Lesson 13-6: Uniqueness (Day 136)

Spring break is definitely on many of our minds right now. As so often happens when I work in two districts, they end up taking different weeks off for spring break. In my new district, it's easy as spring break is always near Easter (that is, Easter Week or Bright Week).

But the blog calendar follows my old district, not my new district. And in that district, spring break works a little differently. First of all, in this district, spring break has nothing to do with Easter. So instead, spring break occurs during the same fixed week each year -- and that's next week. In other words, today is the last day before the vacation on the blog calendar.

We notice that today is Day 136 on the blog calendar, and 136 is about 3/4 of 180. Despite this, spring break is not considered to divide the third and fourth quarters. This is because winter break, which really does divide the first and second semesters, occurred after Day 83, not Day 90. So the start of the fourth quarter is closer to the mathematical midpoint of the second semester, a few days ago.

I'm not sure why the district doesn't simply have spring break right after third quarter. Perhaps it's so that today can be the deadline for submitting the third quarter grades, or maybe today's the day the students actually take home the third quarter progress reports. I'm not sure, since of course I didn't sub in this district today.

But this district doesn't ignore Easter completely. The rule is that Good Friday and Easter Monday are always holidays in this district. In both 2017 and 2019, Easter falls on the third Sunday in April.

Meanwhile, the LAUSD (and hence my charter school from two years ago) always takes off the week before Easter (that is, Holy Week). And just like the blog calendar, LAUSD is also closed an extra day for Cesar Chavez Day. The labor leader's March 31st birthday is on a Sunday, and so it's observed on April 1st instead.

Chavez Day is used to define spring break in the California State University system. Since the actual date falls on a Sunday, campuses are closed on Monday, April 1st instead. Spring break is thus considered to be the week after next-- there are no classes, but the campuses are open every day except Monday, Chavez Day Observed. Thus spring break is a week later this year than last year.

In the University of California system (including my alma mater UCLA), Chavez Day is considered to be the last Friday in March, which is also March 29th. So the UC spring break is next week. In other states spring break may be earlier -- last week, I mentioned that it was spring break at Washington State University.

Finally, I want to point out when spring break is in New York City -- the largest district in the nation and home to many MTBoS bloggers. In the Big Apple, spring break occurs at Passover, which is usually near Easter anyway. This year Passover starts on Saturday, April 20th. But New York, just like the blog calendar, has a rule that Good Friday is always a day off (though Easter Monday isn't part of this rule). And so New York takes off Good Friday and the week after Easter. (In some years, Passover falls a month after Easter -- in such years, Good Friday is a mere three-day weekend.)

As usual, my plans for the blog are to post once or twice during "spring break" as observed by the official blog calendar.

But before we get to spring break, let me write about today's subbing. Yes, in my old district the Big March may be ending, but it's still going strong in my new district.

Today I subbed in a high school history class. It's at the same school as yesterday. Since it's not a math class, I won't do "A Day in the Life" today.

Three of the classes are sophomore World History. These students are currently learning about Stalin and totalitarianism in the former Soviet Union.

The other two classes are freshman history classes. Ordinarily, here in California there is no history class prescribed for freshmen. The official name of this class is "Cultural Geography H/IB/MUN," indicating that the class is for honors, IB, and Model UN students. Indeed, there is an all-day Model UN session scheduled for tomorrow, Saturday. These students are learning about the three Abrahamic religions -- Judaism, Christianity, and Islam.

I only need to write one name on my bad list today. Just like yesterday's teacher, today's regular teacher has a strict cell phone policy -- academic use only. But this policy is only specified in the lesson plan for tutorial only. One girl in tutorial has her phone out and doesn't even try to claim that it's for an academic purpose. Elsewhere in tutorial, I help some students with their math assignments, including several Geometry students who are taking a quiz today. (This is the quiz on Glencoe Lessons 10.1-10.3 that I mentioned in yesterday's post.)

But several students are distracted today when they watch videos -- not on their phones, but on the Chromebooks that they're supposed to have out for their assignments. The third Thursday in March is the start of the NCAA basketball tournament -- and the first two days of the tournament are days when even adults tend to be distracted. Students are especially distracted during periods 4-5, when the only Southern Californian team in the tournament is playing, UC Irvine (a sister campus of my alma mater UCLA). As it turns out, #13 UCI pulls off the upset, defeating #4 Kansas State 70-64.

Meanwhile, I write several names on the good list in all classes -- especially in seventh period, when the Chromebooks must be put away at the end of the day. The regular teacher definitely wanted to make sure that the computers are fully charged upon her return, and so she is offering extra credit to 1-4 students who help put them away. There were three volunteers -- two guys and one girl -- for the extra credit.

Second period is the best freshman class of the day, and sixth period is the best sophomore class. As you may have noticed, these are the periods when the UCI game isn't being played.

You may have noticed that I've added the "music" label to this post. Yes, I'm going to post some computer music in this post, but no, it's not Mocha music. I notice that the Bach Google Doodle is up a second day, and so I don't want to waste this opportunity to post some of my musical creations right here on the blog.

(Yes, once again I'm almost treating today's post like a spring break post, adding extra stuff like music and all that.)

Here's how the Bach player works -- the user is supposed to enter a melody. The computer then adds harmony to the user's melody -- but it's not just random notes that are added. Instead, the computer chooses notes to make the song sound like a Bach piece. It's supposed to analyze all of the famous composer's songs to calculate which notes he would have chosen if he were writing the song.

Bach's songs were based on four-part harmony, SATB (soprano, alto, tenor, bass). The user's melody appears as the soprano part, and the computer supplies the alto, tenor, and bass.

Let's create our songs the same way we did for the Fischinger player two years ago. I'll generate the notes at random. Since I'm supposed to be writing the soprano part, I want to choose notes as high on the staff as possible. I'll use the digits 1-9, where 9 is the highest playable note (a"). Then 1 will correspond to g' (the G above middle C), and 2-8 fill in the octave from a' to g". We'll use 0 as a rest (just as we did in some of the Pi Day songs).

OK, here we go. This the first melody that I created:

Number: .0657977488
Melody: r-  e"- d"- f"- a"- f"- f"- c"- (where the first "r" is a rest)

These are all quarter notes, but I notice there's an option to add eighth notes, too. (To change a quarter to two eighth notes, we move the mouse to the right of the quarter note.)

I press the harmonize button, and this is what is produced:

Soprano: r-  e"- d"- f"- a"- f"- f"- c"-
Alto:       g'---    f'g' a'- a'-  f'-  a'-  re (the last e is a sixteenth note) 
Tenor:     g'- c'- c'#- d'- c'#-d'---    c'-
Bass:      B-  e-  a#- d'- a-   d'---    a-

Hmm, that's interesting. Notice that the my original line was in C major (that is, it contains only naturals with no sharps or flats), yet the computer inserted sharps in the tenor and bass parts. Well, that's Bach for you! Also, I notice that I can't enter sixteenth or half notes, but apparently the Bach player can produce them!

Let me try it again:

Number: .4681002713
Melody: c"- e"- g"- g'- r--- a'- f"-

Perhaps because my song is simpler, the Bach player produced a simpler harmony:

Soprano: c"- e"- g"- g'- r--- a'- f"-
Alto:       f'-  g'f'  e'---    e'-----  f'-
Tenor:     c'-------          c'--- r---
Bass:      a-   c'---      c- c'- c- A-r-

I think it's interesting that even though my melody begins with a C major triad (c"-e"-g"), the Bach player harmonized it as an F major triad (a-c'-f'-c"). This time Bach adds some longer notes such as dotted half and whole notes.

Here's another go at it:

Number: .70838565
Melody: f"- r-  g"- b'- g"- d"- e"- d"-

And here's the Bach version:

Soprano: f"- r-  g"- b'- g"- d"- e"- d"-
Alto:       a'- b'---    g'- g'---      ra'  b'-
Tenor:     d'-------        b---       c'-  f'-
Bass:      d'd g-----       g---       c'- rb

My song began with f", and Bach decided to harmonize is as D minor (d'-d'-a'-f").

Since my original inspiration for converting digits to notes was pi, let's try a pi song. Since our range is from g' to a", I'll code the notes in G major. Also, I'll use eighth notes so I can get more digits in:

Number: 3.14159265358979
Melody: b' g' c" g' d" a" a' e" d" b' d" g" a" f#" a-

Here's how Bach does pi:

Soprano: b' g' c" g' d" a" a' e" d" b' d" g" a" f#" a-
Alto:       g'-   e'  g' f'# g' a'-     a'  g' a'  b' a'-       b'a'
Tenor:     d' r------                    d'--       r--           be'
Bass:       g r-      b a   e   f#e   f#g   a   b  c'#a    d'c'#

This song starts with a G major chord, but ends on an A major chord for some reason.

And of course you knew I had to try Fibonacci:

Number: 11235813213455
Melody: g' g' a' b' d" g" g' b' a' g' b' c" d"- d"-

And here's how Bach does Fibonacci:

Soprano: g' g' a' b' d" g" g' b' a' g' b' c" d"- d"-
Alto:       d' r--       f'#e'  d'-    c' b' g'-    a'-  g'-
Tenor:     d'-    c' b a-      f'#b  c' d' e'-   d'-   b-
Bass:       b-     f#e d   c  B-    Ad# e g  f#-  gd (the last e is dotted eighth, followed by 16th note g)

Let me also try the two songs that are built in. The first is "Mary Had a Little Lamb":

Song: Mary Had a Little Lamb
Melody: e" d" c" d" e" e" e"- d" d" d"- e" g" g"-

Now Bach has a little lamb:

Soprano: e" d" c" d" e" e" e"- d" d" d"- e" g" g"-

Alto:       e'---            b'-     a'-  a'-     b'-  c" a'  a'#-

Tenor:     b-      a-      b-     c'-  d'  c'  g'-  r-       d'-

Bass:       g#-    a  f#  g#-   a g f-      g-   c-      A#-

Only Bach can take a simple C major song, start it with an E major triad, and end it with what's essentially a G minor triad (though A# should really be spelled Bb)!

The other built-in song is "Twinkle Twinkle Little Star":

Song: Twinkle Twinkle Little Star

Melody: c" c" g" g" a" a" g"- f" f" e" e" d" d" c"-

Let Bach twinkle now:

Soprano: c" c" g" g" a" a" g"- f" f" e" e" d" d" c"-

Alto:       g'-     c" a'# a'-    g'-  f'  a'g'a'-    g'---        (a'g' sixteenth notes)

Tenor:     e'---            f'-     e' d' c'---         d'  bf' e'- (bf' sixteenth notes)

Bass:       c---             a  bd'c' b a  g  f-     g-      c    (bd' sixteenth notes)

There are more sixteenth notes and one accidental (a'#), but no unfamiliar chords. Thus this song still sounds recognizably like "Twinkle Twinkle Little Star" (unlike his version of the Lamb). Well, at least Bach beats Mocha -- our Mocha emulator can't play harmony at all.

Last year, on the day I posted Lesson 13-6, I told a little story about music and how I "discovered" the G major scale. Since this is a music post, I might as well retell that story in today's post:

I remember taking a piano class the summer after kindergarten. As it was only a beginners' course, the only major scale taught was the C major scale, which was played on the white keys. But I wondered to myself why one couldn't play a major scale beginning on notes other than C. The following December (either for my birthday or Christmas) I received a small electronic keyboard as a gift, and naturally I tried playing other scales. beginning on D, then E -- but none of them sounded like the proper major scale Do, Re, Mi, etc.

But one of these scales sounded almost right -- the scale beginning on G. The scale G, A, B, C, D, E, F, G sounded correct except for the last few notes. But I didn't know how to make the last part sound like the major scale. Disappointed, I started playing around with chromatic scales -- where I included the black keys as well as the white keys. But as I was still tantalized by the G scale, I sometimes started out by playing the part of the G scale that sounded right -- G, A, B, C, D, E -- and then switched to the chromatic scale. The black key between F and G is called F sharp, or F#, so what I played was G, A, B, C, D, E, F, F#, G, and I often played around with this "scale" for awhile.

Then one day, I accidentally skipped the F note, so what I ended up playing was G, A, B, C, D, E, F#, G. And what I played sounded exactly like the major scale that I had been seeking! And I still remember to this day how excited my seven-year-old self was to "discover" the G major scale! After this, I quickly realized that I could make all the other scales (the D scale, E scale, and so on) sound right by including some of the black keys as well as the white keys on my keyboard.

The next ratio to consider is 6:5. This interval is called a minor third. To produce a minor chord, we take the major chord C, E, G and replace the E with the black key just below E, called E flat, often rendered in ASCII as Eb. So the C minor chord is C, Eb, G.

Why do minor chords sound "sadder" than major chords? Recall that a major chord has its notes in the ratio 4:5:6. Well, a minor chord has its notes in the ratio 10:12:15 -- that is, 5:6 is now the lower ratio and 4:5 is now the higher ratio. Since 4:5:6 is simpler than 10:12:15, the 4:5:6 major chord sounds brighter than the 10:12:15 minor chord. Many popular songs on the radio nowadays tend to be about one of two topics -- falling in love and breaking up. The former songs tend to be in major keys, while the latter songs tend to be in minor keys. Over the past few decades, popular hits in minor keys have increased.

The second keyboard that I owned, when I was in the second grade, had several songs built in, including the classic Greensleeves. This song was in the key of A, but I noticed that it was based on A minor rather than A major, as it contained the chord A, C, E (whereas A major would be A, C#, E). I soon learned that there were minor scales as well as major scales, and tried to deduce what the A minor scale was based on the song. Unfortunately, the version of Greensleeves that was built into my keyboard wasn't truly in A minor, but a scale called the Dorian mode. Instead, I was under the misconception that A, B, C, D, E, F#, G, A was the A minor scale -- that is, A minor was just like G major in that it contained F# rather than F natural.

It was not until the third grade until I took private piano lessons. My third grade classroom had a piano, and the teacher was a pianist. So she taught me piano one a week after school until she left on maternity midway through the year, after which she referred me to her mother to continue my study of the piano. This was when I finally learned that the A natural minor scale was actually the notes A, B, C, D, E, F, G, A -- that is, it has no sharps or flats, just like C major. This minor scale is formed by taking the three Pythagorean intervals (octave, fourth, and fifth) and building a minor chord on each note, rather than a major chord.

Here are a few more links discussing major and minor scales:

http://music.stackexchange.com/questions/22236/why-am-i-always-sticking-to-minor-keys

which in turn links to:

http://www.nme.com/blogs/nme-blogs/the-science-of-music-why-do-songs-in-a-minor-key-sound-sad

Lesson 13-6 of the U of Chicago text is called "Uniqueness." In the modern Third Edition of the text, uniqueness appears in Lesson 5-6. Recall that the lessons of the old Chapter 13 appear in various chapters of the new edition. Uniqueness of Parallels (Playfair's Parallel Postulate) now appears in Chapter 5 so that it can be used to prove Triangle Sum. In the old version, Triangle Sum is proved in Chapter 5 using a slightly different proof, since uniqueness doesn't appear until Chapter 13.

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

The Glide Reflection Theorem only works when the preimage and the image have opposite orientation, not the same orientation. If a figure and its image have the same orientation, then we know that the isometry mapping one to the other is either a translation or a rotation. This case may be a bit tricky -- it could be that the easiest way is simply to translate A to A' and see whether this translation maps B to B' -- if not, then a rotation is necessary. But how do we find the center?

We know that the center of rotation is equidistant from A and A'. Thus it lies on the perpendicular bisector of AA'. For the same reason, the center lies on the perpendicular bisector of BB'. So where these two points intersect is the center of rotation. Notice that if these two perpendicular bisectors are parallel, then the above reasoning constitutes an indirect proof that there is no rotation mapping one to the other -- that is, there is a translation map instead.

Today's worksheet covers all of Lesson 13-6. This means that not only are there questions about the Glide Reflection Theorem, but also about uniqueness in general. A modern form of Euclid's original five postulates are given.

Today is an activity day. Let's look at today's three Exploration questions for Lesson 13-6:

29. What does the word auxiliary mean outside of mathematics?
30. a. Name five things outside of mathematics which are unique.
      b. Name five things which are not unique.
31. Who owns the zebra?

Notice that Question #31 actually directs student to solve the zebra logic puzzle, while last week's Exploration question merely asks students to do the first step. With today being the last day before spring break (on the calendar), it's the perfect day for a puzzle and fun.

Last week, I linked to the following website:

https://sites.lsa.umich.edu/inclusive-teaching/2017/08/16/who-owns-the-zebra/

This link suggests how to set up the "Who owns the zebra?" question for groups. I recommend setting up today's activity in groups just as the link directs us to.