Day, though I would have preferred subbing a math class on this special day.

In fact, I'm still thinking about yesterday's math classes. The special ed classes are spending Pi Day on a field trip to the high school. Actually, the gen ed kids are also thinking about high school -- the teacher whose class I co-taught is giving a diagnostic exam for high school placement tomorrow. She tells me that this is a brand-new test, so I don't know what scores lead to Geometry placement for next year as opposed to Algebra I.

Yesterday I walked past another math classroom. The lesson written on the board was on the circumference of a circle, and so it's perfect for Pi Day. I'm not quite sure what grade the class is since I didn't sub for that class. Pi is typically a seventh-grade topic. It may possibly be an eighth grade Algebra I lesson, now that I see what these students are learning at the other school.

I've heard of the existence of an Honors Math 7 class. Recall that in this district, sixth grade is still considered elementary school, so acceleration can't begin until seventh grade. I think the idea is that Honors Math 7 should cover all of Common Core 7 and half of Common Core 8 -- and since the algebra part of CCSS 8 matches Algebra I content, it's the geometry component of CCSS 8 that would be pushed down into Algebra I.

But I could be wrong, because that plan ignores the SBAC. We know that no amount of acceleration can make the SBAC disappear. So eighth grade Algebra I students must still somehow pass the eighth grade SBAC including its geometry component. In past posts, I suggested that eighth grade Algebra I shouldn't be required to take the SBAC (since after all, ninth grade Algebra I students don't take it), which would make acceleration easier.

At any rate, since I don't fully know the Honors Math 7 curriculum, I can't make any conclusions about what class was learning about pi yesterday. If that was some sort of eighth grade class, the students might have been taking a placement test anyway. The only eighth graders who might have a Pi Day party are the three students to whom I'm bringing pies tomorrow.

As usual, here are some pi-related videos. This time, I wish to link to the videos that I actually showed the morning class today:

1. What Pi Sounds Like:

When I first tell the students that I'd play them some pi music, this is what they have in mind -- songs where the notes correspond to the digits of pi.

2. Mathematical Pi:

No Pi Day is complete without a parody of Don McLean's "American Pie."

3. Pi-namite:

My eighth graders are fascinated that these "little kids" already knew about pi. Of course, the video opening does state that this is an honors fifth grade class.

4. American Pi:

That's right -- here's a second version of "American Pie." It's actually interesting to compare these songs about the history of pi to the video I show my class about the history of film. Both videos begin with the ancient Greeks/Romans, with Ptolemy (moving pictures) and Archimedes (pi). Actually, this pi video also mentions a "Ptolemy," but it's a different Ptolemy -- the geometer Ptolemy lived about a century after the one who looked at moving pictures.

5. Lose Yourself in the Digits:

I've linked to this parody of Eminem's "Lose Yourself" before.

6. Pi Shop:

I mentioned this parody of Macklemore's "Thrift Shop" in previous posts.

7. I Love Pi:

This one's new for me -- a parody of Usher's "OMG."

8. Do the Quad Solve:

This is a parody of Troop 41's "Do the John Wall." This is actually a song about solving quadratic equations, not pi. But it starts playing in my classroom when YouTube is set to Autoplay, and the two Algebra I students enjoy it so much that I let it finish.

Here are a few extra videos that I didn't show my class:

9. 30 Digits: A Pi Song:

At the end of fourth period, I played "Lose Yourself in the Digits." But in the single minute remaining in class from 1:59 to 2:00, I should have played a short song such as this one.

10. Pi as Music (C Major Pentatonic):

The creator of this song says that he has an song about the number "e" in the works. It's too bad that he couldn't finish the "e" song before e Day of the Century, February 7th.

11. Calculating pi by Hand:

According to the video, this is how computers calculate millions of digits of pi. We aren't really meant to use this algorithm by hand -- look at how many mistakes the mathematician makes just to calculate the k = 1 case!

12. Why is pi here? And why is it squared?

This video is all about the "Basel problem" -- why the sum of the reciprocals of squares converges to the strange value pi^2/6.

13. Pi Rant 2018:

Today is Vi Day -- the one day when Vi Hart is guaranteed to post. This time, she muses about what would happen if pi were 3.1418... (that is, if today were Pi Day of the century).

14. How did a $25K bet give rise to Hollywood?

I decided that since I'm posting videos today anyway, I might as well post the video that I played for the film classes today.

Oh, and let's not leave Mocha out of the fun:

http://www.haplessgenius.com/mocha/

10 DIM S(9)
20 FOR X=0 TO 9
30 READ S(X)
40 NEXT X
50 DATA 72,180,160,144,135
60 DATA 120,108,96,90,80
70 N=1
80 FOR X=1 TO 32
90 READ A
100 SOUND 261-N*S(A),4
110 NEXT X
120 DATA 3,1,4,1,5,9,2,6,5,3
130 DATA 5,8,9,7,9,3,2,3,8,4
140 DATA 6,2,6,4,3,3,8,3,2,7
150 DATA 9,5

Here I follow the tradition established by Michael Blake -- stop just before the first zero. I keep the code for zero in the DATA lines anyway, just in case you want to add the zero and more digits.

Once we have completed the song for the major song, there's not much effort needed to change the song to make it fit the minor scale:

50 DATA 30,72,64,60,54
60 DATA 48,45,40,36,32

or the New 7-Limit Scale:

50 DATA 54,105,96,90,84
60 DATA 81,72,70,63,60

or the Bohlen-Pierce Lambda Scale:

50 DATA 21,63,53,49,45
60 DATA 38,35,29,27,23

This is why my New 7-Limit Scale contains ten notes -- I had digit songs like the pi song in mind when I created it.

Meanwhile, the video I posted earlier, Pi in C Major Pentatonic, seems playable in Mocha. Let's look at that program in more detail. At the 0:30 mark of the video, it reads:

• The first number defines the note.
• The next number defines the duration.
• The number after next defines the intonation.

I'm not quite sure what the author means by "the intonation," but let's at least try to code the note and the duration in Mocha. The video tells us that the scale is two octaves of C-D-E-G-A, and the duration is half note (0-1), dotted quarter (2-3), quarter (4-5), eighth (6-7), and sixteenth (8-9).

10 DIM S(9),L(9)

20 FOR X=0 TO 9

30 READ S(X),L(X)

40 NEXT X

50 DATA 162,8,144,8,128,6,108,6,96,4

60 DATA 81,4,72,2,64,2,54,1,48,1

70 N=1

80 READ A,T

90 FOR X=1 TO 32

100 SOUND 261-N*S(A),L(T)

110 A=T

120 READ T

130 NEXT X

140 DATA 3,1,4,1,5,9,2,6,5,3

150 DATA 5,8,9,7,9,3,2,3,8,4

160 DATA 6,2,6,4,3,3,8,3,2,7

170 DATA 9,5,0,2

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

Let's get to our Geometry lesson. We are now working on the AA~ and SSS~ theorems, which complete our study of similarity. There are several ways we can prove these at this point. We can use the original dilation proofs given in the U of Chicago text (Lessons 12-8 and 12-9), or we can use the one similarity theorem we already have (SAS~) plus the corresponding congruence theorems (ASA and SSS, respectively). In the past, I reversed Lessons 12-8 and 12-9 but this year I'm preserving the original order, so no, we can't use SAS~ to prove SSS~.

It's too bad that I can't connect today's SSS~ lesson to Pi Day. In the past I'd move Lesson 8-8 or 8-9 to Pi Day, but this time we're following the book. Of course, there is a connection between the number pi and similarity:

CCSS.MATH.CONTENT.HSG.C.A.1
Prove that all circles are similar.

And the reason that all circles are similar is that there is a constant ratio between the circumference and diameter -- and that ratio is pi.

Happy Pi Day, everybody!