Introduction
Today, of course, is Christmas Day. And tonight is also the last night of Hanukkah -- the night when all eight candles are lit. This combo is often referred to as "Christmukkah." Like Festivus, Christmukkah is generally attributed to a TV show, in this case The OC.
This also marks the second time I've ever posted on Christmas Day -- the first was two years ago. Both Christmas posts were driven by the Yule Blog Challenge and the desire to fulfill it by posting throughout the holidays.
Yule Blog Prompt #7: Something New I Tried (or Still Want to Try) This Year
Well, the one thing I definitely tried this year are my songs. While some of my songs come from previous years, some of them are brand new.
I'll post some of them today. Let's start with "Let's Get Mathematical," a parody of "Let's Get Physical." I wrote this to honor Olivia Newton-John, who passed away during the year:
LET'S GET MATHEMATICAL
Not all of my songs are parodies. For Math III, an original song I wrote was "Transformation," about transforming parent graphs:
This song is written in a brand new scale -- the 10EDL scale. And in other songs I used 12EDL. These scales come from an old computer that I used decades ago. By composing in these scales, I'm able to come up with original tunes for these songs -- but these tunes are tricky to play on a standard guitar.
As the school year began, I wrote that I'd compose a few songs in the new scales first, and then discover which guitar chords sound good with these scales. Now that I'm done so, I'm ready to summarize the various chord riffs I played.
Analyzing 10EDL and 12EDL Tunes
Let's start with the 10EDL scale -- in the key of C, this scale goes C-D-E-F#-A-C'. The song that I listed above, "Transformation," is a simple 10EDL song. Here's the full melody:
F#-C'-F#-C-E-C' (repeated 4x for verse)
E-F#-A (repeated 4x for chorus)
And as I've written many times before, the 10EDL scale is very similar to the major pentatonic scale, except that it contains F# where pentatonic would use a G.
So what chords did I play for this song? Well, since the scale is so similar to C major pentatonic, of course I used a C major chord. But, you might point out, the C major chord goes C-E-G, but we changed the G to F#, so shouldn't C major be invalid?
Well, think about a true pentatonic song. We normally use the I, IV, and V chords -- in this case C, F, and G -- to accompany the song, even though there is no F in C major pentatonic (and even the G chord goes G-B-D, with no B in the pentatonic scale). Thus we're allowed to use notes in the chords even if they don't appear in the melody.
Now we need a secondary chord. We look at the scale again -- C-D-E-F#-A-C' -- and notice that there is a D major chord available, D-F#-A. And we can even add the C' to make it a D7 chord. Thus I've been using D7 often in my 10EDL songs. In fact, the only chords I used in "Transformation" were C and D7.
Of course, we don't use D7 in an ordinary C major song unless it's a double-dominant chord, V/V, that leads to the usual dominant chord G (or G7). But since there's no G in the 10EDL scale, I don't use the G (or G7) chords unless it's over a D in the melody. Only one of my 10EDL songs, "Growth Slope," had enough D's in the melody to justify playing a G7 chord. Therefore of all my 10EDL tunes, "Growth Slope" sounds the most like a traditional C major song.
As for 12EDL, this scale goes A-B-C-D-E-F#-A', so it closely resembles an A minor scale. Remember that the note I'm calling "B" here is a just ratio 12/11 above the root A, so it's about halfway between the notes Bb and B. On the guitar, I might play it as either Bb or B depending on the situation.
Here's an example of one of my simpler 12EDL tunes, "Linear Art," which I played in my Math I classes during that infamous linear art project:
Here's the full melody:
F#-E-A'-C-E-F#-F#-D (4x for verse)
F#-F#-D-F# (4x for chorus)
Notice that unlike C major for 10EDL, a full A minor chord appears in 12EDL, and so I play Am chords in my 12EDL songs. For this particular song, there are so many D's and F#'s in the melody that I had to play yet another D chord.
This song avoids Degree 11, and so the issue of whether to play it as B or Bb doesn't arise. For some of the other songs, such as "Solve for y," I notice that a G chord fits the song the best (even though, once again, there is no G in the 12EDL scale). Then we can interpret Degree 11 as either Bb (to make a Gm chord) or B natural (to make a G major chord). Playing a G power chord (that is, with no third) also works in order to suggest the neutral third in the melody.
In a true Am song, we expect some sort of E chord as the dominant. During classical times, the usual choice would be E7, with a G# (harmonic minor), but in modern times, we might hear an Em chord with a G (natural minor). But once again, while there is an E in 12EDL, there's no G or G#. Once again, we could try an E power chord (but note that the B in the scale isn't a perfect fifth above this E).
By the way, I've gone back and edited the last few songs I posted. When I reflect upon these posts next year (assuming I'm teaching the same classes), I want the lyrics to reflect how I should have taught Math I Chapter 3 (to match my neighbors), rather than how I actually taught the chapter. Then when I see the songs, I'll teach it better next year.
The December 6th-7th song is now a transformation song, since this is the week that my neighbor teachers taught the transformation lessons:
SHORT TRANSFORMATION
The melody is very simple (mainly due to the principal's observation that day) B-B-B-C-A'. Once again, I play some sort of G chord (major, minor, or power) over the Degree 11 notes, then end on Am.
The November 29th-30th song is trickier to change, since it now needs to be a Math III song on exponential and logarithmic functions:
A PRACTICAL USE FOR NONLINEARITY
The changes cause a few problems, though. In particular, I had to replace the four-syllable word "Geometry" with the six-syllable word "nonlinearity." Also, I keep "Practical Use" in the title, even though only one practical use ("population growth") appears in the lyrics. (I might consider shortening the title to just "nonlinearity.")
The other songs I performed in December don't need to change. The "Fraction Busters" lesson should have been given on a Monday -- a non-block (and hence a non-singing) day, so there's no way to make the "Ghostbusters" parody song line up with the lesson. The other December parody, "Sweet Home Algebra," also remains where it is -- it actually fits that day's lesson (multiplying binomials) better.
Rapoport Question of the Day
Today on her Mathematics Calendar 2022, Rebecca Rapoport writes:
Find x.
Once again, we have a Geometry question where are the given information is in an unlabeled diagram, so I must provide all the labels. In Triangle ABC, we have D, E, F on AC, BC, AB respectively. Angle C = 130, AD = AF = BF = BE, Angle DFE = x, and we must find x.
At first I was tricked into thinking that Triangle ABC is isosceles -- but we don't know that, because we aren't told how CD and CE compare. Instead, the isosceles triangles are ADF and BEF (and that aren't necessarily congruent to each other, because we know only SS). So we must perform the necessary angle chase using only the two known isosceles triangles.
From isosceles Triangle ADF, we have Angle AFD = (180 - A)/2.
From isosceles Triangle BEF, we have Angle EFB = (180 - B)/2.
Since AFD + DFE + EFB = 180, this gives us (180 - A)/2 + DFE + (180 - B)/2 = 180, which then simplifies to 90 - A/2 + 90 - B/2 + DFE = 180, or DFE = (A + B)/2
Then from the original triangle, A + B + C = 180, and since C = 130, that gives us A + B = 50. So we obtain DFE = (A + B)/2 = 50/2 = 25. Therefore the desired angle is 25 degrees -- and of course, today's date is the 25th, Christmas Day.
Conclusion
On this Christmas Day, I'm making my fifth Yule Blog post, but the challenge leader Shelli is already on her eighth:
http://statteacher.blogspot.com/2022/12/review-games-i-want-to-try.html
Shelli's eighth post is on games she plays in her class just before a test. She links to a Balloon Pop game from Sarah Carter and states that the true origin of the game is from a decade-old post from a certain other blog. I've seen a version of Balloon Pop on the TV show Survivor, but it never occurred to me to play it in the classroom.
This concludes my post. I hope you were able to enjoy your Christmas -- or Hanukkah -- this year, and that your holiday is starting look like the old holidays we used to know pre-pandemic.
(By the way, an old joke is that Jewish people eat Chinese food on Christmas -- at least in years when it's not Hanukkah. So do they eat Chinese food or latkes today?)
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