Pi Day Post (Day 3.1415...)

Another Pi Day is upon us. This is the sixth ceremonial anniversary of the blog, and so I make every effort to post every year on Pi Day, even if it's on the weekend.

Today on her Daily Epsilon on Math 2021, Rebecca Rapoport writes:

How many integer values of x are there such that there is a triangle with side lengths x, 15, 17 and all angles acute?

Recall that last year, Pi Day was one of three special days when Rapoport wrote a joke (in this case it was "I 8 sum pi") instead of a problems whose answer was the date. This year she again will have three exceptional days, but Pi Day isn't one of them. Instead we have a regular problem that has nothing to do with pi or Pi Day -- but at least it's a Geometry song.

Without the last four words, this would be a Triangle Inequality problem. This tells us that the sum of the lengths of two sides of a triangle is greater than the length of the third side. Applying this to our triangle, we have both x + 15 > 17 and 15 + 17 > x. Combining gives:

17 - 15 < x < 17 + 15

But now we need to worry about those last four words "and all angles acute." While Triangle Inequality appears in Lesson 1-9 of the U of Chicago text, the Pythagorean Inequalities don't appear in at all. In other Geometry texts, we learn that:

a^2 + b^2 = c^2 iff the triangle is right
a^2 + b^2 > c^2 iff the triangle is acute

Applying this to our triangle, we have both x^2 + 15^2 = 17^2 and 15^2 + 17^2 > x^2. Combining gives:

17^2 - 15^2 < x^2 < 17^2 + 15^2

289 - 225 < x^2 < 289 + 225

64 < x^2 < 514

Notice that x can't be 8 (this would produce a right triangle), but it can equal any integer from 9 up to 22, since 22^2 = 484 and 23^2 = 529. The set {9, 10, 11, .., 20, 21, 22} has fourteen elements. Thus there are 14 possible side lengths -- and of course, today's date is the fourteenth of March, Pi Day.

Last year, I subbed in a special ed English classroom on the last three days before Pi Day. This year, I subbed in another special ed class, this time a self-contained one, on Pi Day Adam. In fact, I do more math in the class this year than last year, since the students have several math worksheets -- including the simple circumference/diameter worksheets that I give them.

Let's get to my favorite part of my annual Pi Day post -- the videos:

1. "Pi Day Music Video" by musicnotes online

This video is a few years old, but I include it often in my Pi Day post. It's the only song I know that actually mentions the exact time 3/14 at 1:59, and so I like to play it at exactly 1:59.

2. Parody of "Dynamite" by Taio Cruz

I've watched this video several times before. The students who create this video are 5th graders.

3. "Song from Pi" by Song Scout

I post this one almost every year. Song Scout writes a song where each pitch corresponds to one of the digits of pi.

By the way, I enjoy some of Song Scout's other music videos, even the ones that aren't directly related to the number pi. Here's one of them, about a piano with "two middle Cs::

4. "This Piano's Defect Makes It Impossible to Play":

I also recently watched his "A Piano Piece with Perfect Harmony."

5. Christopher Bill turns pi into music

This is one was first posted last year. Chris Bill uses the digits of pi as Song Scout and so many others have done, except the digits of pi correspond to the lengths, rather than pitches, of the notes.

6. Parody of "Thrift Shop" by Macklemore

This is another one that I post almost every year.

7. "Using Rubik's Cubes to Calculate Pi" by Z3Cubing:

This is a new video for this year. The length of the video is 3:14.

8. "Calculating Pi with Avogadro's Number" by Stand-Up Maths (Matt Parker):

This is also a new video for this year. It's also a great video for Mole Day on October 23rd. I also watched a video about the possibility that pi^pi^pi^pi (that is, pi^^4 or pi tetrated to the fourth) might even be an integer (or at least rational).

9. "Legendary Test Question -- pi is less than 22/7" by MindYourDecisions (Presh Talwalker):

This is also a new video for this year. It's also a great video for Pi Approximation Day on July 22nd (and Talwalker even mentions the PUTNAM exam that college students took last month).

10. Pi Song from Wonder Park:

I've never watched the film Wonder Park, so I didn't know that there even is a Pi Day scene in this movie until it popped up in a Google search. The movie was first released two years ago on the day after Pi Day (the math holiday was on a Thursday that year, and most films come out on Friday). Who knows -- perhaps if I'd known that there was a Pi Day song in this movie, I would have watched it.

11. "How Pi Sounds as Music:

This is yet another one of those songs based on the digits of pi, first posted on Pi Day of the Century.

12. "Pi Day Anthem" by John Sims and Vi Hart

This duet is mostly just reciting digits. The length of the song is also 3:14, although there are a few extra seconds at the end of the video.

13. Vi Hart's Pi Day video 2021

Speaking of Vi Hart, It's Vi Day, the one day of the year when Vi Hart posts a video. This year, Vi connects Pi Day to the start of Daylight Saving Time -- one minute after the AM pi moment (namely 3/14 at 1:59 AM) is 3:00. Vi suggests starting to regain that lost hour by watching this half-hour video -- another song based on the digits of pi, but by grouping some digits together.

Vi begins with 3.1415 (E-C-F-C-G) and its Mobius inversion (A-C-G-G-F -- recall that my old first day of school song at my old charter school was also based on inversion). Then Vi repeats sets of digits that go together, such as ..926.. (D-A-D, a D power chord) and 5-3-5-8 (G-E-G-C, which is C major chord).

By the way, I usually post my annual first day of DST post on the first post after the time change -- which is this post. Since today is Pi Day, I'll save it for tomorrow's post instead.

Oh, and I decided to complete my regained hour of Vi Hart videos by adding this other half-hour video to make a full hour:

14. Twelve Tones by Vi Hart:

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

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

70 N=1
80 FOR X=1 TO 32
90 READ A
100 SOUND 261-N*(18-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

As usual, click on the Sound button before you RUN the program.

This song is based on 18EDL. As I've said before with pi, 0 doesn't first appear until relatively late, which is way some musicians cut off the song just before the first 0.

For 18EDL, we use 0 and 9 as the tonic -- the beginning and ending of the scale. This might sound strange when there's no 0 to represent the root note.

This year, we can change our program so that 1 and 10 are the tonic. This follows the video that Vi Hart posted this year -- Vi tends to treat the digit 0 as 10 (as you can see in that video, where the 0 note is to the right of the 9 note). We'll also fix it so that all 70 digits that appear in the video are in this song:

70 N=1
80 FOR X=1 TO 70
90 READ A
100 SOUND 261-N*(19-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 9,5,10,2,8,8,4,1,9,7
160 DATA 1,6,9,3,9,9,3,7,5,1
170 DATA 10,5,8,2,10,9,7,4,9,4
180 DATA 4,5,9,2,3,0,7,8,1,1

I don't want this to turn into a music post, but the Vi Hart and Song Scout videos are also giving me ideas for finishing some of my other songs that are waiting for tunes. For example, Vi's Twelve Tones strategy can be changed to with the nine tones of 18EDL (counting the two tonics as one).

Song Scout's "Piano Piece with Perfect Harmony" gives hints as to how to give perfect harmony to my songs in 18EDL. Meanwhile, his broken piano with "two middle Cs" doesn't play our standard 12EDO scale well, but it could play other possible EDO scales such as 17EDO (with the black keys divided into sets of 2, 2, 3 just like the piano in the video, leading to a scale of 17 notes). This is an EDO scale that's not compatible with the EDL scales of Mocha.

Meanwhile, another video kept popping up on my YouTube feed while searching for Pi Day songs, even though it has nothing to do with the number pi. It's a new video by Sevish -- the only known musician who has posted an EDL video (he used 10EDL). He uses nonstandard scales often. But unfortunately, this Centaur scale isn't fully playable in Mocha. Still, any Sevish song provides hints for me to finish my songs.

This isn't a music post, so I'm done with scales for now. I'll let you know when I complete these tunes.

Last year, Pi Day marked the start of the coronavirus shutdowns -- Pi Day Eve was the last day for standard instruction, and as we saw, I didn't have in-person students again until October 6th.

Well, this Pi Day represents some positive developments. Today, Orange County officially returns to the red tier which allows schools and some businesses to reopen. And it appears that LA County will reach the red tier tomorrow.

I wrote that nothing less than a coronavirus vaccine will allow crowded LA County to reach the red -- and that's exactly what happened. Governor Newsom announced that the levels for reaching the red tier would become less stringent if a certain vaccination rate was reached -- and it was. Then the virus rates in each county has to stay below a certain level for two weeks -- it's been 14 days in Orange County but only 13 days in LA County, which is why LA has to wait until tomorrow.

And so while last Pi Day marked the beginning of the pandemic, this Pi Day appears to mark the beginning of the end of the pandemic. I hope that by next Pi Day -- which will fall on a Monday -- virus levels will be low enough to allow students to be on campus on Mondays, so that we can have a real Pi Day party once again.

If I timed this post properly, this should be posted on 3/14 at 1:59 PM Pacific Daylight Time. I will be eating my slice of cherry pie within the next hour.

Lesson 12-7: Can There Be Giants? (Day 127)

Today is Pi Day Adam -- the day before Pi Day Eve. Of course, Pi Day Adam becomes significant in years when the actual math holiday falls on a Sunday. In school we frequently celebrate the Adams of holidays that fall on Sundays (such as Mother's Day every year, and Halloween this calendar year).

I was hoping to sub in a math class on this special day, but I didn't. Instead, I subbed in a special ed class in my new district. It's the same class that I recently covered in my February 26th-March 1st posts.

This is one of the reasons that I started giving away pies on Wednesday -- at the time, I already knew that I'd be in the special ed class today, so Wednesday was my last math class before Pi Day. Something similar happened last year -- the last math class before the math holiday was on March 9th. But unfortunately, I taught a zero period that day and didn't have time to get any snacks.

And of course, we all remember what else happened last year. I covered that special ed English class on the last three days -- including Pi Day Eve -- before the coronavirus closure.

Last year, I played Pi Day music on the guitars and ukeleles that teacher had in his classroom. This year, one of the aides is a ukelele player -- but she doesn't bring her instrument to school today. Thus I'm the one who supplies all the music today. I will describe how I attempt to celebrate Pi Day Adam in a special ed class.

For starters, I sing a different Pi Day song each hour on the hour. Sometimes I choose the songs, and at other times I have one of the in-person students select it. Here's what I performed:

8:00 -- "Oh Number Pi"

9:00 -- "Sing a Song of Pi Day"

10:00 -- "Pi Day Serenade" and "Pi Day Carol"

(I sing both songs since they're short. It's more like 10:15, after the speech teacher leaves. This is followed by the daily walk, during which I sing the "Big March" song. Yes, it's still the Big March -- I consider Pi Day to be a bright spot during this tough time for students and teachers.)

11:00 -- "Handy Dandy Pi"

12:00 -- "I'm a Little Number" and "Why of Pi" (since the first song is short)

1:00 -- "Ring Around a Center" (a short song, but the students are now watching a movie)

2:00 -- "American Pi" and "Digit Connection"

(I sing these two long Bizzie Lizzie songs at the end of the day. In addition, I tell the class that Bizzie Lizzie was the one who taught me the first thirty digits of pi, via her "pi major" song -- E-C-F = 3-1-4.)

OK, today's not supposed to be "A Day in the Life," but this post turns into just that as I mention all the Pi Day songs I perform. In that case, I might as well as the "subbing" label.

In addition, just as I did on Wednesday, I give the students the opportunity to earn individual pies to eat from 7-Eleven. Unfortunately, these students are very low-level and haven't learned about pi yet, so how can I ask them questions about that constant for Pi Day?

(That's right -- the two classes to which I sing most of the Pi Day songs are a sixth grade class and a special ed class, neither of which has officially learned about that constant yet. Again, that's just the luck of the draw, as these are the two classes I spend the most time with leading up to the math holiday.)

So here's how I do it -- last night, I searched for the simplest possible pi worksheet. Here's what I found:

https://www.superteacherworksheets.com/geometry/circ-circle-1_TWDBF.pdf

On this worksheet, every question gives the diameter and asks for the circumference. There are no trick questions where the radius is given instead of the diameter, or where the circumference is given, or any of the other questions that we might ask a gen ed class. Nor do I ask for the area (the problem here isn't with the r, but with the "squared"). So all the student has to do is enter the number in the question on a calculator and multiply it by 3.14.

One guy (all the in-person students in this class are guys -- the ones opting out of hybrid are all girls) is already working on a rectangle perimeter worksheet, so this worksheet is not that much of a stretch from his current level. Another guy is doing mostly addition worksheet, but near the end of the day, he needs one last worksheet to earn the pie, and so I give him the pi worksheet. As soon as I tell him to press the times key on the calculator and not the plus key, he figures it out.

In addition to pies, I hand out pencils -- St. Patrick's Day pencils. I usually try to keep my Pi Day and St. Patrick's Day rewards separate, but I usually hand out pencils in special ed classes and since we're within a week of St. Paddy, I might as well pass out the green pencils.

By the way, the movie the students watch at the end of the day is Pixar's Brave. At first, I think to myself how appropriate this film is for St. Paddy, until I remember that Merida is a Scottish princess, not an Irish lass. (Still, I can see how one could get away with passing this off for St. Patrick's Day, as the Scots and Irish share a Celtic heritage.)

Today is Friday, the first day of the week on both the Eleven and Gregorian Calendars. It's the second straight Eleven week that I begin in this particular special ed class.

Resolution #1: We are good at math. We just need to improve at other things.

And the students do see that they are better at math than they think -- at least the two guys who complete the pi worksheets. They learn a brand new concept today.

A long-term sub is supposed to fill this teaching position starting next week. Therefore this should be the last time that I cover this class, at least for now.

Today on her Daily Epsilon on Math 2021, Rebecca Rapoport writes:

In the close-packing of equal spheres, the number of neighboring spheres for each sphere.

As it turns out, this number has a special name -- the kissing number:

https://mathworld.wolfram.com/KissingNumber.html

In 3D, the kissing number is 12 -- and of course, today's date is the twelfth. This is not very easy to prove at all. No, we can't just use the surface area formulas involving pi to find the answer, not even on Pi Day Adam (even though all the spheres are touching, or "kissing," on their surfaces).

Lesson 12-7 of the U of Chicago text is called "Can There Be Giants?" This is one of those "fun lessons" that we can cover if there's time, but in the past we bypassed it to get to 12-8 and the all-important SSS Similarity.

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

As I wrote above, Lesson 12-7 naturally lends itself to an activity. The whole idea behind it is that while dilations preserve shape, they don't preserve stability. This is because of the Fundamental Theorem of Similarity -- a dilation of scale factor k changes lengths by a factor of k, areas by a factor of k^2, and volumes by a factor of k^3. Weight varies as the volume, or k^3, while strength varies only as the area (as in surface or cross-sectional area), or k^2. Therefore, the answer to the question in the title of the lesson is no, there can't be giants because their k^2 strength, couldn't be strong enough to carry their own k^3 weight.

This year, not only does Lesson 12-7 land on our weekly activity day anyway, but it's the day that we observe Pi Day in the classroom. If we can't teach an actual pi lesson on Pi Day (and we can't, since we're always in the middle of Chapter 12 on Pi Day), the next best thing is a "fun lesson" like this one.

All that remains then is a way to make this lesson pandemic-friendly. (Yes, I do give a printed pi worksheet in class today, but this is a special ed class where the students regularly get worksheets.)

There does exist a Desmos activity about a giant. It's called "Marcellus the Giant":

https://teacher.desmos.com/activitybuilder/custom/58093e7b37d6769f0b7fde92

Unfortunately, this doesn't quite capture the full idea of Lesson 12-7 -- Marcellus the Giant can't exist because his surface area-to-volume ratio is too small. Indeed, this activity mentions neither surface area nor volume. And when I force a Google search for biological mentions of surface area-to-volume ratios, the results are mostly about why cells can't be too big, not about why organisms can't be too large.

I'll keep the second worksheet from last year (since there are a few questions about pizza on this Pi Day Adam) -- perhaps we can make an activity out of the Marcellus activity on Desmos and then discuss why he can't really exist. We can even combine it with the "Can There Be Humans?" riddle that I asked last year to make a complete activity. But it would be much more convenient if the entire activity (from the dimensions of Marcellus to the "Can There Be Humans?" riddle) were on Desmos.

Once again, a Brobdingnagian is a giant, while a Lilliputian is a tiny being:

Prove: Brobdingnagians don't exist.

Indirect Proof:
Assume that Brobdingnagians exist. In each of the three dimensions, a Brobdingnagian is 12 times as large as a human. So each bone of a Brobdingnagian would have to carry 12 times as much weight as a human bone. Thus a Brobdingnagian standing is like a human carrying 12 times its own weight. But no human can carry 12 times its own weight, a contradiction. Therefore Brobdingnagians can't possibly exist. QED

But there's a problem with this proof -- it can be used to prove that humans don't exist:

Prove: Humans don't exist.

Indirect Proof:
Assume that humans exist. In each of the three dimensions, a human is 12 times as large as a Lilliputian. So each bone of a human would have to carry 12 times as much weight as a Lilliputian bone. Thus a human standing is like a Lilliputian carrying 12 times its own weight. But no Lilliputian can carry 12 times its own weight, a contradiction. Therefore humans can't possibly exist. QED

The proof is clearly invalid, since humans do exist. My question is, which step is invalid?

Answer: "But no Lilliputian can carry 12 times its own weight, a contradiction."

This step is invalid, because in the real world, Lilliputians can carry 12 times their own weight -- indeed, much more than 12 times their weight. (Real-world Lilliputians are often called "ants.")

Here is the worksheet that I've created for this lesson -- the one that should follow the Desmos lesson above.

END

Lesson 12-6: The Fundamental Theorem of Similarity (Day 126)

Today I subbed in a middle school art class. This is in my new district. Since it's a middle school class, I will do "A Day in the Life" today.

8:45 -- First period arrives. This is the first of two Grade 7-8 art classes.

Every class, no matter what level, begins the same way -- with a "Bellwork" Warm-Up. In all classes, today's Bellwork is to draw a Pokemon character. Some students draw Pikachu or an established character while others create their own.

Then I randomly choose students to present their drawings. If I select an online student (one who is either in the other cohort or opting out of hybrid), that student shows the drawing to the camera, turns on the mic, and then describes the character.

Then the main assignment begins. These students are to create an "agamograph" by drawing two pictures, cutting each into strips, alternate between the first and second pictures, and tape the strips up like some sort of accordion. Then it's possible to see either picture by viewing it at the correct angles.

Each class ends with an Exit Ticket. For these Exit Passes, the students typically attach all work completed today (including the Bellwork) into Google Slides in order to submit them for grading.

For today's song, I choose one I haven't performed in a long time -- "Fraction Fever." In fact, for some reason I've avoided this song ever since I left the old charter school -- a shame perhaps, since the seventh graders at that school especially enjoyed that one. Today, I include the extra verses that I wrote over the summer -- verses that describe how to add, subtract, multiply, and divide fractions.

9:40 -- First period leaves and second period arrives. This is the first of two sixth grade art classes.

After the Bellwork, the main assignment is for the students to make a simple watercolor painting. For most of them, this assignment is very simple.

10:30 -- Second period leaves for break.

10:45 -- Third period arrives. This is the second of two Grade 7-8 art classes.

The regular teacher warns me about this class in her lesson plans -- and I can see why. As you might recall from previous posts, the middle schools in this district tend to have the strictest camera rules for online students on Google Meet. This teacher requires that the cameras be on during attendance -- if the camera is turned off, that student loses participation points. And the teacher makes several camera checks throughout the period, again taking away points from those with camera off. Between the camera checks, she says that it's OK for them to turn the cameras off.

So anyway, I take attendance, and most of the students have cameras off. I tell them that I will mark them as non-participating. But then one girl complains in the chat that the regular teacher only counts the second camera check, and so they shouldn't have to have them on during attendance. A few students agree with the first girl.

In past years, especially at the old charter school, this would have been the seeds of an argument. But then I remember something else that the regular teacher wrote -- the chat feature in Google Meet should be turned off during third period! So I immediately disable the chat -- and that, obviously, is the end of the argument.

By the time I reached the second camera check, only two students have their cameras off. Indeed, the class with the most noncompliant students today is first period, not third. (This has occurred often in middle schools in this district -- after first period, fewer students break the camera rule. It might be because after I see too many students with cameras off in first period, I clamp down and enforce the rule more strictly as the day goes on.)

11:40 -- It is now time for tutorial. As in the other middle schools in this district, students are assigned to a particular period for tutorial each day. Today, they are to report to their fourth period class, which for me is the second of two sixth grade art classes.

And as I've been doing during tutorial at other schools, I sing the entire period. Since I already started performing Pi Day songs yesterday, I might as well continue to do so today. And besides -- middle schools in this district follow Hybrid Plan #2, where Cohort A is in-person Tuesday/Thursday. Thus just like the students who attended the high school in-person yesterday, today is the last day that these particular students are on campus until the Tuesday after Pi Day.

I have the students choose among my Pi Day tunes, and these are what they select:

• This Old Pi
• O Number Pi
• Ring the Bells
• Ludolph the Mathematician
• American Pi (Bizzie Lizzie)
• Digit Connection (Bizzie Lizzie)

I won't link to all of these songs this time -- you can find most of them on Google, except for the Bizzie Lizzie songs whose lyrics I've posted earlier on the blog.

No, I don't bring any 7-Eleven pies for the students today. The only reason I was able to yesterday was that I started the day with conference period (when I can see the attendance rosters and decide how many pies to purchase). Middle schools in my new district don't have conference periods this year -- instead, all teachers have sixth period "conference," and all students have P.E. independent study.

It's a shame though that I had sixth graders during my Pi Day tutorial "concert." Sixth graders haven't learned about pi yet -- pi used to be a sixth grade topic under the California pre-Core standards (which expected eighth graders to take Algebra I), but it's now a Common Core 7 topic. Thus my Pi Day songs would have been better appreciated (and understood) if it had been one of the Grade 7-8 classes.

12:10 -- The sixth graders leave for lunch.

12:50 -- The sixth graders return from lunch. I do sing "Fraction Fever" during this period -- just as I do in all the other classes, despite this class getting so many extra songs during tutorial.

1:45 -- Fourth period leaves and fifth period arrives. This is the only Advanced Art class. Of course, only Grades 7-8 are in this class.

While the Bellwork and Exit Ticket are the same for this class as all the others, the main assignment for them is different. The students are wrapping up a project on linear perspective -- which reminds us of Lesson 1-5 of the U of Chicago text. And the new assignment is on drawing faces, on the computer, using a technique called "vectors" -- and vectors are coming up in Lesson 14-5 of our text. Yes, there is a stronger connection between art and math than we might expect at first.

2:35 -- Fifth period leaves, thus completing my day.

Today is Elevenday on the Eleven Calendar. This is the day that I focus on the millennium resolutions and communication skills.

One sixth grade boy is wearing a USC mask. I tell him that I graduated from UCLA, and that our schools are rivals. I continue to play around with this during all of tutorial and fourth period, but he doesn't respond at all. He appears to be quite shy -- either that, or likely he doesn't follow USC sports as closely as some boys his age might. Perhaps there are better ways I could have tried to engage with this young student.

Elevendays are also for improving my communication skills with adults. During lunch, one teacher (more likely an aide for special ed) I see in the staff lounge is wearing a UCLA jacket. She tells me that she didn't go to UCLA -- she only wore it because she likes its colors. But her son is currently a high school junior, and he's strongly considering applying to UCLA by the end of the calendar year.

I also have the opportunity to communicate on Twitter today. One Texas teacher mentions a Pi Day art project where students make -- you guessed it -- agamographs. (That is, unless you guessed that they make Pi Day watercolor paintings.) Of course I reply that I subbed in an art class today that's also making agamographs, though they have nothing to do with Pi Day. (What was I saying earlier about a connection between math and art again?)

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

Section 12-6 of the U of Chicago text covers the Fundamental Theorem of Similarity. As its name implies, it is the most important theorem related to dilations and similarity. Here is how this theorem is stated in the U of Chicago:

Fundamental Theorem of Similarity (U of Chicago):
If G ~ G' and k is the ratio of similitude, then
(a) Perimeter(G') = k * Perimeter(G) or ...
(b) Area(G') = k^2 * Area(G) or ...
(c) Volume(G') = k^3 * Volume(G) or ...

Notice how I had to rewrite this theorem so that it fits into ASCII. Here the * and ^ symbols denote multiplication and exponentiation, respectively -- these symbols should be recognizable as they appear on TI graphing calculators. The "or ..." sections refer to the text rewriting each equation as a ratio, so that the ratio of the perimeters is k, the ratio of the areas is k^2, and so on, but that is rather awkward to write in ASCII.

David Joyce describes this theorem in his criticism of the Prentice-Hall text:

The proofs are omitted for the theorems which say similar plane figures have areas in duplicate ratios, and similar solid figures have areas in duplicate ratios and volumes in triplicate rations. At least there should be a proof that similar triangles have areas in duplicate ratios; that's easy since the areas of triangles are already known.

Here is the U of Chicago proof:

(a) Perimeter is just the sum of the lengths. Suppose lengths a, b, c, d, e, ...make up the perimeter of G. Then lengths ka, kb, kc, kd, ke, ...make up the perimeter of G'.

Perimeter(G') = ka + kb + kc + kd + ke + ...
                       = k(a + b + c + d + e + ...)
                       = k * Perimeter(G)

(b) Let A = the area of G. Then you could think of the area of G as the sum of areas of A unit squares. Then the area of G' is the sum of areas of A squares k units on a side. Since each square in G' has area k^2,

Area of G' = A * k^2 = k^2 * Area of G.

(c) The argument is identical (except with unit cubes). QED

As I mentioned earlier this week, I'm no longer posting old worksheets that mention the Hung-Hsi Wu proof. Thus I'm rearranging last year's worksheets.

Last year's second worksheet is now the first worksheet, since it mentions the U of Chicago version of the Fundamental Theorem of Similarity.

This is labeled as a so-called "activity." Tomorrow is when I'll post my next pandemic-friendly activity.

Lesson 12-5: Similar Figures (Day 125)

Today I subbed in a high school Precalculus class. It is in my first OC district. Since it's a math class, I'm definitely doing "A Day in the Life" today.

9:00 -- Today even periods meet. But second period just happens to be the teacher's conference.

Before today, I wasn't quite sure how I wanted to celebrate Pi Day this year. With the math holiday on a Sunday this year, we'd ordinarily observe it on the last school day before -- but "last school day" has a different meaning during the pandemic hybrid schedule. For the in-person students in Cohort A, today really is the last time they step on campus until the sixteenth, when it is already past Pi Day.

If this teacher's day didn't start with conference period, I likely buy no Pi Day food today. But since I had the luxury of time with the prep period, I go out and purchase snacks -- but which snacks? The traditional pie that is cut with a knife and shared isn't very pandemic friendly. Instead, I got some individual pies from 7-Eleven for $1 each.

10:10 -- Fourth period arrives. This is the first of two Precalc classes.

This class is studying the six trig functions, in the form y = a sin(bx + c) + d (where sin can be replaced with any of the other functions). These functions are in radians, and thus pi and its multiples are relevant just in time for Pi Day.

These students are preparing for a quiz on the next day this class meets, which is Friday (online for Cohort A and in-person for Cohort B). I hand out four pies today, one to each students who describes a graph or answers a Practice Test question on the board. I also hand out a pie to the student who helps me demonstrate a Putnam problem today --

-- oops, that right! The Putnam exam was delayed from December to February. But on the blog, I just blew right past the date without discussing our usual Putnam problem.

Let's fix this right now. I will go back and edit my February 23rd post and include the Putnam discussion that I usually give on the Tuesday after the exam. (I could just put it in today's post, but in future years I'll want to go back and reread my old posts for Putnam problems. It will be easier for me to find if it's in my post from Tuesday after the exam -- especially if I see the test was February 20th.)

OK, here is my edited February 23rd link:

https://commoncoregeometry.blogspot.com/2021/02/lesson-11-4-midpoint-formula-day-114.html

In this class, I discuss Problem B2 with the students. I find a chessboard in the room, and so I use pawns to represent pegs and chess squares to represent holes. I choose one guy to play the role of "Alice" while I play the role of "Bob." The "Alice" guy figures out quickly that he wins with odd number of pawns (k), but the case where k is even and n is odd is a little tricky.

There is a little time left after the Putnam problem, and so I sing my only song of the day. It is "Sing a Song of Pi Day" (a parody of "Sixpence" -- a song that also happens to mention a pie). I originally found the lyrics at the following website:

https://www.deviantart.com/chibimama/art/Pi-Songs-112578820

Sing a Song of Pi Day
(sung to the tune of "Sing a Song of Sixpence")

Sing a Song of Pi Day
Come celebrate with me
A holiday from math class
Relieves monotony
We've "rounded up" refreshments
An elegant cuisine
The party is approximately March fourteen!

If Pi Day's on the weekend
And we are not at school
We'll celebrate on Friday
A difference miniscule
We'll call it Pi Day Eve and
The party's just the same
With songs and snacks (all circular)
You'll be so glad you came!

11:05 -- Fourth period leaves and sixth period arrives. This is an honors/IB class.

Although this is also a Precalc class, these students are learning something different. Instead of trig, they are learning about conic sections, specifically hyperbolas. Since they just finished ellipses, I mentioned the ellipse area formula A = pi ab in order to draw a connection back to pi. Once again, I hand out four pies to students who help take notes on the board.

Since these students are more advanced, I do Problem A1 with them instead of B2. Unfortunately, I don't get to this problem until there are only about ten minutes left, and so I rush the problem. I truly believe I could have given the students a better feel for this problem if I had slowed down, but hyperbolas are the most difficult of the conic sections, and I needed almost all theperiod to give the hyperbola lesson justice. And there definitely is no time for me to sing anything.

12:05 -- Sixth period leaves for lunch. Ordinarily after lunch is academic support, but it appears that the regular teacher doesn't intend for me to do academic support today, seeing as she scheduled the Zoom meetings to be open to me for fourth and sixth period but closed for academic support. And so my day ends right here.

Today is Tenday on the Eleven Calendar:

Resolution #10: We are not truly done until we have achieved excellence.

And this fits perfectly today. The point of showing the students Putnam problems is to demonstrate what exactly excellence in math looks like. Suddenly, their upcoming tests on trig and conic sections don't look as difficult when compared to the Putnam.

As for the Miller wager, I break even in fourth period with a dozen students (mostly seniors, a few juniors, each worth $2) in-person and two dozen online. But I lose money (that is, Miller wins) in sixth period since there are 18 in-person (mostly juniors, since it's honors) and 13 online.

Of the seven in-person students I see today in fourth period, only one is a girl, the rest guys. But of the ten in-person students in sixth period, only one is a guy, the rest girls. It could be that more girls sign up for the IB program, or it could be just that more IB girls on Cohort A opted into in-person learning while fewer non-IB girls on Cohort A did -- that's simply how the classes happened to be divided.

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

Lesson 12-5 of the U of Chicago text is about similar figures. There is not much for us to change about this lesson from last year, except for the definition of similar itself. Recall the two definitions:

• Two polygons are similar if corresponding angles are congruent and sides are proportional.
• Two figures are similar if there exists a similarity transformation mapping one to the other.

The first definition is pre-Core, while the second is Common Core. The U of Chicago text, of course, uses the second definition. But that PARCC question I mentioned last week must be using the first definition, since it requires that we know what similar means before we can define dilations and ultimately similarity transformations.

In the U of Chicago text, the Similar Figures Theorem is essentially the statement that the second definition implies the first definition. We would actually need to prove the converse -- that the first definition (at least for polygons) implies the second. But the proof isn't that much different -- suppose we have two figures F and G satisfying the first definition of similar -- that is, corresponding angles are congruent and sides are proportional, say with scale factor k. Then use any dilation with scale factor k to map F to its image F'. Now F' and G have all corresponding parts congruent, so there must exist some isometry mapping F' to G. Therefore the composite of a dilation and an isometry -- that is, a similarity transformation -- maps F to G. QED