Today I subbed in an high school special ed English class. It's in my LA County district. Since it's a high school class that isn't a math class and has aides, there's no need for "A Day in the Life" today.
One of the classes is for seniors, while the other is labeled a "reading" class for all grades, but most of the students are seniors anyway. Both classes are reading articles -- the reading students have an article about Jackie Robinson, while the non-reading class seniors have one on birds killed by city lights.
This teacher actually has a conference period -- a rarity in this district under the 4 * 3 schedule. But I must cover a class during fourth period -- it's another special ed class. It appears to be Study Skills. A counselor runs the Zoom completely -- all I need to do is take attendance. This is tricky, though, since she divides the students into breakout rooms right away.
Today is Nineday on the Eleven Calendar:
Resolution #9: We pay attention to math as long as possible.
Let's replace "math" with "English" here. We recall that this is the district where students only have to stay on Zoom for a half hour. In order to make sure the students are paying attention, reading, and answering questions, I decide to choose students at random. One opted-out student doesn't read because he decides to go to the bathroom (at his house). Not only is this soon after a snack break, but he's only on Zoom for a half hour with a full hour between classes. So he really should either go to the bathroom during the break or wait until after the Zoom session -- but how do I really stop someone from going there at their house?
As for today's song, I choose "Roots" -- that's because after I sang it last week, I finally EDL-ize this song and am ready to post a Mocha version. (So I add the "music" label here.) Let's discuss how I did it.
First of all, since I've found my old songbook, I do have remnants of the old pre-EDL tune. This tune consisted of the following notes, from lowest to highest:
D-E-F#-A-B-C-E
So the span of this song is slightly more than an octave. I clearly wrote it so that E is the tonic, and so that low D is a "subtonic" note below the tonic E. Notice that it's ambiguous whether this song is major or minor, since it has neither a major nor a minor third. It probably sounds more minor due to the presence of the minor sixth at C.
The original tune emphasized the perfect fourth at A. So we should choose one of the EDL's that contains the perfect fourth 4/3. One possibility is 12EDL -- the only problem is that 12EDL contains a supermajor sixth 12/7 instead of the minor sixth written above. (If we were to choose 12EDL, the song would sound major despite 12EDL leaning towards a minor scale, since we omit the minor third 6/5.)
But 16EDL is a better choice here. I usually don't use 16EDL due to its awkward neutral third 16/13, but this song doesn't need a third, so we can omit it. If we place E at Degree 16, then F# is Degree 14, A is Degree 12, and C is Degree 10, a just minor sixth 8/5 above E. The only tricky note is B, which we place on Degree 11 -- sounding halfway between Bb and B. But, as luck would have it, the original tune doesn't emphasize B at all, so the subfifth 16/11 is acceptable here.
We might also consider 20EDL here, which also has a perfect fourth. But 20EDL lacks both a perfect fifth and a minor sixth -- I'm willing to give up one of these (as 16EDL does), but not both. And so this leaves 16EDL as the best choice.
As for rhythm, the line "Square root of 1 is 1" sounds best as an eighth note ("Square"), two sixteenth notes ("root of"), two eighth notes ("one is"), and a half note ("one"). This pattern will repeat through the entire song.
And so here is the final Mocha version:
https://www.haplessgenius.com/mocha/
As usual, click Sound before you RUN the program. This is not in call-and-response format (as I might sing it in the classroom), but it does repeat so we can sing both the "square root" and "cube root" verses:
ROOTS
Square root of 1 is 1,
Hey, this is so much fun.
Square root of 4 is 2,
So here's what we should do.
Square root of 9 is 3,
So let's all come and see.
Square root of 16 is 4,
So please show us some more.
Square root of 25 is 5,
And now I feel so alive,
Square root of 36 is 6,
So now there ain't no more tricks.
Cube root of 1 is 1,
Hey, this is so much fun.
Cube root of 8 is 2,
So here's what we should do.
Cube root of 27 is 3,
So let's all come and see.
Cube root of 64 is 4,
So please show us some more.
Cube root of 125 is 5,
And now I feel so alive,
Cube root of 216 is 6,
So now there ain't no more tricks.
Notice that even though I call this a 16EDL song, that low D in the subtonic is Degree 18, so we should technically call it an 18EDL song. This matters -- if we try to lower the song an octave by replacing the first line with 10 N=16, the low D becomes Degree 288, which is out of Mocha range. The largest N can be is 14 (just like any other 18EDL song).
Still, I prefer calling it a 16EDL song because I like to think in terms of the tonic. Thus 16EDL has a perfect fourth and 18EDL doesn't, while 18EDL has a perfect fifth and 16EDL doesn't. So calling it 16EDL reminds me which notes appear in the song relative to the tonic.
By the way, yesterday Rebecca Rapoport acknowledged that an "astute reader" caught her error from yesterday, that the two rhombi aren't necessarily squares. I did tweet about it yesterday, so I like to believe that I'm the "astute reader" who found it for her.
Today's Rapoport problem is also trig, but it's more directly related to the unit circle. Hence it's not a Geometry problem, and so I won't post it here.
This is what I wrote two years ago about today's lesson:
Question 13 of the SBAC Practice Exam is on solving equations:
Emily is solving the equation 2(x + 9) = 4(x + 7) + 2. Her steps are shown.
Part A
Click on the first step in which Emily made an error.
Step 1: 2(x + 9) = 4(x + 7) + 2
Step 2: 2x + 18 = 4x + 28 + 2
Step 3: 2x + 18 = 4x + 26
Step 4: -8 = 2x
Step 5: -4 = x
Part B
Click on the solution to Emily's original equation.
-10.5 -6 -2 0 2 4.5 8
Ah, so this is definitely a strong first-semester Algebra I problem. We must check each step carefully to find an error. For Part A, we see that in Step 2, Emily must combine the like terms 28 and 2, but in Step 3, she writes the sum as 26. (Clearly, she was thinking of 28 - 2 instead of 28 + 2.) Therefore the first step that contains an error is Step 3.
For Part B, let's start from the last correct step (namely 2) and proceed with the sum 28 + 2 = 30:
Step 2: 2x + 18 = 4x + 28 + 2
Step 3': 2x + 18 = 4x + 30
Step 4': -12 = 2x
Step 5': -6 = x
So students must click on the value -6.
My students are likely to make mistakes like Emily's, and so I must be able to correct them. If Emily were a girl in my class, I could help her by praising her last correct step (namely 2) then and ask her, "What's 28 + 2?" Hopefully, she'll realize that it's 30, not 26, and then she can finish solving the equation.
Both the girl and the guy from the Pre-Calc class correctly answer Step 3 and -6 for this question.
Question 14 of the SBAC Practice Exam is on explicit and recursive functions:
Match each recursive function with the equivalent explicit function.
Recursive functions:
f (1) = 18; f (n) = f (n - 1) + 6; n > 2
f (1) = 18; f (n) = f (n - 1) + 12; n > 2
f (1) = 1; f (n) = 6f (n - 1); n > 2
f (1) = 1; f (n) = 12f (n - 1); n > 2
Possible explicit functions:
f (n) = 6^(n - 1); n > 1
f (n) = 12 + 6n; n > 1
f (n) = 12^(n - 1); n > 1
f (n) = 6 + 12n; n > 1
The first two recursive functions are arithmetic sequences, so they might conceivably appear in the first semester of an Algebra I class. The last two recursive functions are geometric sequences. These used to appear in Algebra II, but since advent of the Common Core, I've seen geometric sequences and exponential functions pushed down into the second semester of Algebra I.
Let's look at the two arithmetic sequences first. The first term of each sequence is 18, but in the first sequence the common difference is 6, while in the second sequence it is 12. The coefficient of n in the explicit function is the common difference, so we already know that the first sequence must match to the "+6n" function while the second matches to the "+12n" function.
Even though we don't need it, it's helpful to know how to determine the constant term in each of the explicit functions. Some texts tell the students that the explicit function is f (1) + (n - 1)d (where d is the common difference), and so they must expand (n - 1)d. The U of Chicago Algebra I text has the students work backwards to find a "zeroth term," and this is the constant. So in the first sequence, the zeroth term is 18 - 6 = 12, and in the second, it's 18 - 12 = 6. Therefore the first explicit function must be f (n) = 12 + 6n while the second is f (n) = 6 + 12n.
[2021 update: Recall that this idea also came up during my long-term, when my seventh graders were modeling using linear functions. It helped to consider how many wheels a "size 0 toy train" has.]
Now we move on to the geometric sequences. This time, the trick is that the common ratio must be the base of the exponential function -- that is, f (1)r^(n - 1). In each case the first term is 1, so all that remains is r^(n - 1). Therefore the third sequence is f (n) = 6^(n - 1) while the last is f (n) = 12^(n - 1).
And as for the girl and the guy from the Pre-Calc class, neither one answers this question -- since for some reason, it didn't print on the SBAC Practice packet! At least this packet is much better than what I had at the old charter school two years ago -- where none of the SBAC Practice problems could print.
It's time for a Desmos activity. Here's one I found by Wes Overton, a card sort that includes arithmetic and geometric sequences, including both explicit and recursive formulas:
SBAC Practice Exam Question 14
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