I've stayed away from the traditionalists for most of the summer. But there's an article over at the Joanne Jacobs website, first posted back on Pi Approximation Day, and I simply can't ignore how some of the traditionalist commenters responded.
Only 45 percent of would-be elementary teachers pass state licensing tests on the first try in states with strong testing systems concludes a new report by the National Council on Teacher Quality. Twenty-two percent of those who fail — 30 percent of test takers of color — never try again, reports Driven by Data: Using Licensure Tests to Build a Strong, Diverse Teacher Workforce.
And even though this refers to licensing tests in many different states, California is specifically mentioned later in the article:
California, which refused to provide data for the NCTQ study, will allow teacher candidates to skip basic skills and subject-matter tests, if they pass relevant college classes with a B or better, reports Diana Lambert for EdSource.
The California Basic Skills Test (CBEST) measures reading, writing and math skills normally learned in middle school or early in high school. The California Subject Matter Exams for Teachers (CSET) tests proficiency in the subject the prospective teacher will teach, Lambert writes.
Anyway, I mentioned the CBEST in one of my COVID What Ifs earlier this summer -- and as I pointed out in that post, I failed the CBEST on my first attempt. Of course, I passed the math section with flying colors, and I did well on the reading section. The problem I had was with the writing section.
For the writing section, we had to write two essays in the allotted time. We were given the topics for the two essays. I had no problem with one of the two topics, but I struggled with the other. This was over a decade ago, so I don't remember the exact topic, but it was something like "write about a time when you disagreed with someone famous." And I simply couldn't think of anything to write about.
There was about half an hour left in the testing period, but I didn't know what to write about. I ended up stalling for most of the time, then wrote a few halfhearted sentences about something -- nowhere near the length of a proper essay. I knew that I'd have another opportunity to pass the writing section.
My first attempt was in December, and my second attempt was in February. This time, I found both required topics easy to write about -- and I could use the entire testing period on writing, since I'd already passed the math and reading sections. So this time, I finally passed the writing section, and so I was ready to proceed with my credential.
But let's see what the traditionalists have to say about those who don't pass the CBEST on their first attempt -- and yes, this includes some familiar names.
GL:
Passing the California CBEST is about the equivalent of earning. 900 on the English/Math parts of the SAT (a pretty low bar for a college grad.).
Bruce William Smith:
To avoid our students meeting such teachers, those of us in schools should examine the equivalent of CSET (CBEST should have been tested far earlier, around the end of junior high school) subjects before allowing any teachers’ college trainees to access our classrooms for internships: if states like California refuse to safeguard academic standards, schools offering internships will have to do so instead.
Darren (Right on the Left Coast):
A high school graduate should be able to pass the CBEST •without trying or practicing•. A potential teacher, a college graduate, should pass that test as easily as swatting a fly. If it were up to me, those who fail the CBEST would not be allowed to retake it for a year.
And the other comments are similar. They are all basically saying the same thing -- because I didn't pass the CBEST on my first attempt, then I'm not smart, I'm unable to write at an eighth grade level, and, most importantly, I'm unfit to be a teacher.
And this is all because one of the two particular writing topics at that December examination simply didn't resonate with me. The February topics did resonate with me, which is why I ended up passing the test in February. If only I hadn't taken the test at all in December, so that February was my first attempt instead, then I would have passed it on my first attempt.
Suppose your CBEST topic were something like "What is your favorite color and why?" and you don't have a favorite color. OK, I grant that you might be able to name any random color and then give enough "reasons" why you like that color to pass the test. Maybe there's something else to fill in that blank, "What is your favorite __________ and why?" in a way that it's not so easy to answer on a timed essay unless you already have a favorite "blank."
Or how about this -- since traditionalist Darren Miller is "Right(-wing) on the Left Coast," suppose the CBEST topic were something that right-wingers might struggle with, such as "Explain why equity is more important than equality." I wager that Miller wouldn't pass that test "as easily as swatting a fly."
Ironically, if I had to write an essay on that December topic today, I could do so easily. I can easily write about someone I disagree with -- namely the traditionalists.
On the other hand, notice that the solution here in California is to eliminate the CBEST and CSET as requirements for a teaching credential. I can't agree with this -- and my reason for disagreeing is the bottom line. After all, I've already passed the CBEST and CSET, and so if those tests were no longer required, I'd lose my advantage. I'd suddenly be competing for teaching jobs with candidates who haven't passed that exam.
This forces me in the middle, defending the status quo. I'd rather keep the tests and allow multiple chances to pass them. I oppose both requiring students to pass on the first try (or force students to wait a year before retrying) and dropping the tests altogether.
A Rapoport Geometry Problem
It's also been some time since I posted Geometry problems from the Rapoport calendar. Once again, it's not that she hasn't included Geometry on her calendar -- it's just that most of the Geometry problems keep landing on non-blogging days. This time I finally catch some Geometry.
Today on her Daily Epsilon on Math 2021, Rebecca Rapoport writes:
Find x.
As usual, all of the given info appears in an unlabeled diagram, so I must label all of this myself. In Triangle ABC, D is on AC so that ray BD is an angle bisector. AB = 68, CB = 40, AD = 51, CD = x.
As it happens, it's a straightforward example of a theorem that doesn't appear in the U of Chicago text -- nor in most other high school Geometry texts -- but it does appear on Rapoport calendars:
Angle Bisector Theorem:
If a ray bisects an angle of a triangle, then it divides the opposite side of the triangle into segments that are proportional to the other two sides.
Once we know this theorem, it's easy to set up and solve the proportion:
51/x = 68/40
x = 30
So the desired length is 30 -- and of course, today's day is the thirtieth.
But if we don't know this theorem -- and most high school Geometry students don't -- then this problem is difficult. So let's try to prove the theorem. The fact that it involves proportional sides of a triangle is a big hint that it has something to do with triangle similarity. And dividing a side proportionally reminds us of the Side-Splitting Theorem in particular.
Keeping this in mind, here is a two-column proof of the Angle Bisector Theorem:
Given: Triangle ABC, BD bisects Angle ABC
Prove: AD/CD = AB/CB
Proof:
Statements Reasons
1. bla, bla, bla 1. Given
2. Angle ABD = CBD 2. Definition of angle bisector (meaning)
3. Auxiliary line thru A parallel to BD 3. Existence of Parallels Theorem
5. Angle ABD = BAE 5. Alternate Interior Angle Consequence
6. Angle BEA = BAE 6. Substitution (4 and 5 into 2)
7. AD/CD = EB/CB 7. Side-Splitting Theorem
8. AB = EB 8. Converse Isosceles Triangle Theorem
9. AD/CD = AB/CB 9. Substitution (8 into 7)
Another Rapoport Geometry Problem
There were several other Geometry problems on the Rapoport calendar this week. This one was particularly tricky for me.
Yesterday on her Daily Epsilon on Math 2021, Rebecca Rapoport writes:
Find AD + AE.
This time, the diagram is partly labeled -- only A, D, E are specified. I'll fill in the rest:
In Triangle ABC, D lies on AC, E lies on AB. AD = CB = 16, DC = 10, Angle A = C, Angle AED is right.
Well, we are asked to find AD + AE and we're given AD, so the main task is to find AE. But I was completed stumped as to how to find AE, and unfortunately I had to go to Twitter to see how others are faring on this problem. They came up with several different ways to solve it, but all of them involve drawing an auxiliary line that never occurred to me. This is the altitude to AC through B. Let's label the foot of that new perpendicular line F.
We are given that Angle A = C, and we also have Angle AED = CFB = 90 and AD = CB = 16. Thus Triangles AED and CFB are congruent by AAS, and so AE = CF by CPCTC.
But what is CF? We notice that for Triangle ABC -- which is isosceles since its base angles A and C are congruent (Converse Isosceles) -- the altitude BF that we drew in is also a median. So CF is exactly half of AC, which we know to be 16 + 10 = 26. Thus CF = 13, and so AE = 13.
Now we can finally add the two sides we need -- AD + AE = 16 + 13 = 29. Therefore the desired sum is 29 -- and of course, yesterday's date was the 29th.
OK, that's enough of Rebecca Rapoport for now. Let's get back to Eugenia Cheng's book.
"Molly squeezes through the exit door and finds herself in a gigantic library. Polished wooden bookcases are bursting with books in all shapes and sizes."
There is another note for Molly on this page:
"These are special books that you can read in any order you like, to make a different story. Can you work out how many different possible orders there are? Turn the wheels to make this number, and you'll be able to reach the key to the next room."
There are five books that fell onto the floor, so the question is asking us how many different ways we can order these five books. Before we return to the math, I admit that this is quite interesting -- here we have a five-part story, and we can read the parts in any order yet they still form a coherent story.
For example, Molly's adventures can almost go in any order without disturbing the story -- yes, her room turning inside-out has to be #1, but she can go through 2-6 in any order before reaching the library here in #7. So let's say she goes 4-5-6-2-3 -- to the hall of endless doors, the boiler room, and the carpet room before going up the impossible staircase and through the garden maze. This is an example of an interchangeable story, but there could be a much more interesting one where, say, there could be either a happy or a sad ending based on which of the five stories is last. (This seems like the sort of thing Douglas Hofstadter might have attempted with one of his Godel, Escher, Bach stories with the Tortoise and Achilles.)
OK, so let's get to the author's explanation:
"There are five books, so there are five possibilities for which one you read first. After that there are only four to choose from for the second book, then three left for the next book, then two, and then there's just one left to be the last book. We can multiply these together to get the total number of possible orders: 5 * 4 * 3 * 2 * 1."
Even though Cheng doesn't use the term, this number is 5 factorial. And so our young readers can figure out that the total number of orders is 5! = 120 -- and that's the number that Molly should enter on the combination lock.
And there's also a challenge for the reader:
"The green bookcase is organized by the patterns on the book's spines. Can you spot the odd one out in each row?"
Which one doesn't belong? This one is too visual for me to describe on the blog -- suffice to know that the extra book has an extra symbol on its spine -- two of some symbol instead of one.
Molly's Mathematical Mysteries 8: Symmetry Garden
Let's begin Cheng's eighth adventure:
"The bookcase creaks open to reveal a luscious green garden. Molly has made her way right through the house!"
There is another note for Molly on this page:
"To open the gate, find one butterfly that is not symmetrical. It should match the butterfly on the gates -- and make them open!"
Symmetry is, of course, very important in Geometry. On this page, all the butterflies are folded up, and so the young reader must unfold the butterflies to determine whether their right halves match up with their left halves.
The author explains the symmetry of the butterfly:
"There are several types of symmetry. One kind is where you can fold something in half and it's the same on both sides -- this is called reflectional symmetry."
She goes on to define rotational symmetry -- as well as a third symmetry that we often overlook:
"There's also a kind where you can move something sideways or in another straight line direction and it looks the same, like a fence or the pattern of a brick wall -- this is called translational symmetry."
Objects that have true translational symmetry are necessarily infinite. Tessellations -- which we've seen earlier in Cheng's book -- have translational symmetry.
And so there is a symmetry corresponding to each of the Common Core isometries. Indeed, an object has T-symmetry for some transformation T if its image under T is itself. Technically, this means that "glide reflectional symmetry" is possible. But since the composite of any glide reflection and itself is a translation, any object with glide reflectional symmetry must also have translational symmetry -- and hence is an infinite object.
And there's also a challenge for the reader:
"Can you find all three types of symmetry in the garden?"
We already know that the butterflies have reflectional symmetry. Some of the flowers have rotational symmetry, and the gate itself is translation-symmetric. We'll go through the gate in my next post.
Lemay: More on Chapter 17
In each of these posts, I'm continuing to return to some of the Lemay chapters that we passed over. I now want to look at Chapter 17.
In this chapter, Lemay introduced exceptions. The problem was that all of her examples were just snippets of code, with no complete executable programs.
Let's try a simple Pythagorean Theorem program. We have a method that takes three integer sides and determine whether they are the sides of a right triangle. But this method can throw exceptions, such as some of the sides being zero or negative, or the three sides not being the sides of a triangle (that is, violating Triangle Inequality).
public class PythagoreanTest { public boolean IsARightTriangle (int a, int b, int c) throws NonPositiveSide, NotATriangle { if (a<=0 || b<=0 || c<=0) throw new NonPositiveSide(); if ((a+b<=c) || (a+c<=b) || (b+c<=a)) throw new NotATriangle(); return ((a*a+b*b==c*c) || (a*a+c*c==b*b) || (b*b+c*c==a*a)); } public static void main (String args []) { PythagoreanTest x = new PythagoreanTest(); try { if (x.IsARightTriangle(3, 4, 5)) System.out.println("3-4-5 is a right triangle"); else System.out.println("3-4-5 is not a right triangle"); if (x.IsARightTriangle(1, 2, 3)) System.out.println("1-2-3 is a right triangle"); else System.out.println("1-2-3 is not a right triangle"); } catch (NonPositiveSide np) { System.out.println("Not all sides are positive"); } catch (NotATriangle nt) { System.out.println("This is not a triangle"); } } } class NonPositiveSide extends Exception { public NonPositiveSide () {} public NonPositiveSide (String msg) { super(msg); } } class NotATriangle extends Exception { public NotATriangle () {} public NotATriangle (String msg) { super(msg); } }
Output:
3-4-5 is a right triangle This is not a triangle
Conclusion
Track and Field at the Olympics have finally started, and I'm definitely enjoying it. At the time stamp of this post, I've watched many of the preliminary heats in the sprint events. The first final with medals awarded is in the 10,000 meters. But the 10K final first aired on Peacock streaming, which I don't have, and are airing on delay on NBC, starting just after the timestamp of this post.
2. Molly and the Mathematical Mysteries 5: Escape the Boiler Room
3. Molly and the Mathematical Mysteries 6: Carpet Weaving
4. Lemay: More on Chapter 16
5. Conclusion
Introduction: Tokyo Games
The Olympic Games are now in full swing. But as I mentioned in my last post, the pandemic continues to cast its shadow over the competitions in Tokyo. And even before the coronavirus, some people have complained about the increasing cost and hassle of the Games, to the point that they ask, should the Olympics be abolished.
Notice that many sports already have their own international tournaments -- and in many cases, these tournaments are more prestigious than their Olympic counterparts. The most well-known example, of course, is soccer. More fans pay attention to the World Cup than the Olympic tournament -- and compared to even continental championships like the recently completed Euro Cup and the ongoing CONCACAF Gold Cup, the Olympics are an afterthought. So if the Olympics were abolished, there are still major international tournaments for soccer teams.
As a former high school track athlete, I often pay special attention to Track and Field. There are World Championships in Track and Field, ordinarily held in odd-numbered years so that they don't clash with the Olympics. (This time, worlds will be in 2022 due to the Olympics this year.) But unlike with soccer, most fans don't pay attention to Track and Field until the Olympics -- we watch Usain Bolt in Beijing, London, and Rio, but not at worlds. So if the Games were abolished, there would still be worlds in Track and Field, but hardly anyone would watch them.
This year, I've turned on the Olympics from time to time. The sports that have aired during my viewing sessions include beach volleyball, swimming, skateboarding, synchronized diving, mens basketball, and mens triathlon. Swimming is in the same boat as Track and Field -- there are World Championships in odd-numbered years, but how many people watched Michael Phelps or Katie Ledecky during those odd-numbered years? So abolishing the Olympics would relegate swimmers to relative obscurity.
As for basketball, this sport ought to be more like soccer, since there's also a basketball World Cup held every four years. But the 2019 World Cup was mostly ignored by Americans, whose team lost unceremoniously to France in the quarterfinals. So international basketball -- just like track and swimming -- is also ignored until the Olympics. Then again, basketball -- like soccer -- has domestic leagues (like the NBA) that fans do enjoy outside of international competition. Basketball players don't need the Games to be famous.
But consider an athlete like Simone Biles. During the runup to the Rio Games, some commentators were already proclaiming her to be the greatest gymnast of all time -- the GOAT -- but others were wondering, how can this "overhyped" gymnast be the GOAT when she hadn't done anything yet? Of course, she had dominated the World Artistic Gymnastics Championships the three previous years, but her sport is just like track and swimming -- only the Olympics matter.
The prime of a gymnast's career is often considered to last eight years -- from her sixteenth birthday to her twenty-fourth. Indeed, the minimum age to compete as a gymnast in the Games is 16, and while gymnasts often do participate past their mid-20's (including a 46-year-old Uzbek who was at Tokyo this year), they usually aren't competitive.
Since a gymnast's prime lasts eight years, we expect there to be two Olympics during that time. But for Biles, who was born on Pi Day 1997, there was only edition of the Olympics during her prime. She was too young for the London Games, and the Tokyo Games are slightly beyond her prime years.
Had Biles been born a mere three months earlier (as calendar years count rather than birthdays), she likely would have participated in the London Games, and possibly would have dominated. Then she wouldn't have been disparaged as a "nobody" leading up to Rio. Then London and Rio would have been her two Olympiads -- dominating there would have cemented her reputation as the GOAT. She then could retire after the 2019 Worlds, especially once the pandemic began.
Instead, 1997 turns out to be a terrible year for a gymnast to be born, especially one who has her eyes set on becoming the GOAT. Since Biles was too young to compete in London, she ends up pushing herself to compete in Tokyo, where the Games are delayed due to the pandemic. She ends up competing past her 24th birthday, making her older than most of her opponents.
And her age showed -- during last night's team competition, Biles struggled. Team USA had to settle for the silver medal, and gymnast herself specifically cited having mental issues. She said that she wasn't having fun in Tokyo -- and I bet she would have had fun in London had she been allowed to compete.
And all of this is because Biles was under so much pressure to win at the Olympics -- because her victories at worlds aren't considered good enough. If the Olympics were abolished, she would definitely be the GOAT based on her performances at worlds and other competitions -- but then she wouldn't be as well-known, since her sport would largely be ignored.
Molly and the Mathematical Mysteries 5: Escape the Boiler Room
Let's begin Cheng's fifth adventure:
"Molly finds the right door, but it swings shut behind her with a loud CLANG. It won't open again! She'll have to find another way out of here and on through the rest of the house."
There is another note for Molly on this page:
"The door needs steam to help it open. Fold the pipes into cubes to make them connect up. To open the exit door, find the net that joins up with no overlaps or gaps."
This puzzle is similar to the Hidden Shapes maze, but I was a bit more confused. It took me some time to figure out that the folding needs to be three-dimensional. The key word in the clue is "cubes." Here the author explains:
"In this room, the folding shapes are made from six squares, which fold up to make cubes. You might wonder what other flat patterns of six squares fold up to make a cube. Not all of them will work...but if it does, it is called a net. We can look for nets of other 3D shapes, too. There are 11 possible nets that make a cube."
Recall that nets appear in Lesson 9-7 of the U of Chicago text, on making surfaces. They are also emphasized in the Common Core Standards, in our Math 6 classes.
And there's also a challenge for the reader. It's to identify which of two possible patterns of squares is really the net of a cube. While I enjoy this type of Geometry problem, it's not possible for me to post such visual problems to the blog.
Molly and the Mathematical Mysteries 6: Carpet Weaving
Let's begin Cheng's sixth adventure:
"The steam-controlled door bursts open...The next room Molly enters is filled with wonderful tapestries. There are carpets all over the floor, the walls, and even the ceiling!"\
"Weave the strips over and under so that only shades of green are showing. It should match the pattern on the loom! If you do this correctly, a door will magically appear. Push it to enter the next room!"
I can see what weaving needs to be done -- indeed, back when I was a young first grader, all of us created an art design based on weaving. The problem is that these aren't my own strips of paper, but strips connected to the page in the book. I fear that by trying to weave the strips, I might accidentally rip them and ruin it for the next reader -- perhaps a young girl waiting to be awed and inspired. So instead I leave the weaving strips alone.
Here is the explanation from the author:
"Take a look at the hanging carpets above. The pattern only shows each shade of green or pink once in each row and once in each column. Both carpets do this, but the patterns are different. This is called a Latin square."
The most common Latin square in our daily lives, by the way, is a Sudoku puzzle. Unlike the 4 * 4 Latin squares in this book, a Sudoku is a 9 * 9 Latin square that satisfies other properties as well (such as the 3 * 3 sub-square rule).
And there's also a challenge for the reader:
"Look closely at other hanging carpets on the wall. Can you find somewhere Molly hasn't visited yet? Perhaps that's where she's headed next!"
OK, I recognize the Hidden Shapes maze, impossible shapes, a self-similar tree fractal, and an Escher impossible staircase. So we'll have to wait until next time to find out Molly's next destination.
Lemay: More on Chapter 16
In each of these posts, I'm continuing to return to some of the Lemay chapters that we passed over. We already spent two posts on Chapter 15, and so I want to look at Chapter 16.
In this chapter, Lemay introduced packages and interfaces. I'm hoping to write code to implement a Quadrilateral Hierarchy using interfaces. In C++ we'd use multiple inheritance to code this hierarchy, but according to Lemay, this is not possible in Java. So we must use interfaces instead.
Let's start by writing a basic quadrilateral class. What information do we need to keep track of in order to determine our quad? If this had been a triangle instead, we might keep track of the three sides -- then we could use the Law of Cosines to find each angle. Or we might keep track of ASA instead -- then the third angle is found using Triangle Sum and then the other two sides using the Law of Sines. Come to think of it, any triangle congruence theorem can be used -- SSS, SAS, ASA, AAS.
This suggests that for quads, we should use a quadrilateral congruence instead. We've discussed these on the blog before -- two such congruences are SASAS and ASASA. But which is better? To answer this, we must keep in mind that we're eventually going to write interfaces for all of the special quads in the hierarchy -- isosceles trapezoids, kites, parallelograms, and so on.
Let's try SASAS first. How can we tell, given SASAS, that a quad is a trapezoid? Well, the sides don't help us, but the angles do -- if two adjacent angles are supplementary, then it's a trapezoid. (This is the Trapezoid Angle Theorem, Lesson 5-5, U of Chicago text.) Notice that the two A's in SASAS need not be the supplementary ones -- that is, if in Quad ABCD the given parts are AB, Angle B, BC, Angle C, CD, then B and C aren't necessarily supplementary angles. It could be that A and B are supplementary (and hence C and D as well) instead. But that might not matter for the program just yet -- when implementing a trapezoid, we might force B and C to be the supplementary angles.
Now let's consider isosceles trapezoids. And that's where our problems begin -- if ABCD is an isosceles trapezoid with B and C as the supplementary angles, then the congruent sides are BC and AD -- rather than the desired given AB and CD (unless the isosceles trapezoid happens to be a rectangle). Given the SASAS condition, we're more likely to make AB and CD the congruent sides, which then makes B and C be congruent angles. It turns out that this is indeed sufficient for ABCD to be an isosceles trapezoid (with the proof given years ago on the blog).
But then we'd be in an odd situation -- for trapezoids, B and C are supplementary, while for isosceles trapezoids, B and C are congruent. So our computer would conclude that an isosceles trapezoid isn't a trapezoid necessarily, despite the name. This is the opposite of what we want.
The solution is to use ASASA instead of SASAS. The three given angles are now A, B, C, and the given sides are AB, BC. This solves the problem -- for trapezoids, either A, B or B, C are supplementary, and for isosceles trapezoids, A, C are supplementary with B congruent to either A or C. We also proceed down the hierarchy -- for parallelograms, A, C are congruent with B supplementary to both, and for rectangles, A, B, C are all 90 degrees. We can see why ASASA is superior to SASAS -- the angles provide us more information (particularly about parallel sides) than the sides, and so it's better to be given more angles.
This leaves us with the kite and the figures below it in the hierarchy In fact, rhombi (respectively squares) are easy -- we just use parallelograms (respectively rectangles) with AB = BC. As for kites, sufficient information for a kite is to have A, C congruent and AB = BC. The only problem is that we could have a kite with angles B, D congruent and BC = CD instead. This is analogous to what would have happened with trapezoids under SASAS, and our solution here is similar -- when implementing a kite, we force A, C to be the congruent angles and B, D to be the ends.
OK, so let's start writing code. Since we're learning about packages, let's write our Quad class as part of a quadrilateral package. All five side and angle measures will be of type double. Let's start by having accessor methods for each of the variables, and a constructor:
package quadrilateral; public class Quad { privatedouble A, AB, B, BC, C; public double getA () {return A;} public double getAB () {return AB;} public double getB () {return B;} public double getBC () {return BC;} public double getC () {return C;} public void setValues (double X,double XY,double Y,double YZ,double Z){ A = X; AB= XY; B = Y; BC = YZ; C = Z; } }
Notice that there's one more method we can easily add here -- a getD method. This is easy since the angles of a quad add up to 360:
public double getD () {return 360-A-B-C;}
Conclusion
Right now (as of the time stamp of this post), I'm watching the Olympics. There is an interview on right now with three gymnasts -- the three teammates of Simone Biles. Once again, they are discussing the problems their famous teammate is having with the competition this year.
I look forward to watching Track and Field, which will start in a few days. Five years ago at Rio, the Olympics started later in August, and many of track finals were past the first day of school. I missed many of the races because I was busy at the old charter school, trying to begin my career as a math teacher. Many people will watch our sport for ten days -- and then ignore our sport until the Paris Games in 2024, despite there being World Championships in 2022 and 2023.
Some people think that the Tokyo Games should have been delayed yet again to 2022. Of course, there will also be the Beijing Winter Games next year -- but then again, having the summer and winter Games in the same year was the norm through 1992. Some people think that all Olympics should be cancelled until the pandemic is "over" -- but when will that be? As I wrote in my last post, even the vaccines aren't enough to end the pandemic. There might be a new Greek-letter variant of the coronavirus that breaks out just in time to ruin the Paris Games in 2024, or even our local Olympics here in Southern California in 2028. If the pandemic doesn't complete end until 2031 (as I feared in my last post), then all Olympics would need to be cancelled until the Brisbane Games in 2032.
Or, as some people have suggested even before the pandemic, the Olympics simply should be abolished once and for all. So instead of ignoring Track and Field for the next three years, sports fans would be free to ignore our sport forever.
Some anti-Olympians might point out that if a fast sprinter wants eyeballs, then he should just quit Track and Field and become a running back in the NFL, where he'll get all the attention he wants. But that doesn't really help our female Olympians. The Games are their only stage -- if we abolish the Olympics, then thus ends their path to fame and fortune.
I've been told that when I was a young three-year-old, I was excited to hear about and watch the Olympics -- the last time they were held in my hometown. And so I, as a former high school track athlete, will continue to defend the Games.
Today is Pi Approximation Day, so-named because July 22nd -- written internationally as 22/7 -- is approximately equal to pi.
It also marks seven full years since my very first post here on the blog. Yes, I've declared Pi Day of the Century -- March 14th, 2015, to be the ceremonial start date of this blog. But the chronological start date was July 22nd, 2014. I like celebrating both Pi Day and Pi Approximation Day -- including eating pie both days -- and so I celebrate both of them as launch dates for the blog.
Last year, I used the anniversary to create a FAQ about my blog and myself. I do the same this year -- and this year's FAQ is very different from last year's because I'm focusing on many coronavirus-related questions. As usual, let me include a table of contents for this FAQ:
1. Who am I? Am I a math teacher? 2. How will the coronavirus affect my current employment? 3. Who is Eugenia Cheng, and what is the Garden of Hidden Shapes? 4. What is the Hall of Endless Doors? 5. What is the actual reopening plan in my OC districts? 6. Who is Laura Lemay, and why am I learning Java? 7. What's "Mocha music"? 8. Who is Rebecca Rapoport? 9. What is Shapelore? 10. What is a COVID What If? 11. How will the coronavirus school plans affect this blog (and Twitter)? 1. Who am I? Am I a math teacher? I am David Walker. I'm trying to start a career as a math teacher, but I must admit that my career isn't off to as good a start as I'd like.
Five years ago was my first as a teacher at a charter middle school, but I left that classroom. By now, I fear that I won't be hired to teach this year at all. If I'm not hired, then I'll remain a substitute teacher. But this will make the launch of my teaching career that much more difficult. It's not that I'm having trouble finding work as a sub -- it's that I'm struggling to find work as a regular teacher. As I explained on the blog last month, I didn't always make the best choices inside and outside of the classroom. These decisions have weakened my resume, and so the human resources departments at most schools and districts will reject my application.
So the answer to this question is no, I'm not currently a math teacher, and I'm losing hope that I'll be one again soon.
2. How will the coronavirus affect my current employment?
I currently work as a substitute teacher in two districts here in Southern California. Both of these districts are in Orange County.
At this time last year, I worked in two districts -- one in LA County, the other in Orange. But my LA County district had always called me much less often than my OC district -- and this was even before the pandemic. During the pandemic, my OC district reopened in late September while my LA district remained closed, so at that point all of my subbing calls were from OC.
At that point, a second OC district hired me for a long-term subbing position. I covered both Math 7 and Math 8 from late September to early January. Once that position ended, I was still in the system for both OC districts, and so I continue to work for both of them.
My LA County district finally reopened in early April, and so during April and May I was working in three different districts. But subbing in three districts is awkward, and so I wish to return to two districts in the new year.
And even though I received many calls from LA County after the pandemic, I just can't trust the district to provide me with regular work in the new year -- especially considering how few calls I received the first four years that I worked there.
And so my final decision is for me to keep accepting calls in both of the Orange County districts and forget about LA County entirely.
3. Who is Eugenia Cheng, and what is the Garden of Hidden Shapes?
Eugenia Cheng is a mathematician and author. She has written five books -- the first four are mainly for adults, but her most recent book, Molly and the Mathematical Mysteries, is for children. And her children's book is also our side-along reading book for the second half of summer vacation. So let's go back to where we left off from my last post.
Let's begin Cheng's third adventure, called "The Garden of Hidden Shapes":
"Phew...Molly is glad she found a way out of that Impossible Staircase. Now she's in a huge garden. It's some kind of maze. From her starting point, Molly can see a bright triangular flower bed, a patterned pond of swimming fish, and an odd sculpture. But where is the way out?"
There is another note for Molly on this page:
"This challenge is all about shapes. Can you unfold the hexagon to make a big triangle in two different ways? One of them will make a path to the house."
This is still technically a Geometry blog, and so I definitely enjoy this page. By the way, the "triangular flower bed" is shaped like Sierpinski's triangle, and the "odd sculpture" is an impossible structure, not unlike the Impossible Staircase that Molly just left on the previous page.
But the emphasis here needs to be on solving the maze, which I can't fully describe on the blog. The author tells us more about the hexagon that the reader must unfold to produce a maze solution:
"Sometimes shapes fit together to make other shapes. In this maze, six triangles fit together to make a hexagon. The triangles are special ones where all three sides are the same length: these are called equilateral triangles."
She proceeds to describe tessellations. Tessellations, or "tiling the plane," appear in Lesson 8-2 of the U of Chicago Geometry text.
And there's also a challenge for the reader:
"This hexagon is made up of six tiled triangles. How many other tiling shapes can you find in the garden?"
In fact, the "patterned pond of swimming fish" mentioned earlier is a tessellation of fish.
4. What is the Hall of Endless Doors?
Let's begin Cheng's fourth adventure, called "The Hall of Endless Doors":
"Molly makes her way through the maze and reaches the front door of the house. She steps inside and looks around in awe. In front of her is a grand staircase and a dizzying hallway full of doors. Where could they all lead?"
There is another note for Molly on this page:
"To move forward, choose a door carefully. But watch out -- some doors lead to hallways that go on forever...you don't want to pick one of those!"
The author's math lesson for this page is "self-similarity":
"The doors in this hallway look like they go on forever! Each door pattern has a smaller door inside it, with an even smaller door inside that, and so on...They're a bit like nesting dolls. We could keep zooming in forever...providing the doors keep going on forever."
The name "self-similarity" is weird, but it makes sense. We know that a figure has rotational symmetry if there exists a rotation mapping it to itself. Anyway, a figure has "dilational symmetry" -- more commonly known as self-similarity -- if there exists a dilation mapping it to itself. And just as the trivial rotation of 0 (or 360) degrees doesn't count, neither does the trivial dilation of scale factor 1. The scale factors of all the dilations on this page are slightly less than 1, so that the effect is gradual.
"In math, something that keeps going forever is called infinite. Let's imagine an infinite set of dools. You could keep opening them forever without running out of dolls, but they'd get so tiny iy would be very hard to see them."
Infinity -- or should I say, beyond infinity -- was the theme of Cheng's second book if you recall.
And there's also a challenge for the reader:
"How many nesting dolls can you find hidden in the scene? Can you match up three sets of five dolls?"
One of the three sets is right on the table/dresser, but the other two sets are spread out on the page. I also enjoy looking at the numerous other designs on the screen. One of them is the infinite tree that we programmed Logo to draw back in late March (Lesson 13-8 of the U of Chicago text).
Doors 1, 2, 3 are all infinite dead ends. Door 4 leads to Molly's next adventure -- and my next post.
5. What is the actual reopening plan in my OC districts?
Both districts are hoping to bring all students back for a full five days per week, with the overwhelming majority of these students being in-person. This should end the hybrid plan once and for all. But not everything will be completely back to normal. At the very least, students and teachers must continue to wear masks in the classroom, according to Governor Gavin Newsom.
The reason for the continued masks can be summarized in one word -- Delta. And that's not "delta" as the mathematical symbol for "change," often used in the slope formula. Though it does have something to do with change -- as in the coronavirus changing. The coronavirus numbers had been decreasing until about a month ago, when the Delta variant of the virus fueled a sharp increase in cases.
With the rise of the Delta variant, there's no way that we can remove masks. Indeed, LA County has even declared a return to masks for all indoor public places -- even for vaccinated people.
Governor Newsom has already ended color-coded tiers on June 15th -- just before the Delta surge. But it was pointed out that had the color tiers still existed, LA County -- which had advanced all the way to the yellow tier at the time the tiers ends -- would have regressed all the way back to the red tier. Recall that in the red tier, schools can reopen, but most of them would be open in some sort of hybrid. My LA County district would be in Stage 2, with one day of attendance per week -- and LAUSD itself would have 2-3 days of in-person "Zoom in a room," and so on.
Despite returning to the equivalent of the red tier, most districts in LA and Orange County plan on reopening for in-person attendance five days per week. This could be because of the ineffectiveness overall of the hybrid plans -- the at-home students never succeeded as well as the in-person students.
Last year at this time, I was both overly pessimistic and overly optimistic about the pandemic. While I was hoping to be vaccinated by Tau Day 2021, I was pessimistic that it could take several years for a vaccine to be developed. As it turns out, the vaccines were announced in November, and I received both doses of the Moderna vaccine in May, thus beating my vaccination goal by a few weeks.
But I was optimistic that once the vaccines were developed, we could distribute it widely, and then the pandemic would soon be over afterward. What I didn't imagine was the current situation -- the vaccines are out, and yet the pandemic is still going strong with the Delta variant. It's often stated that we needed to reach a 70% vaccination rate in order to achieve herd immunity. But we're just shy of that level -- because less than 70% are willing to be vaccinated. There are just enough people choosing to remain unvaccinated to allow the Delta variant to spread.
I'm thinking about the Tokyo Olympics -- the opening ceremonies are set for tomorrow. Had the Olympics gone on last summer as scheduled, it would have been a disaster -- a few athletes would have tested positive for the virus each day, and the entire Games would have had to be set in a "bubble" without fans (not unlike the 2020 NBA playoffs that were contested at around the time the Olympics should have been). So instead, we waited until this year to hold the Games -- and now a few athletes are testing positive for the Delta variant each day, and the entire Games are now in a bubble without fans.
When the delay was first announced, there was a possibility that it would have been only about nine or ten months, to spring 2021. Notice that had the Games been contested in May-June, we could have vaccinated the athletes, fill the stadiums with plenty of fans, and returned the athletes safely to their home countries all before Delta had a chance to spread. But by waiting a full 12 months, that gave just enough time for Delta to spread and ruin the Olympics. Indeed, now many people (including the Japanese themselves) are questioning the wisdom of holding the Games at all.
Indeed, this is sort of like what happened with high school sports in California. The state cancelled all sports until December, when schools can start holding practices for January contests. But then there was a November virus surge -- and sports had to wait for February practices and March contests. On the other hand, we were between virus waves in August-October -- so fall sports could have started at their normal time, before the third wave. Most likely, playoffs and state championships (scheduled for November) would still have been cancelled -- but recall that fall playoffs were really cancelled anyway, so it wouldn't have made a difference, except for the regular seasons being played at their proper time.
In last year's annual FAQ, I wrote about how I often divide my life into two- and four-year chunks:
We need to be thinking in terms of periods longer than a year. Earlier on this blog, I mentioned the period of time I call a quaver, or half a quarter (named after a British musical eighth note). I also use musical names for periods longer than a year -- I call two years a breve (a double whole note) and four years a longa (a quadruple whole note).
I first came up with dividing my life into breves and longae when I was in college. I noticed that I had first enrolled in my new high school on November 3rd, 1995, was transferred to a Chemistry class (and ultimately to the magnet program) on November 3rd, 1997 (exactly one breve later), and started my first day of work at the UCLA library on November 3rd, 1999 (exactly one longa later).
Thus I divided my life into breves and longae, beginning with November 1995. Indeed, I've given names to these time periods:
1. Nov. 1995-1999: The High School Longa a. Nov. 1995-1997: The Academy Breve b. Nov. 1997-1999: The Magnet Breve 2. Nov. 1999-2003: The Bruin Longa a. Nov. 1999-2001: The Undeclared Breve b. Nov. 2001-2003: The Math Major Breve 3. Nov. 2003-2007: The Between Longa a. Nov. 2003-2005: The Gap Breve b. Nov. 2005-2007: The Local Library Breve 4. Nov. 2007-2011: The Preliminary Longa a. Nov. 2007-2009: The Preliminary Start Breve b. Nov. 2009-2011: The Preliminary End Breve 5. Nov. 2011-2015: The Clear Longa a. Nov. 2011-2013: The Clear Start Breve b. Nov. 2013-2015: The Clear End Breve 6. Nov. 2015-2019: The Teaching Longa a. Nov. 2015-2017: The Charter Breve b. Nov. 2017-2019: The Two County Breve
I actually announced the end of the fifth longa here on the blog.
Breves and longae have significance outside of my life. Longae are especially important on the Julian and Gregorian Calendars since Leap Days occur approximately once per longa -- about four months after the start of the longa. (Breves occasionally occur on Reform Calendars with 8-10 days per week, since 730 is closer to a multiple of these week lengths than 365 is.)
Presidential elections occur once per longa. Congressional elections occur once per breve, with Election Day about halfway through each breve.
The Summer Olympics usually occur once per longa, while each breve contains either a Summer or Winter Olympics. Even with Tokyo delayed until tomorrow, these Games are still occurring before the end of the current breve.
The longae are named after what diploma I was trying to earn at the time -- high school, college, preliminary credential, or clear credential. Unlike the first breve and longa, my major life events didn't always happen on November 3rd, so the names are approximate. For example, the California BTSA program to clear my credential takes two years -- that is, one breve. But that time doesn't line up with any of the breves listed above -- it does fit fully in the fifth longa. Meanwhile, I was only at the charter school for under a year, but it does fit fully in Breve 6A.
But what shall we call the current seventh longa and breve?
And that takes us back to 2021. I was going to name it after the pandemic -- the length of the pandemic would determine whether it counts as a breve or a longa. But in that post, I kept equating the arrival of a vaccine with the end of the pandemic. So I wrote things like:
7. Nov. 2019-2023: The ??? Longa a. Nov. 2019-2021: The Coronavirus Breve
if a vaccine is developed this breve and:
7. Nov. 2019-2023: The Coronavirus Longa
if there's no vaccine. What I didn't foresee is that we'd have a vaccine here in Breve 7A, and yet the pandemic stretches into 7B due to the Delta variant and many people rejecting the vaccine.
Of course, the breves and longae are supposed to be named after the stages of my own life -- not the pandemic or anything else occurring in the world. Then again, the pandemic has caused me to make some life decisions. The main event of Breve 7A for me is the long-term subbing position -- and the fact that it apparently won't lead to a full-time teaching career will force me to make a huge decision in Breve 7B -- perhaps even a new career:
7. Nov. 2019-2023: The Coronavirus Longa a. Nov. 2019-2021: The Long-Term Breve
b. Nov. 2021-2023: The New Career Breve
I'll make another decision about breve naming this November, as that's when the breve is set to change to a new breve.
As for the pandemic, last year I worried that we might not have a vaccine until Tau Day of the Century in the year 2031. It could be that we have the vaccine now, yet the pandemic still doesn't end until 2031 due to more Greek-letter variants (including the Lambda variant which is just now emerging) -- including several vaccine-resistant strains, and new vaccines to battle those strains resistant to the old strains (also known as "booster" shots).
And of course, if all else fails, it could me that the only way for me to make any money is to leave the field of education altogether.
6. Who is Laura Lemay, and why am I learning Java?
Laura Lemay is the author of an online book about the Java computer language. Here's a link to her online text:
If the schools are shut down for an extended period of time -- and it becomes impossible for subs like me to get work -- then I must seriously consider looking for a job other than teaching.
The cliche response to someone who can't get a job is "learn to code." Well, that's exactly what I'm trying to do here. I chose Java because it's currently the language taught in AP Computer Science -- if AP considers Java to be a language worth learning, then I do as well. I assume this means that there are coding jobs out there where Java is the language of choice.
I've currently involved myself in a number of summer projects that I'm describing on the blog. The most important of these by far is my learning Java project.
So far I've reached Lesson 18 of Lemay. But recently I've been going back a few lessons to redo a few programs that I missed or had trouble figuring out. Today, let's return to Lesson 14:
I've tried out most of these programs, but I skipped Listings 14.5 and 14.6 because both of them involve websites that existed back in the 1990's when Lemay first wrote this book -- these websites might not necessarily exist these days. Today I at least wish to do Listing 14.5:
Listing 14.5. Bookmark buttons.
1: // Buttonlink.java starts here
2: import java.awt.*;
3: import java.net.*;
4:
5: public class ButtonLink extends java.applet.Applet {
6:
7: Bookmark bmlist[] = new Bookmark[3];
8:
9: public void init() {
10: bmlist[0] = new Bookmark("Laura's Home Page",
11: "http://www.lne.com/lemay/");
12: bmlist[1] = new Bookmark("Gamelan",
13: "http://www.gamelan.com");
14: bmlist[2]= new Bookmark("Java Home Page",
15: "http://java.sun.com");
16:
17: setLayout(new GridLayout(bmlist.length,1, 10, 10));
18: for (int i = 0; i < bmlist.length; i++) {
19: add(new Button(bmlist[i].name));
20: }
21: }
22:
23: public boolean action(Event evt, Object arg) {
24: if (evt.target instanceof Button) {
25: linkTo((String)arg);
26: return true;
27: }
28: else return false;
29: }
30:
31: void linkTo(String name) {
32: URL theURL = null;
33: for (int i = 0; i < bmlist.length; i++) {
34: if (name.equals(bmlist[i].name))
35: theURL = bmlist[i].url;
36: }
37: if (theURL != null)
38: getAppletContext().showDocument(theURL);
39: }
40: } //ButtonLink.java ends here
41:
42: //Bookmark.java starts here
43: import java.net.URL;
44: import java.net.MalformedURLException;
45:
46: class Bookmark {
47: String name;
48: URL url;
49:
50: Bookmark(String name, String theURL) {
51: this.name = name;
52: try { this.url = new URL(theURL); }
53: catch ( MalformedURLException e) {
54: System.out.println("Bad URL: " + theURL);
55: }
56: }
57:} //Bookmark.java ends here
But no, it doesn't work. For some reason, line 5 produces a syntax error:
5: public class ButtonLink extends java.applet.Applet {
There is absolutely nothing wrong with this line -- it comes directly from Lemay, and this is the correct first line for an applet class. Yet there's something about this line that's preventing me from running this applet now.
This is the sort of thing that I need to figure out before I can launch a post-teaching career. I'm not a coder unless I can get the computer to work -- and right now, I can't do that yet.
7. What's "Mocha music"?
In many recent posts, I refer to something called "Mocha music." This is a good time to explain what Mocha music actually means.
When I was a young child in the 1980's, I had a computer that I could program in BASIC. This old computer had a SOUND command that could play 255 different tones. But these 255 tones don't correspond to the 88 keys of a piano. For years, it was a mystery as to how SOUND could be used to make music. Another command, PLAY, is used to make music instead, since PLAY's notes actually do correspond to piano keys.
Last year, I found an emulator for my old BASIC computer, called Mocha:
When we click on the "Sound" box on the left side of the screen, Mocha can play sounds, including those generated by the SOUND command. So finally, I could solve the SOUND mystery and figure out how the Sounds correspond to computer notes.
I discovered that SOUND is based on something called EDL, equal divisions of length. We can imagine that we have strings of different lengths -- as in a string instrument or inside a piano. The ratio of the lengths determine their sound -- for example, if two strings are in a 2/1 ratio, then the longer string sounds an octave lower than the shorter string.
The key number for SOUND is 261, the "Bridge" (or end of the string). Mocha labels the Sounds from 1 (low) to 255, so we subtract these numbers from 261 to get a Degree ranging from 260 (long string) to 6 (short string). The ratios between the Degrees determine the intervals. I found out that the Degrees corresponding to powers of 2 (8, 16, 32, 64, 128, 256) sound as E's on a piano, with Degree 128 being the E just above middle C (that is, E4).
Let's say we were to play the following two notes on Mocha:
10 SOUND 51,8
20 SOUND 86,8
The second number 8 indicates a half note, since 8 is half of 16 (the whole note). But we want to focus on the first numbers here, which indicate the pitches (tones).
We first convert the Sounds to Degrees. Since 261 - 51 = 210, the first note is Degree 210. The Degree of the second note is 261 - 86 = 175. Now the ratio between these two Degrees is 210/175, which reduces to 6/5. This is the interval of a minor third, so the two notes are a minor third apart. As it turns out, the two notes sounds as G and Bb -- "rugu G" and "rugugu Bb."
Let's try another example:
30 SOUND 144,8
40 SOUND 196,8
Warning -- we don't attempt to find the ratio 196/144 (which is 49/36 by the way). We only find the ratios of Degrees, not Sounds. The Degrees are 261 - 144 = 117 and 261 - 196 = 65. Thus the interval between the notes is 117/65 = 9/5, a minor seventh. (Using Degrees instead of sounds makes a big difference, since 49/36 would be an acute fourth or tritone, not a minor seventh.) The names of the two notes played by Mocha are "thu F" and "thugu Eb."
Where do all these strange color names like "gu/green" and "thu" come from? Actually, they refer to Kite's color notation, and the colors tell us which primes appear in the Degree:
white: primes 2 or 3 only
green: prime 5
red: prime 7
lavender: prime 11
thu: prime 13
su: prime 17
inu: prime 19
Kite's color notation also uses colors such as yellow, blue, and so on. But these are "otonal" colors, while EDL scales/lengths of string are based on "utonal" colors only.
The website where Kite explains his color notation is here:
Of course, the whole purpose of this is to compose music. When I was at the old charter school, I used to compose my own math songs for the class, but this was before I discovered that there was a Mocha emulator. Now that I know about Mocha, I can generate random music on Mocha and then make new songs.
By the way, you might notice that I'm not using Java to compose these songs. Deep down, I was hoping that perhaps Lemay wouldd teach us how to do everything above in Java instead of BASIC. But instead, Lemay only tells us how to include sounds from files into our applets -- and just as I don't have access to the CD containing images, I can't access any of these sounds either.
Then again, how would I play any of these songs in an actual classroom (if I ever get back there)? It may be possible to play some Mocha music if the school computers can access Mocha. But in general, I'd like to play some of these songs on my guitar. In fact, in previous posts, I started thinking about how we might refret a guitar so that it plays 18EDL music (that is, by spacing the frets equally apart, 1/18 of the string length, so that it really is an "equal division of length").
Unfortunately, my current guitar is broken. After leaving the old charter school, the tuning knob on one of the strings, the D string, doesn't turn properly. I can tune the string down, but not up.
Thus the note played by the string gradually became flatter. For a while, the note played by the string was at least a third-tone -- nearly a semitone -- too flat. In Kite's color notation, if the correct tuning of the string is white D, it became tuned to red C# instead.
I tried tuning all the other strings so that they were in tune with the flattened D string. But this meant that they were no longer in tune with the outside world. In fact, last Christmas I watched a holiday special, Grandma Got Run Over by a Reindeer. I had the sheet music for the title song, and I tried playing my guitar along with the song during the special. But my guitar wasn't in tune -- the show played the song in the key of E major, while my guitar was more like Eb major.
I was so upset that I tried to tune my D string back to D -- but instead, I made it worse. My D string is now about a whole tone too flat, sounding more like a C.
Afterward, I remembered that there was a guitar shop near one of the schools where I sub. I was thinking about going in to replace the tuning knob one day -- perhaps around the last day of school, and definitely by next Christmas. But you already know what happened -- the coronavirus shut everything down before I had a chance to go in.
It's also possible for me to tune all the strings down a whole tone -- and instead of a tuning knob, I could get a capo and place it at the second fret to recover the original tuning. But I'm tired of having an out-of-tune guitar -- in fact, I used to have perfect pitch, where I could hear a note and identify which note it is. But my out-of-tune guitar corrupted my perfect pitch -- if a note sounded like my D string, I'd want to identify it as D, when in reality it was a C# (or even a C).
Notice that I can kill two birds with one stone here -- I might find a retuning so that my D string can play as the C that it's stuck at, and also anticipate a retuning that fits with Mocha music and a possible 18EDL refretting (that is, whenever I can reach a guitar store again).
In a previous post, I mentioned several possible 18EDL tunings. Unfortunately, all of those tunings involving keeping the D string as D. Here's a possible retuning:
EACGAE
I like to keep as many strings tuned to standard tuning as possible. This tuning keeps four of the six strings -- in addition to the broken D string, the B string is tuned down to A. With four strings tuned correctly, I hope that I'll be able to regain perfect pitch.
In the tunings that I've mentioned earlier, all the open strings are assumed to be white. But for this tuning, the C and G strings are green while the rest are white. (Or if you prefer, keep the C and G white and tune the other strings yellow, which is equivalent.) Using this tuning, the open strings form a just A minor 7th chord (Am7/E).
Here are the notes playable in EACGAE if C and G are green and the rest white:
wa E | su E#, wa F#, gu G, ru G# wa A | su A#, wa B, gu C, ru C# gu G | sugu G#, gu A, gugu Bb, rugu B gu C | sugu C#, gu D, gugu Eb, rugu E wa A | su A#, wa B, gu C, ru C# wa E | su E#, wa F#, gu G, ru G#
I'm not quite sure how this tuning will play out in practice, though. For example, it's easy to see how to play a C major chord (xx0030), but A minor chord is tricky. The strings are tuned to Am7 and we'd like to get rid of the G to make it Am. But the A at the second fret on that string is green A, while the open strings are white A. It might sound better just to play Am7 instead of Am. Likewise, an Em chord would need to be played as Em7 (022020).
Meanwhile, G major would be playable except that there aren't quite enough open strings. Instead, we must settle for G/B (x22023) or G/D (xx2023), or perhaps a G6 (322020) or Gadd9 chord (323003, but the add9 here is 20/9 instead of the usual 9/4). While D major isn't playable, there is a playable D7 chord (as xx2232).
(If we prefer all the strings to be white, we might try EABEAE -- one of the tunings I mentioned in a previous post, though it wasn't preferred. But if I tune my D string down to B, then I won't be able to tune it back up to C, since I can only tune it down and not up. I don't wish to commit to this tuning, especially since only three strings would match standard tuning rather than four.)
During my long-term position, I played songs using this tuning, EACGAE (standard fretting, not any EDL fretting.)
So instead, let's just code a pi song based on 16EDL, similar to the song we played for Tau Day:
NEW 10 N=16 20 FOR X=1 TO 32 30 READ A 40 SOUND 261-N*(17-A),4 50 NEXT X 60 DATA 3,1,4,1,5,9,2,6,5,3 70 DATA 5,8,9,7,9,3,2,3,8,4 80 DATA 6,2,6,4,3,3,8,3,2,7 90 DATA 9,5
As is traditional, I stop just before the first zero. Then digits 1-9 map to Degrees 16 down to 8, with the lowest note played on E (line 10, N=16). We can change the value of N to any value from 1 to 16 to change the key.
Here's an actual song converted to 12EDL, a simpler EDL scale, the Sailor Pi theme song:
NEW 10 N=13 20 FOR X=1 TO 26 30 READ D,T 40 SOUND 261-N*D,T 50 NEXT X 60 DATA 8,4,8,2,9,4,9,2,10,4,11,4,9,12 70 DATA 9,4,9,2,10,4,10,2,11,4,12,4,10,12 80 DATA 12,4,12,2,10,4,10,2,8,4,6,4,7,12 90 DATA 8,4,9,4,10,2,11,6,12,16 Only the main verse is coded here. The "bridge" part -- which is instrumental in both the original Sailor Moon and Lizzie's Sailor Pi song -- is too hard for me to code without sheet music.
Here are the lyrics for the first verse -- the part which we coded above:
Fighting fractions by moonlight Perplexing people by daylight Reading Shakespeare at midnight She is the one named Sailor Pi.
Speaking of Bizzie Lizzie, today's a great day to post lyrics after all -- the lyrics to her "Digit Connection," a parody of "Rainbow Connection," which I posted on the blog 2 1/2 years ago:
THE DIGIT CONNECTION Copyright 1998 by Elizabeth Landau, aka Bizzie Lizzie
1st Verse: Why are there so many debates about pi? And what's on the other side? Pi is a ratio of random proportions. Its digits have nothing to hide. So we've been told and some choose to believe it, But I know they're wrong, wait and see! Someday we'll find it, the digit connection, Mathematicians, logicians, and me.
2nd Verse: Who said that everything has some sort of pattern, Consisting of nothing but math. Somebody thought of that, and someone believed it. Now we're all caught in its wrath.... What's so hypnotic in something chaotic, And what do we think we might see? Someday we'll find it, the digit connection, The optimists, the theorists, and me.
All of us under its spell, We know it must be math-e-magic...
3rd Verse: Have you been half asleep? And have you heard voices? I've heard them calling my name. Is this the sweet sound that calls the young sailors, The voice might be one and the same.... I've heard it too many times to ignore it, Irrational, random, and free. Someday we'll find it, the digit connection, The lovers, the dreamers, and me.
3.1415926535 dot, dot, dot!
Bizzie Lizzie also had an "American Pi" song, but there's just one problem. I copied down the lyrics from her old website just before that site disappeared -- into my notebook. That's right -- the notebook that I took out on Pi Day Eve to sing to the class I subbed that day, and just left in the classroom.
Fortunately, I finally found that book, and I posted the lyrics to the song in time for Pi Day.
Bizzie Lizzie's e song was a parody of "Sugar, Sugar," by the Archies. I think I recall the refrain:
e (2.718) Ah, number number (281828) You are my natural log, And you got me calculating. e (2.718) Ah, number number (281828) You are my derivative, And you got me calculating. But alas, I can't remember the rest of the song. Well, since it's e Day, let me supply extra lines. Some of these lines are from my faint memories of Landau's original song -- I made up all the lines that I couldn't remember.
1st Verse: I just can't believe the loveliness of graphing you. I can't believe you're more than two. I just can't believe the loveliness of graphing you. I can't believe you're more than two. (to Refrain)
2nd Verse: I just can't believe your digits go forever now. As long as a number can be. I just can't believe your digits go forever now. As long as you're the number e. (to Bridge)
Bridge: Put a little cash in the bank, money. Put a little cash in the bank, baby. I'll make more next year, yeah, yeah, yeah! Put a little cash in the bank. 100% interest on my money. Compound it continuously, baby. I'm gonna take the limit now, yeah, yeah, yeah! My cash is multiplied by you, e. (to Refrain)
The bridge is mostly mine -- Landau didn't mention anything about money in her song. I chose to include money since it rhymes with the original Archies lyrics ("honey") as well as retell the story of Jacob Bernoulli's discovery of this constant.
8. Who is Rebecca Rapoport? Rebecca Rapoport is the author of Your Daily Epsilon of Math calendar. In most years, she produces a calendar that provides a math problem. The answer to each question is the date. I will post the Rapoport questions for each day that I blog. Traditionally I would post only her Geometry questions, but ever since the schools closed, I've been posting non-Geometry as well. The question for July 22nd this year is:
8!/ sigma(n=1)(60)n (round)
The word "round" suggests that the answer isn't an integer -- and so we should use a calculator, rather than expect all the factors to cancel leaving the answer.
Still, we know that 8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 and the summation gives (60 * 61)/2:
(8 * 7 * 6 * 5 * 4 * 3 * 2 * 2)/(60 * 61)
= (8 * 7 * 6 * 4)/61
= 22.03...
The final answer rounds to 22 -- and of course, today's date is the 22nd. (By the way, 7pi = 21.991, which is why today is Pi Approximation Day.)
Let's do one more problem from later this week, since it's an actual Geometry problem:
Segment AB is tangent to circle C with length 24sqrt(3). Find x.
(Here is the given info from the diagram: A is the point of tangency, B is also the center of a circle, and both circles are tangent to each other and have radius x.)
The key is to consider Triangle ABC. There is a right angle at A by the Radius-Tangent Theorem (the radius is perpendicular to the tangent line at the point of tangency). One of the legs, AC, is known to be the radius x, and another, AB, given to be 24sqrt(3)
That leaves us with the hypotenuse BC. The two circles are tangent to each other -- that is, they have a common tangent line and point of tangency. By Radius-Tangent both radii are perpendicular to this tangent line, and so the two radii are collinear since 90 + 90 = 180. Thus two radii add up to the hypotenuse -- that is, the hypotenuse is 2x.
Now we use the Pythagorean Theorem:
x^2 + (24sqrt(3))^2 = (2x)^2
(24sqrt(3))^2 = 4x^2 - x^2 = 3x^2
24^2 = x^2
x = 24
So the desired radius is 24 -- and of course, the date of this problem is the 24th. Notice that the longer leg is exactly sqrt(3) times the shorter leg -- in other words, ABC is a 30-60-90 triangle. The sudden appearance of 60-degree angles explains why exactly six circles surround a seventh circle so that the circles all touch each other and the central circle (that is, why the 2D kissing number is 6.
9. What is Shapelore?
When we teach Geometry, sometimes I fear that students will struggle over the vocabulary. This includes not just longer words/phrases such as "Alternate Interior Angles" and "Reflexive Property," but even the shorter words for the trig functions "sine," "cosine," and "tangent." More often than not, these confusing words come from complex Greek and Latin roots.
And so this project seeks to replace some of these words with roots that come from Old English. It was inspired by the Anglish website, where Anglish is the language we get by removing all Greek and Latin words from our language, leaving behind pure English:
In fact, the name "Shapelore" is itself in Anglish -- here -lore refers to something that is studied, so it's the Anglish equivalent of the Greek suffix "-ology":
In Shapelore, "Alternate Interior" becomes Otherside Inside and "Reflexive Property" becomes Selfsame Law. I've decided to keep some words as simple even if they are from Latin or Greek -- for example, most people know what a "square" is, so I keep it even though the word is from Latin. I'm also keeping the word "angle," also from Latin. The Anglish website proposes nook for "angle," but I'd rather keep "angle." Thus I refer to this as "Plain English" instead of "Pure English" or Anglish.
The new names for the trigonometric functions are wheelex for "cosine," wheelwhy for "sine," and wheelslope for "tangent." The wheel part indicates that these are the wheel (or circular) functions. So instead of tan(67.5) = 1 + sqrt(2), we'd write ws(67.5) = 1 + sqrt(2). The word "root" is Old English and I already said I'm keeping "square," so sqrt doesn't change, but tan becomes ws, wheelslope.
So far, I've been translating the U of Chicago Geometry text into Plain English. In 1991, the University of Chicago School Mathematics Project published a series of secondary mathematics texts. I was able to purchase both the Algebra I and Geometry texts for two dollars each at a local public library. One thing I discovered was that the U of Chicago Geometry text is based on the same transformations that appear in the Common Core Standards -- even though the U of Chicago published the text nearly two decades before the advent of the Core.
I might return to Shapelore at some point, but I don't know when.
1. Draw Curiosity
Notice that this video, from five years ago, actually acknowledges Pi Approximation Day.
2. Sharon Serano
Well, this gives 20 facts about pi.
3. Michael Blake
I like the idea of using music to teach math, so here's a favorite, Michael Blake's "What Pi Sounds Like." I tried to find a way to incorporate songs such as this one into the classroom. 4. Coding Challenge #140
Today is Pi Approximation Day, and so this video is all about approximating pi. 5. Ki & Ka
This video came out last year.
6. Sen Zen
This is yet another video that uses the Archimedes method of approximating pi.
7. Stand-Up Math
This video uses the Leibniz series, but all work is done by hand.
8. Think Twice
This video uses a Monte Carlo approximation technique. I attempted to do something similar using Java in an earlier post.
9. NKC Insight
This video came out last year, but I must have missed it.
10. Yearn 2 Learn:
In this new video, here are some classroom activities for either Pi Day or Pi Approximation Day.
11. hurfcharacterlimit:
I had to include this new video since it has the famous Pi Approximation integral from Calculus.
10. What is a COVID What If?
This a fictional story that I made up in order to imagine how my life would have been different if the pandemic had occurred earlier than COVID-19, which began in December 2019 and affected most of us by March 2020. It's a way for me to put myself in my students' shoes -- I can't truly understand what it's like to be a young person during the pandemic until I imagine placing the pandemic in my own youth.
I have written six separate versions of the COVID What Ifs. Each one places the pandemic in a different year -- COVID-86, COVID-91, COVID-93, COVID-97, COVID-08, COVID-14. In each case, the numbered year refers to the December when the pandemic ultimately begins -- so just as COVID-19 started in December 2019, COVID-n starts in December n. The schools don't close until March of the year n+1. We are currently in the year n+2, and so n+2 for each of the six What Ifs will resemble the year 2021 under COVID-19.
So far, what we know about Fall 2021 is that masks will be required in schools. This means that my young self is also masked in each Fall n+2. This means that masks extend into my second grade year for COVID-86, my seventh grade year under COVID-91, my freshman year for COVID-93, my first year at UCLA for COVID-97, and my year at the old charter school for COVID-14.
I'll only continue writing the COVID What Ifs if something out of the ordinary (that is, beyond masks) occurs in 2021 or 2022 (mapping back to all the n+2 and n+3 years). Otherwise, I've already completed the main part of the What If stories, focusing on the respective n+1 years.
11. How will the coronavirus school plans affect this blog (and Twitter)?
On this blog, I've always posted one lesson of Geometry per school day -- and these school days follow the calendar in one of my Orange County districts. I still haven't which one, but note that the two calendars are similar.
It's not so much the coronavirus that will affect my blog plans, but my career. If I get a job that's outside of education, then I won't need to have an education blog any longer, and so I'll end the blog.
By the way, last year I started an education Twitter account:
But once again, without a teaching job, I won't need an education Twitter account either.
It's just that I don't want to wait years, breves, longae, or decades for my next paycheck. Many people are suffering, not just subs. And it will be difficult for me to find work. (Again, if I do get a job outside of education, I'll no longer post to this blog.) That's the main thing that will be on my mind over the next few months, during these tough times of uncertainty.
And so I wish everyone a happy Pi Approximation Day!
