Easter Saturday Post

Table of Contents

1. Introduction & Rapoport Problem of the Day
2. Spring & Summer Plans for the Blog
3. Mocha Music: Crab Canon
4. Finishing Crab Canon's Eighth Notes
5. More on EDL and EDO Music
6. April 15th Rapoport Problem Using Tangent Half-Angle Formula
7. April 15th Rapoport Problem Using Pure Geometry?
8. Geometry and Scale Diagrams
9. Cosmos Episode 12: "Coming of Age in the Anthropocene"
10. Conclusion

Introduction & Rapoport Problem of the Day

Today on her Daily Epsilon of Math 2020, Rebecca Rapoport writes:

How many triangles can be made by connecting three points on the grid above?

[Once again, all the givens are in an unlabeled diagram. The important thing here is that there are six points in two horizontal rows.]

I decided to post today's question since it's about triangles and hence Geometry. But we can truly solve this problem without thinking (much) about Geometry.

There are six points and we wish to choose three of them, and 6 choose 3 is 20. But there are two forbidden combinations -- the three in the first row and the three in the second row. Subtracting these two combos leave us with 18 triangles.

If we want, we can label the points in the first row A, B, C and the second row D, E, F. Then three of the possible triangles are DEB, DFB, and EFC. But the specific triangle names don't matter here. We already know that there are 18 triangles -- and of course today's date is the eighteenth.

Today is the Saturday after Easter -- also known as Easter Saturday. Many people believe that Easter Saturday was the day before Easter Sunday, but it isn't -- that was Holy Saturday.

In general, the days from Palm Sunday to Easter Sunday have "Holy" in their names. No day can have Easter in its name that's earlier than Easter Sunday itself, so this the last day of Easter Week. I also point out that tomorrow will be Orthodox Easter.

(In fact, I saw this debate appear on Facebook. A mother who is homeschooling due to the virus is happy that this week is spring break/Easter vacation -- apparently, her children regularly attend a school that takes the week after Easter off, so she's following the same calendar. But then another commenter complained that "spring break" is just a euphemism to remove Christianity from the calendar and that real "Easter week" contains Maundy Thursday, Good Friday, Easter Sunday. But this is false -- on the liturgical calendar, the week he's describing is Holy Week, not Easter Week!)

And this week, as I've already long suspected would happen, my old district declared that there will be no in-person classes the rest of the year. The State of California has extended the stay-at-home orders through Friday, May 15th, and if school were to resume the following Monday, it would already be Day 170. It would seem pointless to resume school that day even if were just review for finals, especially considering that the district has already stated its grading policy. (Meanwhile, my new district still hasn't declared what to do about semester/trimester grades.) Day 131 will forever be my last day of the 2019-2020 school year.

There is still some hope that the Class of 2020 will have some semblance of a graduation ceremony if the stay-at-home orders are lifted on May 15th. The seniors already won't have a prom.

At this point, I'm no longer labeling these posts as the nth "spring break" post or whatever. Indeed, as far as we subs are concerned, it's already summer. There is no chance for me to enter a classroom -- or earn any money -- until the first day of school in the fall.

Much of this post, therefore, is devoted to my plans for the blog. My mind is still spinning -- I now have to start thinking in terms of a summer-like schedule for blogging. I'll continue to post once or twice as I've been doing each week since Pi Day -- and as I've done every summer. Indeed, earlier this year, I was already making plans for my summer blog posts. And now it's here a few months earlier than expected.

Spring & Summer Plans for the Blog

Traditionally, the last three days of school (Days 178-180) have followed a pattern -- the first day is when I post the final exam (and a traditionalists' post), the second day is for visiting math teacher members of the MTBoS, and the third day is a preview post (for summer and the year to come).

As it turns out, my March 12th-14th posts almost follow that pattern already. On March 12th I posted the last test of the year (albeit the Chapter 12 Test, not the final), on March 13th I wrote a little about how other teachers are dealing with the closure, and on Pi Day I discussed future blog posts (at least up through today's post, expecting that the schools would have reopened by now).

The only thing I need to do is consolidate all my MTBoS links into the March 13th post -- and, of course, add the "MTBoS" label. Here is a link to that new Pi Day Eve post:

https://commoncoregeometry.blogspot.com/2020/03/lesson-13-1-making-conclusions-day-131.html

Meanwhile, I think back to the end of the 2016-2017 school year, after I left the old charter school, when I started posting some Chapter 12 lessons and finished with Chapters 13-15. I'm now declaring all Chapter 12 posts from 2017 to be defunct, so that only Chapters 13-15 count. That way, you can refer back to that year to see material from Chapters 13-15 that we missed this year.

(Note: I didn't delete those posts in the Great Posting Purge of 2017 because I'd already declared one of those posts to be my milestone 500th post, and I didn't wish to interfere with the post count. But to keep things simple from this point on, I'll count Chapter 13 at the first chapter after leaving the charter school, not Chapter 12. It also means that I should have posted every lesson exactly five times, although I did jump around the first two years of this blog.)

Here is a link to the Lesson 13-1 post from 2017:

http://commoncoregeometry.blogspot.com/2017/04/lesson-13-1-logic-of-making-conclusions.html

My hope is that these two links (my posts for Chapters 13-15 and links to other math teachers via the MTBoS) will help my readers who are desperate for Geometry material during the school closures.

My other annual post is the day when I list my most popular posts of the year. I'll post this during the first half of May, as usual.

The main thing I said I wanted to post is on computer programming. As I explained earlier, I took coding classes at UCLA, and now I'm considering returning to coding as an alternative career since the future is uncertain for subbing and in-person education. My plan is to finish Cosmos first, and with the season finale coming up on April 20th, expect coding in the last post of this month and extending at least through May.

And speaking of "cosmos," I wrote that I was considering reading Ian Stewart's Calculating the Cosmos during the summer -- after all, I kept quoting from it one chapter at a time, so it's high time that we read it cover-to-cover as a side-along reading book. Indeed, this is the probably the best time to do so, especially considering that I own a copy of the book, whereas with libraries closed for the foreseeable future, I won't be checking out or purchasing for $1 any books soon.

There's actually nothing stopping me from starting Stewart's book now -- my intention was to start it after the last day of school, and unexpectedly, it is after my last day of school. But I don't want anything else to distract me from starting my computer coding. So I'll at least do some coding first, and most likely begin Stewart's book right when I was expecting to start it, in June.

And there was one more thing in the back of my mind that I was planning, perhaps for the summer of some other time. Something that I believe makes higher math more daunting is the use of many unfamiliar vocabulary terms. Geometry and Trigonometry are notorious for their tricky vocabulary.

So I was considering rewriting some Geometry chapters using simple language. Since we skipped Chapters 13-15, these are the chapters I might rewrite during the summer. I haven't quite decided how this project might go, but I was encouraged to do this when I once read a possible alternative name for Trigonometry itself -- "threenooklore." I don't recall where I originally found the word, but a Google search reveals the following link:

https://anglish.miraheze.org/wiki/Talk:Telcraft

(Apparently, there's some debate here as to whether "threenooklore" or "threesidelore" is better. We'll discuss this more when I actually decide to begin this.)

One last blogging project will be a reblog of all my old songs. Unfortunately, on my last day before the school closures, I took out my songbook and started playing Pi Day Eve songs on the guitars and ukeleles that were in the classroom. The only problem is that I left that songbook in the classroom -- and now the earliest I can see the inside of that classroom again is August! I'd suddenly realized that I didn't have the book after I'd been driving away from the school for about 20-30 minutes -- and I really didn't want to arrive at home 40-60 minutes late just to retrieve the book. (When I was leaving the classroom, I was thinking more about getting the empty Pi Day pizza boxes out of the room than my songbook, until it was too late.) This marks the second time that I left song lyrics in a room to which I no longer have access -- this is exactly what happened on the day I left the old charter school.

This time, I'll gradually post lyrics over the summer so at least they'll be easier for me to find on my blog when it's time to sing them. I think I'll also save them on my hard drive -- if I even get another songbook, I'll just print and glue the lyrics in (like an interactive notebook).

Mocha Music: Crab Canon

Everything I've described so far refers to upcoming posts. So what will I post today?

So far, I've been reblogging or completing posts from the corresponding days in previous years. I've never posted on Easter Saturday before -- but I have posted on April 18th before, including last year.

Last year at this time, we were in the middle of reading Douglas Hofstadter's book. In the chapter that I posted on April 18th of last year, I took a song -- Bach's Crab Canon -- mentioned by Hofstadter and converted part of it to Mocha.

(I know -- I say I want to learn a new coding language, but then I keep going back to Mocha and BASIC over and over):

There's also Crab Canon -- the name of both an Escher drawing and a Bach piece. (In the reversible Dialogue, the Tortoise mentions the Escher drawing and Achilles mentions the Bach piece.) The artwork is reflection-symmetric, and the piece can also be played forwards and backwards.

This is the sort of song that I'd like to try coding in Mocha. Unfortunately, BASIC doesn't allow us to read notes from the DATA lines backwards. Instead, we place the notes and their lengths in DIM arrays and then use FOR loops to play it forwards and backwards (STEP -1 is backwards):

https://www.haplessgenius.com/mocha/

10 DIM D(20),L(20)
20 FOR X=1 TO 20
30 READ D(X),L(X)
40 NEXT X
50 DATA 18,8,15,8,12,8,11,8,19,12
60 DATA 12,8,13,8,14,8,15,8,16,8
70 DATA 17,8,18,4,19,4,24,4,19,4
80 DATA 16,4,15,8,16,8,18,8,16,8
90 N=9
200 FOR X=1 TO 20
210 SOUND 261-N*D(X),L(X)
220 NEXT X
230 FOR X=20 TO 1 STEP -1
240 SOUND 261-N*D(X),L(X)
250 NEXT X

[As usual, turn on "Sound" before you RUN the program. Also, I intentionally changed the 100's to 200's in the line numbers.]

Bach wrote his song in the key of C minor. I wrote my version in 18EDL (white D minor) and then used N=9 to bring it down to C minor to match Bach's version.

I ad-lib a little when Bach plays a few notes not available in 18EDL. Actually, there is also a second line in eighth notes. Due to Hofstadter's Law, I don't have time to code it all out -- plus it includes a few more notes unavailable in 18EDL.

Finishing Crab Canon's Eighth Notes

In the program above, I skipped the 100's so that we can fill in the eighth notes to represent the second part of the song:

110 DATA 12,2,13,2,13,2,9,2,12,2,15,2,16,2,15,2
120 DATA 13,2,12,2,11,2,10,2,9,2,15,2,13,2,12,2
130 DATA 11,2,16,2,15,2,13,2,12,2,13,2,15,2,16,2
140 DATA 15,2,13,2,12,2,11,2,10,2,11,2,12,2,13,2
150 DATA 12,2,11,2,10,2,9,2,8,2,10,2,11,2,12,2
160 DATA 11,2,10,2,9,2,8,2,7,2,9,2,10,2,11,2
170 DATA 10,2,9,2,8,2,7,2,6,2,8,2,12,2,8,2
180 DATA 9,2,8,2,7,2,6,2,7,2,8,2,9,2,10,2
190 DATA 9,4,12,4,15,4,18,4

We added eight bars of eighth notes (64 notes total) and one bar of quarter notes, and so these 68 extra notes means that all the 20's should turn into 88's:

10 DIM D(88),L(88)
20 FOR X=1 TO 88
200 FOR X=1 TO 88
230 FOR X=88 TO 1 STEP -1

Note: in coding, it's usually considered bad style to change all of these so-called "magic numbers" one at a time -- in a modern language, we'd define a constant variable "NotesInSong" and set it equal to 20 for the short version and 88 for the extended version. If we tried that in Basic, it would look something like this:

5 NS=88
10 DIM D(NS),L(NS)
20 FOR X=1 TO NS
200 FOR X=1 TO NS
230 FOR X=NS TO 1 STEP -1

So now all we'd do is change Line 5 to NS=20 for the short version and NS=88 for the long version.

All of those DATA lines are tedious to type -- it would be nice if there was a way to omit all of those 2's to denote eighth notes. It might be worth rewriting the program as follows:

25 READ C
30 D(X)=INT(C)
35 L(X)=(C-INT(C))*20+2

Instead of reading the Degree and length directly, we now read a special code C. The integer part (INT) of this code is the Degree, while the decimal part gives the length via a formula. The digit in the tenths place is one less than the correct length as a number of eighth notes. For example, a quarter note is .1 (quarter note = 2 eighth notes) and a dotted half note is .5 (dotted half = 6 eighth notes).

The code for a single eighth note is .0, so there's no decimal part at all. This allows us to write the eighth note DATA lines using only Degrees:

110 DATA 12,13,13,9,12,15,16,15

while the DATA lines for the longer notes contain a single decimal place, so it's not more difficult to type than the original line (unless, of course, we confuse periods and commas when typing):

50 DATA 18.3,15.3,12.3,11.3,19.5

The eighth note line requires notes in the higher octave. The song contains the notes C, D, Eb, F (plus a stray Db) in the upper octave, but the song plays these as C, D, E, G. The G doesn't sound too much out of place, but that E is a red E (supermajor 3rd), which doesn't belong in a song in C minor.

It might be helpful to use 20EDL here instead of 18EDL. To avoid having to rewrite the DATA lines, let's simply add two to each degree:

90 N=8
210 SOUND 261-(N*D(X)+2),L(X)
240 SOUND 261-(N*D(X)+2),L(X)

The root is now green C, so we might as well change N=9 back to N=8. The lowest notes remain more or less the same, although the green Eb has become a su D# (17u D#). The ratio 20/17 (called a suyo 2nd in Kite's color notation) is narrower than a green 3rd (but still wider than a blue 3rd).

And unfortunately, this still doesn't get the high F right -- instead of being too sharp and sounding as a G, it's now too flat and sounds as an E. We can't expect songs written in standard 12EDO to sound the same when played in EDL scales.

More on EDL and EDO Music

By the way, that last sentence is a bit strange there. Our standard scale is called 12EDO (Equal Divisions of the Octave), but the letters EDO look a lot like EDL (Equal Divisions of Length). At some point I wonder whether I might write about converting 12EDO to 18EDL and a reader might think I'm converting from 12EDL to 18EDL (that is, from one EDL to another).

Perhaps to avoid this problem, I should write our standard scale as 12TET instead of 12EDO. The notation 12TET means "12 Tone Equal Temperament." So what's equal about 12TET are the intervals between the notes (if written as cents -- 1 semitone = 100 cents). For EDL scales, what's equal are the lengths of vibrating string between each of the Degrees.

Let me reblog an old program that states how to play some EDO scales (or TET scales) using Mocha's EDL scale:

NEW
10 INPUT "EDO";N
20 DIM R(N), D(N)
30 FOR S=0 TO N
40 R(S)=2^(S/N)
50 NEXT S
60 M=N
70 FOR L=12 TO 260 STEP 2
80 PRINT "TESTING EDL";L
90 E=0
100 FOR S=0 TO N
110 D=L/R(S)
120 E=E+ABS(D-INT(D+.5))
130 NEXT S
140 IF E>=M THEN 190
150 FOR S=0 TO N
160 D(S)=INT(L/R(S)+.5)
170 NEXT S
180 M=E
190 NEXT L
200 CLS
210 PRINT "PLAYING SCALE"
220 FOR S=0 TO N
230 PRINT "DEGREE";D(S);"SOUND";261-D(S)
240 SOUND 261-D(S),4
250 NEXT S
261 END

(Note: Line 40 requires an up-arrow where you see the ^ caret.)

Here's how the program works -- the user inputs the desired EDO in line 10. Lines 30-50 set up the correct ratios for your EDO -- so if you choose 12EDO, then the ratios are 2^(0/12), 2^(1/12), 2^(2/12), all the way up to 2^(12/12).

Lines 70-190 set up a loop to test every EDL, from the shortest (12EDL) to the longest (260EDL). It calculates the error E between the correct ratio and closest possible degree available (notice how INT(D+.5) is used to round D to the nearest degree). The IF statement in line 140 is used to see whether E is less than the minimum error M encountered so far. If E>=M, then we skip to line 190 and check the next EDL. If E<M, then we have a new "best EDL so far," and lines 150-170 set up a loop to replace the list of "degrees of the best EDL so far" with the new list.

Afterwards, the final loop from lines 220-250 actually play the best scale on the computer. (Don't forget to click the "Sound" box in order to hear the sound!) The final END line isn't needed, but I include it just as an excuse to honor Bridge 261 with a line 261.

Here are the best possible scales the computer found for some EDO's:

5EDO: use 256EDL (Degrees 256, 223, 194, 169, 147, 128 -- Sounds 5, 38, 67, 92, 114, 133)
7EDO: use 156EDL (Degrees 156, 141, 128, 116, 105, 95, 86, 78)
10EDO: use 256EDL (add Degrees 239, 208, 181, 158, 137 in between 5EDO)
12EDO: use 232EDL (Degrees 232, 219, 207, 195, 184, 174, 164, 155, 146, 138, 130, 123, 116)
16EDO: use 214EDL (Degrees 214, 205, 196, 188, 180, 172, 165, 158, 151, 145, 139, 133, 127, 122, 117, 112, 107)

I actually recommend not going past 12EDO, since the error E rises sharply. If you must try a higher EDO, then use 16EDO as it has the least error. Other multiples of 4 (20EDO, 24EDO, 28EDO) have lower errors for their size, while 22EDO and 31EDO (mentioned in earlier posts) also have lower errors for their size. Beyond 31EDO, all scales are about equally bad (with the odd EDO's marginally better than the even EDO's).

It's interesting that 232EDL is the best for 12EDO, since the EDL link I mentioned in an earlier post recommended 120EDL and 196EDL (among others) to make a 12EDO scale.

But to me, making EDO scales is such a waste. The computer uses EDL's, which are based on exact divisions of length, so we should be playing sounds in just intonation, not EDL's.


April 15th Rapoport Problem Using Tangent Half-Angle Formula

That's right -- that Rapoport problem from Wednesday is still bugging me. It annoys me that I had to use threenooklore -- sorry, I meant trig -- in what I suspect should have been pure Geometry, and I was worried about rounding errors.

Let's look at the problem once again:

Line AP is tangent to Circle O at Point A. Point Q is between A and P. And these lengths are shown: PQ = 4, BQ = 6, BP = 8, and radius = r. What is r^2?

Oh, and let me add the new info from last time. Point B isn't on Circle O. Instead, Line BP is tangent to Circle O at Point C.

We were able to figure out without trig that BC = 1, and hence CP = 9. The place where we needed trig was where we were working with angles -- first, we used the Law of Cosines to find Angle BPQ, since all three sides of Triangle BPQ are known:

cos BPQ = (BP^2 + PQ^2 - BQ^2)/(2 * BP * PQ)
cos BPQ = (8^2 + 4^2 - 6^2)/(2 * 8 * 4)
cos BPQ = 11/16

To avoid rounding errors here, instead of finding the inverse cosine of BPQ, let's just call the unknown angle theta:

cos theta = (BP^2 + PQ^2 - BQ^2)/(2 * BP * PQ)
cos theta = (8^2 + 4^2 - 6^2)/(2 * 8 * 4)
cos theta = 11/16

Then we used the tangent ratio in a right triangle, that we know now is correctly labeled CPO (with the right angle at C). We found out earlier that Angle CPO is half of BPQ, so we write:

tan CPO = OC/CP
tan(theta/2) = r/9
r = 9 tan(theta/2)

So what we need is tan(theta/2) and we already know what cos theta is. Fortunately, there exists a half-angle formula for tangent -- one I can never remember and must always look up:

tan(theta/2) = +/- sqrt((1 - cos theta)/(1 + cos theta))

We need r to be positive, so we can forget about the plus-or-minus. Let's just plug in cos theta:

r = 9sqrt((1 - 11/16)/(1 + 11/16))
r = 9sqrt(5/27)
r = 9sqrt(15)/9
r = sqrt(15)
r^2 = 15

And voila -- we obtain our desired value without any rounding errors at all.

April 15th Rapoport Problem Using Pure Geometry?

OK, so we eliminated the need to round. But we still needed the Law of Cosines and the tangent ratio, so we still used trig. And so I still want to know whether we can avoid trig altogether and solve the problem in pure Geometry.

Let's look at the trig lines again:

cos theta = (BP^2 + PQ^2 - BQ^2)/(2 * BP * PQ)
tan(theta/2) = OC/CP
tan(theta/2) = sqrt((1 - cos theta)/(1 + cos theta))

What would happen if we eliminated theta altogether and just write the equation with side lengths? I will try it now:

OC/CP = sqrt((1 - cos theta)/(1 + cos theta))
           = sqrt((1 - (BP^2 + PQ^2 - BQ^2)/(2 * BP * PQ))/(1 - (BP^2 + PQ^2 - BQ^2)/(2 * BP * PQ)))
           = sqrt((2 * BP * PQ - BP^2 - PQ^2 + BQ^2)/(2 * BP * PQ + BP^2 + PQ^2 - BQ^2))
           = sqrt((BQ^2 - (BP - PQ)^2)/((BP + PQ)^2 - BQ^2))

But that's as far as I can get. What I was hoping was for more terms to cancel -- perhaps both expressions BP - PQ and BP + PQ each work out to equal a single length, which would then leave (something)^2 - (something else)^2 in both the numerator and denominator. Perhaps these could both be simplified using the Pythagorean Theorem as (hypotenuse)^2 - (leg)^2 = (other leg)^2, leaving only a simple ratio after taking the square root. If that ratio isn't OC/CP itself, then maybe it's equal to OC/CP via some hidden similar triangles.

There's one more thing about that weird square root expression. Since both the numerator and denominator are differences of squares, they can both be factored, leaving:

OC/CP = sqrt(((BQ + BP - PQ)(BQ - BP + PQ))/((BP + PQ + BQ)(BP + PQ - BQ)))

This sort of reminds me of Heron's formula for the area of a triangle. Indeed, if in Triangle BPQ we let a = BQ, b = PQ, c = BP, and s be the semiperimeter, then we obtain:

OC/CP = sqrt(((s - b)(s - c))/(s(s - a)))

(Each factor is equal to 2 times one of s, s - a, and so on, but all the 2's cancel.)

This isn't quite Heron's formula for the area of Triangle BPQ. But it does hint that perhaps our pure Geometry solution involves areas of triangles and other figures.

Indeed, if we use the known lengths of BPQ (4-6-8) and use Heron's formula, we can find its area to be 3sqrt(15) square units. The desired radius turns out to be r = sqrt(15), so perhaps we can show that a rectangle of base r and height 3, or a triangle of base r and height 6, has the same area as BPQ.

In fact, this could be way Rapoport asks for r^2 and not r -- not only because she wants the answer to be a natural number (the date), but also because a square of radius r and its area comes up. The fact that we almost get a formula for the area of BPQ does not mean that Heron's formula is needed -- it means only that we might indirectly use the area of BPQ and other figures in the solution.

Geometry and Scale Diagrams

Remember that my troubles with this problem all began when I misinterpreted the diagram on April 15th and believed that the point labeled B is on Circle O.

Actually, Rapoport herself finally addressed the problem with her diagram. On her website, she often posts hints and corrects errors. Here is a link to her website:

http://rapoports.net/rebecca/2020calendar.php

Well, now you can finally see what this diagram looks like. Once again, Rapoport doesn't label the points -- I'm the one who came up with A, B, C, O, and so on.

So now we see that the point B (where the sides of length 6 and 8 meet) really isn't on the circle. But now we might ask, how far away is point B from the circle? You're right -- BC = 1, but I want to know the shortest distance from B to the circle, not the tangential distance. In other words, if Z is the point where OB intersects the circle, then what is BZ?

We can actually use Power of a Point to find this distance, now that we know r = sqrt(15). The power of B is the square of the tangential distance, so it's 1. This is equal to the product of BZ (the desired length) and the distance from B to the other side of the circle (that is, BZ + diameter):

BZ(BZ + 2r) = BC^2
BZ(BZ + 2sqrt(15)) = 1
BZ^2 + 2sqrt(15)BZ - 1 = 0

Using the Quadratic Formula, we find that BZ is about 0.127.

And now we see why her diagram is so difficult to interpret -- BZ is so small, especially as compared to the longest length in the circle, CP = 9. Thus the longest length is about 70 times the shortest.

Any scale that allows B and Z to be easily distinguished will lead to CP not fitting in the diagram. On the other hand, any scale that allows CP to fit on the diagram will make B and Z appear identical. No scale allows both BZ and CP to be sized appreciably and reasonably. Yes, Rapoport does so on the diagram on her website, but she can't do it in the tiny boxes on the calendar. I wonder whether this is the sort of diagram that simply shouldn't be drawn to scale.

I admit that I've complained about diagrams not being to scale earlier on the Rapoport calendar. I was especially upset when I saw an apparently acute angle labeled 105 degrees, or an apparently obtuse angle labeled 70 degrees.

But, as I now find out, there are some problems where a scale diagram make a problem worse, not better, than a non-scale diagram. If the April 15th diagram hadn't been to scale, it could distinguished B from Z without making CP too long to fit. (On such a diagram, ironically, Angle BQP might have appeared acute, even though we found its measure to be about 105 degrees.)

If it weren't for the virus and schools were open, it's unlikely that I'd still be discussing the problem on the blog three days after its scheduled date. Indeed, if I'd had a full day of subbing on April 15th, I might have skipped the problem altogether or simply blogged "The diagram is confusing or wrong," especially if something interesting happened in the classroom that day (or it was a math class). It's only because schools are closed that I can spend so much time on a single problem.

Cosmos Episode 12: "Coming of Age in the Anthropocene"


Here is a summary of Cosmos Episode 12, "Coming of Age in the Anthropocene"

• A baby -- one of 360,000 born today -- has a bright future ahead of her, all because of science.
• Earth's history was chaotic -- an asteroid hitting our planet became our moon.
• Small cyanobacteria in the ocean converted the CO2 into oxygen/ozone, making the sky blue.
• Homo erectus didn't arrive until after 11 PM New Year's Eve on the Cosmic Calendar.
• By 11:58:30 (40,000 years ago), our Neanderthal and Denisovan cousins became extinct.
• Our epoch is called the Anthropocene, or "era of man," due to our influence on the earth.
• Some believe the Anthropocene began 10,000 years ago, when we first made a species extinct.
• In China about 4,000 years ago, coal was discovered, and a millennium later, rice was planted.
• Perhaps the Anthropocene began during the age of nuclear weapons, around Tyson's birth year.
• The stories that always endure are myths, such as Homer's Iliad and the Trojan War.
• For Cassandra, knowing the future was both a blessing and a curse.
• Tyson tells the baby another story -- a time without refrigerators, when we had to use iceboxes.
• The first refrigerators contained chemicals called chlorofluorocarbons to keep food cool.
• But two UC Irvine scientists, Molina and Rowland, found that Freon destroys the ozone layer.
• Due to a global outcry, CFC's are banned, and the layer will return by the baby's 50th birthday.
• A Japanese scientist, Suki Manabe, once asked, what is keeping the earth's climate constant?
• Suki was invited to immigrate to the US, where he continued to study climate and its changes.
• In 1967, he accurately predicted the rise in temperature now known as global warming, and it will destroy the baby's future unless we do something to stop it.

Obviously, if I'd tried to teach the content of this episode at the old charter school, much of it would have been covered as part of the Green Team project that should have come at the end of the year, in the days leading up to (April) Earth Day.

Indeed, I recall that "Human Interactions" (with the environment, that is) occurred at the end of all three middle school grades on the Study Island website. (Unfortunately, that's one of the few things I remember from that website). It means that all grades under the NGSS get some of this content. Even though the grandfathered classes (Grades 7-8 my first year, eighth grade only my second year) don't need this material, they'd get it anyway if Green Team was successful. (If Green Team would have disappeared in my second year, then that class of eighth graders would be the only class to go a full year without environmental science.)

I'm completely aware that (anthropogenic) "global warming" or "climate change" is a mainstream, yet controversial, part of science. I don't wish to get into politics here. Of course, Neil DeGrasse Tyson makes it clear in this episode which side of the issue he's on.

As a science teacher, I was to teach to the standards regardless of the political implications. Here is the main middle school standard that refers to global warming (both anthropogenic and natural):

MS-ESS3-5. Ask questions to clarify evidence of the factors that have caused the rise in global temperatures over the past century. [Clarification Statement: Examples of factors include human activities (such as fossil fuel combustion, cement production, and agricultural activity) and natural processes (such as changes in incoming solar radiation or volcanic activity). Examples of evidence can include tables, graphs, and maps of global and regional temperatures, atmospheric levels of gases such as carbon dioxide and methane, and the rates of human activities. Emphasis is on the major role that human activities play in causing the rise in global temperatures.]

This would have been considered a sixth grade Earth Science standard under the old model, but is spread out over all three years under the Preferred Integrated model.

Conclusion

Our main traditionalist, Barry Garelick, blogged today. This post is mostly about a podcast, and there's only one commenter so far. I might make it a traditionalists' post next time, especially if more commenters show up at the Garelick blog.

Meanwhile, I appreciated yesterday's Google Doodle for teachers, even if I am only just a sub.