Go Formative: Unit 4 Test (Day 78)

Yesterday was my non-blogging day. My school followed Friday's block-hybrid schedule, and so I followed Friday's posting schedule -- I tweeted instead of blogged.

Today is an ordinary Tuesday. And so I'm following Tuesday's posting schedule -- I write on the blog about the seventh graders.

Originally, Math 7 was supposed to work on APEX Unit 5 this week. Unit 5 isn't really a unit -- it's more like a semester final. But earlier this month, the department chair decided to cancel Unit 5. She figures that there's no point in giving a semester final, since most of the students will fail it. Instead, she decided to expand Unit 4 an extra week, as the students definitely need it.

But even with the extra week, the students still struggle with this unit. No teacher at our school is enchanted with the way that equations are taught in APEX. While the Math 8 teachers gave the APEX Unit 4 Test anyway, the department head decided to write her own test for Math 7. Just like Math 8, the Practice Test is in Go Formative, but now the actual Unit 4 Test is in Go Formative as well. It tests equations much better than APEX. Still, the students are struggling. The test has 18 questions, and while a few students have succeeded, most students have single-digit scores.

Part of the blame goes to the Benchmark Tests. Once again, the schedule has us continuing to teach lessons after the Benchmarks are given each day. There are certain types of equations that the students are just barely seeing before the test (for example, equations with fractional coefficients) and there's no opportunity to practice them. APEX certainly isn't providing them with this practice.

In the end, the department head decided that the Unit 4 Test won't count. We're still giving the exams this week, but their scores won't be included in the second quarter grade. After second period yesterday, I was favoring letting the test scores count anyway. The top students did well on the test and will get at least an A- in the class, while for the weaker students, the test isn't hurting them and they're getting at least a C-. Part of this was because of multiple APEX retakes -- students couldn't advance unless they earned 4/5 on their quizzes. This boosted their grades enough so that this non-APEX test doesn't hurt.

But after third period today, I agree with the chair that they shouldn't count. Many students today simply did no work, period -- not even the Practice Test for which I was telling all the answers. Yes, yesterday's class worked harder even though they had more special ed students. But much of the blame here lays at the feet of the online students, including kids who are normally in person but stayed home today -- all but one of the students doing no work today was at home.

Instead, the last assignment that will count is the Solving Inequality activity (Google slides). I have the students work on this during tutorial. But this means that the students are spending 35 minutes on an assignment that counts and then 100 minutes on assignments that don't count. Struggling students who need to improve their grades might wish to get help on the activity, but instead I must go over the Practice Test -- they get confused, shut down quickly, and work on nothing.

Meanwhile, after long last, the missing Cosmos episode has finally aired. I was obviously hoping that this episode would air on a night when there was no school. But it originally came on National Geographic in March just days before the schools closed for the pandemic, and then it came on FOX last night just days before the schools close for winter break.

I'm adding the "Neil DeGrasse Tyson" label and summarizing the episode, as follows. Here is a summary of Cosmos Episode 2: "The Fleeting Grace of the Habitable Zone":

• In our galaxy there may be spaceships from other worlds, learning more about our planet.
• Venus was once in our solar system's habitable zone -- now our planet is in the habitable zone.
• Earth can remain habitable only if humans can stem the tide of global warming.
• The sun, now a yellow dwarf, will someday become a red giant, cooking our planet.
• At this point, the habitable zone will turn towards Mars, Jupiter, and the Jovian moons.
• Eventually we'll have to search for new worlds, which we can -- because we've done it before.
• The Voyagers of prehistoric China ventured from their homeland, heading first for Taiwan.
• For some reason, they left Taiwan and explored the mysterious unknown of the ocean.
• They travelled the Philippines, Indonesia, and ultimately Polynesia via nayar, or sail.
• Likewise, we will someday cross our interstellar ocean to new stars and planets.
• The sun can act as a gravitational lens to make a sort of telescope to magnify 100 billion times..
• We already have the technology to build it, and so there's no reason to start building it now.
• The nearest star, Proxima Centauri is four light years away -- 70,000 years at Voyager's speed.
• Instead, we will need a new sort of spacecraft that can be pushed by photons to near-light speeds.
• With a Alcubierre drive, we can exceed light speed and settle on an planet 100 light years away.
• A young Carl Sagan imagined a voyage to far-off planets and stars.
• To him, a trip to Proxima Centauri would change our species for the better.
• Beings are quarantined from one another, and traversing the stars lifts this quarantine.

Quarantine -- that's the last thing we want to hear about these days. But yes, we are quarantined on our planet until we gain ways to travel to other planets.

Tyson mentions Proxima Centauri b -- the habitable planet to nearest to ours. Chapter 13 of Ian Stewart's Calculating the Cosmos -- the chapter we were about to read for side-along reading until my summer abruptly ended and I took this long-term job -- is also about habitable planets, although Proxima Centauri b hadn't been discovered at the time Stewart wrote his book. (I'd already started typing the post at the time I found out about subbing. I might finally post that draft soon, especially now that I finally watched this missing episode.)

And with that, I finally finish my springtime project of summarizing Cosmos episodes, nine months after I began. It remains to be seen whether there will be a fourth season of this show.

For music break today I choose "Rudolph the Statistician," one of the Christmas parodies that I found on a website, but never posted on the blog until now:

https://musingsofamathteacher.wordpress.com/2010/12/20/funny-christmas-math-songs/

RUDOLPH THE STATISTICIAN
(Tune: “Rudolph the red-nosed reindeer”)

Rudolph, the statistician
Sampled data everywhere.
Mean heights and heart conditions
Calculated with great care.

All of his population
Gathered from coast to coast
Sought out his information;
He would always give the most.

Then one foggy Christmas eve
Santa came to say,
“Rudolph, with your pencil write,
What gifts I should take tonight.”

Quickly he polled the children,
Then Rudolph shouted out with glee:
“Stats did provide the answer:
Give Nintendo and Barbie!”

I was considering doing Vi Hart's "Twelve Days of Christmath" -- or at least my version of it with "Gimme a high five!" (for the "Five golden rings!" line). But we can't give high fives during the pandemic -- and besides, a few eighth graders started singing Rudolph in the middle of class, so I humored them by singing this song.

Both "Rudolph the Statistician" for Christmas and "Ghost of a Chance" for Halloween refer to the last strand of the Common Core -- statistics and probability -- that the students will get with the regular teacher at the end of the year. I need to come up with some holiday songs that refer to units that are actually taught in October and December, instead of SP.

By the way, I never did formally assign the "Whodunnit" assignment on expressions, despite singing a song about it. (Then again, I'm not supposed to determine what assignments to give the students based on what songs I want to sing, as tempting as it may be.)

Now that we completed Unit 4, and since we won't do Unit 5, we should be ready to start Unit 6 after winter break -- the geometry strand at long last. But now the department head is considering doing one more mini-test for Unit 4 -- one that counts -- after winter break.

Recall that this long-term position lasts for only one more week in January, and then the regular teacher is scheduled to return. It was already planned for Math 8 to continue in Unit 4 through January, and so my only ticket to G, my favorite strand, was through Math 7. I was already considering blogging only Math 7 for that week in January (rather than both Math 7 and Math 8) in order for me to cover more of the geometry strand on this Geometry blog.

The chair says that she will discuss it with me and make the final decision on Friday (the day that there are no students, hence this week's schedule change). What I fear is that she'll say, "We'll prepare the kids for the mini-test on the Monday we return, give them the mini-test on their block day, and then we'll start geometry the second week." Yes, it's convenient to start a new unit with a new teacher (just as he himself left me to start with Unit 2) -- but I really want to teach our favorite strand, geometry, even if just for one week.

Then again, she's wondering what geometry will look like in a pandemic situation -- anything that involves shapes and graphs will be a challenge to teach when so many students are online.

Lesson 2-6: Unions and Intersections of Figures (Day 26)

Before we begin today's lesson, let's continue to take a peek at that seventh grade class I subbed for last week -- the one whose Google Classroom notifications I'm still getting in my email.

Well, today another IXL lesson was assigned. The due date isn't until next week, so it appears that this teacher is already looking ahead to next week. I notice that while one class only has to reach a maximum score of 80, the other class has to reach a score of 90 -- the honors class. It takes a minimum of nineteen questions to reach a score of 90 if all of them are correct. As we already know about IXL scores, getting from 80 to 90 is harder than going from 70 to 80, yet easier than getting from 90 to 100.

A new lesson for next week has also been posted -- one-step equations, multiplication and division. As we may recall, in the year at the old charter school, I didn't reach one-step equations until February. I sang the song "Solving Equations," and the important verse for one-step equations is, "Whatever we do to the left side, the same done to the right side. A letter alone on the left side, a number alone on the right side. That's all you have to do to solve it!"

Meanwhile, let's finally get to that first episode of Neil DeGrasse's science show -- the one I missed six months ago on National Geographic but didn't watch until last night on FOX.

Episode 1 of Cosmos: Possible Worlds is called "Ladder to the Stars." Here is a summary:

• Humans were originally hunters and gatherers, until we became voyagers to new worlds.
• Our "ship of the imagination" can take us anywhere through space and time.
• NASA's Voyager 1, the most distant object in space, is currently traveling in interstellar space.
• Our first destination is a pair of black holes that will cause a huge collision that will slow time.
• We can watch Cosmos through our devices because we can manipulate electromagnetic waves.
• If we could manipulate gravitational waves, we'd understand the start and future of the universe.
• On the Cosmic Calendar, life began (on September 15th) four billion years ago.
• DNA -- the ladder of the double-helix -- evolved from materials formed in the middle of stars.
• On the Cosmic Calendar, mammals (on December 26th) developed a neonatal cortex in the brain.
• About 9,000 years (20 seconds) ago, the first cities were built, yet society was still egalitarian.
• The first maps were devised, and humans were able to locate "home" on their maps.
• In the 17th century, Enlightenment philosophers began to study the world around them.
• Dutch-Jewish philosopher Baruch Spinosa wrote The Book of Nature -- to him, God is nature.
• There are two kingdoms in the world -- those who build the world and those who destroy it.
• Plants and insects developed a symbiotic relationship that lasts for millions of years.
• Every third bite of food we eat ultimately goes back to insect-pollinated plants.
• It's up to us to prevent the sixth great extinction event, the Anthropocene.
• A few decades from now, a fleet of starships will depart from the driest Atacama desert.
• Their destination is Alpha Centauri -- our first exploration of a world outside our solar system.

As the first episode of the season, this is in some ways a summary of the season to come. We've already watched episodes about the Cosmic Calendar, and the Anthropocene extinction event last spring, and traveling to Alpha Centauri is the topic of the second episode -- which, unfortunately, was not the second episode that aired on FOX last night.

Instead, for some reason FOX aired Episode 11 last night, "Shadows of Forgotten Ancestors." I already described this episode on the night it aired on National Geographic, in my April 13th post.

This is one reason why I like to watch episodes when they first air, rather than be on the mercy of networks who don't rerun their shows properly. I didn't watch the first two episodes in March because the schools were still open that week (and Cosmos was something I decided to watch during the lonely weeks of the coronavirus shutdown), and now there's still an elusive Episode 2 that I haven't seen.

And who knows when Episode 2 will finally air on FOX? It certainly won't be this upcoming Tuesday, because that's when the first of the Presidential Debates will air (and there was a commercial for those debates during Episode 1) -- oh well!

Lesson 2-6 of the U of Chicago text is called "Unions and Intersections of Figures." (It appears as Lesson 2-5 in the modern edition of the text.)

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

Lesson 2-6 of the U of Chicago text focuses on unions and intersections. This is, of course, the domain of set theory.

In many ways, set theory is the basis of modern mathematics, and so many textbooks -- including higher math such as Precalculus and beyond -- mention set theory early on. Of course, the focus in this text is on unions and intersections of geometric figures. In particular, unions are used to define both polygon and angle, while intersections are used to define parallel lines.

The first three examples in the text, where the underlying sets contain natural numbers, real numbers, and points -- are OK. But I didn't like the fourth example, on airlines. I've decided to throw this one out -- if we want a non-mathematical example, why not just use sets of letters, such as {a, e, i, o, u}, the set of vowels?

One of the most important sets in mathematics is the null set, or empty set. According to the text, this set can be written as either { } or an O with a line through it (often called O-slash by students). Once again, since I can't represent that symbol on Blogger, let's use the strikethrough instead: O.

Now the text mentions that the intersection of two sets might be the empty set. But it doesn't mention what happens when one finds the union, or intersection, of the empty set and another set. As it turns out, the union of the empty set and another set is that other set -- so the empty set acts as the identity element for union, just as 0 is the identity for addition and 1 is the multiplicative identity. But the intersection of the empty set and another set is the empty set -- so the empty set acts as the absorbing element for intersection, just as 0 is the absorbing element for multiplication.

One question students often ask is, if { } is the empty set and O is the empty set, what's {O}? When I was young, I once heard a teacher point out that this is not the empty set because it's no longer empty -- it contains an element.

50th Anniversary of the Second Earth Day

Table of Contents

1. Introduction

2. A Blog Omission
3. What I Still Remember About That Science Class
4. Reblogging About the Green Team
5. What Actually Happened with the Green Team
6. An Interesting Rapoport Problem
7. Cosmos Episode 13: "Seven Wonders of the New World"
8. Earth Day and the Coronavirus
9. Revisiting the Fourth Dimension Yet Again
10. Conclusion

Introduction

Today is April 21st. Lately, I've been naming these posts after the special day or holiday on which I post them. But today falls right in between several holidays as observed on different calendars:

• Yesterday was Patriots' Day. It is celebrated mostly in New England -- especially Boston, with two special sporting events: the Boston Marathon and the 11AM Red Sox game. (Of course, due to the coronavirus, neither of those events are taking place today.) Because of this, spring break in this area is tied to Patriots' Day, not Easter. (Last year, Patriots' Day fell during Holy Week, but this is rare.)
• On the Hebrew Calendar, today is Holocaust Remembrance Day, although technically the holiday ended earlier this evening, at sunset. Through Cosmos, I've recently blogged about some Jewish scientists who were victimized by the Holocaust.
• On the Islamic Calendar, Ramadan starts later this week. The new moon is on Wednesday, but like all Muslim months, Ramadan will start with the sighting of the crescent. This might happen tomorrow evening in some time zones, but it's more likely to be Thursday night.
• And tomorrow is Earth Day. It will be the 50th anniversary of the second Earth Day, since the anniversary of the first Earth Day was a month ago, on March 21st.

Three years ago today, my old charter school held the culmination of the Green Team project. The reason it was on Earth Day Eve and not Earth Day itself was that the 22nd fell on a Saturday.

On the first Earth Day, I posted mainly about science, and I'll do the same in today's post. I think I'll begin with the story of a time I subbed in a science classroom, but never blogged about it until today.

A Blog Omission

It was four years ago this week when I interviewed for the position at my old charter school, and I was hired there by the end of April. Subbing, meanwhile, continued throughout May and the first few days in June.

Interestingly enough, during that last month of subbing, I had two multi-day assignments. One was during the second week in May, which I spent in a high school math class. I described it extensively on the blog. But the other long assignment, in a middle school science class during the week leading up to Memorial Day, has not been blogged.

Why didn't I mention this class on the blog? Well, that's an easy one -- at the time, I blogged about math classes and no other subbing, not even multi-day assignments. This was, after all, a math blog, and I was just hired as a math teacher. So why should I have bothered with that science class that had nothing to do with the math that I'd be teaching soon?

By now, you already know what ended up happening. At my new position at the charter school, I'd be expected to teach science in addition to math. And ironically, this middle school science class had more in common with what I'd need to teach three months later than that high school Geometry and Algebra II class.

Part of this is my own fault. I didn't need to be so stuck up about not blogging non-math classes. Even if I was subbing for a class of which there was no chance in the world that I'd be teaching soon (such as Spanish, considering that I don't speak it), blogging about it would have allowed me to reflect on other things, such as classroom management (which matters no matter what subject I'm teaching).

The other part falls on my new school. At the time I was hired as a math teacher, the thought was that a separate teacher would cover science -- indeed, such a teacher was there for the current 2015-16 school year. I suspect what happened was that she decided at the last minute not to return for the following 2016-17 school year, and no science teacher was found to replace her.

I have no idea when the administrators realized that I would need to cover science. I signed my contract in mid-June, and there was still no mention of any science class. Still, I wonder how things might have been different if I'd been informed of the possibility that I might have to teach science at the time I was hired.

For starters, I would have mentioned the full week of subbing science on the blog, as well as any other science classes I subbed for that May. Recall that there was a blogging challenge that year to post 30 times in May, so these science classes might have been something to blog about. It's doubtful that I would have described the science classes in full "A Day in the Life" format (since that wasn't part of the May blogging challenge), but at least I would have posted more details.

I also would have paid more attention to the science classes themselves. I could have looked at the walls for examples of science labs and other assignments, as well as check out the science texts to see what I'd be expected to teach (although the impending transition to NGSS standards might have affected any info I would've gleaned from the texts).

And I could have asked other science teachers for pointers. As I'd soon find out at the new school, the only other math/science teacher was at our sister charter (and she didn't teach eighth graders), so using other teachers as a resource would be difficult. On the other hand, while subbing for science, there were most likely two other science teachers right next door to me. And I'd be justified in asking them for help, especially at the school where I spent a full week teaching science (and even more so if I told them that I was a newly hired math teacher who might have science thrust upon me).

The idea, of course, is that if I could have done these things in May, I would have had a much better experience teaching science at the charter school in August. And perhaps with that better experience, I wouldn't have needed to leave the school when I did.

But of course, there was nothing I could do when I didn't know I'd be teaching science. As far as I know, during the week of May 23rd (when I was subbing science), the administrators at my new school hadn't the slightest inkling that their current science teacher wouldn't return the following year.

This is the reason why, ever since I left the old charter, I made a point to blog more about subbing in classes other than math, particularly middle school science classes, as well as days when I had to make key classroom management decisions. I want to keep a record of any ideas that might unexpectedly come in handy if I ever return to teach my own class someday.

What I Still Remember About That Science Class

Unfortunately, it's been four years since I subbed in that class. And I've subbed in hundreds of classrooms since that week. Thus I remember very little about the class. I don't even recall whether it was a sixth, seventh, or eighth grade class -- or perhaps different grade levels each period.

I do remember that some sort of written assignment (perhaps a lab) was due one day. The regular teacher told me not to collect any late assignments -- and I let the students know this. Still, one student tried to turn his assignment in late. I'd told him that if he gave he a late assignment, I'd just rip it up -- and I did. No, I didn't rip it up completely, but I left a huge half-tear in the packet, with a note to the teacher that I, the sub, intentionally ripped it because it was late. (It was up to the teacher to decide what to do about the late work.)

The main thing I do remember about this class relates to classroom management. There was an incident where several students appeared to bully a certain girl in one of the classes. I do recall that the girl was an immigrant (or the daughter of immigrants) -- but not from Mexico. I remember looking up the history of her last name and concluded that she was Ghanaian.

Often when bullying occurs, I can't tell who started it. It could be that the group was harassing the girl (perhaps because of her nationality). But it was also possible that the girl started it -- she started saying bad things about the others, and they became upset with her.

What did I do in that situation? Most likely, I said something like, "I don't care who started it, but I want all of you to stop it!" There were clear times when it was obvious that the girl was the victim -- for example, one day we watched some video about animals (which is why I suspect this was a seventh grade Life Sciences course), and one of the others remarked that the girl looks like one of the creatures from the video. But I also recall the girl insulting the others as well. The one thing I knew at the time was that I was three months away from starting at the charter school -- and the middle school kids I'd be seeing soon were likely to bully others, just as I was seeing right then.

What should I have done in that situation? As a sub, when it comes to classroom management, there are two types of conflicts that might occur -- student vs. teacher, and student vs. student. In the first situation, student vs. teacher, I'm directed in the lesson plans to give or collect some assignment or enforce some unpopular rule, and the students tell me that I'm wrong to collect the work or enforce the rule. This often led to arguments (and, unfortunately, yelling). I've found that one way to avoid arguments here is to be more subtle when enforcing the rules (that is, "pick my battles").

But if it's student vs. student, I should not be subtle when it comes to handling the situation -- I must be more emphatic. In this case, the kids want me to intervene, as opposed to the student vs. teacher situation when they want me to let them do whatever they want. And such intervention is even more necessary if it's student vs. students -- we have a single student on one side of the conflict and a group on the other, and I can't tell whether the single student is a jerk or a victim.

With that in mind, here's what I should have done: I begin by taking the side of the lone girl -- I treat her as a victim and tell the other students to leave her alone. But I warn her that I'll be monitoring the situation closely -- if at any point it appears that she's the aggressor rather than the victim, then I'll quickly side with the group against her.

One thing that might have helped -- especially considering that I'd see these kids all week -- is to speak with the girl in private. Unfortunately, I don't recall what the bell schedule was, with so much time having elapsed since the incident. Depending on the regular teacher's schedule, I might ask to see the student after class. But if this was, say, the last period of the day, she'd have wanted to go home and considered my keeping her after class to be a punishment (and hence refused to stay).

The best possible case was if her period was followed immediately by my conference period. Then she wouldn't have seen my keeping her as a punishment. (I could easily send her to her next class with a note that she was "helping out the sub.") During this time I have a heart-to-heart conversation by asking her, "Why do you think the other students bully (or don't like) you?" Then I can reassure her that I'll make every effort to protect her as long as she doesn't become the aggressor in return.

(Actually, I suspect that this week might have been part of SBAC testing -- the SBAC is often given in March or April in high schools but not until May in middle schools. Even though this was the year before the first California Science Test, the state math and ELA tests would have necessitated a block schedule that week. With a block schedule, her class was much more likely to be followed by a break than a conference period, so I wouldn't have been able to keep her.)

Assuming that this girl was indeed a seventh grader, she would have been a few months away from starting the eighth grade. There would be a few eighth grade girls at my new charter school who were often bullied, including one victim who then turned around and bullied a sixth grader herself. If I had written down (on the blog) the earlier bullying incident and learned how to handle it, perhaps I could have fared better with the bullies and victims in my new school. Maybe then the victim at my new school wouldn't have become a bully the next day -- and with better management and bully-handling skills, maybe I wouldn't have needed to leave the old school, even if I never completely figure out how to teach science properly.

Reblogging About the Green Team

Each time I make a science post, I keep referring to "Green Team" over and over again. So now you might ask, what exactly does this "Green Team" refer to?

Obviously, it refers to something that was supposed to happen at the old charter school. Let me go back and reblog parts of the November 2016 post where I first learned about the Green Team:

So what exactly is this Green Team, anyway? Here is a quote from the brochure:

"Let's Move Nation & Beyond's" Green Team Program is designed to engage students and their families; school faculty, administration and leadership; and the broader community on the importance of energy conservation topics. It will focus on campus-wide awareness, direct education to children, and Green Team leader development."

As I wrote earlier, I met with the leader of the Green Team this afternoon. She tells me a little more about how this program will work:

-- There will be a monthly newsletter for all students K-8 informing them about ways they can help their families save energy and water. We're hoping that the first newsletter will be December 1st.

-- The next step is to introduce this unit in science with a Pre-Assessment. This Pre-Assessment should be given just before winter break.

-- In January, the science unit will begin in earnest. Students can bring in copies of their energy and water bills. By the end of the unit, we're hoping that students have implemented enough saving tips to see a noticeable reduction in their bills. The grade with the largest reduction can win prizes, which could be anything from a classroom pizza party to even some sort of field trip.

-- There will also be a science project or fair associated with this unit. If everything goes right, the projects will be completed by Earth Day, which is Saturday, April 22nd.

In order for this to work out, as the Green Team leader explains, there should be at least ten schools participating in the program. So far, five schools have signed on, counting my own. All of these schools are either charters, like my own, or small private schools. Unfortunately, our sister charter can't be included, because it's not actually within the Los Angeles city limits and thus isn't served by the LA Department of Water and Power (one of the sponsors of the program).

Originally, the Green Team program was intended for Grades 5-7. But it's awkward to include sixth and seventh grade at a middle school but not eighth, so now it's extended to eighth grade as well.

On the other hand, fifth grade usually isn't included at most middle schools. But we're a K-8 school, and so we can include fifth grade all we want. I've spoken with our fifth grade teacher on our PD day, and she's enthusiastic about the program as well. After all, even the fifth grade class has only the Illinois State text and thus the science is limited to only the STEM projects. Both of us know that both fifth and eighth grades have NGSS tests in May, so let's include both grades as this could be the only science they'll see. I might even drop by our school on Wednesday and let the fifth grade teacher know what I learned today -- since, if you recall, she'll be at the school that day to decorate!

As I said earlier, I definitely look forward to working with the Green Team in my room. I hope that it will help my students out with science -- and even save them a little money.

What Actually Happened with the Green Team

Obviously, the Green Team didn't go quite as planned. As I wrote above here, the success of the Green Team was dependent on other schools signing up for it -- and most likely, not enough schools signed up for it quickly.

The so-called "Pre-Assessment" was indeed just before winter break, but it wasn't formal -- it was just about four short questions about the students' habits (turning off lights and water, and so on).

The creator of the Green Team did visit us in January, but there was no science unit. Instead, she had the students fill out some information on the website, and two students would be randomly chosen to get T-shirts and have their pictures printed in the aforementioned newsletter. The winners turned out to be a sixth grade boy and a seventh grade girl.

And as for the science project/fair, that might have been close to starting right around the time that I left the school. I still had access to email, and saw that there would be a class visit to the LA County Arboretum on a certain Saturday in March -- a place where I once took a field trip as a young sixth grader as well.

(By the way, the Arboretum has made the news recently due to the coronavirus. It is one of the few places that has remained open during the outbreak. Meanwhile, Santa Anita Park for horse racing has been shut down, despite being in the same city -- Arcadia -- as the Arboretum.)

Thus it's possible that the Green Team science project proceeded as intended. But unfortunately, most of the action took place after I left the school.

The Green Team led me to make some questionable decisions regarding my teaching. First of all, November, when I first learned about the Green Team, was right around the time that I finally got printed teachers' editions of the Illinois State science texts (for Earth and Physical Sciences, though not for Life Science). Yet I didn't try then to start using these texts, because I thought that the Green Team would take care of my science curriculum.

Then in December, I submitted a lesson plan that reserved an entire day for the "Pre-Assessment" for the Green Team, namely the Tuesday (and penultimate day) before winter break. And this would lead to three problems. First, since the "Pre-Assessment" was just four questions, there was no real science pretest or lesson that day. Since I'd already told the students that there would be science that day, I quickly made up some science questions from the Study Island Unit 1 ("Pretest"). The students grew upset since it seemed that I was always giving them science "pretests" and without ever teaching any actual science.

The second problem was that one eighth grade girl had missed the previous week of school and tried to make up the math she'd missed. I told her not to worry about it -- there'd be no time to make it up due to the science pretest. And of course, that fizzled out, leaving her well-behind with her math (and her math grade).

And finally, it ruined my assignment for Wednesday, the last day before winter break. Because there was no real assignment on Tuesday due to the pretest fiasco, it led students to believe that there would be no work the entire week, so kids were playing around on Wednesday. One seventh grade girl would get in trouble that day for playing around with her phone. (Then again, the lesson I tried to give on Wednesday was a bad one anyway. I'd been inspired by previous days of subbing to assign a holiday graphing worksheet just before breaks, but this makes more sense in high schools when students are more comfortable with graphing, not in middle schools.)

I already wrote a month ago what I should have done here -- I should have taught science and acted as if there was no Green Team. Plan a curriculum as described (Physical Science for eighth grade, Life Science for seventh, Preferred Integrated for sixth) and just ignore Green Team activities until they actually begin. Don't include the Green Team in lesson plans until I'm completely sure that I have material for Green Team lessons.

An Interesting Rapoport Problem

There are no Geometry problems on the Rapoport calendar this week. But these days, I've been blogging Rapoport problems anyway, mainly because it's the only math I've been doing lately. (When it's not a Geometry problem, I bury it deeper in my post, as opposed to Geometry problems that I include right at the top of the post.)

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

The number of subsets of {1, 2, ..., 6} that contain no consecutive numbers.

Before we begin, don't forget that sets (and thus subsets) can contain 0 or 1 element -- and of course, every such set fails to contain two numbers (much less any two consecutive numbers). Thus we already know seven subsets that work -- the empty set, {1}, {2}, {3}, {4}, {5}, and {6}.

It's easy to see that any subset with four or more elements won't work. And so all that remains are to check the subsets with 2-3 elements. There are only 6 choose 2 + 6 choose 3 = 15 + 20 = 35 such subsets, so it wouldn't take that long to check all of them.

But if we did that, we wouldn't learn as much as if we looked for patterns instead. Let's define:

f (n) = the number of subsets of {1, ..., n} that contain no consecutive numbers.

Then our goal now is to find f (6). There are two initial cases here:

f (0) = 1 (the empty set)
f (1) = 2 (the empty set and {1})

So f (2) is the first non-trivial case here. We know that f (2) is at least 2, since the two sets in f (1) work for f (2) as well. And likewise f (3) is at least as great as f (2), since any set in for f (2) works for f (3) as well. Since we're looking for patterns, let's write this as:

f (2) = f (1) +
f (3) = f (2) +
f (4) = f (3) +
f (5) = f (4) +
f (6) = f (5) +

Looking at f (2) again, we wish to include the new element 2. There are two sets from f (1) we have so far, empty and {1}. We can't adjoin 2 to the set {1} since the new set {1, 2} contains consecutive numbers, but we can adjoin it to the empty set to obtain {2}. We adjoined 2 to the empty set -- which, if you recall, is the only set in f (0). So we have:

f (2) = f (1) + f (0)
f (3) = f (2) +
f (4) = f (3) +
f (5) = f (4) +
f (6) = f (5) +

Looking at f (3) again, we wish to include the new element 3. There are three sets from f (1) we have so far, empty,{1}, and {2}. We can't adjoin 3 to the set {2} since the new set {2, 3} contains consecutive numbers, but we can adjoin it to the empty set to obtain {3} and to the set {1} to obtain the new set {1, 3}. We adjoined 2 to the empty set  and to {1} -- which, if you recall, are the only two sets in f (1). So we have:


f (2) = f (1) + f (0)
f (3) = f (2) + f (1)
f (4) = f (3) +
f (5) = f (4) +
f (6) = f (5) +

And by now the pattern is obvious. To find f (4), we start with the sets in f (3) and then adjoin 4 to each of the previous sets that don't contain a 3, which are exactly the sets in f (2). And we do the same with 5 and 6:

f (2) = f (1) + f (0)
f (3) = f (2) + f (1)
f (4) = f (3) + f (2)
f (5) = f (4) + f (3)
f (6) = f (5) + f (4)

That's right -- the values of f (n) are simply the Fibonacci numbers! And since f (0) is the second Fibonacci number and f (1) is the third Fibonacci number, f (6) must be the eighth Fibonacci number, which is 21. And of course, today's date is the 21st.


Cosmos Episode 13: "Seven Wonders of the New World"


Here is a summary of Cosmos Episode 13, "Seven Wonders of the New World":

• Carl Sagan, the first Cosmos host, was drawn to science at the 1939 World's Fair in New York.
• That World's Fair was where television was first introduced.
• A young Neil DeGrasse Tyson visited that same World's Fair a quarter century later.
• The fair presented a vision of the future filled with higher technology, such as computers.
• There were also models of spacecraft that could be used for both exploration and warfare.
• Now imagine what the New York World's Fair would look like in the year 2039
• A young girl named Nguyen (no, not Fawn) is seen contemplating about her future.
• A new colossal building, made from carbon dioxide, is at the entrance to the World's Fair.
• In the Pavilion of the Searchers, famous scientists are virtually brought to life.
• In the Pavilion of the Fourth Dimension (time), you can go anywhere on the Cosmic Calendar.
• "Saco," who lived half a billion years ago, was the last common ancestor of nearly all animals.
• Botanists discover a plant, otherwise green, that turns red if a landmine is nearby.
• They breed a new poplar tree that converts dangerous TCE gases into harmless chloride.
• In the Pavilion of the Lost Worlds, you can revisit civilizations that disappeared millennia ago.
• The Nok lived in ancient Nigeria, and the Indus Valley flourished, but little is known of them.
• In the Pavilion of Worlds Still to Come, you can see the depths of the solar system and beyond.
• It is like an Encyclopedia Galactica listing the distant worlds and their survival probability.
• Our survival probability is only 50% per 100 years, unless we embrace our position on the Tree of Life and learn to respect nature and the world around us.

These last two episodes of Cosmos are timed perfectly with Earth Day. In Episode 12, Tyson talked about the sixth mass extinction event -- the Anthropocene extinction -- and stressed that it's caused by humans ruining the environment. That episode was a warning that unless we change our habits now, we will lose thousands of species.

Episode 13, in contrast, gives a reason for optimism. The timeline imagined by the little girl Nguyen shows carbon dioxide levels finally returning to their pre-industrial levels, for example. Tyson expresses the hope that we humans will realize the error of our ways and ultimately change our habits in order to protect our environment. This optimism is, of course, the reason that Earth Day was created fifty years ago.

The Worlds' Fairs are used in this episode since the fairs in years past were where futuristic forms of technology were first introduced. Tyson mentions that he himself, as a young child, was inspired by the 1964 fair, as was the original host Carl Sagan at the 1939 fair. Thus the girl Nguyen represents a future scientist who is likewise drawn to save the environment by what she sees at the 2039 fair -- exactly one century after Sagan's fair.

It's possible that I could have taught some of these ideas at the old charter school. Once again, some of these ideas fit with the Green Team that was supposed to start in March three years ago. No, the Green Team Science Fair wouldn't have been anywhere near as extravagant as any of the Worlds' Fairs were or will be.

But our fair could have at least inspired our students to think about their ecological futures. Perhaps some of our students might go on to solve the major environmental problems of today.

Earth Day and the Coronavirus

A month ago, I mentioned that some tips for protecting our environment (throwing stuff away keeps our landfills from overflowing), are at odds with tips for protecting our health during the coronavirus (throwing stuff away gets rid of germs more easily). As I promised, I'm posting some links discussing what to do about protecting the environment during the virus outbreak:

https://www.theverge.com/2020/4/2/21204094/plastic-bag-ban-reusable-grocery-coronavirus-covid-19

With grocery stores being one of the few places still open during COVID-19 lockdowns, disposable plastic bags are making a comeback as some people fear that reusable bags could spread the disease. Before the pandemic, a growing number of governments banned single-use plastic bags in an effort to cut down on waste. But as the novel coronavirus has spread around the globe, people have gotten leery about coming in close contact with other people and their possessions, including reusable bags. On March 31st, New Hampshire became the first state in the US to temporarily ban reusable bags during the pandemic.

Like pretty much everything else right now, reusable bags should probably be handled more carefully to minimize the risk of transmitting disease to other people. At the same time, there has been no evidence so far that using reusable grocery bags have been responsible for spreading the novel coronavirus.

But protecting public health doesn’t have to be at odds with efforts to stem the flood of plastics filling landfills and collecting in the ocean, environmental advocates say. “If stores, particularly workers, want to keep themselves as safe as possible and limit the bags coming in because they don’t know if people wash them, certainly a temporary pause on that, I think that’s understandable,” Ivy Schlegel, a senior research specialist for Greenpeace USA, says. But that “pause” shouldn’t be permanent, says Schlegel. She has followed the plastics industry’s history of fighting environmental reforms by claiming that reusable bags are unsanitary. She sees the industry seizing the opportunity now to push its own agenda.

The following Scientific American link, published in time for Earth Day, discusses how air pollution makes us more susceptible to the virus:

https://blogs.scientificamerican.com/observations/air-pollution-covid-19-and-earth-day/

Fifty years ago, on April 22, 1970, millions took to the streets on the first Earth Day, demanding clean air and environmental protection. Massive citizen participation, some 10 percent of the U.S. population at the time, led to the Clean Air Act, and other landmark environmental protection laws.

Because COVID-19 attacks our lungs, air pollution makes us more vulnerable to the virus. It’s no surprise that COVID-19 deaths are high in places with poor air quality. Recent research from Harvard University documents this well. In fact, the study’s findings, the researchers noted, “underscore the importance of continuing to enforce existing air pollution regulations to protect human health both during and after the COVID-19 crisis.” So why are we doing the opposite?

On the other hand, the World Economic Forum notes that due to the coronavirus lockdowns, fewer people are traveling -- and this has a positive effect on our environment:

https://www.weforum.org/agenda/2020/04/coronavirus-lockdowns-air-pollution

The COVID-19 pandemic has caused industrial activity to shut down and cancelled flights and other journeys, slashing greenhouse gas emissions and air pollution around the world. If there is something positive to take from this terrible crisis, it could be that it’s offered a taste of the air we might breathe in a low-carbon future.

The World Health Organisation (WHO) estimates that about 3 million people die each year from ailments caused by air pollution, and that more than 80% of people living in urban areas are exposed to air quality levels that exceed safe limits. The situation is worse in low-income countries, where 98% of cities fail to meet WHO air quality standards.

Revisiting the Fourth Dimension Yet Again

Three year ago, on April 21st, 2017, the Green Team Fair should have occurred, but unfortunately, I'll never know whether there really was a fair or not. I'd already left my old school back then?

What did I blog that day? I wrote about the fourth dimension again, this time from the perspective of Eugenia Cheng (and referred back to Rudy Rucker as well). Since Neil DeGrasse Tyson briefly mentioned the fourth dimension in this last episode, let me reblog the fourth dimension from Cheng:

Chapter 12 of Eugenia Cheng's Beyond Infinity is "Infinite Dimensions." She begins:

"Would you like to be able to travel in time? It sounds exciting, but it also sounds terrifying because of the dire consequences if you even slight interfere with your own past."

In this chapter, Cheng writes about the concept of infinitely many dimensions. But this is tricky as it's difficult to imagine what it means for a world to have four dimensions, much less infinitely many.

Last year at around this time (between Round It Up to Pi Day and April Fool's Day 2016), I was blogging about Rudy Rucker's The Fourth Dimension: Toward a Geometry of Higher Reality. As the title implies, this book discusses the fourth dimension. Cheng writes about the fourth dimension as well, albeit en route to even higher dimensions.

By the way, last year I mentioned Rucker's other book -- Infinity and the Mind. That's right -- it's yet another book about infinity. We can see how Rucker and Cheng apparently both love infinity and higher dimensions. Clearly I share that love as well, since I keep blogging about their books.

Let's get back to Eugenia Cheng. Anyway, Cheng begins by describing several examples of one-, two-, and three-dimensional objects. A line is one-dimensional, but so is a circle, since someone living on a circle can only go one way or the other, just like on a line. (This is why we often refer to a circle as S^1.) Likewise, a plane and a sphere are both two-dimensional. In both cases we need a pair of coordinates to specify our location, but even on a plane, we can use circular (polar) coordinates. Of course, our ordinary space is three-dimensional.

One example of a possible fourth dimension is time. She writes:

"If you find this hard to understand, you can imagine time as a fourth dimension. This is a valid and in fact powerful way to think of time, put to great use in theoretical physics, but it is just one way of thinking of a fourth dimension, not the only way."

Rucker also mentions time as an example of a fourth dimension. Cheng also gives color as a possible fourth dimension, where you're only in the same place as something if you're the same color. Again, Rucker gives the same example in his book.

According to Cheng, dimensions should be independent. So when she gives her location as "4 miles east, 2 miles north, and sqrt(18) miles northeast," only two of those dimensions are independent. So she doesn't need to provide a northeast coordinate.

Cheng proceeds:

"As my research is in higher-dimensional category theory, I spend a lot of time talking to people about dimensions, both formally and informally."

I already knew that category theory is Cheng's specialty from her first book, How to Bake Pi. So of course she finds a way to discuss category theory in her new book as well. Indeed, the next chapter of her book, Chapter 13, is titled "Infinite-Dimensional Categories."

And just as in her first book, Cheng likes to give food analogies. She writes that in order to make French macaroons, she needs to decide how much icing sugar, caster sugar, and ground almonds to include in the recipe, so that's three "dimensions" to consider. The oven temperature is the fourth dimension, and the fifth through ninth dimensions are all temporal -- how long to whisk the whites, fold the batter, pipe the circles, let them stand, and finally bake them.

She continues:

"You might think that this is all very well, but is there ever a need to study those higher-dimensional spaces? Do we even need to know that they're there?"

Her answer is an emphatic "yes" -- we use seven dimensions every time we move our arms. Two of these are shoulder movement and two more are wrist movement. The remaining three dimensions are all angles -- the elbow angle, the angle the hand is facing, and the upper arm angle. She points out that as humans we don't think about all of these dimensions, but if we were to program a robotic arm, we must consider all of them.

But Cheng admits:

"Thinking about higher-dimensional space is difficult. That's why there are whole branches of math devoted to it."

Now Cheng provides several examples of improperly reducing dimensions. The dominant example is when filling out surveys that ask, "Do you prefer socializing in large groups or one-to-one?" as if one can't like both equally or hate both equally. She adds that for years, people thought of politics as being one-dimensional (left-wing or right-wing) when two dimensions represent them better. Thus she provides the following link to a two-dimensional political survey:

http://www.politicalcompass.org

She wraps up the chapter with the following:

"Sometimes when I'm doing a 'pros and cons' type evaluation of a situation, I discover that the criteria are too difficult to separate out. For example, you might be thinking about short-term benefits and long-term benefits."

In other words, there are too many variables to consider -- and sometimes we can't even tell how to divide the criteria into separate variables! And so this is a job for -- infinitely many dimensions. As Cheng points out, "No wonder decisions are difficult."

Conclusion

Even though this was Neil DeGrasse Tyson's last episode of Cosmos, we're not done with it yet. We missed the first two episodes, and so I'll recap those as soon as they air on FOX, which should be at some point this summer.

I wish that Tyson would give us a 14th episode on the one scientific issue that's on everyone's mind right now -- the coronavirus. But recall that these episodes were produced last year (and were delayed due to allegations against Tyson), so there's no way there could be a coronavirus episode. I did find a link to a YouTube podcast where Tyson discusses the virus and the environment, so this is the best we can do for now:

Speaking of podcasts, I never did blog about the traditionalists' podcast I mentioned in my last post, since I wanted this post to be about Earth Day and science. Unfortunately, I didn't avoid politics completely, since much of the environmental debate is political.

With Cosmos done for now, I'll be making plans for computer/coding in my next post.

Meanwhile, I notice that another math blogger, Vi Hart, made a cororavirus-related video:

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.