Thursday, August 31, 2017

Lesson 1-2: Locations as Points (Day 12)

This is what Theoni Pappas writes on page 243 of her Magic of Mathematics -- a table of contents for Chapter 10, "Mathematics & Architecture":

-- Buckminster Fuller, Geodesic Domes & the Buckyball
-- 21st Century Architecture -- Spacefilling Solids
-- The Arch -- Curvy Mathematics
-- Architecture & Hyperbolic Paraboloids
-- The Destruction of the Box & Frank Lloyd Wright

And so these are the topics we'll be discussing in Pappas throughout the next two weeks as we discover the link between math and architecture.

Lesson 1-2 of the U of Chicago text is called "Locations as Points." (It appears as Lesson 1-1 in the modern edition of the text.) The main focus of the lesson is graphing points on a number line. Indeed, we have another description of a point:

Second Description of a Point:
A point is an exact location.

Yesterday I made a big deal about the first description of a point -- the dot -- since many of our students are interested in pixel-based technology. Locations as points aren't as exciting -- but still, the second description is something we think about every time we find a distance. The definition of distance is highlighted in the text:

The distance between two points on a coordinatized line is the absolute value of the difference of their coordinates.

Other than this, the lesson is straightforward. Students learn about zero- through three-dimensional figures, but of course the emphasis is on one dimension. One of the two "exploration questions," which I included as a bonus, is:

-- Physicists sometimes speak of space-time. How many dimensions does space-time have?

Hey, we were just discussing this in Pappas! The answer, of course, is four -- even though there might be as many as ten dimensions in string theory. We ordinarily only include Einstein's four dimensions and don't consider the extra six dimensions of string theory as part of "space-time."

Here's the other bonus question:

-- To the nearest 100 miles, how far do you live from each of the following cities?
a. New York
b. Los Angeles
c. Honolulu
d. Moscow

Well, part b is easy -- I worked in L.A. last year and my daily commute obviously wasn't anywhere near 100 miles, so my distance to L.A. is 0 miles to the nearest 100 miles. The U of Chicago text gives the distance from L.A. to New York as 2451 miles as the crow flies, but 2786 miles by car. I choose to give the air distance in part a, in order to be consistent with parts c and d (for which only air distance is available). We round it up to 2500 miles. My answers are:

a. 2500 miles
b. 0 miles
c. 2600 miles
d. 6100 miles

Hmmm, that's interesting -- I'm only slightly closer to New York than to Honolulu.

Today is the last day of August -- hence the last day of Blaugust. I don't consider any of my posts to be Blaugust posts, since I'm not currently a teacher.

The most dedicated Blaugust poster is Illinois high school teacher Jackie Stone. She made an amazing 26 posts during the month. And as she's a Geometry teacher, it's interesting to compare her posts to what I write about on this blog.

In this post, Stone begins:

Today I gave pre-assessments in all of my classes.  It is a necessary evil in this world of data driven decisions.

And of course, I posted Benchmark Tests earlier this week. But Stone's post isn't actually about the pre-assessments, but about quick activities to do when there are five minutes left in class. (Last year, my activity for the last five minutes of class was called "Exit Pass.")

Of course, just because Stone is from Chicago, it doesn't mean that she uses the U of Chicago text -- or, for that matter, the Illinois State text.

Wednesday, August 30, 2017

Lesson 1-1: Dots as Points (Day 11)

This is what Theoni Pappas writes on page 242 of her Magic of Mathematics:


That's because this is the opening page of Chapter 10, "Mathematics & Architecture." There is nothing on this page except a photo captioned, "The Oracle office building, Redwood City, CA."

Now that we've finished the science chapter in Pappas, there's no need for me to write daily about my science class from last year. I know that to the readers, writing about my science class is like crying over spilled milk -- I didn't teach the class the way I wanted, and it's over. Still, I continue to keep that class in mind. It's the reason that I'm not teaching in a class right now -- and if I ever want to teach again, I must avoid the mistakes I made in that class. So I'll be continuing to write about my old class from time to time (but not in today's post, at least).

I'll post the table of contents for Pappas Chapter 10 tomorrow. Notice that we discussed a little on architecture in Chapter 0 of Serra. But once again, the Pappas and Serra topics miss each other on the blogging schedule.

Instead, our focus is now the U of Chicago text. Just like the Serra text, it's an old Second Edition (1991), and there are newer editions in which the chapters are ordered differently. Since my plan this year -- unlike past years -- is to follow the order strictly, let's revisit the chapter order in my text:

Table of Contents
1. Points and Lines
2. Definitions and If-then Statements
3. Angles and Lines
4. Reflections
5. Polygons
6. Transformations and Congruence
7. Triangle Congruence
8. Measurement Formulas
9. Three-Dimensional Figures
10. Surface Areas and Volume
11. Coordinate Geometry
12. Similarity
13. Logic and Indirect Reasoning
14. Trigonometry and Vectors
15. Further Work With Circles

Let's compare this to the modern Third Edition of the U of Chicago text. The first thing we notice is that the new text has only 14 chapters, not 15. We observe that the first twelve chapters are more or less the same in each text, and so it's Chapter 13 that is omitted in the new version. Instead, the material from the old Chapter 13 has been distributed among several different chapters.

You might recall that in the past when I used to juggle the lessons around, it was Chapter 13 that I moved around the most. So you could argue that when I was breaking up Chapter 13, I was actually adhering to the order in the new Third Edition -- unwittingly, of course!

Let's look at Chapter 13 in the old text, and I'll give the lesson in the new text to which the old Chapter 13 material has been moved:

-- Lessons 13-1 through 13-4 (on indirect proof) are now the first three lessons of Chapter 11, just before coordinate proofs. (Lesson 13-2, "Negations," is no longer a separate lesson in the new text.)
-- Lesson 13-5, "Tangents to Circles and Spheres," is now Lesson 14-4, in the circles chapter.
-- Lesson 13-6 through 13-8 (on exterior angles of polygons) have been incorporated into Lessons 5-6 and 5-7 (on Triangle Sum).

Some of these changes are those I once made by myself -- for example, including tangents to circles with the other circle lessons.

Besides the breakup of Chapter 13, here are the other major changes made in the Third Edition:

-- Chapters 4 through 6 exhibit many changes. In my old version, reflections appear in Chapter 4, while the other isometries don't appear until Chapter 6. In the new version, all isometries are defined in Chapter 4. With this, the definition of congruence (and some of its basic properties) have now moved up from Chapter 6 to Chapter 5. Only Triangle Sum remains in Chapter 5 -- the properties of isosceles triangles and quadrilaterals have been pushed back to Chapter 6.
-- With this, Chapter 3 has a few new sections. Two transformations are actually introduced in this chapter, namely rotations and dilations. This may seem strange, since rotations are still defined as Chapter 4 as a composite of reflections in intersecting lines -- and reflections themselves don't appear until Chapter 4. It appears that the purpose of rotations in the new Lesson 3-2 is to introduce rotations informally, as well as tie them more strongly to the angles of Lesson 3-1. (Rotations appear before reflections in Hung-Hsi Wu, but Wu does for different reasons.) Arcs also now appear in Lesson 3-1 instead of having to wait until 8-8. Meanwhile, the new Lesson 3-7 on dilations (which are still called "size transformations") is essentially the old Lesson 12-1 and 12-2. Again this is only an intro -- dilations are still studied in earnest only in Chapter 12.
-- Chapter 7 is basically the same as the old text, especially the first five sections (except that SsA in Lesson 7-5 now has an actual proof). The new Lesson 7-6 is the old Lesson 8-2 on tessellations. I see two new lessons in this chapter, Lesson 7-9 on diagonals of quadrilaterals and Lesson 7-10 on the validity of constructions. (David Joyce would approve of this -- but he'd take it a step forward and not even introduce the constructions until this lesson.) Meanwhile, the old Lesson 7-8 on the SAS Inequality (or "Hinge Theorem") no longer appears in the new text.
-- Chapter 8 has only one new section -- Lesson 8-7, "Special Right Triangles," is the old 14-1. This is so that special right triangles are closely connected to the Pythagorean Theorem.
-- Chapter 9 was always a flimsy chapter in the old book -- it's on 3D figures, yet most of the important info on 3D figures (surface area and volume) don't appear until Chapter 10. Now surface area has moved up to Chapter 9, reserving Chapter 10 for volume (except for the surface area of a sphere, which remains in Chapter 10). The old Lesson 9-8 on the Four-Color Theorem has been dropped, but that was always a lesson that was "just for fun."
-- The last section of the old Chapter 12 (side-splitter) is now the first section of Chapter 13, which is the new trig chapter. Lesson 13-2 is a new lesson on the Angle Bisector Theorem, and Lesson 13-4 is a new lesson on the golden ratio. I've actually seen these ideas used before -- including on the Pappas Mathematical Calendar -- but this is the first time I've seen them in a text as separate lessons. This is followed by lessons on the three trig ratios. Vectors, meanwhile, have moved up to Lesson 4-6, so that they can be closely connected to translations.
-- Chapter 14 should be like the old Chapter 15, but there are a few changes here as well. Ironically, I, like the text, moved tangents to circles to this chapter (Lesson 14-4) so that it would be closer to the other important circle theorem, the Inscribed Angle Theorem. But inscribed angles have been moved up in the new text to Lesson 6-3. This places that lesson closer to the Isosceles Triangle Theorem, which is used in the proof of the theorem. Meanwhile, Lesson 14-6 technically corresponds to 15-4 ("Locating the Center of a Circle") of the old text, but it has been beefed up. Instead of just the circumcenter, it discusses the other three concurrency theorems (important for Common Core) as well as the nine-point circle of a triangle.

Meanwhile, of immediate concern are Chapters 1 and 2 of the new text. Unlike the others, these chapters haven't changed much from the old text. The only difference in Chapter 2 is that Lesson 2-3, on if-then statements in BASIC, has been dropped. (After all, who uses BASIC anymore, except on computer emulators, as in yesterday's post?) In its place is a new lesson on making conjectures.

Two of the lessons of Chapter 1 have been dropped. One of them is actually today's Lesson 1-1, as its material has been combined with the old Lesson 1-4. Meanwhile, Lesson 1-5, on perspective, has been delayed to Chapter 9 (which makes sense as perspective is definitely related to 3D). The last lesson in Chapter 1 is on technology -- a "dynamic geometry system," or DGS. (That's right -- goodbye BASIC, hello DGS!) Officially, it still corresponds to the last lesson of the old Chapter 1, since this lesson still introduces the Triangle Inequality Postulate (but now students can test out this postulate for themselves on the DGS).

On the blog, I'll continue to follow the old Second Edition of the U of Chicago text. But if I ever get to sub in a classroom again, the classroom has priority over the U of Chicago order. In this case, if an important lesson is skipped, I could sneak the lesson in by following the Third Edition order instead.

Okay, without further ado, let's finally start the U of Chicago text!

Lesson 1-1 of the U of Chicago text is called "Dots as Points." This isn't a lesson that I covered on the blog before, since for the first three years of this blog, we always began with Lesson 1-4. But I did mention one important idea from this chapter -- the first description of a point:

First description of a point:
A point is a dot.

This is the start of a new school year. Many students enter Geometry having struggled throughout their Algebra I class. Now they come to us in Geometry, and after all the frustration they experienced last year, the first question they ask is, "Why do we have to study Geometry?" Well, the answer is:

A point is a dot.

The old U of Chicago text writes about dot-matrix printers. This isn't relevant to the 21st century, and indeed they don't appear in the modern edition. But here's another question to ask students -- if you didn't have to take math, what would you do at home instead of math homework? And if the answer is "play video games," then guess what -- video game graphics consists of millions of dots. Or, more accurately, they consist of millions of points, since:

A point is a dot.

Images on video games don't come out of nowhere -- someone had to program in the millions of dots, treating them as points -- therefore using Geometry. So without Geometry, video games don't exist. If you want to answer that question -- "What would you do if there was no math?" -- then next time choose something that doesn't require math to build.

In the modern version of the text, there is a brief mention of pixels as part of both computer images and digital camera images. Again, it's not emphasized as much, since "dots as points" must share the new Lesson 1-3 with "network nodes as points."

Tuesday, August 29, 2017

Benchmark Tests (Day 10)

This is what Theoni Pappas writes on page 241 of her Magic of Mathematics:

"They [John Schwartz and Michael Green] had been working on it for over a decade in spite of little encouragement from colleagues, who found the 10-dimensional world hard to accept. Their published paper finally caused physicists to take their idea seriously."

This is the last page of the section on string theory, and the final page in the science chapter. Pappas concludes this chapter by writing:

"These are just the beginning discoveries and applications of an emerging mathematical field -- knot theory."

Today I'm posting the Benchmark Tests. Just as I've done in the past, I'll be using these test days for tying up loose ends, including the traditionalists label.

Actually, the traditionalist debate has been quiet lately. The most prominent traditionalist, Barry Garelick, hasn't posted in a week (probably because he's focused on his first day of school).

Here is the link to Garelick's most recent post:

This post contains a link to an article, which I'll link here as well:

The article asks a typical question students ask in math class, "When will I ever use this?" The article suggests that engagement is the key to answering this question. Naturally, Garelick disagrees:

I usually find that students ask this when they are frustrated and/or having difficulty with a particular type of procedure or problem. When they are capable of doing the procedure, they tend to be just as engaged as they would with any activity. Not to mention that the prevalence of this question is helped along by TV sitcoms that may feature such a situation. The question is met with the predictable laugh track as the camera zooms in to a close up of teacher’s frustrated expression.

Well, I partly agree here. I've written before that students don't complain about tasks that are easy, fun, or high-status. So when they ask "When will I ever use this?" it really means that the math is hard, but as soon as they are "capable of doing the procedure" so that it's no longer hard, the complaint vanishes.

In my old class, a different sort of complaint appeared -- when I tried to get the students to be quiet using call-and-response and other ideas, they complained that this was "juvenile." But they had no problem with sing songs during music break, which is arguably even more juvenile. That was because singing songs was fun, but being quiet during class isn't.

Garelick mentions TV sitcoms here, and alluded to "tropes" earlier in his post. The famous website TV Tropes lists "Everybody hates mathematics" as the relevant trope here.

There is one comment on Garelick's post here -- and of course, it's SteveH, the blog co-author:

ALL STEM career paths require individual focus and success on p-sets [problem sets-dw]. That IS a one size fits all requirement. If it doesn’t fit the student, then engagement is no alternate approach. Engagement can only be built on top of high expectations and an emphasis on individual homework. These educators NEVER talk about the importance of individual homework – the missing component in K-6. They can’t justify full inclusion and low expectations by claiming that “engagement is key.”

Here's what I think is meant by "engagement is key" -- many students only do assignments if they find them enjoyable. If they don't want to do it, then they don't do it. Yes, students are wrong to believe this, but that doesn't affect the fact that they do believe it. Engagement is the attempt to make students want to do math so that they'll actually do it.

I've said it once and I'll say it again -- students may learn very little from doing projects and activities, but they learn absolutely nothing from p-sets and homework assignments that they leave blank. I've even seen students leave the worksheets I give them on their desks -- they don't even bother to put them in backpacks since they have no intention of doing it. "Very little" trumps "absolutely nothing."

SteveH mentions "full inclusion" here -- the opposite of "full inclusion" is "tracking." He proceeds to write that tracking still occurs today, except it's parents and tutors who track, not teachers or schools.

Now there is something that I do wish to discuss in this traditionalists-labeled post -- the debate over Common Core and testing. In the past, I've written that I was of two minds on this issue. I could see why many people would oppose testing, but I conceded that it has its uses.

But here's the thing -- I went through my first testing season as a teacher last year. And as I continue to reflect on what went wrong in my first year, I'm starting to believe that standardized testing is much to blame for my struggles last year. In fact, I might not have left my classroom had it not been for the state test.

Last year, I wrote that I couldn't survive the Big March -- the stretch of the year between President's Day and Easter when there are no off days from school. It's not the longest holiday-free stretch -- in fact, as Columbus Day isn't a holiday in California, many schools in our state have no days off between Labor Day and Veteran's Day. But the Big March feels much worse -- and part of the reason is that the Big March is right in the middle of test-prep season, whereas Labor Day to Vets Day isn't as pressure-filled.

Test prep at my school began at the start of second "semester" -- that is, the mathematical midpoint of the year. As I mentioned before, we already lack conference periods at our school due to the lack of staff -- instead, our only quasi-break had been during P.E. time. But P.E. time most days of the week was replaced with SBAC Prep for sixth and seventh grades, while eighth graders had their test prep on Wednesday mornings.

Moreover, consider the problems I had with classroom management. The students decided that they only had to obey my support staff member and didn't have to listen to me. For all three grades, SBAC Prep took place at times when my support staff member wasn't present -- Grades 6-7 had it after she left while eighth grade had it before she arrived. Therefore the students misbehaved the most in my class during SBAC Prep time.

Then, of course, there was the issue that many students didn't remember what they were supposed to have learned, so they were resistant to doing the reviews. SBAC Prep time turned into copying down the questions and answers that I wrote on the board, which of course isn't learning at all.

The whole point is that had there been no Common Core, SBAC, or standardized testing, students might have been better behaved the second half of the year. There would have been P.E. time for the students and planning time for me, instead of my having to figure out what to do for SBAC Prep with no planning time. And I might not have left my school.

Traditionalists are split when it comes to standardized testing. Some of them favor standardized testing in order to combat grade inflation and increase learning.

Here's what I mean -- suppose a student is failing a class. The parent then complains to the teacher -- and the gist of the complaint is, "please make the grading scale easier." The teacher acquiesces to the parent by lowering the standards -- the accumulation of such lowering leads to grade inflation.

Now suppose the parent decides to complain to a tutor instead of a teacher. Now the gist of the complaint can't be "make the grading scale easier" since tutors don't give grades. So instead they end up saying, "please teach my kids more math" (so they can pass the teacher's tests). Traditionalists like this, since it results in the students learning more math.

So standardized testing makes teachers into tutors -- there will be fewer requests to "make the grading scale easier" and more requests to "teach the kids more math," since teachers don't grade the standardized tests. In fact, some traditionalists would like to see standardized test grades override teacher grades. This is to reduce grade inflation and increase learning.

Of course, while may they favor standardized testing, they still oppose the Common Core tests. In the post above, SteveH writes again about how the highest score on the PARCC and SBAC corresponds to "can pass a college Algebra course" instead of "can pass a high school Calculus course." Standards that don't encourage AP Calculus are said to decrease learning rather than increase it.

Another issue I had in my class was with computerized testing. During SBAC Prep time, I was required to have the students take the practice tests from the CAASPP website. Some 8th graders -- specifically those who couldn't remember how to do the problems -- refused to do it. Meanwhile, in Grades 6-7, there weren't enough laptops. Some students who didn't want to take the test just passed the laptop to another student -- and there were no consequences for this, since there was nothing else for them to do except take the online test. Also, sometimes the CAASPP website didn't work -- and many students tried to take advantage of this.

For these reasons, traditionalists oppose computer tests like PARCC and SBAC. They believe that any test worth its salt must be taken with paper and pencil.

I like the idea of a computer-adaptive test like the SBAC. For those who complain that a standardized test is "one size fits all," the point of an adaptive test is to avoid being "one size fits all." But that doesn't erase the issues I had with computers this year.

I remember once when I was subbing at a school during SBAC time (not just prep for the SBAC, but the actual test). School was set up for two-hour blocks to accommodate the SBAC. I was in an English class on a day when math tests were being administered. The regular teacher mentioned in her notes that students are not allowed to go to the restroom, period. She made no exceptions for emergencies, probably because students would lie about having an emergency. (And she wrote the bell schedule in her notes, so it's not as if she was unaware of the block schedule.)

Sure enough, one girl claimed that she had to go to the restroom for an emergency. This led to a long, heated argument. Regardless of what you think I should have done in that situation, the truth is that the whole argument is much less likely had there not been two-hour blocks forced by the SBAC.

I stated before that the entire state math test should be only 30 minutes long -- especially in middle and high school where students attend class by periods. With only a half-hour test, plus leeway to set up and put away the laptops, the entire exam fits within a single period, so there would be no need for a block schedule. With only one period required for math, if the computers break down, the test can simply be given the next day. This isn't possible under a block schedule, where the test must be given on the day specified by the block schedule.

I'm not sure what to do about the other problem -- the need for test prep months before the test. In theory, there should be no need for test prep, as simply following the curriculum ought to prepare the students for the test. Perhaps there should be fewer confusing test questions, since much of test prep time is to teach students how to decipher them.

Oh, and if there's going to be a computerized test, then scores should be released instantly. That's one of the main advantages of taking tests on computers, but in reality, scoring still takes months. Part of this is the existence of "performance tasks" that can't be graded electronically.

The other issue currently being discussed in California is the need for remedial classes at our Cal States and community colleges. Of course, traditionalists oppose these -- they want to see more students taking Calculus in high school, not Algebra in college.

There are a few more things that I wish to discuss in this post. I mentioned earlier that I'm posting the Benchmark Tests, so let me include a full version of the Benchmark Test song:

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?
It's the end of the year so let's
See how much we know, know know!

It's all old stuff on Benchmark Tests.
There's no need to take a guess.
We leave none blank, oh no, no, no!

The teacher sees our Benchmark Tests,
Just like the SBAC more or less.
That's the way to go, go, go!

I'm still in the process of updating the Fraction Fever song, to include a new verse or so on addition, subtraction, multiplication, and division of fractions. In doing so, I found something that will help me greatly -- a YouTube video of the actual 1980's Fraction Fever game!

Unfortunately, the player starts the game before the opening song (on which I based my Fraction Fever song) is completed. But I believe that only the last few notes are missing -- and it's the part that was easiest for me to remember from the past, anyway. It's the last line, where I keep repeating "Fraction! Fever!" over and over -- and it's part that my students remember the most as well.

If you watch the actual video, you might notice that no addition of fractions is actually needed to play this game. I want to include arithmetic in my version of the game, since that's what middle school students are actually learning. On the other hand, I retain the basic rules -- right answers=elevators to go up a floor, wrong answers=holes to go down a floor, and so on. The player points out that the scoring system is confusing. In my version, the floor (or level) itself is the team score. Oh, and I post my answers on a wall, and students have to jump up to reach them. This corresponds to the fact that in this game, you must jump (using a pogo stick) on the right answers.

While I was searching for Fraction Fever, I stumbled on another old website:

This website actually emulates my old 1980's computer. Notice that several games are available (but unfortunately, not Fraction Fever).

Anyway, recall that earlier this summer when I was writing about my old computer (in the Pappas music posts), I mentioned that there was a SOUND command (in BASIC) that played musical notes (numbered 1-255). But I never tried to play music with it, especially as there was also a PLAY command that played notes lettered A-G (as in real music). Now with this emulator, I can finally figure out what the Notes 1-255 actually correspond to. If so, then this emulator can be used to play microtonal music (that is, other than 12EDO).

I had made two guesses regarding the conversion of SOUND numbers to musical notes:

-- The SOUND notes are an n-EDO system for some n. Then Notes n, 2n, 3n, 4n, and so on would sound an octave apart.
-- The SOUND notes correspond to frequencies (hertz). Then Notes n, 2n, 4n, 8n, and so on would sound an octave apart.

As it turns out, both guesses are wrong. On the emulator, let's click "Sound" under the screen on the left side of the page. Then we type in:

SOUND 1,10

This plays Note 1 for a length of 10 "SOUND-units." I believe that on my old computer, one SOUND unit is about 1/15 second. On this emulator, it doesn't play a continuous note but beeps instead -- but right now we're more concerned with the note. (Of course, if we're trying to play microtonal music, that beeping would be annoying!)

Now, let's try the highest note:

SOUND 255,10

This clearly sounds several octaves higher than Note 1. But let's try the middle note:

SOUND 1,10
SOUND 128,10

Now Notes 1 and 128 sound just about an octave apart. But Notes 128 and 255 still sound much more than an octave apart. Instead, let's try:

SOUND 128,10
SOUND 192,10

Note 192 (which is around 3/4 of the range) now sounds near an octave above Note 128. This should now remind us of a string -- if we play half of its length, it sounds an octave higher, and if we fret it at the 3/4 mark (so the vibrating string is 1/4 of the original string), it sounds two octaves higher.

And now, after 30 years, I finally know the secret of SOUND. The numbered notes correspond to EDL (equal divisions of length), as if we had a string and were dividing its length up into equal parts.

We can test this out by writing a short program to create a major scale. On page 176 of Pappas, she tells us how to create a musical scale:

"For example, starting with a string that produces the note C, then 16/15 of C's length gives B, 6/5 of C's length gives A, 4/3 of C's gives G, 3/2 of C's gives F, 8/5 of C's gives E, 16/9 of C's gives D, and 2/1 of C's length gives low C."

Notice that the common denominator of all the fractions mentioned here is 90. So let's try using an octave with 90 equal divisions of length (180EDL). Now since Note 255 is the highest note, let's think of 256 as the "end of the string" -- the bridge, to use a guitarist's term. (After all, 255/256 still plays a note, but 256/256 means that we have no string vibrating, hence no note.)

So 90 notes below the bridge is Note 166, and 90 notes below this is Note 76. These notes are an octave apart, and we can easily divide this into a major scale using the ratios above.

Note          Ratio          Out of 180          From Bridge 256
C               1/1              90                     166
B               16/15          96                     160
A               6/5             108                   148
G               4/3             120                   136
F                3/2             135                   121
E               8/5              144                   112
D               16/9           160                   96
C               2/1              180                   76

Finally, we write a program to play all of these notes:

10 SOUND 76,10
20 SOUND 96,10
30 SOUND 112,10
40 SOUND 121,10
50 SOUND 136,10
60 SOUND 148,10
70 SOUND 160,10
80 SOUND 166,10

At the end of the program, we type in RUN to run the program. This does sound like a major scale, but let's try adding another note an octave above the last note:

90 SOUND 211,10

This last note sounds a little bit narrower than a true octave. In fact, when we played Notes 128 and 192 together earlier, it also sounds slightly narrower than an octave.

This seems to imply that maybe 256 isn't the bridge after all. Unfortunately, we can't play a true octave unless we know what the exact bridge is. Notice that if 256 were the bridge, then Notes 254 and 255 would be a full octave apart, but they clearly aren't. Using trial and error, the true bridge seems to be around 261 or 260. Let's use 261 as our bridge:

Note          Ratio          Out of 180          From Bridge 261
C               1/1              90                     171
B               16/15          96                     165
A               6/5             108                   153
G               4/3             120                   141
F                3/2             135                   126
E               8/5              144                   117
D               16/9           160                   101
C               2/1              180                   81

In case you're curious how I determined the bridge to be 261, type NEW for a new program, then try the following:

10 B=261
20 FOR X=3 TO 7
30 SOUND B-2^X,10

(Here the ^ symbol is actually an up-arrow -- press the arrow on the keyboard.) The played notes now sound like descending octaves. If we change line 10 to B=260 or B=262, the octaves are slightly off.

Let's type in NEW for a new program, then we run the following:

10 SOUND 81,10
20 SOUND 101,10
30 SOUND 117,10
40 SOUND 126,10
50 SOUND 141,10
60 SOUND 153,10
70 SOUND 165,10
80 SOUND 171,10
90 SOUND 216,10

This time, the last two notes sound like a true octave. This means that the first eight notes do form a major scale -- a justly tuned major scale, rather than the 12EDO major scale.

Of course, we've been calling the top note a C, but is it actually a C? It appears that this note is closer to concert B than to C -- we can confirm this by using PLAY:

90 PLAY "B"

The last two notes sound alike, indicating that this is really a just major scale on B. But in practice, we can call this a "transposing instrument" and just pretend that the note is C, since we'd rather write a C major scale than a B major scale with all of its sharps.

At this point, you may wonder why we don't just use PLAY, where we can play a true C note? The whole point of using SOUND is that we can actually play notes in just intonation, whereas PLAY will just give you the 12EDO scale.

Before we leave, suppose we wanted to play a septimal interval, such as the harmonic seventh. We can't play this exactly, since 90 isn't divisible by seven. The desired ratio is 8/7, and if we multiply this by 90, we can round this to 103 (or Note 158 as measured from Bridge 261). This isn't exact (but it's at least more accurate than 12EDO).

To obtain a just harmonic seventh, we must use a division other than 90EDL. The closest multiple of seven is 91 -- this produces a harmonic seventh between Notes 79 and 157. But of course, with 91 we lose all of the 3-limit and 5-limit intervals. The correct octave to use depends on what other intervals we're using in the song -- for example, 126EDL gives us a no-fives tuning.

The following is a link to music played in EDL systems:

Naturally, no scale larger than 255EDL can be played on the emulator. Don't forget to subtract the notes from Bridge 261 -- so the first three notes of 120EDL -- given as 120, 114, 108 -- correspond to Notes 141, 147, and 153 on SOUND.

OK, let's finally post the Benchmark Tests. These are based on old finals posted to the blog. I admit that the tricky thing about Benchmark Tests in Geometry is that the students are coming off of a year of Algebra I, when they've thought little about Geometry at all. This is different from Benchmark Tests in middle school or Integrated Math, where there should be some continuity from year to year.

Monday, August 28, 2017

Lesson 0-9: Chapter Review (Day 9)

This is what Theoni Pappas writes on page 240 of her Magic of Mathematics:

"Ten dimensions seems out of the question. But if one thinks of dimensions as descriptive numbers that pinpoint the locations and characteristics of an object in the universe, they become more comprehensible."

Here Pappas is taking us from knot theory to string theory. According to string theory, the universe contains more than just the three ordinary spatial dimensions and one time dimension. Instead, as Pappas writes, there could be nine spatial dimensions and one time dimension, for a total of ten.

I've alluded to string theory in the past on the blog, in posts devoted to the works of David Kung and Mario Livio. And of course, Eugenia Cheng wrote about higher dimensions. Like Pappas, Cheng informs us that a "dimension" is nothing more than a "descriptive number" -- a coordinate -- so the idea of there being ten of them isn't as far-fetched.

Pappas tells us that string theory is related to the possibility of a TOE -- theory of everything. Of course, I've written about the movie Theory of Everything here on the blog before. This movie is the story of Stephen Hawking, a physicist who studied string theory and sought out a TOE.

Pappas continues:

"The ideas of TOE have been evolving for over 20 years [as of 1994]....The results have been most convincing."

Here is a link to a string theory website:

And this is what it has to say about the extra dimensions:

It's not so hard to construct higher dimensional worlds using the Einstein equations. But the question is then: WHY BOTHER?
   It's because physicists dream of a unified theory [a TOE -- dw] : a single mathematical framework in which all fundamental forces and units of matter can be described together in a manner that is internally consistent and consistent with current and future observation.
   And it turns out that having extra dimensions of space makes it possible to build candidates for such a theory.

In string theory, the extra dimensions are very tiny -- in other words, they're "rolled up" in a certain way into "strings."

String theory is obviously not a middle school science topic, nor is it a high school science topic. As of now, string theory still lies at the frontier of our knowledge. Even though we now see how knot theory ties in to science, it still has nothing to do with the science I should have taught last year.

Lesson 0.9 of Michael Serra's Discovering Geometry is labeled "Chapter Review." In the Second Edition, chapter reviews have their own lesson numbers, but in the modern editions (just as in the U of Chicago text), the chapter reviews are unnumbered.

At this point, we may wonder, should there be a Chapter 0 Test? If there were a Chapter 0 Test tomorrow, then this would allow us to start Lesson 1-1 of the U of Chicago text on Day 11, which would be Wednesday.

Some teachers may point out that Chapter 0 consists of just introductory activities and so it shouldn't be tested -- and besides, a Chapter 0 test would be so soon after the first day of school, when many students are still requesting schedule changes from their counselors.

On the other hand, without a Chapter 0 Test, the first test would be the Chapter 1 Test on Day 20. At some schools, grades must be submitted every quaver (i.e., twice a quarter). Day 20 would be very close to the end of the first quaver -- and at many schools, grades are due a few days before the mathematical end of the quaver. So whether or not there should be a Chapter 0 Test depends on how often a school issues progress reports, as well as whether a teacher wants to give a solid test before those first progress reports are issued.

As far as this blog is concerned, my decision is to follow what my old school did last year. If you recall from last year, the first test I gave my students was called a "Benchmark Test." This was, of course, a diagnostic pre-test to determine what the students already knew, and what they would need to learn in the coming year.

Therefore tomorrow I will post some Benchmark Tests for Geometry. It will preview lessons to be covered the entire year. Nonetheless, today's worksheet is based on review questions from Lesson 0.9 of Serra's text.

In all these posts, I've been continuing to think about how I should have taught science last year. No, I wouldn't have taught string theory, but thinking about last year's Benchmark Tests are causing me to think about how the class was organized again.

Recall that the basic structure of the class, as envisioned by Illinois State, looked like this:

-- Monday: Coding
-- Tuesday: STEM Project
-- Wednesday: Traditional Lesson
-- Thursday: Learning Centers
-- Friday: Weekly Assessment

But there were several problems with this weekly plan. Most notably, seventh graders didn't attend my class on Wednesdays, when the traditional lesson ought to be taught. This usually meant that the traditional lesson was pushed back to Thursday -- the day before the assessment on Friday. Clearly, this isn't learning at all.

A better weekly plan would look like this:

-- Monday: Coding
-- Tuesday: Traditional Lesson
-- Wednesday: Learning Centers
-- Thursday: Science Project
-- Friday: Weekly Assessment

Not only do the seventh graders now have a traditional lesson, but all the grades benefit from having an extra day between the lesson and the assessment. Meanwhile, the project is now explicitly listed as a science project. This keeps the math class more or less traditional while the science projects prepare students for laboratory science -- required in high school under the a-g requirements for college. All the seventh graders end up missing are the learning centers -- which, frankly, are less important than the Tuesday math lesson and the Thursday science lesson.

So now I've established Thursday as my science day. It's also likely that I would have included some science as part of the Wednesday learning centers, but for the most part science is on Thursday.

Now science for Grades 7-8 are based on the old California science standards, and so I take projects from the life and physical science texts found on the Illinois State website. But sixth grade is based on the new NGSS standards, which I find on the Study Island website. I no longer have access to these websites, so I can't quite be sure how I would have divided the class into units.

But if I recall correctly, the sixth grade curriculum was divided into nine units, with a pre-test in Unit 1, the main curriculum in Units 2-8, and a post-test in Unit 9. This corresponds roughly to months, so Unit 1 is in August, Unit 2 in September, all the way up to Unit 9 in April (and then May can be used for test review). Notice that the Unit 1 Pre-Test in August fits nicely into Benchmark Testing Week -- and in fact, I could give that pre-test on Thursday of that week, to fit the pattern to be established.

Science projects can be given every other Thursday, to meet the requirement that we submit project photos to Illinois State every fortnight. I admit that the idea of having to prepare so many projects intimidated me a little. It might have been good to give the mousetrap car project that first Thursday in September (as it appears in the math STEM texts for all three grades), but then switch to science projects only, starting with our second project.

On the non-project Thursdays, I have the students take Foldable notes. And on the last Thursday of the month, I give the science test. Most likely, the first real science test (for Unit 2) wouldn't have been given until the last Thursday in September, due to other distractions earlier in the month (holidays, a field trip, and Back to School Night). The only problem is that this was after grades for the first progress report were due (echoing the problem I mentioned earlier in this post). Fortunately, at that point there were no separate grades for math and science (just a "STEM grade"), so I could get away with having no science grades until after the first progress report.

Yes, it's tricky to figure out how to grade these assignments. We weren't told about PowerSchool and the weighting percentages until October, and so I probably would have winged it with the grades until being made aware of the weights. At that point, I most likely would have squeezed all science into the 40% reserved for "Formal Assessment and Projects." Of course, late in the second trimester, PowerSchool was changed again so that I needed to give separate grades for math and science.

In weeks when there is a formal assessment in science on Thursday, the weekly math assessment on Friday changes to a Dren Quiz instead, so that the students don't have two hard tests in a row. The Dren Quiz number can correspond to the Study Island unit number, so the first real science test is followed by the 2's Dren Quiz. The pre-test during Benchmark Testing Week could be followed by a 1's Dren Quiz (which doesn't count, since no tests during that week count in the grades), or if that's too goofy, I could keep the 10's Dren Quiz (which was my first actual Dren Quiz).

In fact, the idea that the second week of school should follow the same pattern as the rest of the school year extends back to the first week of school. For example, the Opening Week Activity for Thursday should lean more towards science, and so on.

But of course, this is all a dream. This is what my class could have looked like if I had taught science properly, but unfortunately I didn't teach it properly.

As of today, the Japanese artist mentioned on the worksheet, Kunito Nagaoka, is still alive.

Friday, August 25, 2017

Lesson 0-8: Perspective (Day 8)

This is what Theoni Pappas writes on page 237 of her Magic of Mathematics:

"What distinguishes mathematical knots from the everyday knots one ties, is that they have no ends. They are a closed type of loop, which cannot be formed into a circle."

We've returned to the section on knots. Here Pappas provides the mathematical definition of a knot as a closed loop that can't be deformed to a circle (the "unknot"). Therefore the ring structures from Serra are not knots, since they aren't a single closed loop. The Olympic rings are five circles, not one.

Pappas tells us that knots can't exist in more than three dimensions. The reason is that all closed loops in 4D can be deformed to the unknot. Instead, we can knot a plane in 4D. One example of a knotted plane is the Cartesian product of any 3D knot and any line segment.

According to Pappas, the simplest knot is the trefoil knot. Here is a Wolfram link to this knot:

The "tre-" in trefoil means three, and indeed this knot has three crossings ("crossing number 3"). She tells us that the trefoil is the only knot with crossing number 3. The trefoil is closely related to its reflection image. Even though there is an isometry mapping one to the other, no isotopy (continuous deformation) exists.

Pappas tells us that there is only one knot of crossing number 4 (the "figure-eight knot") and two knots of crossing number 5. But the number of knots grows rapidly -- she states that there are over 12,000 knots of crossing number at most 13.

As usual, there is a sequence on OEIS -- the number of prime knots with n crossings. (I mentioned the OEIS, the online sequences website, on the second day of school last year.)

The final example Pappas gives on this page is the composite of a knot and its reflection. She tells us that to our surprise, this cannot deformed to the unknot. She writes that if we try to deform it:

"They simply pass through one another and remain unchanged."

There's not much on this Pappas page that is teachable in a middle school science class at all. So instead, let's go straight into Serra.

Lesson 0.8 of Serra's Discovering Geometry is called "Perspective." This is the second of two sections appearing in the Second Edition but not in the modern editions. Serra begins:

"Many of the paintings created by European artists during the Middle Ages were commissioned by the Roman Catholic Church. The art was symbolic; that is, people and objects in the paintings were symbols representing religious ideas."

Unlike Lesson 0.5 on mandalas, which we choose to include on the blog even though it's "missing" from the modern editions, Lesson 0.8 can just be left out altogether. This is because we'll be starting the U of Chicago text next week, and that text already has a lesson on perspective (Lesson 1-5), so Day 15 would just be a repeat of Day 8.

Then again, we recall that in my class last year, the students in all grades had trouble drawing cubes even though those were on isometric paper rather than in true perspective). So we might wish to teach perspective on both Day 8 and Day 15. True perspective drawings should most likely be completed on plain unlined white paper, with a straightedge to draw lines toward the vanishing point. Lined notebook paper for one-point perspective drawing may also be acceptable -- but not for two-point perspective (the subject of today's worksheet).

The worksheet below comes from "marcandersonarts" and "Daisuke Motogi."

Thursday, August 24, 2017

Lesson 0-7: Islamic Art (Day 7)

This is what Theoni Pappas writes on page 235 of her Magic of Mathematics:

"This famous drawing by Leonardo da Vinci appeared in the book De Divina Proportione, which Leonardo illustrated for mathematician Luca Paoli in 1509. Leonardo wrote an extensive section on the proportions of the human body in one of his notebooks."

This is the first and only page of the section "Secrets of the Renaissance Man." Of course you can't see what drawing Pappas is referring to here, so let me provide a link:

Pappas explains:

"In his book, he also made reference to the works of Vitruvius, the Roman architect (circa 30 B.C.) who also dealt with the proportions of the human body."

The title, De Divina Proportione, refers to the "divine proportion," which is also known as the golden ratio or Phi. Both Vitruvius and Leonardo believed that Phi = (1 + sqrt(5))/2 appeared in certain ratios of the human body. This is explained at the following link:

  • In the distance from the Da Vinci’s guide line drawn at the hairline to the guide line at the foot, the following are all at golden ratio points:
    •  the navel, which is most often associated with the golden ratio of the total height and not the height of the hairline
    • the guidelines for the pectoral nipples
    • the guidelines for the collar bone
  • In the distance from the Da Vinci’s guide line drawn at the elbow to the guideline at the fingertips
    • the base of the hand is at a golden ratio point.
Pappas concludes:

"Leonardo adds, The length of a man's outspread arms is equal to his height."

This estimation appears in a Square One TV song, "Rule of Thumb" by Kid 'n Play. The rappers are trying to measure the length of the floor. One member knows that his height is about six feet, so he concludes that the length of his outspread arms from fingertip to fingertip is also six feet. There isn't a separate video on YouTube for this song, but it does appear at the start of this YouTube video on Math Talk, a spin off of Square One TV. This was recently posted a few months ago:

How can I connect this back to the science class I taught last year? Well, the actual ratios of the human body isn't part of the curriculum, although the human body itself is.

Under the old California standards, here's how a seventh grade life science class was organized. It began with a little chemistry with an emphasize on the elements required for life (hydrogen, carbon, oxygen, and so on). Then the lessons focus on cell structure, DNA, and genes. Next would be evolution and the history of life on earth. This is usually followed by biodiversity, with lessons first on microbes and fungi, plants, and then animals. Within the animal unit, typically invertebrates are covered first, then the various orders of vertebrates -- fish, amphibians, reptiles, birds, mammals. So humans appear last in this section -- but then this is followed immediately by the human body, also known as anatomy. Since I should have followed the California standards for seventh grade, this meant that the unit on the human body should have appeared at the end of the year.

Lesson 0.7 of Michael Serra's Discovering Geometry is called "Islamic Art." This is in the Second Edition -- in the modern editions, "Islamic Art" is Lesson 0.6. Serra begins:

"Islamic art is rich in geometric forms. Islamic artists were familiar with geometry through the works of Euclid, Pythagoras, and other mathematicians of antiquity, and they used geometric patterns extensively in their art and architecture."

Yesterday, I rearranged the Pappas pages so that her first page on knots would fall on the same day as the knots lesson in Serra. But the result of this change is that today's Pappas page is on the human body in art, while Serra's lesson on Islamic art has nothing to do with the human body:

"Many of [Muhammad's] followers interpreted his words to mean that the representation of humans or animals in art was forbidden. Therefore, instead of using human or animal forms for decorations, Islamic artists used intricate geometric patterns."

As usual, the questions I derive from Serra's text instruct the student to create Islamic-style art. This art is based on tessellations.

There are a few interesting things in this lesson. First, Serra includes a sidebar called "Improving Reasoning Skills -- Bagels I." As it turns out, Bagels is an old 1980's computer game. I never played it on my old computer, but as a young child, I actually had an old toy (Speak & Math) which included a version of Bagels (called "Number Stumper"). Here's a link to a modern version of Bagels:

During the Responsive Classroom training at my old school, the presenter actually suggested Bagels as an opening week activity. In her version of the game, the word "Bagels" was replaced with "Nada," but the words "Pico" and "Fermi" were retained (so she called the game "Pico, Fermi, Nada"). Again, I don't post any version of "Pico, Fermi, Bagels/Nada," but if you want, you can use it in your own classroom instead of the "Islamic Art" lesson.

I do however include Serra's project for this lesson, "Geometry in Sculpture." This isn't directly related to Islamic art, though. Instead, he writes about Umbilic Torus, a sculpture. It was created by Helaman Ferguson and used as a trophy for the Jaime Escalante award -- named, of course, for the world's most famous math teacher.

Wednesday, August 23, 2017

Lesson 0-6: Knot Designs (Day 6)

This is what Theoni Pappas writes on page 236 of her Magic of Mathematics:

"From the time Alexander the Great cut the Gordian knot, knots and their many shapes and forms have pervaded many facets of our lives. Magicians, artists, and philosophers alike have been intrigued by such knots as the trefoil knot, which has no beginning or end."

This is the first page of a long section, "Knots in the Mysteries of Life." You may be wondering, what happened to page 235, since today, August 23rd, is the 235th day of the year? Here's the explanation:

Lesson 0.6 of Michael Serra's Discovering Geometry is called "Knot Designs." And there you go -- a natural reading would put today's Lesson 0.6 one day away from the knot section in Pappas -- so tantalizingly close to lining up.

And so I couldn't resist starting the Pappas knot section today. We'll go back to page 235 tomorrow, before returning to knots with page 237 on Friday.

The Pappas section on knots focuses on a branch of mathematics called "knot theory." She tells us that "knot theory is a very recent field of topology." I've written about topology on the blog before, and in fact Pappas herself discusses topology in her book (see my May 10th post). Two figures are topologically equivalent if one can be bent, stretched, tied, or untied to form the other.

During this entire Pappas science chapter, I've been writing about the science class I taught -- and thinking about how I should have taught these lessons in my own class. Of course, knot theory isn't a suitable topic for middle school science.

Then again, Pappas points out that knot theory is related to the sciences. She tells us that 19th century physicist Lord Kelvin -- of temperature fame -- once believed that atoms were knots in the ether:

"Although his theory was not true, the mathematical study of knots is a very current topic today."

...and may ultimately have a link to physics after all. But Pappas will get to that later in this section.

Meanwhile, let's see what Serra has to say about knots. This is Lesson 0.6 in my old Second Edition, while it's Lesson 0.5 in the modern editions, as those editions omit my 0.5.

Serra begins:

"Knots have played very important roles in cultures all over the world. Before the Chinese use ideograms, they recorded events by using a system of knots."

The Pappas section and Serra lesson have much in common. Both books depict a Celtic knot. As Serra explains, the ancient Celts carved various knot designs in stone.

But Serra has a more inclusive definition of knot than Pappas. He writes:

"Knot designs are geometric designs that appear to weave in or interlace like a knot."

By Serra's definition, the Olympic rings form a knot. The Pappas definition is more in line with the mathematical definition of knot, but we will "not" get to that definition until Friday's post.

Today's worksheet is based on Serra's definition, so there is much emphasis on rings. Of the three questions I selected, one of them has the students draw a knot using a compass (so the shapes will end up being rings). The other two are puzzles involving interlocking rings.

One of the puzzle questions asks students to sketch five rings linked together such that all five can be separated by cutting open one ring. This is easy -- just link four rings to a center ring. The other question is a classic -- the Borromean rings are three rings such that all three are linked, and yet no two of them are linked.

Here is a link to a solution to the Borromean ring puzzle:

According to the link above, the Borromean rings are physically impossible -- unless the rings deviate slightly from perfect circularity. Hence the link labels this as an "impossible figure" not unlike the op art from Monday's lesson. Of course, we can draw op art, including perfectly circular Borromean rings, on paper with a compass.

Actually, I found a few interesting links involving Borromean rings. Here is Evelyn Lamb, who writes a math column for Scientific American once or twice per month ("Roots of Unity"):

Recall that knots and science are related. Here is a Borromean ring consisting of three atoms:

And here's a link to Borromean onion rings, courtesy one of my favorite mathematicians, Vi Hart:

Pappas introduce knots by writing about Alexander the Great and the Gordian knot. Actually, Serra mentions the Gordian knot as well, even though I didn't include this question on my worksheet.

The recreational math website Cut the Knot is actually named after the Gordian knot. Here is a link to Alexander (Bogomolny) the Great, who explains why he chose that name:

Tuesday, August 22, 2017

Lesson 0-5: Mandalas (Day 5)

This is what Theoni Pappas writes on page 234 of her Magic of Mathematics:

"We have heard of body language created by the various messages we give off by the way we move or posture our bodies. But what's body music?"

This is a one-page section in Pappas, simply titled "Body Music." The subject of this page is DNA -- apparently, it's possible to make music out of its strands. Pappas explains:

"A musical link becomes apparent, when one considers the recurring sequences [of bases] as recurring melodies of a song. In fact, many of them have been put to music, both in an octave and in other intervals."

Here it appears that just as the digits of pi and the Fibonacci numbers mod seven are used to make songs (as I mentioned back in my Pi Approximation Day post), it's possible to take the bases of DNA to create body music.

I can only imagine how this works. Recall that the four bases are adenine (A), cytosine (C), guanine (G), and thymine (T). We notice that A, C, and G are already musical notes, so all we have to do is change T to a valid note and we'll have a song. An interesting choice is F, because then F-G-A-C are the four playable notes on the Fischinger Google Doodle from two months ago (if we play the scale in F major rather than C major).

Moreover, DNA bases appear in pairs, with A paired with T, and C paired with G. If we play two-part music to represent the two strands, then all the intervals are consonant, with F-A as a major third (and its inversion the minor sixth), and C-G as a perfect fifth (and its inversion the perfect fourth). If we still had the Fischinger player available, we might choose random numbers from 1 to 4 (F-G-A-C), create a "base bass line" using the complementary bases, and play some "body music."

So far in these Pappas science posts, I've been writing about how I should have taught the subject in my own class. Well, I already said that I should have taught DNA to my seventh graders. Making "body music" probably isn't a suitable project in a school, as students would suddenly have to learn more music than science. The only way I'd do it is if every student had a Fischinger player to make the notes, so all the students have to do is pair the bases correctly to create the song.

But even though I wouldn't play body music in my class, I did play music in my class. Just yesterday I wrote about two of my songs, "The Dren Song" and "Earth, Moon, and Sun." I regret many things about my first year of teaching, but the idea of having a music break is not one of them. Yet often the students didn't know the purpose of music break -- and sometimes neither did I.

Recall that according to Fawn Nguyen, it helps to have a vision of the classroom you want. So what exactly was my "vision" for music break? To me, music break served two purposes:

-- To break up the monotony of 80-minute blocks
-- To give the students something to help them remember difficult topics

For example, I wondered whether I should let the students simply ignore my song and talk all the way throughout the break. According to the first purpose, this is okay -- why not let the students talk if I'm giving them a break?

But according to the second purpose, this is no good. When I first came up with music break, I thought about the famous "Pop Goes the Weasel" parody for the Quadratic Formula. If students sing the song, it might enter their heads when it's time for the Quadratic Formula test. But if they're not paying attention, then the song would be of no use come test time.

Too often, my students talked right through music break. But then again, they had already neutered the "no talking" rule -- if they were talking throughout the lesson, why would they suddenly stop when it's break time? It would only be for a particularly funny line in the song -- and then they would stop to laugh at my singing, not to remember anything mathematical.

One of my most successful songs for jogging the memory was "Measures of Center" -- a "Row, Row, Row Your Boat" parody about the mean, median, and mode. One of my sixth graders who transferred from another school told me how her former teacher had sung her this song. Later on, the coding teacher asked the seventh graders about mean, median, and mode -- and they couldn't remember what they were until I started singing this song.

But there was a blown opportunity for me to sing a song to jog student memories. During SBAC Prep time, I set up a practice test on the computers, and the first few questions were on exponents. Many of the eighth graders  didn't remember the laws of exponents -- or possibly, they never learned them because they were talking throughout the lesson. I'd written a song about these laws -- a parody of the UCLA fight song -- and I could have sung it just before they began their test, but I didn't. The students were then became frustrated when they couldn't answer a single question on the practice test.

Why didn't I sing the song then? Well, when I first covered exponents in October, I'd posted the lyrics on the wall, but then I took them down and packed them away in the closet when it was time for a new lesson and new song. Indeed, I'd forgotten that I even had an exponent song -- recall that this wasn't really a song about exponents, but a tack-on verse of the song "Unit Rate," which was geared towards my sixth and seventh graders. It's easy to remember "U-N-I-T Rate! Rate! Rate!" but more difficult to recall the second verse on exponents.

Since I left my school, I decided to collect all my songs into one notebook. Then I could whip out the notebook and sing the songs whenever I needed them -- especially in preparation for cumulative exams such as the SBAC.

Some of my songs were more popular than others. Parodies were often more popular, since it's much easier for me to change the lyrics than invent a tune from scratch. Of course, Square One TV songs were even more enjoyable, since I didn't have to create the tune or the lyrics.

One song that was surprisingly popular, especially with the seventh graders, was "Fraction Fever." I point out that, just like a parody, I didn't have to create a tune -- the music ostensibly came from the old computer game from 30 years ago. But in practice, I actually did ad-lib a few notes, since I couldn't remember all the notes from so long in the past.

Many seventh graders enjoyed singing "Fraction Fever." The problem was that the lyrics of the song were all about playing Fraction Fever. They weren't about anything that could help them on a test, such as adding, subtracting, multiplying, or dividing fractions.

I should have taken advantage of the song's popularity and adding verses about fraction arithmetic. I could even keep some of the lines about game rules and combined them with arithmetic lines -- perhaps rhyming "elevator" (to get to the next level in the game) with (common) "denominator."

Indeed, I could even have sung this song during Learning Centers. One of the DIDAX manipulatives involved fractions, and so whenever we used them I could sing the song. Students who tend to groan whenever they see fractions might now look forward to seeing them, knowing that I'd be about to sing "Fraction Fever." Unfortunately, the class that liked the song the most -- seventh grade -- was also the class I saw the least, and so I never tried Learning Centers with them.

But in general, during music break the students should either be silent or sing along with me. Music break really is a break from writing, but it's still class time, so they should be listening to me.

Pappas concludes:

"Therefore, it is not unusual for scientists and mathematicians to seek out the music of the body."

...and the same is true for science and math teachers.

I didn't want this to turn into another classroom management post -- I was planning to finish with management during the summer posts. But the success of my "music break" ultimately depended on classroom management to keep the kids quiet and listening to my songs. (Again, I can never have too much management after what happened last year.)

I've said that management is a rarely blogged topic. But another MTBoS challenge has started -- not Blaugust, but something called "Sunday Funday." Julie Reulbach, a North Carolina high school teacher, is the leader of this challenge. Each Sunday, she posts a topic, and participants write about the challenge during the week.

Last week's Sunday Funday topic was classroom management. I assume that she was strongly influenced by that Fawn Nguyen post when she chose this topic -- indeed, Nguyen is listed as the first post, and several other participants quote Nguyen in their own entries.

Sunday Funday is another challenge in which I'm not worthy to participate until I hopefully make my return to the classroom. That doesn't stop me from linking to other participants, though. Here are a few key entries that I found in Reulbach's list:

Peggy Bondurant (state not listed) is one of the newest MTBoS bloggers. After acknowledging Nguyen as the originator of this prompt, she writes:

I enlist the help of a time-watcher or two to help make sure there is time for clean-up at the end of class.  I teach how to put away our laptops - log-off all websites, shut down (not sign-off), place in #'ed slot and plug-in power chord.  Anytime there is an algorithm for a daily or weekly task, I teach it.  My HS Ss do well with these, though the laptops and restroom procedures get re-taught often!

Of course, Bondurant's post reminds me of my own problems with IXL and laptop procedures. We see that her slots were "#'ed" (i.e., numbered). That's the main reason I had problems -- I should have numbered the slots before the first day of school. On the other hand, my class didn't have enough power chords, so that wouldn't have been part of my procedure.

Brianne Beebe, a New York high school teacher, is already familiar as one of the participants of "Day in the Life." (Her daily posting day was the seventh.) She writes:

I rarely contact parents about student misbehavior.  Most student misbehavior that I encounter can be handled in the classroom.  The worst thing my students do most of the time is talk at the wrong time.  (I hope I'm not jinxing myself for this year.  *Knock on wood)  My typical discipline hierarchy is as follows:  first a warning, second a small consequence (taking phone for the period, moving seat, etc), third a referral with phone call home.  I think I've only made it to the third step once in five years.  Typically, when I call parents about student misbehavior it's because the student's behavior has been repeated for a few days and the consequences in the classroom have not deterred it.

Beebe is what I call the "ideal classroom manager." The ideal manager reaches the higher levels of the hierarchy very rarely -- she says that she only needed one referral in five years. She's also a "natural classroom manager," particularly if those are her first five years of teaching.

I, meanwhile, had to give many referrals and phone calls in just five months of teaching. And it's easy to see why -- look at that second step, where Beebe confiscates phones and changes seats. In my class, my students would either refuse to surrender their phones, or as we've seen, even claim that it's just a phone case and not the actual phone. And they'll refuse to change seats when I tell them to. So I'd have no choice but to make the referral or phone call with the reason "refusal to surrender phone or change seats after being asked to do so more than four times."

Since Beebe is a natural, I assume that she has a strong teacher tone. When she speaks in that teacher tone, the students surrender their phones and change seats when asked. Since I lack teacher tone, the students refused to do what I ask. Therefore I must reach higher levels on the hierarchy than a teacher with a strong tone like Beebe.

Tara Daas, a Georgia high school teacher, is also familiar from "Day in the Life." (Her daily posting day was the 26th.) She writes:

Though I have not really had a lot of issues with classroom management in the years past the start of my career, I feel that many methods I tried and used lacked so much purpose and meaning.  In reflection, I feel most of it was driven by re-action to past problems rather than pro-active measures.  Though I had become pro-active with parent communication, the pro-active switch on everything else was stalled.  The pro-active parent communication helped behind the scenes of my classroom early in my teaching career, and it kept amount of communication less frequent and more positive.  When I began to implement more pro-active and positive measures in the main stage of my classroom with students in later years, I never really gave classroom management much thought anymore – it managed itself.

The distinction between reactive and proactive management is also made by Lee Canter. I've never thought of it that way before -- proactive interactions with students (and parents, as Daas points out) tend to be more positive.

Okay, that's enough with music and management. Let's finally get to Geometry.

Lesson 0.5 of Michael Serra's Discovering Geometry is called "Mandalas." This is the first of two sections included in the old Second Edition yet omitted from the modern editions.

But what, exactly, is a mandala? Serra explains:

"A mandala is a circular design arranged in layers radiating from the center. The word mandala comes from Hindu Sanskrit, the classical language of India, and means 'circle' or 'center.'"

As Serra points out, other cultures had mandalas, not just the Hindus. The Aztec calendar, for example, was constructed as a mandala.

Many mandalas exhibit threefold or sixfold symmetry. They are related to the regular hexagon, and so the compass and straightedge can be used to construct them. At any rate, the compass should at least be used to draw the circle that is the base of any mandala.

All the mandalas on these pages come from a Google image search. There is no project in this section, but of course "draw your own mandala" is a natural question for this section.