I wonder whether there's much point in posting the last "A Day in the Life" today. Unlike yesterday, when there were history and co-teaching classes, today is nothing but giving the final. Well, I'll guess I'll post it for completeness, but there's not much to say.
8:25 -- After homeroom (which I don't have), fifth period arrives to take the final.
9:40 -- Fifth period leaves for first snack.
9:55 -- Sixth period arrives to take the final.
11:05 -- Sixth period leaves for second snack, which leads into first period conference. Therefore my day essentially ends here.
Unlike yesterday, there are no behavior problems today. Thus the best two classes from Wednesday (second and fourth periods) are the ones that struggle to behave on Thursday.
Today is Sixday:
Resolution #6: We ask, what would our heroes do?
Again, I keep the same message on the board as yesterday -- calculators are allowed, but we should be able to do basic math without one (just as our 1955 heroes did).
Oh, and there's one more thing I do that's related to this resolution -- the song. As I like to do during these multi-day math classes, I come up with a new song to perform. And since last time I found a song from YouTube ("Compound Interest"), today's song is original. I'll write more about this song in today's post.
After each class takes a final, the aide takes them to the machine to be graded. Today's scores are just as dismal as yesterday's, with only three students passing in each class. A couple of students earn B's, while a couple get 23/39 (which would have been passing if there were a 40th question for them to get correct, 24/40 = 60%).
Actually, my day ends a little after 11:05, as one guy stays behind to finish his test. He asks for help on an equation that he struggles on as it contains fractions. I tell him that I can't really help him since it's a final, and time's basically up anyway. So he gives his test to the aide for grading, and I help put the desks back into rows (after spreading them out for the final). She returns just in time to tell him that his grade is 35/39 -- the top score in the class.
Once again, 36/40 is exactly 90%, so 35/39 is short of 90% -- and again, it demonstrates why there ought to be 40 questions on the test, not 39. Then again, some teachers actually use 4/5 rounding and count 89.5% as an A -- 35/39 is 89.7%. (As for the other tests of length ending in -9, 26/29 is also 89.7% rounded to an A, 44/49 is 89.8% rounded to an A, and 39/49 is 79.6% rounded to a B.)
For other teachers, there's no rounding -- only the digit in the tens place determines the grade, so if the tens digit is an 8 rather than a 9, the grade is B. Sometimes it depends on the grading software (like PowerSchool at the old charter) whether a teacher at a school rounds or not.
(By the way, 19 is still a cruel number of questions to have on a test, even with rounding. A student who earns 17/19 has a grade of 89.47%, which rounds to 89% to the nearest percent but 89.5% to the nearest tenth of a percent. This actually happened to someone I knew when I was a young high school student -- the software prints the grade as 89.5% B, and the student asks, why didn't 4/5 rounding raise his grade to an A? The answer is that the 5 is itself rounded -- the tenths digit is actually a 4, and by 4/5 rounding, 89.47% rounds down to 89%. We avoid this sort of argument simply by adding a twentieth question to a test with exactly 19 questions.)
Meanwhile, there are too many students who either sit there without finishing the test, or just fill it all in randomly and say they're done. Thus there are way too many scores close to 25%. I understand that these students are special ed, but still, many of them need to put in more effort if they wish to be successful now and in high school next year.
OK, so let me post the new song. I decided to let Mocha generate a new song in our usual preferred Mocha scale, 12EDL.
Notice that this week at the middle school, I didn't sing "Big March," even though with trimester finals, this is perhaps the Big March-iest week of the Big March. Because it was a math class, all of the songs I perform today are related to the math that appears on the test.
Once again, I decide to have Mocha generate some quick sixteenth notes in addition to eighth, quarter, and dotted quarter notes. In vocal music, sixteenth notes are great when singing words with many syllables, but they are tricky when trying to learn the tune (as I am, since I generated this tune just last night).
When I composed the tune, I used Mocha's random number generator. But one huge problem with this is that computers can't generate truly random numbers, only "pseudorandom" numbers. Indeed, if you run the pseudorandom generator several times, exit the Mocha window, and then enter and run it again, you'll end up generating the exact same numbers -- and hence the exact same song.
I usually play with random many times whenever I run Mocha -- before running the 12EDL code -- to prevent the same song from being composed by the computer. But last night, I couldn't help but notice that my "Whodunnit" song was included as part of the new pseudorandom tune. The note lengths were different because L had a different value at the start -- instead of eighth and dotted quarter notes, there were sixteenth and dotted eighth notes. But the tune was recognizable.
To fix the song, I simply replaced this bar with a single whole note, D. I also made other changes by hand after the song was generated, mostly changing low A with high A when it fits the tune better.
https://www.haplessgenius.com/mocha/
10 N=8
20 FOR X=1 TO 35
30 READ A,T
40 SOUND 261-N*A,T
50 NEXT X
60 DATA 10,4,12,2,8,2,9,4,11,1
70 DATA 12,2,11,1,7,2,8,2,11,4
80 DATA 9,6,10,2,7,2,6,1,7,1
90 DATA 6,4,7,4,7,4,9,16,8,4
100 DATA 12,4,7,4,8,4,8,2,8,2
110 DATA 12,2,9,2,10,6,9,2,6,4
120 DATA 6,4,6,4,8,3,7,1,6,16
As usual, click on Sound before you RUN the program. Here are the scale and the notes used:
12EDL scale:
Degree Note
12 white A
11 lavender B (or Bb)
10 green C
9 white D
8 white E
7 red F#
6 white A
C-A-E-D-B-A-B-F#-E-B-D-C-F#-A'-F#-A'-F#-F#-D-E-A-F#-E-E-E-A-D-C-D-A'-A'-A'-E-F#-A'
Notice the notes "D-E-A-F#" in the middle of the song. This is actually part of the "Whodunnit" song that the pseudorandom generator produced. I decided to keep it anyway, since I'd changed these notes back in "Whodunnit" -- back then I changed low A to high A, but here I keep low A. Also, I keep F# here, but back then I changed F# to D (so that the notes D-E-A-D would appear in the middle of that murder mystery song).
Now let's add some lyrics. I just take the final hints that aide has already written on the board (plus my calculator comments for the sixth resolution) and then make it into song lyrics:
The Slope Song
Delta-y over delta-x,
Rise over run and,
y equals mx plus b.
Calculator,
Use it if you need, but,
Try in head just for me.
y minus y all over,
x minus x, and,
Keep, change, change, yes you see.
Calculator,
Use it if you need, but,
Try in head just for me.
Notice that more teachers are starting to use "delta" when teaching slope. Once again, as a young student I once tried to impress my Algebra I teacher by using delta in my homework, and now suddenly teachers are using delta for real. Meanwhile, "keep, change, change" is a mnemonic used to remember how to subtract integers (keep the first number, change the subtraction to addition, and finally change the sign of the second number). Since the slope formula requires subtraction in the numerator and denominator, "keep, change, change" is relevant here.
Of course, this is a fairly short song that I wrote in one night. If this were a full song to perform on the day that slope is actually taught (as opposed to just being reviewed and tested), a slope song should probably explain it all better. So the delta expression and "rise over run" equal the slope, while y = mx + b is the slope-intercept form, not just the slope. And of course, "y minus y" is really short for y_2 - y_1, which we'd really need to explain better. Perhaps one of these days I'll expand this into a full song that can be used to teach slope.
This is what I wrote last year about today's lesson:
Section 12-6 of the U of Chicago text covers the Fundamental Theorem of Similarity. As its name implies, it is the most important theorem related to dilations and similarity. Here is how this theorem is stated in the U of Chicago:
Fundamental Theorem of Similarity (U of Chicago):
If G ~ G' and k is the ratio of similitude, then
(a) Perimeter(G') = k * Perimeter(G) or ...
(b) Area(G') = k^2 * Area(G) or ...
(c) Volume(G') = k^3 * Volume(G) or ...
Notice how I had to rewrite this theorem so that it fits into ASCII. Here the * and ^ symbols denote multiplication and exponentiation, respectively -- these symbols should be recognizable as they appear on TI graphing calculators. The "or ..." sections refer to the text rewriting each equation as a ratio, so that the ratio of the perimeters is k, the ratio of the areas is k^2, and so on, but that is rather awkward to write in ASCII.
David Joyce describes this theorem in his criticism of the Prentice-Hall text:
The proofs are omitted for the theorems which say similar plane figures have areas in duplicate ratios, and similar solid figures have areas in duplicate ratios and volumes in triplicate rations. At least there should be a proof that similar triangles have areas in duplicate ratios; that's easy since the areas of triangles are already known.
Here is the U of Chicago proof:
(a) Perimeter is just the sum of the lengths. Suppose lengths a, b, c, d, e, ...make up the perimeter of G. Then lengths ka, kb, kc, kd, ke, ...make up the perimeter of G'.
Perimeter(G') = ka + kb + kc + kd + ke + ...
= k(a + b + c + d + e + ...)
= k * Perimeter(G)
(b) Let A = the area of G. Then you could think of the area of G as the sum of areas of A unit squares. Then the area of G' is the sum of areas of A squares k units on a side. Since each square in G' has area k^2,
Area of G' = A * k^2 = k^2 * Area of G.
(c) The argument is identical (except with unit cubes). QED
As I mentioned earlier this week, I'm no longer posting old worksheets that mention the Hung-Hsi Wu proof. Thus I'm rearranging last year's worksheets.
Last year's second worksheet is now the first worksheet, since it mentions the U of Chicago version of the Fundamental Theorem of Similarity.
Today is an activity day. There's no point in using the Lesson 12-6 Exploration question from the U of Chicago text since it asks students to take the scale model from Lesson 12-5 (in other words, we can't avoid 12-5 anyway) and then weigh it (which students probably can't do since most math classrooms don't even have scales for weighing objects).
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