But since it's math again, I will indeed make today's "Day in the Life" post. The focus resolution will once again be:
4. Begin the lesson quickly instead of having lengthy warm-ups.
That's because this regular teacher not only wants me to go over the homework -- she wants me to assign a grade to the homework. The grading is simple -- just a five-point scale -- but going around to each student and checking the HW is bound to take plenty of class time.
8:15 -- There's no zero period today -- instead I begin with homeroom at the regular time.
8:30 -- This is one of the middle schools where all six classes rotate, and hence the rotation schedule has nothing to do with the day of the week. Today the rotation begins with second period, which is an eighth grade math class. The students are learning about perimeter, area, and volume. I try to grade all the homework quickly at the start of class, but it still takes about fifteen minutes or so.
The teacher has prepared a lesson where the students find the volumes of prisms, but some of the questions require the students to find the volumes of pyramids. Fortunately, I'm a math teacher, so I can improvise and create a pyramid volume lesson on the fly. The most difficult problems are the volumes of triangular pyramids. All triangular faces on today's worksheet are isosceles, and in these problems, all three side lengths of each triangular base as well as the altitudes are given -- along with the height of the pyramid, of course. And with all three sides being whole numbers, the altitudes (derived from the Pythagorean Theorem) -- which are needed in the formula -- are decimals. So it's no wonder that these problems are difficult.
9:20 -- Second period ends and third period begins. This is another eighth grade math class. I'm only slightly faster at grading the homework than the previous class. Still, I consider this to be the second best behaved class of the day.
10:15 -- Third period ends and the students head for snack.
10:30 -- Fourth period begins. This is the period when I'm finally hoping to get the hang of grading the homework -- except there's another twist. This is actually an Algebra I class. And as it turns out, the previous night's assignment consists of five worksheets. Yes, I know that I assigned my students packets at one point last year, but the packets were never a one-night assignment.
On some of the pages, the students only had to complete the even-numbered problems, while on other pages they were assigned to complete all of the problems. Moreover, some pages of the packet are clearly shrunk down (or dilated!) from 11" * 17" size and hence had two page numbers, while others pages were unnumbered. And so the students are confused as to which questions are assigned.
This makes the homework grading take a long time -- do I give the same grade to someone who finished all the problems as someone who did only the evens? And then when I finally finish the grading, it's now time to go over the answers. Many of the students have questions after they check the answers -- mainly because the answer key contains so many wrong answers! The grading and correcting of homework take up about half the period.
By the way, the homework assignment contains pages on both transformations and perimeter/area (the same pages that are assigned to the Math 8 classes). This is the tricky thing about Algebra I in eighth grade -- the students must still take the Common Core Math 8 SBAC exam. Both today's Algebra I class and yesterday's are scrambling to teach topics that aren't usually taught in Algebra I yet appear on the eighth grade SBAC. (In fact, this is why the teacher is out today -- all eighth grade math teachers are at a meeting, presumably to discuss the SBAC.) The main lesson of the day is on transformations, in particular dilations. It's like Lesson 12-1 of the U of Chicago Geometry text.
I do like to add that one of the homework problems my class corrects today asks for the area of an isosceles triangle given only the lengths of its sides. The teacher's answer key declares that the area is impossible to find. But the front page of the packet actually provides the formula -- if the base is a and the legs are b, then the area is (a/4)sqrt(4b^2 - a^2). The formula is clearly derived from the Pythagorean Theorem, which is used to calculate the height of the isosceles triangle. I suspect that the teacher prepared the answer key hastily -- she's swamped with all the extra work to get the class prepared for SBAC as well as make it to today's meeting.
11:20 -- Fourth period ends. Fifth period is the teacher's conference period, which leads to lunch.
1:00 -- Sixth period arrives. This is another Algebra I class, but this is the school that always begins with silent reading right after lunch.
1:15 -- Silent reading ends. I begin grading the homework. I still use about half the period to grade and correct the homework, but this time it goes faster since the answer key is correct -- I use the extra few minutes I save to explain the isosceles triangle area formula that fourth period uses earlier. This class runs more smoothly, and hence it's the best behaved class of the day.
2:05 -- Sixth period leaves and first period arrives. This is another Math 8 class. So at least the homework grading takes as long as second period -- they don't expect me to go over the answers (even though they're working on the same packet that contains the isosceles triangle area formula).
But unfortunately, this class is more talkative than the others, especially when eighth grade photos are distributed near the end of this class. I'm not as sure that these kids understand as much of the volume assignment as the others.
2:55 -- First period leaves, and my day ends.
As I reflect back on today, I wonder whether I could have saved time with the homework. For example, in sixth period I could have started grading the homework during silent reading time. But I'm not sure whether this would have been a good idea. In all classes, the students get louder when I walk around the classroom to grade HW, so if I grade papers during silent reading, then SSR would have been ruined -- and all for naught, since sixth period goes so smoothly today anyway.
It's first period where I feel I could have managed the class more smoothly. At the start of the day, the office manager warns me that the photos are being distributed in first period -- and I've already seen firsthand how easily distracted these students are. In hindsight, a cleverer idea presents itself -- just start the period with the volume assignment and then when the photos arrive, start grading last night's HW (as the students are distracted by now anyway).
One guy in first period is particularly disruptive. I decide to call on students randomly to answer the volume questions -- just like yesterday, except the numbers come from the class roster (so no blue tape on the desks is needed). So of course, the disruptive student is chosen to answer one of the triangular prism problems. He doesn't want to answer, and so he starts claiming that his number is 32 when there are only 31 names on the roster. Then he tells me that he doesn't know the answer because he's just returned from the restroom. I end up walking him through the problem, but I suspect that other students just tell him the answer.
There are a few things I could have done with this student. Since by this time the class photos are ready to be picked up, I could have sent this student out to pick them up. Normally I wouldn't want someone who just returned from the restroom to walk out again, but teachers often don't mind if talkative students like him are out of the room.
Of course, if I'd started the class with volume as I should have, then I would have chosen this guy to answer a question well before photo time. In this case, the best thing to do after he claims "I don't know" is to get back to him and ask him the next question on the worksheet -- which happens to be the volume of a cube with side length 2. He might have been able to figure this question out without the other students helping him, and he would have gained more confidence in answering.
The other question to take away from today is, how should I handle homework in my own classes, particularly the upcoming summer math class? We know that many math teachers recommend not even assigning HW any more. I'd rather wait until after the first training session (in two weeks) before making a decision.
But this is a good place to segue into our traditionalists topic, since today is test day. Over the weekend (on Cinco de Mayo), Barry Garelick posted. He made a more recent post since then, but let's link to the post that has drawn more comments:
https://traditionalmath.wordpress.com/2018/05/05/ncsm-nctm-annual-conf-dept/
Every year the National Council of Teachers of Mathematics has its annual conference, complete with celeb speakers, vendor booths, instructional seminars, and the usual array of topics that pass for effective practices.
From what I hear from a friend who teaches high school math, this year’s was no different. Her report follows below:
And let's just skip to the conclusion:The bottom line is that it was all pretty bad. My school district paid a few thousand dollars to send my colleague and me to this stuff. We were there for the last day of NCSM and the first day of NCTM. Needless to say, I was glad to get home!!
This post has drawn seven comments, including three from SteveH. Let's see what he has to say:
SteveH:
Let’s see, the CCSS math slope starting from Kindergarten only targets no remediation for College Algebra at the end of 12th grade. In 7th grade, most schools define two or more tracks where the only STEM track is the one where students get to a proper algebra class in 8th grade. However, they don’t tell students and parents what has to now be done at home or with tutors to get to that track and they don’t ask us parents of the best students what we had to do. That sounds like a gap in their critical thinking and self-assessment.
Of course, I've seen both of those tracks today, with three sections of Math 8 and two of Algebra I.
SteveH:
“Pre-AP Algebra 1 [9th grade] focuses deeply on the concepts and skills that are most essential for college and career success, so mastery of linear relationships is a major focus of this course. Linear functions and linear equations are the basic building blocks of many advanced topics in math. Pre-AP Algebra 1 is streamlined to give students the time and space to thoroughly master these concepts and skills.”
“streamlined to give students the time and space”? Cover less material? “Space”? Linear is emphasized and quadratic is well, streamlined out? The academic gap is bridged with BS. Well, at least SteveH explains more about this "Pre-AP" class here. This is a tough one. On one hand, the unit on quadratic equations is when students start asking questions like "When are we ever going to use this in real life?" Even linear equations appear to be more useful than quadratics. The NCTM is trying to devise courses in which the question "When will we use this in real life?" is never answered.
On the other hand, we can argue that "Because you need to know it to prepare for AP Calculus" is an acceptable answer to this question in a class explicitly designated as a "Pre-AP" class. In a general class, some students have no intention of taking Calculus and wish to reach the point as soon as possible where they no longer have to take a math class. But "Pre-AP" students, by definition, intend to take an AP math course in the future.
One problem with eighth grade Algebra I is that eighth graders are still required to take the SBAC in Common Core Math 8. Even if eighth grade material is covered in seventh grade Honors Math for students headed for eighth grade Algebra I, the fear is that there would be 12 or months between the material being taught and the material being tested. And so teachers must scurry to review this material right before the test, resulting in five-page assignment packets where students don't know which questions to answer. Therefore the Common Core does indirectly discourage eighth grade Algebra I through its administration of the SBAC.
A obvious solution presents itself -- eighth grade Algebra I students shouldn't be required to take the SBAC math exam. After all, ninth grade Algebra I students don't have to take the SBAC, so why should eighth grade Algebra I kids?
SteveH:
What are they thinking? Let’s take an ordinary proportion math problem and make it into something that isn’t math? Let’s make a big deal about this and waste a lot of time so we can ignore all of the other interesting textbook homework p-set variations of proportionality and weighting factors. There is no one right answer? Um, look at Consumer Reports with all of their weighting factors. That’s what people do – translate non-exact choices into exact ones. You might disagree, but math helps you define different merit or objective functions.
Here SteveH is referring to a particular math problem that was presented at the NCTM meeting. In past posts, I pointed out that the love affair with questions for which "there is no one right answer" is to avoid telling weaker students "You're wrong!" After all, "You're wrong!" only makes students defensive -- it doesn't change weak math students into strong math students.
That part about using math to define different "merit or objective functions" reminds me quite strongly of Wayne Wickelgren's state evaluation functions. Traditionalists like to assume that people prefer thinking in terms of quantity rather than quality. Often this is true, but for many people -- the ones who struggle in math classes -- this is definitely not the case.
SteveH:
[This talks about high school math in general. It’s not posed as leaving the AP Calculus STEM track alone, but offering other useful tracks than CCSS pseudo-algebra II and no remediation in college algebra. There is also a lot of other baggage here, such as “…beyond a focus on college and career readiness.” I would be more open to other tracks if they were geared to specific degree, career, and life paths, AND they first fixed up the low expectation, no ensured mastery in K-6.]
Here SteveH is referring to a book called Catalyzing Changes. OK, I guess that I can agree with what SteveH writes here. On tracks geared to other life paths, I hope that the question "When will we use this in real life?" will have a more obvious answer. And I already agree with more traditionalist math in the lowest elementary grades.
Let's return to Mocha music. So far, we've been investigating the 12EDL scale:
The 12EDL scale:
Degree Ratio Note
12 1/1 white A
11 12/11 amber B
10 6/5 green C
9 4/3 white D
8 3/2 white E
7 12/7 red F#
6 2/1 white A
Today our goal is to observe properties of this scale, in preparation for composing music.
We begin by reviewing Kite's color notation. The color "white" means that the degree contains only factors of 2 or 3 (Pythagorean). "Green" denotes the factor 5, "red" denotes the factor 7, and "amber" denotes the factor 11.
These are Kite's utonal colors. He's also defined the corresponding otonal colors -- "yellow" for the factor 5, "blue" for the factor 7, and "jade" for the factor 11. But EDL music is utonal -- that is, it's based on harmonic undertones, not overtones. In fact, it's easy to program Mocha to play a fundamental note followed by its undertones:
10 INPUT D
20 X=261
30 X=X-D
40 SOUND X,4
50 GOTO 30
This program is based on a loop that stops as soon as the undertones are out of range (which is indicated with a message ?FC ERROR). According to Kite, otonal intervals sound more consonant than utonal intervals. This is why major scales, which are otonal, sound happy while minor scales, which are utonal, sound sad. Since Mocha music is utonal, it's much easier to play a just minor scale in Mocha than a just major scale.
Indeed, we notice that the 12EDL scale contains a just minor triad, A-C-E. Moreover, the first five notes are somewhat like a minor scale -- A-B-C-D-E. Since the full A natural minor scale would be A-B-C-D-E-F-G-A, only the red F# sounds out of place.
One thing about EDL scales is that as we ascend them, the intervals between the notes increase. In fact, we see that the interval between white A and amber B is 12/11 (151 cents) while the interval between red F# and white A is 7/6 (267 cents).
This is the exact opposite of the concept of leading tone, where the interval between the last two notes is reduced. For example, the A minor scale is often played as A harmonic minor, where the scale goes A-B-C-D-E-F-G#-A. So G# is the leading tone that leads to A. Even the Lambda scale (Bohlen-Pierce) was designed to start with "a hearty whole tone" and end with a leading tone. But our EDL scale goes against the natural tendency to end with a leading tone.
You might notice that the note we call "amber B" is 151 cents above white A -- that is, it's about halfway between a minor second and a major second (hence the name "neutral second"). You might wonder why I call it "amber B" if it's halfway between B and Bb. Actually, I call it "amber B" because I'm combining two different just intonation notations -- Kite and Helmholtz-Ellis.
Kite himself uses his color notation in two different ways. One of them is to name intervals. Here the interval 11/8 is called a "jade fourth," while 16/11 is an "amber fifth." There are two different types of neutral thirds -- the "jade third" of 11/9, and the "amber third" of 27/22.
But Kite also uses colors to name the notes. These names consist of a color followed by the note name such as B or Bb. Suppose we were to begin on white G and try to build both a jade third and an amber third above the white G. Should a jade 3rd above G be jade Bb or jade B? And should an amber 3rd above G be amber Bb or amber B? (At this point I'm discussing just intonation in general, as this has nothing to do with Mocha-playable notes. After all, no jade note is playable in Mocha.)
Kite tells us that the note names must be consistent. We notice that a jade third plus an amber third equals a white fifth (3/2, the perfect fifth). This means that the jade 3rd and amber 3rd above G can't both be B's (since major 3rd + major third isn't perfect fifth), nor can they both be Bb's (since minor 3rd + minor 3rd isn't perfect fifth). Thus one of them must be B and the other Bb. Here Kite leaves open which one to name B and which to name Bb -- he tells us that both ways have advantages as well as disadvantages.
We notice that a jade third is 347 cents, while an amber third is 355 cents. This is one good reason to call the notes "jade Bb" and "amber B" -- at least the Bb is a few cents lower than the B.
But the reason I prefer jade Bb and amber B goes back to Helmholtz-Ellis. This notation uses special accidentals to name notes in just intonation. In H-E notation, a jade third above G would be labeled as "Bb quarter-sharp" and an amber third above G would be "B quarter-flat." In an EDO system (such as 24EDO), Bb quarter-sharp and B quarter-flat would be enharmonic, but these are not considered enharmonic in just intonation. The quarter-flat symbol resembles a "d" (just as the ordinary flat symbol resembles a "b"), so we can write B quarter-flat in ASCII as "Bd." I haven't decided how to write quarter-sharp in ASCII -- but it doesn't matter for us since all quarter-sharp notes are otonal (hence unplayable in Mocha). The quarter tone accidentals represent the interval 33/32.
OK then, so why do I call the note "amber B" rather than "amber Bd," since the note really is a quarter tone flat from B? The answer is that "amber Bd" is redundant -- the "d" accidental already tells us that the note is amber. There are H-E accidentals to denote other commas, but Kite's color notation makes those accidentals redundant as well. This includes + for the syntonic comma 81/80 and > for the septimal comma 64/63. In past posts I used "7" for the septimal comma, but I've since decided that ">" is better. Hence the entire 12EDL scale can be written as:
A, Bd, C+, D, E, F#>, A
But Kite's colors make all accidentals redundant except the classical accidentals # and b. We can see why # and b are still necessary -- if we take F#> (red F#) and lower it by seven perfect fifths, we'll eventually reach F> (red F) -- which is far outside of Mocha range. Likewise, we could raise jade Bb seven perfect fifths to reach jade B. Still, this tells us that the note an amber third above white G should be "amber B," since this is another way of writing "Bd."
The note a jade third above white G is called "jade Bb." This is another way of saying Bbq (or whatever the quarter sharp symbol is). The color "jade" replaces the non-classical quarter sharp accidental, but no colors can replace the classical flat accidental. Notice that the interval between white F and jade Bb is 11/8. This tells us that the jade fourth must be spelled like a perfect fourth.
OK, so let's return to 12EDL. Our EDL scales don't have a leading tone as we ascend from F# to A, but in some ways EDL scales do have a descending leading tone, from amber B to white A. This suggests that we could create a song where the last two notes are amber B-white A. In fact, all the notes of the scale are part of two triads: wA-gC-wE and aB-wD-rF#. Perhaps we might be able to use the latter triad to lead into the former triad (the A minor triad).
Once we write a song in 12EDL, we can play it in 21 different "keys" -- that is, there are 21 possible root notes for 12EDL. Any of the 21 Mocha-playable undertones of Degree 12 (with the last one being Degree 252 = 12 * 21) can be used as the root note of a 12EDL scale. This is a far cry from the scales of just major (only one Mocha-playable key) and just minor (only three Mocha-playable keys).
Possible 12EDL root notes in Mocha:
Degree Note
12 white A
24 white A
36 white D
48 white A
60 green F
72 white D
84 red B
96 white A
108 white G
120 green F
132 amber E
144 white D
156 ocher C
168 red B
180 green Bb
192 white A
204 17esque G#
216 white G
228 19esque F#
240 green F
252 red E
Here are the Chapter 15 Test answers:
1. 144pi - 288 square units.
2. 178 degrees.
3. Arc DE = 65 degrees.
4. Many answers are possible, for example Angle A = 47.5 degrees.
5. 14 degrees.
6-7. These are visual, so I can't put the answers here.
8. 21.
9. a. (-2, 9). b. 7. c. Many answers are possible. To find lattice points on the circle, we go right, left, up, and down seven units, to obtain (5, 9), (-9, 9), (-2, 16), and (-2, 2).
10. a. (0, 0). b. sqrt(72). c. This time, sqrt(72) = 6sqrt(2), so we can go diagonally to find lattice points on the circle, to obtain (6, 6), (-6, 6), (6, -6), and (-6, -6).
11. This is the complete the square question -- included because such problems are on PARCC!
x^2 + y^2 - 8y = 9
x^2 + y^2 - 8y + 16 = 25
x^2 + (y - 4)^2 = 25
So this gives us:
11. a. (0, 4). b. 5. c. (5, 4), (-5, 4), (5, 9), and (5, -1).
12. a. A circle with radius 20 feet. b. 40pi feet.
13. Draw any circle.
14. About 1.68%.
15. 15.
16. Cavalieri's Principle. Take that, traditionalists!
17. a. When the line and circle intersect in a point. b. When the line is perpendicular to the radius at the point of tangency. PARCC contains a few tangent problems, and all of them appear to involve angle measures, so that right angle is important.
18. a. 36 degrees. b. 18 degrees. PARCC also contains problems on inscribed angle measure -- possibly in the same question as tangents.
19. a. About 2.5 or 2.6 cm. b. The ratio to the circumference to the diameter is -- what else -- pi. We see that we estimate pi as either 3.08 or 3.2 using this measurement. Interestingly enough, 3.14 is almost exactly halfway between these two estimates.
20. a. 24 square units. (The height is 4, using the Pythagorean Theorem) b. About 38.5 square units.
No comments:
Post a Comment