Table of Contents

1. Introduction & Civil Unrest
2. Calculating the Cosmos Chapter 1: Attraction at a Distance
3. Shapelore Learning 13-1: The Witlore of Making Findings
4. Shapelore Learning 13-2: Notsayings
5. Some Rapoport Math Problems
6. Lemay Chapter 6 Part 1: Creating Classes and Applications in Java
7. More on the Point Class in Java
8. Mocha Music: "The Dren Song," "Count on It," "Benchmark Tests," "Fraction Fever"
9. Reblogging: Summer School and Edgenuity
10. Conclusion

Introduction & Civil Unrest

This is my first true summer post. The last day of school, Day 180, in my old district would have been three days ago, on Wednesday. Thus today would have been the first post of summer vacation -- and because of this, I'm starting my summer projects in today's post. It's a bit strange to think that the span of time that has passed since my last day in the classroom is just about the same length as a normal summer vacation, so it almost feels as if I should be back in class in a matter of days. On the other hand, it doesn't feel like much of a summer at all, since nearly everything I'd do during a typical summer has also been shut down by the coronavirus.

But in my last post, I briefly mentioned a current event -- civil unrest in Minnesota. That day, I wrote that I didn't wish to discuss this on the blog, especially since it's a racially polarizing event. Usually, I only mention race on the blog in connection with education, and the Minnesota event isn't directly related to education.

But as I've said before, sometimes I don't truly understand how significant a current event is until it causes things in my daily life to be closed or cancelled. The Minnesota civil unrest didn't just stay in Minnesota -- it expanded all the way to essentially my backyard, Southern California. And it has resulted in nightly curfews, beginning last Saturday night for LA City and Sunday for LA County, with other counties here having similar curfews. Only then did I truly understand the seriousness of the Minnesota incident -- even when compared to similar protests.

And this has been a pattern. I didn't truly understand how serious the coronavirus was until first the sports leagues, and then the schools, shut down. I didn't truly understand how serious 9/11 was until first a UCLA librarian yelled at me for leaving a bag unattended, and then sports leagues shut down.

And I, as a young fifth grade student, didn't understand at first how serious the Rodney King riots in LA were -- the most comparable situation in my life to the present. On May 1st, 1992 (which I know was a Friday -- since 1992 and 2020 are exactly 28 years or one solar cycle apart, their calendars are identical), I heard from the news that the LAUSD had cancelled school that day. But I lived in a much smaller district, so its school schedule wouldn't make the TV news. And so that morning, I wondered whether there would be school that day.

Of course, if this had been 2020, it would have trivial just to go to the district website and find out whether school was open. But this was 1992 -- the only computer I had was the one the Mocha emulator is based on, and we wouldn't get our first computer with Internet for another five years. So finding out the school schedule wasn't easy back then.

In our school folder it said, "Office Opens: 8:00" -- and this was tricky, since I would normally start walking to school at 8 AM or a few minutes later. I had to get dressed, eat breakfast, and prepare just in case there was school. We waited those last few minutes -- 7:58, 7:59, 8:00 -- until the office was open so that we could call the school. But there was no answer. I don't remember whether it was a busy signal or simply that no one picked up. And so I had no choice but to walk to school.

But it was eerily quiet on the way there -- I didn't see very many other students walking the route to our elementary school. So when I was a few houses away from the school, I ran the last few steps -- and sure enough, the school was like a ghost town. Thus I turned around and walked back home.

Later on that day, a friend in my class informed us that his family had called the school and found out that it was cancelled. To this day, I'm not certain what happened -- either the school was swamped with parents and so we couldn't connect, or the administrators had been answering calls before 8:00 (thereby assuming that most parents would ignore "Office Opens: 8:00" in their folders) and had left to go home themselves by 8 AM.

Sports were cancelled that day as well, at least in LA County. Both the Lakers and Clippers had made the playoffs and were on the verge of being eliminated -- the Clippers game had been moved to nearby Anaheim, while the Lakers had to play in Las Vegas, Nevada. In hockey, the LA Kings had also qualified for the playoffs, but they'd been eliminated in Game 6 the night before the riots. And in baseball, the Dodgers games were postponed. These were made up with doubleheaders later on in the season -- a rarity here in Southern California as it almost never rains here during baseball season.

As a young 11-year-old, I was certainly aware that different races existed. Our school was definitely integrated in that members of all major races attended there. Three months earlier was Black History Month, and our homeroom teacher had given us some worksheets on famous individuals -- I say "homeroom" because the Path Plan was about to be implemented, and so we had more than one teacher during the day.

I don't recall whether there were any race-related violence at our school during the riots. But I do recall that there were some huge fights involving the sixth graders that led to major punishments. The usual Grade 5-6 recess was cancelled, to be replaced with a fifth grade recess and then each sixth grade class had its own separate recess. (I'd gotten so upset when the ringing bells didn't match the length of recess that I started to tear up my shirt.) The Path Plan was cancelled and we had to stay in homeroom all day, except when it was time for music. And indeed, the chorus teacher had to implement what he called the "deadly boy-girl" -- we were forced to alternate boy-girl-boy-girl as we sat in the auditorium. (I actually didn't mind that much, as both girls I had to sit by were friends.)

There was also a tragedy that year. During one of the sixth grade recesses -- some time before they were to graduate to middle school -- a boy passed away on the basketball court. (I believe that he had a preexisting condition, possibly asthma.) We'd been neighbors the previous year. Because of this, I remember more about this than the riots and their immediate aftermath.

Returning to the present, there's a very good chance that had it not been for the coronavirus, school would have been cancelled over the George Floyd protests. Just imagine this -- Friday afternoon would have been Day 177, and just before the final bell rings, we tell the students "Good luck on your finals next week." Then suddenly Days 178-180 are cancelled due to the civil unrest, and we never see the students again. It's one thing to know three months in advance (due to coronavirus) that finals and graduation are cancelled, but it's another to know it only three days in advance..

(And here in Los Angeles, Game 7 of the Western Conference Finals between the Lakers and Clippers might have been cancelled due to the unrest -- and they can't just move it to another city, because the protests are happening in all 50 states. An Angels-Yankees series would also have been postponed since there are also protests in Anaheim.)

It's also possible that had there been no virus, on Days 174-177, the students might have been too distracted by the Minnesota incident to concentrate on anything else. There could have been a discussion of the incident -- and some students might have even asked me (as a sub in their class) for my opinion. This is the sort of thing that I'm loath to write about on the blog -- and I'm not always comfortable talking about these issues in person with students either.

The last week of school before the coronavirus closure was unusual in that not only did I sub all five days that week, all in high schools, but that I had at least one class of seniors all five days. I recall that on the last day, two girls were starting to grow concerned about their prom and graduation. At that point, maybe they needed some, an adult, just to talk to and commiserate with them. And I definitely did feel sorry for them -- I know I'd be upset if there had been an outbreak in 1999, after seeing so many classes graduate ahead of me and wondering when it would finally be my year. But I didn't say anything to the girls when they were standing in front of me -- I just kept on singing my Pi Day Eve songs.

I'm just not that sort of "touchy-feely" (as in verbally expressing feelings) type of person. And if I had trouble talking about a politically neutral subject like a pandemic, how much more difficulty would I have had discussing the politically charged Minnesota unrest?

For now, let me link to teacher-bloggers who are more eloquent than I am when it comes to this sort of political topic. First, our queen, Fawn Nguyen, has spoken:

http://fawnnguyen.com/gardening-and-teaching/

To change a community, you have to change the composition of the soil…
We are the soil.

– Ron Finley, The Gangsta Gardener

I got this far on the post, then the next day, watching the video of George Floyd’s horrific suffering that resulted in his senseless death leaves me limp. This “Gardening and Teaching” post becomes stupid, my blog pointless. Zoom is unbearable, even with mic and camera off. A bowl of oatmeal in the morning is all I can manage to make and swallow until I repeat 24 hours later. I was afraid I couldn’t hold it together with the students yesterday because I’ve been weeping steadily like a garden hose left on slow trickle.

Nguyen had already planned and typed her post on gardening, then interrupted it to comment on George Floyd. She posted this the same day that I made my last post -- she found a way to mention Floyd in her post, but I didn't.

Sam Shah (of Everywhere Continuous, Nowhere Differentiable) has had much to say. His school had a virtual conversation about Floyd on Zoom:

https://samjshah.com/2020/06/03/its-tuesday-evening/

Meanwhile, Sara VanDerWerf lives in Minnesota, near Floyd's hometown. She did comment about the situation on Twitter, though not on her blog as of yet.

Another influential teacher who has commented on the unrest is Jose Vilson:

https://thejosevilson.com/a-justice-letter-to-educators-of-color-and-conscience/

I don't want this to become a traditionalists' post since I just had such a post last week. And so I'll end it right here. But once again, if schools had been open last week and some students asked me of my opinion of the Floyd case, what would I say? It's not simple to give the statistical argument that I mentioned above, or start talking about magical red buttons, to young students sitting in front of me (regardless of their race, or what side of the debate they're on). Students are scared when they hear stories like the Floyd incident, and what they need from me is someone to comfort them -- not someone to talk about cold-hard numbers or magical buttons (that don't even provide a complete answer to anything).

I'll never know what I would have said had the schools been open and there was no virus. But then again, some people believe that had there been no virus, there would have been no unrest. The argument is that many people are already on edge due to the virus -- with many people losing their lives and others their jobs -- and then on top of that uneasiness, we add the Floyd incident. If there had been no virus, the Floyd incident still might have occurred, and there still might be some protests, but they wouldn't have been so violent as to require a curfew or the National Guard to stop them.

If this is true, then it is more about feelings than about statistics or magical buttons. And that's all the more reason not to resort to stats when discussing the Floyd incident. But again, I'd much better at discussing numbers than discussing feelings.

This is the longest "introduction" section of a post that I've ever written. But I had to say what I had to say -- and arguably, I should have said this in my last post. If Fawn Nguyen found a way to mention Floyd in her gardening post, then I could have written about Floyd in my last post -- especially since it was a post about race anyway due to the traditionalists' debate and the movie I watched that week.

So let me finally get on with the summer projects that I wanted to begin in today's post.

Calculating the Cosmos Chapter 1: Attraction at a Distance

Our first summer project is in Ian Stewart's Calculating the Cosmos. It's a book I purchased over three years ago -- back when I was still teaching at the old charter school. It's a book about math, but also a book about science -- and I needed to think more about science since I was supposed to be teaching science.

Since then, I've bought and described other Stewart books on the blog, and I did discuss some chapters from Calculating the Cosmos here on the blog. So it's about time that I finally read this book, especially now since the local library is still closed. Thus Calculating the Cosmos is officially our side-along reading book for the summer. All of these posts will receive the "Stewart" label.

Today we'll read the first chapter, but it's proceeded by a lengthy prologue. So let's begin there with the prologue. This prologue, like all of Stewart's chapters, begins with a qutoe:

"Why, I have calculated it."
-- Isaac Newton's reply to Edmond Halley, when asked how he knew that an inverse square law of attraction implies that the orbit of a planet is an ellipse. Quoted in Herbert Westren Turnbull, The Great Mathematicians.

The prologue proper begins:

"On 12 November 2014 an intelligent alien observing the solar system would have witnessed a puzzling event. For months, a tiny machine had been following a comet along its path round the Sun -- passive, dormant."

Stewart here is describing the first landing of a space probe on a comet. He explains the reasons why we'd want to do such a thing:

"Comets contain material that goes back to the origin of the solar system, so they can provide useful clues about how our world came into being. Astronomers think that comets are dirty snowballs, ice covered in a thin layer of dust."

At first, it was believed that some of earth's water came from comets, but not any more:

"Asteroids are a better bet. The Rosetta mission is just one example of humanity's growing ability to send machines into space, either for scientific exploration or for everyday use."

Here the author provides a picture of the "rubber duck" comet 67P, imaged by Rosetta. He now explains why space exploration -- and studying math -- important:

"As far as everyday life is concerned, our ability to create accurate mathematical models of bodies interacting under gravity has given the world a range of technological wonders that rely on artificial satellites: satellite television, a highly efficient international telephone network, weather satellites, satellites watching the Sun for magnetic storms, satellites keeping watch on the environment and mapping the globe -- even car satnav, using the Global Positioning System."

Here Stewart provides a picture of Pluto -- on July 14th, 2015, NASA's New Horizons space probe sent this historic image to Earth, the first to show clear features on the dwarf planet. He continues:

"Twentieth-century rocketry pioneers pointed out that the first stage of a multistage rocket would be able to left the second stage and its fuel, while dropping off all of the excess weight of the now-exhausted first stage."

I wrote about the next part of the prologue on the blog nearly three years ago -- in August 2017, when the solar eclipse was occurring:

"Conversely, astronomical phenomena have influenced the development of mathematics for over three millennia, inspiring everything from Babylonian predictions of eclipses to calculus, chaos, and the curvature of spacetime."

So now we see who the first eclipse predictors were -- ancient Babylonian mathematicians.

"By thinking about the geometry of the solar system, astronomical pioneers realized that the Earth goes round the Sun, even though it looks the other way round from down here."

The author now describes the steady-state theory in which the ancients believed:

"But there was a widespread belief that nothing had changed, did change, or would change in any dramatic manner over countless eons."

But of course, they were wrong:

"Clockwork has given way to fireworks. From a cosmic viewpoint, the solar system is merely one insignificant bunch of rocks among quadrillions."

Three years ago on the blog, I wrote:

"Einstein saw his theories verified by two of his own predictions: known, but puzzling, changes to the orbit of Mercury, and the bending of light by the Sun, observed during a solar eclipse in 1919."

I remember watching this in an episode of Genius, the series whose first season was on Einstein. He actually tried to send a colleague to observe an eclipse five years earlier, in 1914. The path of totality passed through Russia, but World War I derailed Einstein's plans. The colleague was arrested by the Russians as a possible spy!

And of Einstein, Stewart continues:

"But he couldn't have realized that his theory would lead to the discovery of some of the most bizarre objects in the entire universe: black holes, so massive that even light can't escape their gravitational pull. He certainly failed to recognize one potential consequence of his theory, the Big Bang."

The author tells us that this idea is what his book is all about:

"Namely: there are mathematical patterns in the motions and structure of both celestial and terrestrial bodies, from the smallest dust particle to the universe as a whole."

And it all began with Isaac Newton. Stewart closes the prologue with a poem:

"Nature has laws.
They are mathematical.
We can find them.
We can use them.

"Of course, it wasn't that simple."

Chapter 1 of Ian Stewart's Calculating the Cosmos is called "Attraction at a Distance." As usual, it begins with a quote:

Macavity, Macavity, there's no one like Macavity,
He's broken every human law, he breaks the law of gravity.
-- Thomas Stearns Eliot, Old Possum's Book of Practical Cats

Chapter 1 proper begins:

"Why do things fall down? Some don't. Macavity, obviously. Along with the Sun, the Moon, and almost everything else 'up there' in the heavens."

This chapter is all about the force of gravity, as first formulated by Isaac Newton. He discovered that the force that causes his famous apple to fall to earth is the same force that keeps the moon in orbit:

"Taking their different masses into account, the forces turned out to be identical. This convinced him that the Earth must be pulling both apple and Moon towards it."

That force, of course, is gravity. But long before Newton, the ancient Greek philosopher Aristotle had has own idea of why objects fall down:

"Having reached its natural rest state, the body remains there, moving only when a force is applied. As theories go, these aren't so bad."

Three years ago, I blogged the following:

"People calculated aspects of the cosmos, such as eclipses, for millennia before anyone realized that gravity existed. But once gravity's role was revealed, our ability to calculate the cosmos became far more powerful."

And indeed, Aristotle's theories were proved to be wrong by Newton:

"The force of gravity, and the manner in which bodies respond to forces, lie at the heart of most cosmic calculations."

We return to the ancient astronomers and the patterns they observed in the sky:

"Why do some movements show patterns, while others break them? The Sumerians and Babylonians provided basic observational data."

But during the Dark Ages, European knowledge of astronomy was dominated by Aristotle's theories:

"The universe was believed to be geocentric, with everything revolving around a stationary Earth. The torch of innovation in astronomy and mathematics passed to Arabia, India, and China."

During the Renaissance, European scientists returned to the forefront, including Johannes Kepler, who sought out the shape of the orbits of the planets -- which are symmetrical, yet not circles:

"Kepler didn't expect this, and for a long time it persuaded him that an ellipse must be the wrong answer. The shape and use of an ellipse are determined by two lengths: its major axis, which is the longest line between two points on the ellipse, and its minor axis, which is perpendicular to the major axis."

At this point, the author provides a picture of the conic sections -- the ellipse, parabola, hyperbola -- and the basic features of an ellipse, including its axes, foci, and center. Since the orbits are indeed ellipses, these features are known as the orbital elements:

"A major goal of early astronomy was to calculate the orbital elements of every planet and asteroid that was discovered."

Another scientist who discovered some of the properties of gravity was Robert Hooke:

"In a lecture to the Royal Society in 1666, Robert Hooke said that all bodies move in a straight line unless acted on by a force, all bodies attract each other gravitationally, and the force of gravity decreases with distance by a formula that 'I own I have not discovered.'"

All of these scientists had to work with mathematical formulas:

"Newton's laws of motion and gravity triggers a lasting alliance between astronomy and mathematics, leading to much of what we now know about the cosmos."

And we continue to work with these formulas with the aid of computers:

"Instead of calculating approximate formulas for the motion, and then putting the numbers into the formulas, you can work from the beginning with the numbers."

Three years ago, I blogged the following:

"So exquisitely accurate are those rules that astronomers can predict eclipses of the Sun and Moon to the second, and predict within a few kilometers whereabouts on the planet they will occur, hundreds of years into the future. These 'predictions' can also be run backwards in time to pin down exactly when and where historically recorded eclipses occurred."

And here's what we discovered about gravity in recent years:

"In 1993 Cris Moore used numerical methods to show that three bodies with identical masses can chase each other repeatedly along the same figure-8 shaped orbit, and in 2000 Carles Simo showed numerically that this orbit is stable, except perhaps for a slow drift."

Stewart shows us a picture of this Fig-8 three-body orbit. (I mentioned numerical analysis in many previous posts -- in particular, I once considered getting a Ph.D. at UCLA in numerical analysis.) One possible example of a three-body system is a binary star and a planet:

"There's tentative evidence that Kepler-16b, a planet orbiting a distant star, might be one of them. One aspect of Newton's law bothered the great man himself; in fact, it bothered him more than it did most of those who built on his work."

And that's the title of this chapter -- "attraction at a distance." How exactly does the force of gravity work on two bodies far apart? It took until Einstein before this problem was addressed:

"At the speed of light -- if that were possible -- it would be infinitely thin, have infinite mass, and time on it would stop. Mass and energy are related: energy equals mass times the square of the speed of light."

So the Theory of Relativity ultimately solved the gravity problem:

"No action at a distance is needed: spacetime is curved because that's what stars do to it, and orbiting bodies respond to nearby curvature."

The author shows us a picture of the effect of curvature or gravity on a particle passing a massive star or planet. Indeed, our GPS systems must take the Theory of Relativity -- both general and special --into effect:

"Without these corrections, within a few days the satnav would place you in the middle of the Atlantic."

Stewart concludes the chapter by reminding us that Newton's physics has its limitations:

"Indeed, general relativity must be invoked, ably assisted by quantum mechanics. And even those two great theories seem to need extra help."

Shapelore Learning 13-1: The Witlore of Making Findings

This marks the beginning of our second summer project -- Geometry in Plain English.

I remember the one time at the old charter school when a student complained about the academic vocabulary, or "big words," used in my classes. If I recall correctly, the phrase that she complained about was "waning gibbous" -- one of the phases of the moon -- in science class.

Most of those "big words" that we use in math and science classes come from Greek or Latin, with some of the Latin words passing through French. (For example, gibbous comes from Latin.) I mentioned this in my last post, when I was writing about the Greek/Latin words used at the spelling bee and the French I took in high school. It turns out that very few Modern English words can actually be traced back to Old English.

I wonder, especially in math, whether these hard words make our subject more intractable. The example I kept giving is "trigonometry" -- how many adults ten years after graduation still remember that trig has anything to do with triangles? Instead, "trig" simply means "that class I hated in high school," while "sine," "cosine," and "tangent" simply mean "buttons on a calculator."

The word "trigonometry" comes from Greek -- it means "three angle measure." And so we wonder, perhaps people would remember what trig is if we replaced the Greek with English roots -- such as "threenooklore" or "threesidelore." These names comes from the following website:

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

The name of this website, "Anglish," means pure English -- that is, English without any words derived from Latin or Greek. Here is a full description:

https://anglish.miraheze.org/wiki/The_Anglish_Moot:What_is_Anglish%3F

Anglish is English with fewer words borrowed from other languages. Beyond this goal, there are a wide range of beliefs on what should make up Anglish. Some may only wish to use native English words when they are available, avoiding borrowed words where possible, but accepting them where needed. Others may wish to take out borrowed words altogether. Where the latter find no living native words, they will seek to bring an older word back to life or come up with a wholly new word in its stead.

I've been watching many more game shows lately during the closure, including Jeopardy! tournament for teachers. Earlier this week, there was a category about words that have been derived from Old English -- words listed there were knight, slake, dale, eke, bight. (That last word means "curve" or "indentation" in a river, and is not related to bite.) These are the sorts of words that appear more in Anglish than in Modern English.

One fan of Anglish is Wendy Krieger -- our huge base 120 (or "twelfty") enthusiast. She named one of her twelfty concepts in Anglish -- a "twistaff" is a pair of digits used to represent a single place in base twelfty, so 12:34 -- which is 12 * 120 + 34 = 1474 in decimal -- 12 and 34 are "twistaves."

A famous author who was also a fan of Anglish was George Orwell -- which is interesting, since much of Anglish sounds a bit like Newspeak (from his book 1984). It's true that some Newspeak words, such as bellyfeel, blackwhite, and ownlife, have pure English roots. But some other words aren't -- "doubleplusungood" isn't Anglish, since "double" and "plus" are both French. If Orwell had wanted a pure English word here, he might have chosen something like twicemoreungood.

The name of our subject, Geometry, comes from Greek. In Anglish, it becomes "Shapelore," which means the study of shapes. In Modern English, the word "lore" appears only in the word "folklore," but in Anglish, anything we study is called a "lore."

But our goal here isn't to write everything in pure English or Anglish. Instead, we wish to eliminate the more difficult Latin and Greek words that students have trouble remembering. For example, let's look at the Shapelore page on the Anglish website:

https://anglish.miraheze.org/wiki/Shapelore

Shapelore is the lore and meting of nibs, threads, boards, and shapes, and how they can be reckoned and scored along each other. The kinship between and among the many deals of shapelore is also learned of and reckoned by those wise men who teach this lore.

The first sentence reads, "Geometry is the study and measurement of points, lines, planes, and shapes, and how they can be analyzed and counted along each other." The undefined terms point, line, and plane all come from French, so in Anglish they're be replaced with pure English terms nibs, threads, and boards. But to me, point, line, and plane are such simple words that I'd like to keep them, even if they're not pure English.

In fact, this is why I'd rather call my project "Plain English," not "Pure English" or "Anglish." There are simple words like point and line that come from French -- and replacing these with nibs and threads is likely to add confusion. On the other hand, one of the two words that tripped up my student, waning, actually comes from Old English. It's just that most of the time, we expect students to have an easier time understanding what they read if we avoid complex Latin/Greek roots.

In this project, I'll be translating some Geometry lessons into Plain English. Since we left off at the start of Chapter 13 when the schools closed for the virus, we'll start with that chapter. I'll cover two lessons per day through Chapter 13 so we can hurry up and get to Chapter 14 -- that's the chapter on trig/"threenooklore"/"threesidelore" that I really want to reach.

Lesson 13-1 of the U of Chicago text is called "The Logic of Making Conclusions." In the title of this lesson, we have two non-English words to deal with, logic and conclusions. We'll look at the Anglish website for some hints:

https://anglish.miraheze.org/wiki/English_Wordbook/L
https://anglish.miraheze.org/wiki/English_Wordbook/C

The suggested word for logic is "witcraft." As for conclusions, the two words suggested there are "ending" and "finding." "Ending" sounds better for the conclusion of a film, or the conclusion of this blog post. The conclusions of Geometry are more like "findings," so that's the word we can use.

Of course, we don't have to use these words. The word logic is simple enough to keep -- most people know what logic is. As for conclusions, recall that the U of Chicago text, in its two-column proofs, uses the words conclusions and justifications for "statements" and "reasons." Even though these two words are Latin/French, they might be simple enough for us to keep.

Oh, and we can't call them lessons -- we already know that's a French word from my last post. The suggested words are "teaching" and "learning." Let's call it "learning" since we wish to emphasize what the students need to do -- learn. And "learning" and "lesson" both start with L anyway.

And so without further ado, let's begin Learning 13-1 of our Shapelore book. I'll write our new text in italics, with extra commentary on my part in parentheses. Remember that we're not translating all of the words into Anglish/Plain English -- just the Geometry vocab.

To begin the study of logic, we examine a proof similar to some you have written before. In this case, however, we have identified each statement with a letter so that the logic becomes clearer. (Here I keep the word logic since it's simple. I also keep proof as too simple to change -- but if we were to change it, the suggested word is "witness.")

Let's skip the proof for now and proceed with the next paragraph:

Recall that a reason for a finding is a true if-then saying with the finding as the aftermath. (Here I use reason for "justification" even though it's French. The suggested word for statement is saying, and I use finding for "conclusion." The word "consequent" definitely needs to be replaced, and the first suggested word for "consequence" is aftermath. Notice how this idea is used in step 1 of the above proof. p => q is the reason for finding q from the given p. The general logical principle is called the Law of Detachment because q is "detached" from p => q. (The usual name for this law, "modus ponens," is Latin. In the name Law of Detachment, detachment is French.) It is a generalization of the common sense idea that when p => q is a true statement and p is true, q follows.

Hey, since q follows from p, perhaps Law of Following should be our Plain English name instead:

Law of Following:
If you have a saying or given information p and a reason of the form p => q, you may find q.

Findings can be beforemaths for making other findings. (Believe it or not, the suggested word for "antecedence" really is beforemath. I like that a lot -- changing "antecedent" and "consequent" to beforemath and aftermath.) In step 2, from the finding q and the reason q => r, the finding r follows. Having steps 1 and 2, you can find that p => r. The proof can be diagrammed as follows. (I won't show the diagram here. Also, diagram is Greek, but it's used enough in non-mathematical English for us to keep it.)

The logical idea is the Law of Overfaresomeness.

The suggested word for "transitive" is overfaresome, but I wonder whether this word would be easier for students to remember than the original word "transitive." But consider this -- Transitivity is in a way just like Detachment, except that there are three statements instead of two. So let's call this something like Threefollowing:

Law of Threefollowing:
If p => q and q => r, then p => r.

Some of the examples here are non-mathematical, so I'll skip those. One of the examples is algebraic, but I'm not sure whether it's worth renaming "Algebra" -- which comes from an Arabic word. The suggested name here is Frothering. Yes, I know of the Voldemort principle -- "fear of a name increases fear of the thing itself" -- and that many students begin to hate math when they hear scary sounding names like "Algebra." Yet there's no point trying to pretend we aren't doing math by changing "Algebra" to Frothering or even "Geometry" to Shapelore. I'll keep on calling it Shapelore, even though we'd still call the class Geometry. I believe that even years after taking a class, students still remember that Geometry has something to do with shapes, even as they forget Trigonometry and what it means.

Let's go back to the proof we skipped over. It's worthwhile to see what some of the earlier Geometry terms (taught in the first twelve chapters of the book) are converted to:

Given: BE | | CD
Prove: Triangle ABE ~ ACD

Proof:
Findings Reasons
1. Angle ABE = ACD, 1. | | lines => corr. angles =
Angle AEB = ADC
2. Triangle ABE ~ ACD 2. AA Similarity Theorem

OK, let's convert this proof into Plain English. First of all, the suggested word for "angle" is nook. I am tempted to keep "angle" since I believe that it's simple enough for students to remember.

A "triangle," of course, is a shape with three angles. If nook means "angle," then it follows that "triangle" must be threenook. On one hand, even a kindergartner learns what a "triangle" is, so we might wish to keep this word. But on the other, we know looking head that many students confuse words like "hexagon" and "heptagon," so it could be helpful to rename these along with "triangle."

Also, "gon" is Greek for angle, so "hexagon" and "heptagon" should be sixnook and sevennook. But the word "side" is English, and these are more recognizable as sixside and sevenside. Once again, our goal here isn't to be be consistent with the original Greek/Latin roots, but to make these easier for students to understand. Thus names with side are preferable to those with any word for "angle."

The suggested word for "parallel" is akin, but once again, "parallel" might be plain enough. But I definitely want to change "corresponding," since students do confuse these. The suggested word for "corresponding" is matching, which I like.

In the name of the AA Similarity Theorem, I'm keeping Angles. But "similar" might be worth changing here -- the suggested word is alike, but something like shapesame actually captures the mathematical meaning more closely. For "congruent," allsame can work here. And "theorem" can be changed as well -- students need to know that theorems are statements that have been proved. One word that works is truesaying -- though since we're keeping "proof," provedsaying migh be better.

So here's what our proof looks like now, indicating which words we keep and which ones we change:

Given: BE | | CD
Prove: Threeside ABE ~ ACD

Proof:
Findings Reasons
1. Angle ABE = ACD, 1. akin lines => matching angles =
Angle AEB = ADC
2. Threeside ABE ~ ACD 2. AA Shapesameness Truesaying

Example 2 is a Geometric example that's worth looking at:

Example 2: What findings can be made from the following?
(1) Every rhombus is a kite.
(2) The diagonals of a kite are perpendicular.
(3) BUSM is a rhombus.

Let's start with "perpendicular." The suggested words are straight and upright. I like upright better, especially since it contains the word right -- perpendicular lines, of course, form right angles. The suggested words for "diagonal" are leaning, sloping, and slanted. I think lean is the best here -- the importance of a diagonal of a polygon is that it joins vertices that aren't adjacent -- so the diagonals do lean inside the shape.

That leaves us with the names of the quadrilaterals. Notice that on the Anglish website, the word fourside is often used to mean "square." But I disagree -- fourside would more accurately mean "quadrilateral," following the pattern of threeside "triangle," sixside "hexagon," and so on.

There is actually a suggested name for a rhombus -- a rout. There are comparisons to Dutch and German listed here -- and you might ask, if we're avoiding Latin/Greek, why use Dutch/German? The reason is that Old English is actually related to Dutch and German, more so than to Latin. Thus it's thought better to use a Germanic word than a Latin word. But I wonder whether our students would really remember what a rout is more easily than what a rhombus is.

As for other quadrilaterals, rectangle is also given a Dutch name -- righthook. I'd have thought that rightnook is better if we're keeping threenook for "triangle" -- otherwise rightside is best. Since we're using akin for "parallel," the suggested word for "parallelogram" is an akinside. Meanwhile, trapezoid has the suggested name boardshape.

Fortunately, kite is already an English word. Finally, square is well-known, so we'll keep it. Thus here's our new version of Example 2:

Example 2: What findings can be made from the following?
(1) Every rout is a kite.
(2) The leans of a kite are upright.
(3) BUSM is a rout.

We might even decide to throw the suggested names and name the quadrilaterals even more systematically, perhaps adding the word "four" to indicate that all four sides have a given property:

"quadrilateral" = fourside
kite (already English)
"trapezoid" = akinside
"parallelogram" = fourakinside
"rhombus" = foursameside
"rectangle" = fourrightside
square (grandfathered in)

It's now obvious that every parallelogram is a trapezoid -- every fourakinside is an akinside. But (1) now says that every foursameside is a kite, which still isn't as obvious.

Shapelore Learning 13-2: Notsayings

Lesson 13-2 of the U of Chicago text is called "Negations." There are several suggested words for "negate" on the Anglish website -- gainsay, unsay, withersay, and so on. But I think that the best word to use here is notsaying -- let's emphasize the word not, since we negate a statement by adding that particular word.

And so the lesson begins:

Unstraight reasoning is based on the idea of notsaying. (The lesson starts with "indirect reasoning" -- "reasoning" is grandfathered in. Something that's direct is straight, and so something that's indirect must be unstraight.) The notsaying of a saying p, called not-p, is a saying that is true whenever saying p is false and is false whenever saying p is true. (True is English but "false" is Latin. But the word "false" is common enough to be grandfathered in.)

Once again, we'll skip to the Geometric examples:

p: The fourside ABCD is not an akinside (parallelogram -- I haven yet to make fourakinside official).
not-p: The fourside ABCD is an akinside.

p: Threeside ABC is sidesame (isosceles).
not-p: Threeside ABC is unsidesame (scalene).

t: Angle A is a right angle.
not-t: Angle A is inward (acute) or Angle A is outward (obtuse).

The names inward and outward come from the original Shapelore page.

Recall that from any if-then (conditional) p => q, you can form its otherway (converse) q => p. (A suggestion for "conversely" is otherwise, which doesn't quite capture our meaning of "converse." I like otherway because that's what q => p is -- p => q written the other way.) Let p = "a threeside is allsidesame" (equilateral) and q = "a threeside has three inward angles."

Original p => q: If a threeside is allsidesame, then it has three inward angles.
Otherway q => p: If a threeside has three inward angles, then it is allsidesame.

Here the original is true, the otherway false.

Notsaying both sides of the original if-then gives a new if-then of the form not-p => not-q, called the notway (inverse) of the original. (Again, inverses have "not" in them, so let's emphasize this.)

Notway not-p => not-q: If a threeside is not allsameside, it does not have three inward angles.

The notway is false. There are threesides which are not allsameside but do have three inward angles.

However, if both parts of the original are notsaid and the beforemath and aftermath are switched, a second true saying appears. This saying, of the form not-q => not-p, is called the othernotway of the original. (Yes, othernotway is a perfect description of what a "contrapositive" is.)

Othernotway not-q => not-p: If a threeside does not have three inward angles, it is not allsameside.

Law of the Othernotway:
A saying (p => q) and its othernotway (not-q => not-p) are either both true or not false.

Our final example is Geometric, so let's give it here:

Example 2: Given are two sayings.
(1) Every square is a kite.
(2) Fourside POTS is not a kite.
What finding can be made using both statements?

Solution: In if-then form, (1) is: If a shape is a square, then it is a kite. This is a true saying; thus, the othernotway is also true and may be said as follows: if a shape is not a kite, then it is not a square. Now, since (2) POTS is not a kite, you can use the Law of Following to find that POTS is not a square.

My hope is that students will understand this last example much more easily than the original, where they're likely to get their conditionals, converses, and contrapositives all mixed up.

Some Rapoport Math Problems

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

sigma _n=0 ^infinity 2^(-n)n^2 (Yes, I know that sigma notation looks weird in ASCII.)

This clearly isn't a Geometric figure problem. And it isn't even a geometric series problem -- if it weren't for that n^2, the series would be 1 + 1/2 + 1/4 + 1/8 + ..., a geometric series with sum 2. But instead, this series is 0 + 1/2 + 4/4 + 9/8 + ..., which isn't geometric. So it's not as obvious what we must do to find its sum.

Hmm, what should we do about those squares? Well, since we're adding terms in a series, why don't we begin by rewriting the squares as sums? Note that perfect squares are the sums of odd numbers -- so we have 1 = 1, 4 = 1 + 3, 9 = 1 + 3 + 5, and so on. There should be some way to take advantage of this with our series.

Let's ignore the 0 term -- we might as well have begun the sum with n = 1. Then we'll rewrite all squares from 1^2 as a sum of odd numbers:

1/2 + 4/4 + 9/8 + 16/16 + 25/32 + 36/64 + ...

= 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ... (= 1)
+ 3/4 + 3/8 + 3/16 + 3/32 + 3/64 + ... (= 3/2)
+ 5/8 + 5/16 + 5/32 + 5/64 + ... (= 5/4)
+ 7/16 + 7/32 + 7/64 + ... (= 7/8)
+ 9/32 + 9/64 + ... (= 9/16)

= 1 + 3/2 + 5/4 + 7/8 + 9/16 + ...

This still doesn't give us a simple geometric series. But hold on a minute -- just as perfect squares are the sums of odd numbers, odd numbers are the sums of 2's (with an extra 1 thrown in). So we write:

1 + 3/2 + 5/4 + 7/8 + 9/16 + ...
= 1 + 1/2 + 1/4 + 1/8 + 1/16 + ... (= 2)
+ 2/2 + 2/4 + 2/8 + 2/16 + ... (= 2)
+ 2/4 + 2/8 + 2/16 + ... (= 1)
+ 2/8 + 2/16 + ... (= 1/2)

= 2 + 2 + 1 + 1/2 + 1/4 + 1/8 + ...

Except for the initial 2 (which arises from the extra 1 that produces odd numbers), the rest of this is, finally, a geometric series:

2 + 2 + 1 + 1/2 + 1/4 + 1/8 + ... = 2 + 4 = 6

Therefore the desired series sum is 6 -- and of course, today's date is the sixth.

Here are some actual Geometry problems on the Rapoport calendar from earlier this week:

Problem #1: How many faces does a noble disphenoid have?

Yes, it's a Geometry problem, but it's purely a research question -- until we learn what a noble disphenoid is, we can't even begin to solve this problem. There is one hint here -- "how many faces" suggests that a noble disphenoid is probably a polyhedron.

OK, that's as far as we can get without Google, so let's just look it up:

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

A tetrahedron with identical isosceles or scalene faces.

Oh, so a disphenoid is a tetrahedron. And we know from Lesson 9-7 of the U of Chicago text that a tetrahedron has four faces, so our desired answer is fourth -- and this problem was from the fourth.

By the way, the "noble" part means that all the faces and vertices are congruent. Because of this, all disphenoids are noble, and so "noble disphenoid" is redundant.

This definition is a bit confusing -- is "nobility" the same as "regularity"? No -- if a polyhedron is regular, then it is noble, but the converse is false. In fact, we can define "regular" in terms of "noble" using a recursive definition:

An n-dimensional polytope is regular if it is noble, and each of its faces is a regular (n-1)-dimensional polytope (recursive).

The initial case is 1D -- all line segments are regular. Therefore all noble 2D-polygons are regular (since they can't have any non-regular sides). But not all noble 3D-polyhedra are regular, since they might have non-regular triangles as faces.

There's actually an error in the U of Chicago text, in the Exploration Question for Lesson 9-7:

A regular polyhedron is a convex polyhedron in which all faces are congruent regular polygons and the same number of edges intersect at each of its vertices.

That's not the definition of "regular polyhedron" -- that's the definition of "Platonic solid." If a polyhedron is Platonic, then it is regular, but the converse is false. The problem is "convex" -- there exist regular polyhedra (called "star polyhedra") that aren't convex and hence not Platonic. But all regular polygons in 2D are convex. And I admit that I never knew any of this until I had to research "noble disphenoid."

So in conclusion, we have:

If a polytope is Platonic, then it is regular.
If a polytope is regular, then it is Platonic.

And the converses are true in 2D but false in 3D.

We've just started our Plain English project, and that name "noble disphenoid" is just begging for a Plain English equivalent. The word "disphenoid" is Greek -- "di-" means "two," and "sphenoid" means something like "wedge." Since "wedge" is Old English, twowedge is an acceptable Anglish calque for "disphenoid." (A "calque" means that we naively translate each part of the original word into Old English.) As for "noble," the suggested word from the Anglish website is highbred. Thus a "noble disphenoid" is a highbred twowedge.

As we've seen above, polyhedra are much more complicated than polygons. But since the tetrahedron is the simplest polyhedron, it might be worth it someday to come up with a Tetrahedron Hierarchy similar to our Triangle and Quadrilateral Hierarchies in 2D.

For starters, we should come up with a new name for "tetrahedron." We can't use fourface because "face" is French (though we'll probably grandfather in "face" as a simple word.).We might try going back to the suggested word for "plane" -- board. Then a "tetrahedron" is a fourboard.

A cube is a sixboard, but we'll probably grandfather "cube" in. An octahedron becomes eightboard, and so on. We've already used highbred for "noble." A suggested word for "regular" are everyday -- but I like the second word better, steady. "Platonic" is obviously Greek since it was named after an ancient Greek man, Plato. Even Anglish can preserve proper names, perhaps replacing only the suffix (changing "Platonic" to Platonish), but I have no reason to do so in my Plain English project.

Problem #2: What is the area of a right isosceles triangle with hypotenuse 2?

Answer: A right-isosceles triangle is a 45-45-90 triangle. Its hypotenuse is 2, and so its leg must be 2/sqrt(2), which is sqrt(2). Then its area is (1/2)sqrt(2)sqrt(2) = (1/2)2 = 1. Therefore the desired area is 1 -- and of course, this problem was from the first. We haven't quite covered 45-45-90 triangles yet since we never made it to Chapter 14.

Problem #3: For a convex octagon, the ratio of the sum of the interior angles to the sum of the exterior angles.

Answer: An octagon is an eightside. The sum of its interior angles is (8 - 2)180 = 1080 and the sum of its exterior angles is 360. Therefore the desired ratio is 1080/360 = 3 -- and of course, this problem was from the third. Interior angle sums appear in Lesson 5-7, but exterior angle sums don't show up until later in Chapter 13, so we haven't covered them yet either.

Lemay Chapter 6 Part 1: Creating Classes and Applications in Java

Even though my other summer projects are beginning, we're still learning Java. But I will return to dividing each lesson into parts.

Here's the link to today's lesson:

http://101.lv/learn/Java/ch6.htm

Lesson 6 of Laura Lemay's Teach Yourself Java in 21 Days! is called "Creating Classes and Applications in Java." Here's how this chapter begins:

In just about every lesson up to this point you've been creating Java applications-writing classes, creating instance variables and methods, and running those applications to perform simple tasks. Also up to this point, you've focused either on the very broad (general object-oriented theory) or the very minute (arithmetic and other expressions).

The first thing we do in this lesson is create classes. As Lemay explains, classes derive from others:

By default, classes inherit from the Object class. If this class is a subclass of another specific class (that is, inherits from another class), use extends to indicate the superclass of this class:

class myClassName extends mySuperClassName {
...
}

The first thing we'll want to put in our classes are some variables, known as instance variables:

On Day 3, "Java Basics," you learned how to declare and initialize local variables-that is, variables inside method definitions. Instance variables, fortunately, are declared and defined in almost exactly the same way as local variables; the main difference is their location in the class definition. Variables are considered instance variables if they are declared outside a method definition. Customarily, however, most instance variables are defined just after the first line of the class definition. For example, Listing 6.1 shows a simple class definition for the class Bicycle, which inherits from the class PersonPoweredVehicle.

And this takes us right to our first listing:

Listing 6.1. The Bicycle class.

1: class Bicycle extends PersonPoweredVehicle {
2:     String bikeType;
3:     int chainGear;
4:     int rearCogs;
5:     int currentGearFront;
6:     int currentGearRear;
7: }

I won't even bother to enter this class into my compiler -- it lacks a main method, or any other method for that matter. And on Day 1, when the author gave her PersonPoweredVehicle class, I never wrote it, so the class extends a nonexistent class.

The next example here is constants:

A constant variable or constant is a variable whose value never changes (which may seem strange given the meaning of the word variable). Constants are useful for defining shared values for all the methods of an object-for giving meaningful names to objectwide values that will never change. In Java, you can create constants only for instance or class variables, not for local variables.

The keyword for defining constants in Java is final. This is different from C++, and as usual, I'll include all "technical notes" from Lemay when she compares Java to C++:

The only way to define constants in Java is by using the final keyword. Neither the C and C++ constructs for #define nor const are available in Java, although the const keyword is reserved to prevent you from accidentally using it.

Now that our classes have instance variables, let's add some methods:

Method definitions have four basic parts:

The name of the method
The type of object or primitive type the method returns
A list of parameters
The body of the method

And Lemay adds:

In other languages, the name of the method (or function, subroutine, or procedure) is enough to distinguish it from other methods in the program. In Java, you can have different methods that have the same name but a different return type or argument list, so all these parts of the method definition are important. This is called method overloading, and you'll learn more about it tomorrow.

Notice that those "other languages" don't include C++, since its methods are similar to Java. I suspect that those other languages that don't allow overloading are Pascal and BlooP.

We're now ready for the next listing:

Listing 6.2. The RangeClass class.

 1: class RangeClass {
 2:     int[] makeRange(int lower, int upper) {
 3:         int arr[] = new int[ (upper - lower) + 1 ];
 4: 
 5:         for (int i = 0; i < arr.length; i++) {
 6:             arr[i] = lower++;
 7:         }
 8:         return arr;
 9:     }
10:         
11:     public static void main(String arg[]) {
12:         int theArray[];
13:         RangeClass theRange = new RangeClass();
14: 
15:         theArray = theRange.makeRange(1, 10);
16:         System.out.print("The array: [ ");
17:         for (int i = 0; i < theArray.length; i++) {
18:             System.out.print(theArray[i] + " ");
19:         }
20:         System.out.println("]");
21:     }
22: 
23: }

We now move on to the this object, which is similar to C++ in that it refers to the current (in other words, this) object. The only difference is that this is a pointer in C++, but not Java:

t = this.x;          // the x instance variable for this object
this.myMethod(this); // call the myMethod method, defined in
                     // this class, and pass it the current
                     // object
return this;         // return the current object

In many cases you may be able to omit the this keyword entirely. You can refer to both instance variables and method calls defined in the current class simply by name; the this is implicit in those references. So the first two examples could be written like this:

t = x          // the x instance variable for this object
myMethod(this) // call the myMethod method, defined in this
               // class

The next example is about the scope of a variable -- that is, where the variable may be used:

When you declare a variable, that variable always has a limited scope. Variable scope determines where that variable can be used. Variables with a local scope, for example, can only be used inside the block in which they were defined. Instance variables have a scope that extends to the entire class so they can be used by any of the methods within that class.

And this leads to our final listing for Lesson 6 Part 1:

Listing 6.3. A variable scope example.

 1: class ScopeTest {
 2:     int test = 10;
 3: 
 4:     void printTest () {
 5:         int test = 20;
 6:         System.out.println("test = " + test);
 7:     }
 8:     
 9:     public static void main (String args[]) {
10:        ScopeTest st = new ScopeTest();
11:        st.printTest();
12:     }
13: }

This code prints test = 20, because the local variable defined in Line 5 hides the object variable in Line 2. Lemay tell us that it's much better just to give the variable in Line 5 a different name.

More on the Point Class in Java

Now is the time when I normally create my own Java code based on today's lesson. Now I have enough to create a hierarchy of classes based on the Quadrilateral Hierarchy. (Or should I call it the Fourside -- uh, what's another word for "hierarchy" again?)

But I've actually been waiting to enhance our PointyThings class for a while now, so let me fix it right now. This is what it looks like so far:

import java.awt.Point;
import java.lang.Math;

public class PointyThings {
public static void main(String args[]) {
Point pt1 = new Point(10,10);
Point pt2 = new Point(5,15);
float slope = ((pt2.y - pt1.y)/(pt2.x - pt1.x));
System.out.println("The slope is " + slope);
double dist = Math.sqrt((pt2.x - pt1.x)*(pt2.x - pt1.x) + (pt2.y - pt1.y)*(pt2.y - pt1.y));
System.out.println("The distance is " + dist);
}

}

First of all, slope and dist are just begging to be written as methods, so let's do so:

import java.awt.Point;
import java.lang.Math;

public class PointyThings {
float slope(Point pt1, Point pt2) {
return (pt2.y - pt1.y)/(pt2.x - pt1.x);
}

double dist(Point pt1, Point pt2) {
return Math.sqrt((pt2.x - pt1.x)*(pt2.x - pt1.x) + (pt2.y - pt1.y)*(pt2.y - pt1.y));
}

public static void main(String args[]) {
PointyThings p = new PointyThings();
Point pt1 = new Point(10,10);
Point pt2 = new Point(5,15);
System.out.println("The slope is " + p.slope(pt1, pt2));
System.out.println("The distance is " + p.dist(pt1, pt2));
}

}

But I've been thinking of adding some more methods to this class as well. Another thing we do with points, besides finding slopes and distances between them, is perform transformations. We can write methods for certain translations, rotations, reflections, and dilations:

Point transl34(Point pt) {
Point im = new Point(pt.x + 3, pt.y + 4);
return im;
}

Point rot90(Point pt) {
Point im = new Point(-pt.y, pt.x);
return im;
}

Point reflx(Point pt) {
Point im = new Point(pt.x, -pt.y);
return im;
}

Point dil2(Point pt) {
Point im = new Point(pt.x * 2, pt.y * 2);
return im;
}

These methods take a Point object and return another Point object. To display these points, we might write the following line in main:

System.out.println("pt2 dilated is (" + p.dil2(pt2).x +"," + p.dil2(pt2).y + ")");

Output:
pt2 dilated is (10,30)

It's works, but it's annoying that we can't simply println our Point objects. Hey, that gives me another idea -- let's write another method that takes a Point object and returns a String that we can print:

String ptstr(Point pt) {
return "(" + pt.x + "," + pt.y + ")";

}

And in main:

System.out.println("pt2 dilated is " + p.ptstr(p.dil2(pt2)));

There's one more thing that we might want to do here. Recall that we tried to write a midpoint method earlier, but we didn't because the coordinates of Points are ints, yet the midpoint of two lattice points need not be a lattice point.

The midpoint is guaranteed to be a lattice point if both coordinates of both endpoints are even -- and hey, we just wrote a method of dilation of scale factor 2 that guarantees just that! So why don't we write our midpoint method, but then dilate the points in main before feeding them to midpoint.

Here is the final version, with our new midpoint method:

import java.awt.Point;
import java.lang.Math;

public class PointyThings {
float slope(Point pt1, Point pt2) {
return (pt2.y - pt1.y)/(pt2.x - pt1.x);
}

double dist(Point pt1, Point pt2) {
return Math.sqrt((pt2.x - pt1.x)*(pt2.x - pt1.x) + (pt2.y - pt1.y)*(pt2.y - pt1.y));
}

Point transl34(Point pt) {
Point im = new Point(pt.x + 3, pt.y + 4);
return im;
}

Point rot90(Point pt) {
Point im = new Point(-pt.y, pt.x);
return im;
}

Point reflx(Point pt) {
Point im = new Point(pt.x, -pt.y);
return im;
}

Point dil2(Point pt) {
Point im = new Point(pt.x * 2, pt.y * 2);
return im;
}

Point midp(Point pt1, Point pt2) {
Point im = new Point((pt1.x + pt2.x)/2, (pt1.y + pt2.y)/2);
return im;
}

String ptstr(Point pt) {
return "(" + pt.x + "," + pt.y + ")";
}

public static void main(String args[]) {
PointyThings p = new PointyThings();
Point pt1 = new Point(10,10);
Point pt2 = new Point(5,15);
System.out.println("The slope is " + p.slope(pt1, pt2));
System.out.println("The distance is " + p.dist(pt1, pt2));
Point pt1pr = p.dil2(pt1);
Point pt2pr = p.dil2(pt2);
Point ptm = p.midp(pt1pr, pt2pr);
System.out.println("Midpoint of dilation " + p.ptstr(ptm));
}

}

Mocha Music: "The Dren Song," "Count on It," "Benchmark Tests," "Fraction Fever"

The last of my declared summer projects is to post all of my songs on the blog -- both the lyrics and in Mocha. This has become all the more desirable once I left school on that last day before the coronavirus closure and realized that my notebook of songs was still locked in the classroom.

Again, I took it out because I wanted to perform the songs on the guitars and ukeleles that were in that particular classroom -- especially the Pi Day songs as it was the last three days before Pi Day. I actually left the book in that class two out of the three days I was there -- but leaving it the first day was harmless, since I'd be back the next day.

On Pi Day Eve after school, I'd been driving for ten minutes when I realized that I'd left the notebook was still in the classroom. Indeed, as I was cleaning up to leave, I was concerned more about getting the empty pie and pizza boxes (from the Pi Day Eve party) out of the room, not my songbook. I knew that it would take another ten minutes to get back to the school, plus at least ten more minutes to convince the administrators to unlock the door, walk to the room, and walk back. Thus I would have arrived at home at least half an hour late, only because I'd remembered the book a little too late. So I decided the songbook behind, even though I knew the schools were about to be shut down.

And now, of course, it will have been at least five months before I can get in that room again. I'm already assuming that it's been thrown away -- I won't even try to get it back.

So here's my plan -- I'll cut-and-paste the lyrics of each song in these summer posts -- I think I'll include two weeks' worth of songs in each post. For today's post, we'll look at the first four songs that I sang at the old charter school during Weeks 1-2 -- "The Dren Song," "Count on It," "Benchmark Tests," and "Fraction Fever."

I've never posted the tunes of some of these songs before, and so I'll do so for the first time now. I'll post the tunes in Mocha code so you can play them yourselves.

Of course, it's odd to post Mocha BASIC code in the middle of our Java lessons. Actually, I wonder whether there's a way to code music in Java -- perhaps as an applet for the Internet, since some website do have background music. Well, we'll have to wait for Lemay to tell us so.

Let's start with our first song, "The Dren Song."

The Dren Song -- by Mr. Walker

I don't know why I take math.
I'm all caught up in its wrath.
I'd rather just be a dren.
I would be so happy then.
Tell me what would happen when,
I'm no longer just a dren.
What if I were great at math?
What would be my future path?
Customers won't think it's strange,
When I figure out their change.
Algebra and calculus,
Get me in a cool college.
Once I finish my degree,
Future employers will see,
Of my strong background in STEM.
I know that will impress them.
Reach the moon, be a hero!
I won't just be a zero!
I'll be great, or it may seem,
That this all is just a dream,
'Cause my math skills are so bad.
I can't subtract! I can't add!
I can't multiply by ten.
I will always be a dren.
Now I know why I take math.
Help me find a better path!
I would be so happy then.
But I'm just a dren.

I'd come up for the tune for this song over the previous summer -- it was just something that was stuck in my mind. Keep in mind that I didn't know about the Mocha emulator or EDL scales until after I left the charter school, so these songs won't perfectly fit the EDL scales.

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

10 N=8
20 FOR X=1 TO 12
30 GOSUB 200
40 NEXT X
50 RESTORE
60 FOR X=1 TO 12
70 GOSUB 200
80 NEXT X
90 RESTORE
100 FOR X=1 TO 4
110 GOSUB 200
120 NEXT X
130 END
200 FOR Y=1 TO 7
210 READ A
220 IF X=2 THEN A=20-A
230 IF Y=7 THEN T=8 ELSE T=4
240 SOUND 261-N*A,T
250 NEXT Y
260 RETURN
300 DATA 12,11,10,12,13,12,11
310 DATA 13,12,11,13,12,11,10
320 DATA 12,11,10,12,13,12,11
330 DATA 13,12,11,13,12,11,12
340 DATA 10,9,8,10,11,10,9
350 DATA 12,11,10,12,13,12,11
360 DATA 10,9,8,10,11,10,9
370 DATA 11,10,9,11,10,9,10
380 DATA 10,9,8,10,9,8,7
390 DATA 11,10,9,11,10,9,8
400 DATA 10,9,8,10,9,8,7
410 DATA 9,8,7,9,8,7,8

As usual, don't forget to click on "Sound" before you RUN the program.

I originally wrote this song in a minor key, so I used 12EDL to simulate a minor key. The lowest note of my song is a leading tone (which I simulated using Degree 13) and the highest note is a sixth (which is Degree 7). But of course, these aren't exact -- 13/12 is a bit wider than a leading tone, and 12/7 is a supermajor sixth, not a minor sixth. But at least Degrees 12-10-8 are a minor triad.

The key to this song is that it's symmetrical. The second part of the song is the same as the first, except that the low notes become high notes and vice versa. To code this in Mocha, we have three different FOR X loops, each of which has its own GOSUB 200 subroutine. On the second loop X=2, Line 220 subtracts the Degree from 20, so that Degree 13 becomes 7 and vice versa, Degree 12 becomes 8 and vice versa, and so on.

It's almost as if the song is being played on a Mobius strip, which has only one side. Instead of a "second side," we arrive at the first side except that the notes are arranged in the other order. The first part is Lines 1-12 of the song, the second part is Lines 13-24 (reversing low/high), and the third part is Lines 25-28 (repeating the first four lines of the first part for the last four lines of the song).

Vi Hart has a video about a Mobius music box, if you need to know exactly what I mean here:

It's likely that Vi Hart's video was at least a partial influence on my song.

Suppose we preferred using 18EDL instead of 12EDL to represent our minor key. Then our leading tone and minor sixth become Degrees 19 and 11. Anyway, 19/18 does sound very much like a leading tone, and while 18/11 is still slightly sharp (a neutral sixth), it surely beats supermajor 12/7.

On the other hand, we no longer have a perfect fourth in 18EDL. In order to keep our Mobius symmetry (subtracting from 30 instead of 20), the fourth must be Degree 14 (9/7 = supermajor third), since the second is at Degree 16 (9/8).

The second song for today is "Count on It." Here are the lyrics, courtesy the following link:

http://wordpress.barrycarter.org/index.php/2011/06/07/square-one-tv-more-lyrics/#.V7UYlSgrKUl

Count On It

Lead vocals by Larry Cedar

Sooner or later, you’re gonna see some math
You can count on it
Sooner or later, those numbers cross your path
You can count on it

You may be hoping it will go away
But let me tell you, math is here to stay
You can count on it, hoo, yeah
You can count on it

Everywhere you look, they’re measuring the action
You can count on it
Everywhere you look, they’re even using fractions
You can count on it

They’re keeping time, and they’re keeping the score
They draw the line, and they’re running the store
You can count on it, hoo
Yeah, you can count on it

Look at the dial; look at how far
Look at how much; look where we are
Look at the gauge; look at the graph
Check out the numbers; you’ve got the last laugh

‘Cause it ain’t mystery; there’s nothing tough about it
You can count on it, that’s right
Soon you’re gonna see that you couldn’t live without it
You can count on it, hoo

Don’t take a genius or a great magician
To make a pretty good mathematician
You can count on it, hoo, yeah
Yeah, you can count on it, whoo
Oh, you can count on it, whoo
Baby, you can count on it

(fade out over Larry singing skat)

Since this song comes from Square One TV, I don't post a Mocha version of it. If you wish to know the tune of the song, you can get it directly from Square One TV:

The third song for today is "Benchmark Tests." I've written several verses for this song and posted them over the years, so let me post it all here in one spot:

Benchmark Tests -- by Mr. Walker

Verse 1:
Why do we take Benchmark Tests?
It's the start of the year so let's
See how much we know, know know!

It's much new stuff on Benchmark Tests.
If we don't know it, we take a guess.
We leave none blank, oh no, no, no!

The teacher sees our Benchmark Tests,
Knows what to teach more or less.
That's the way to go, go, go!

Verse 2:
Why do we take Benchmark Tests?
The first trimester is done so let's
See how much we know, know know!

It's some new stuff on Benchmark Tests.
If we don't know it, we take a guess.
'Cause there's still time to grow, grow, grow!

The teacher sees our Benchmark Tests,
Knows what to teach more or less.
That's the way to go, go, go!

Verse 3:
Why do we take Benchmark Tests?
The second trimester is done so let's
See how much we know, know know!

It's some new stuff on Benchmark Tests.
If we don't know it, we take a guess.
'Cause there's still time to grow, grow, grow!

The teacher sees our Benchmark Tests,
Knows what to teach more or less.
That's the way to go, go, go!

Verse 4:
Why do we take Benchmark Tests?
The school year is almost done so let's
See how much we know, know know!

It's all old stuff on Benchmark Tests.
We know it now, we don't take a guess.
We leave none blank, oh no, no, no!

The teacher sees our Benchmark Tests,
We're ready for the state test, more or less.
That's the way to go, go, go!

I actually once wrote a Mocha version of "Benchmark Tests" and posted it on the blog. But's it's not the original version that I sang in class -- instead of EDL, it was based on a just Pythagorean scale (that is, 3-limit or "wa"/white in Kite colors), and also had a Mobius inversion pattern similar to "The Dren Song" above.

The original version is what I wrote for the Google Mbira Doodle a few weeks ago. So this is the version that we'll write in Mocha today. It's in a major key, which we'll simulate using 18EDL:

NEW
10 N=8
20 FOR V=1 TO 3
30 FOR X=1 TO 4
40 FOR Y=1 TO 7
50 READ A
60 IF Y=7 THEN T=8 ELSE T=4
70 SOUND 261-N*A,T
80 NEXT Y
90 RESTORE
100 NEXT X,V
110 DATA 18,16,18,16,18,18,18
120 DATA 18,16,14,16,18,16,18
130 DATA 16,11,9,11,12,12,12
140 DATA 16,11,9,11,12,12,12

This code plays a single verse, with the third line of each stanza repeating as the fourth line. In reality, I'd only ever sing one verse at a time, depending on what time of year the Benchmarks are given.

The fourth and final song today is "Fraction Fever." I only ever sang the first verse in class -- I've added extra verses on the blog after I left the charter school:

FRACTION FEVER

First Verse:
Hey, if you've never
Played Fraction Fever
Here's how to get the action
You gotta get the right fraction!

Choose the wrong one and down you fall
(Down you fall!)
Through the hole and that's not all!
(That's not all!)

If you find the right one later
(Right one later!)
You'll go up in the elevator!
(Elevator!)

When you get to Floor 20
(Floor 20!)
You'll win plenty!
(Win plenty!)

Fraction! Fever!
Fraction! Fever!

Second Verse:
Hey, if you've never
Played Fraction Fever
Here's how to do addition
And also subtraction!

To go up in the elevator
(Elevator!)
Find a common denominator
(Denominator!)

Add or subtract the numerators
(Numerators!)
That will lead you to the elevators!
(Elevators!)

When you get the answer, always try
(Always try!)
To simplify!
(Simplify!)

Fraction! Fever!
Fraction! Fever!

Third Verse:
Hey, if you've never
Played Fraction Fever
Here's how to multiply
And also to divide!

Multiply the numerators
(Numerators)
Multiply the denominators!
(Denominators)

Don't forget when you divide
(You divide)
Flip the second one down upside!
(Down upside)

When you get the answer, always try
(Always try!)
To simplify!
(Simplify!)

Fraction! Fever!
Fraction! Fever!

This song was based on the old game of the same name that I played on my old computer -- you know, the one that Mocha emulates. So it should be a cinch to write a Mocha version of the song. The problem is that it was based on a cartridge that we inserted in the computer -- it's impossible to see BASIC code for cartridges. On the emulator, the "Load Bin" button is how we insert cartridges -- but unfortunately, "Fraction Fever" isn't one of the games listed there.

I did find a YouTube video of the old game right here:

The song begins at 1:30, but unlike my seventh graders, the creator of the YouTube video doesn't like the song. And he presses the START button just seconds before their favorite part -- where I repeat the last two lines "Fraction! Fever!" If I recall correctly, "Fraction!" was played as a very high C, while "Fever!" was a leading tone, a high B. (This seems a bit odd for an ending since the rest of the song was in D minor, but oh well.) So you can sing the song to that tune.

Actually, here's another "Fraction Fever" video. Only the first few seconds of the song are played -- but for some reason, the high C and B notes are playing in harmony the first lines of the song. (Most of the time, this computer couldn't play music in harmony at all, so this must be a slightly different computer from Mocha.)

Reblogging: Summer School and Edgenuity

Now that summer is here, there isn't much need for me to reblog any more, since reblogging isn't what I normally do in summer posts. But today I have a more interesting post to reblog here.

The last time I blogged on June 6th was two years ago -- that date was Day 180 that year. At the time, there was a possibility that I'd be hired to teach a summer school class. Most of the class involved using Edgenuity software.

Even though I found out a few days after June 6th that I wouldn't get the summer job, it's important to see another example of educational software in the wake of the coronavirus. Most districts, if they're having summer school at all, are continuing to use distance learning -- and we still have no idea what schools will look like in the fall. So it's possible that online software like Edgenuity will continue to grow in importance.

OK, here is the reblogged post:

Today is the last day of school in my old district. It isn't the last day of school in my new district, where it is only Day 174. But the blog is following the old calendar.

Usually, today is when I post a preview of the upcoming school year, but of course, summer school is on my mind right now. In my new district, today is the last summer school training meeting.

As of now, I still don't know whether enough students will sign up for summer school. Remember that even if my summer class is canceled, there won't be a Great Post Purge of 2018 -- that is, I won't go back and delete every post that mentions summer school. Even when I first found out about summer school, I knew that it was dependent on there being enough students to sign up for the class.

Assuming that the summer school class happens, then what are my plans? Of course, the class doesn't start next week (since this is only Day 174), but the week after. The classes are three weeks, four days per week, and I'm scheduled to teach two classes, each a little less than two hours.

When I see my fellow summer Algebra I teacher today (yes, she's the current student teacher at one of the district high schools), we confirm the following plans. She continues to think in terms of the Glencoe text that she uses during the school year, even though we're actually using Edgenuity, an online curriculum:

First Week: Chapters 1-2
Second Week: Chapters 3-4
Third Week: Chapter 5 and District Final

In the name of purity, I should use the names of the actual units in Edgenuity:

Solving Linear Equations
Introduction to Functions
Analyzing Functions
Linear Functions
Point-Slope Form and Linear Equations
Solving Equations and Inequalities

The first unit, Solving Linear Equations, actually corresponds to Chapter 2 of Glencoe. Again, Chapter 1 of Glencoe doesn't actually appear in Edgenuity (since Chapter 1 is based on middle school standards in Common Core), and so we'll teach them Chapter 1 material without a computer -- mostly Order of Operations and the Distributive Property.

In fact, it's possible that my supplemental lessons could come the U of Chicago Algebra I text. The Order of Operations is taught well in Lesson 1-4 of the text. Unfortunately, the Distributive Property is spread out among several lessons in Chapter 6. The distributive property is first introduced in Lesson 6-3, but this is mostly about combining like terms -- as in 2x + 2x = (2 + 2)x = 4x -- as well as discount and markup questions -- as in x - 0.25x = (1 - 0.25)x = 0.75x for 25% off.

More general examples of the Distributive Property appear in Lesson 6-8, whose title is "Why the Distributive Property Is So Named." Examples of distributing a negative value appear in the next lesson, "Subtracting Quantities." All three of these Chapter 6 lessons contain equations to solve, since the U of Chicago text introduces solving equations before the Distributive Property.

Along with us Algebra 1A teachers, the Algebra 1B and Algebra II teachers continue to be concerned with how Edgenuity presents some of the lessons. In fact, when I explored some Edgenuity lessons, I observe that in the videos, fraction and decimal equations are solved directly without clearing the fractions or decimals first. But then the ensuing quiz asks students to identify the number by which they must multiply both sides to clear the fractions or decimals -- and there are two or three such questions on a ten-question quiz! The Algebra II teachers notice similar problems on their respective quizzes -- to the extent that they're seriously considering foregoing the Edgenuity quizzes and just creating and printing their own tests.

This just goes to show us that we teachers shouldn't blindly assume that Edgenuity is teaching all of the students properly. Throughout the entire process, we must continue to monitor the students to see whether they are actually learning -- or are the videos too confusing.

Conclusion

Today a parade of cars passes near my house. At first I thought it was one of the makeshift graduation parties that you're seeing these days, or maybe even a birthday party. And indeed, there's a neighbor celebrating a birthday party today, but that's not what the parade is for.

It's a protest parade. And since I stepped outside my house and waved to one of the drivers inside, it means that I technically participated in the protest parade.

I am now officially a protester. And I did it from the comfort of my front porch -- and so I'm able to participate while maintaining social distancing.

I will say the following about the George Floyd incident. I recall reading on Facebook that the entire incident happened due to a counterfeit $20 bill, and Floyd was described as "fitting the description" of the man who passed the fake bill. I, of course, have no idea whether Floyd really is the one who passed the bill or not -- and we'll likely never know.

So far, I haven't said anything about race yet. But let's look at this from a mathematical, statistical point of view. First, we know that blacks are likely to have a lower income than whites. Second, it's plausible that low-income people are more likely to pass counterfeit money than high-income people, since after all, the latter have enough real money to avoid the need to resort to passing fake money. So it's possible that blacks, having a lower income, are more likely to pass fake money -- which would mean that innocent blacks are more likely to fit the description (a physical description, which includes skin color) than whites are. (Again, we don't know whether Floyd is innocent or guilty of the counterfeiting charge.)

And this is what people, including a white police officer, sees -- a black suspect fitting the description of someone who committed a crime. Of course, this doesn't justify what happened the following nine minutes at all. But it does raise the underlying issue -- should people be treated differently, based not on whether they've done something wrong, but based on whether they fit the description of someone who's done something wrong? As always, this question has no simple answer.

And this extends far beyond Floyd, beyond even other incidents involving the police. The entire racial traditionalist tracking debate is based on this -- should someone be placed on a low track, not because his/her own scores are low, but because he/she fits the description -- that is, because he/she is of the same race -- as someone who really does belong on the low track? And I've made analogies in previous posts about this same topic -- including analogies about magical red buttons. As usual, this is a question that has no simple answer.

But I do know one thing. As I mentioned in my last post, most of my students at my old charter school were black (with the rest Hispanic). And while I like to remain neutral on highly charged political issues, I know whose side I'm on now -- my students' side. I hope that the result of these protests will be a safer world for them.

Geometry, Common Core Style

Saturday, June 6, 2020

Stewart Chapter 1: Attraction at a Distance

Count On It

Lead vocals by Larry Cedar

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