But first, let me write in those programs that create Halloween Mocha music -- again, I post it today so that next year, I can just click "October 2018" and be led straight to the program. Our first one codes 18EDL again, but with an emphasis on open guitar strings:
http://www.haplessgenius.com/mocha/
10 CLS
20 N=8
30 FOR A=0 TO 6
40 B=4
50 X=A-INT(A/2)*2
60 D=20-RND(2)*5
70 PRINT D;
80 L=RND(B)
90 SOUND 261-N*D,4*L
100 IF L>1 THEN FOR I=1 TO L-1:PRINT " ";:NEXT I
110 B=B-L
120 IF B>0 THEN D=19-RND(10):GOTO 70
130 PRINT
140 NEXT A
150 PRINT 18
160 SOUND 261-N*18,16
Here's how the new Line 60 works. The fundamental note of 18EDL is white D, and since this is Halloween music, we want the scale to be D minor. Let's look at the 18EDL scale again:
The 18EDL scale:
Degree Ratio Note
18 1/1 white D
17 18/17 17u D# (su D#)
16 9/8 white E
15 6/5 green F
14 9/7 red F#
13 18/13 thu G
12 3/2 white A
11 18/11 lavender B
10 9/5 green C
9 2/1 white D
The open notes of the guitar are E, A, D, G, B, E, while C is also an easy open chord to play. On the other hand, our one open B chord (B7) fits with E minor (or major), not D minor. I've decided that I don't want both B7 and C to appear together. C major is more closely related to D minor than B7 is, and so we choose to include C rather than B:
The 18EDL scale (guitar open strings):
Degree Ratio Note
18 1/1 white D
16 9/8 white E
13 18/13 thu G
12 3/2 white A
10 9/5 green C
Of course, there's no reason to exclude, say, green F, since as the minor third, green F sounds harmonious over a Dm chord. But for these simple songs, I want the first note of the measure to be played over a chord with that note as the root, to make it easier for me to remember.
The Degrees in this list are 18, 16, 13, 12, 10. This is so close to being an arithmetic sequence, which would be easier to include in Line 60. We notice that Degree 14 is red F#, while Degree 13 is thu G (formerly known as "ocher G"). Neither red F# nor thu G (which is almost G#) actually matches the string G on a guitar. But technically, white G is closer to red F# than thu G -- the former is 28/27 (about 63 cents) away, while the latter is 27/26 (about 65 cents) away.
So I feel justified in using Degree 14 in the program (which Line 60 codes for), but simply playing it as G on the guitar. In 12EDO, the 7-limit is only slightly better represented than the 13-limit -- and indeed, playing 14 as G makes 14/11 (about 418 cents) sound as G-B, a major third, rather than a perfect fourth. And besides, F# isn't really needed in a D minor song, since D-F# is a major third.
So this gives us:
The 18EDL scale (guitar open strings):
Degree Ratio Note
18 1/1 white D
16 9/8 white E
14 9/7 red F# (played as G on the guitar)
12 3/2 white A
10 9/5 green C
Since we're playing Degree 11 as B, the scale is more Dorian (with a major sixth) than minor. But we prefer B since G major, unlike G minor, is an open guitar chord.
If we write our songs in 12EDL (my originally intended scale, based on Degree 12 or A) rather than 18EDL and play Degree 7 (red F#) as G, then the scale resembles Kristen Lawrence's minor scale (minor seventh, no sixth). Earlier, I considered keeping red F# as F# and play the songs over two different chords, Am and Bm (or Bm7, which like B7 is an open guitar chord). But I didn't try out any 12EDL songs this year, so I don't know which way is better.
The randomizer program as written above creates songs with many whole and dotted half notes. The long notes make it more difficult to add lyrics. Over the summer, I wrote another program that includes more short notes, such as eighth and sixteenth notes, so that we can add lyrics. Let me include my latest version of these programs:
(18EDL)
NEW
10 CLS
20 N=8
30 FOR A=0 TO 13
40 B=2
50 L=RND(B)
60 BB=4
70 LL=RND(BB)
80 D=19-RND(10)
90 P=A*32+(4-B)*8+(4-BB)*L*2
100 D$=RIGHT$(STR$(D),2)
110 PRINT @ P,D$
120 SOUND 261-N*D,L*LL
130 BB=BB-LL
140 IF BB>0 THEN 70
150 B=B-L
160 IF B>0 THEN 50
170 NEXT A
180 PRINT @ 448, "";
190 SOUND 261-N*18,16
For 12EDL, make these changes:
80 D=13-RND(7)
190 SOUND 261-N*12,16
This program makes the rhythm fit 2/4 time (which can be interpreted as 4/4 time), with note lengths ranging from sixteenth to half notes (with a whole note at the end of the song). But unfortunately, this code no longer places guitar-friendly notes at the beginning of each measure.
Finally, I played Degree 18 (white D) on my computer, and it matches D on my musical keyboard at my home. I'm not sure why "white D" matched middle C on the piano in the classroom last week -- perhaps Mocha sounds on that laptop play as different tones.
OK, let's get to today's subbing and "A Day in the Life":
7:55 -- Second period (again, at high schools "first period" is more like zero period) is the first of the three math classes. It's Basic Math, and the students have a Pizzazz worksheet on factoring -- they must list all the factors of a number.
Anticipating a special ed math class, I bring my Number Talks book with me. But Cathy Humphreys writes more about multiplying than factoring. In fact, in her Chapter 3, one of the strategies for multiplying is "factor a factor," which presumes that students find factoring easy. Students are provided with a 25 * 25 multiplication table to assist with the factoring.
Without any Number Talks strategies available, I come up with my own strategies so that students can circle all the correct factors of a number from a given list:
- Always circle 1 and the number itself.
- If the number is even, circle 2. If the number is odd, don't circle 2.
- Never circle 0.
- Never circle any choice greater than the number.
These tips, of course, fit this particular Pizzazz worksheet, which often multiples of the number (including zero) as possible choices.
8:50 -- Second period leaves and third period arrives. This is another math class, except it's Algebra 1B, the second semester of Algebra I spread out to a year for special ed students.
These students are learning about exponents -- indeed, they have the same worksheet that the other special ed class was completing back on October 18th (at another school). And so I already know which problems I need to emphasize (including that infamous -2y^7/(14y^5) question).
9:50 -- Third period leaves for snack.
10:00 -- At this school, tutorial is after snack. A handful of students attend, including one student who stays just long enough to tell me that he's heading out for Cross Country -- my sport! His best time for three miles is just over 16 minutes -- he beats my best time of around 18 minutes (and that was almost exactly 20 years ago). I wish him luck at his League Finals.
10:30 -- Fourth period is the one science class. The class is labeled as Integrated Science. We know that California is encouraging middle school integrated science, but I still occasionally see high school integrated science from time to time (and recall that I as a young high school student -- back when I was still running XC races -- took integrated science). These students take a test on physical science, including the phases of solid, liquid, and gas (and plasma).
11:30 -- Fourth period leaves and fifth period arrives. This is a "Study Skills" class, but today they are watching Nightmare Before Christmas -- how festive for Halloween.
12:25 -- Fifth period leaves for lunch.
1:10 -- Sixth period arrives. This is the final math class, another Algebra 1B. This time, I have both third period and the October 18th to help me choose which problems to go over.
2:05 -- Sixth period leaves. Once again, this teacher has five straight classes, and so there's no need for me to stay for seventh period.
Since this is a special ed class, aides cover most of the classes. And so in this post, I'll focus on the two classes without aides, third and fourth period.
Third period has some of the hardest workers -- in fact, most students finish the first worksheet and a few finish the second as well. In fact, I name this class to be the best of the day.
Fourth period science, however, is a different story. It's always tough being a sub on a day when the students must take a test, since it's hard to make them be silent.
And today, two students take over the class. One student -- the lone senior in the class -- claims that he is a TA. Usually, TA's are clearly marked on the roster, but I've covered classes (often art or music) where the TA's aren't labeled. Since he's the only senior, it's plausible that he's a TA.
The other misbehaving student is a sophomore. I've subbed at this same school on October 19th -- this guy was one of the trio who leaves class early. He claims that he can't take today's test because he was absent all week -- instead, I should let him have a free day to study on his phone (as there are no extra science texts in the classroom).
I stop him from having a free phone day, but then both he and the senior continue to tell jokes, make noises, and generally make fun of me the entire period. Many students can't concentrate and are unable to take the test. The two troublemakers start throwing tennis balls at the test takers. I say nothing, since they'd consider yelling and getting upset at them to be a reward.
In the end, I inform the fifth period aide. She writes a referral for the senior, while I write up the younger student. But I'm worried, because to the other students during the actual class, it appears that I'm doing nothing about the pair of troublemakers. She also tells me that normally, the senior almost never attends class, yet he just happens to show up today.
And so I wonder what I could have done better in this class. Perhaps I should have just let the pair play on phones the whole time -- at least they might not have thrown objects at the others. But then when the teacher returns, they might claim "The sub told us we didn't have to take the test!" and so it makes me look bad.
I'd better figure something out quickly. This is the first day of a two-day assignment, and I worry about how I'm going to deal with fourth period tomorrow. Those students still have a test to finish, and the two troublemakers will be there.
Meanwhile, normally when I sub for science, I compare this class to what I should have taught two years ago at the old charter school -- but it's difficult to think about science because I'm so worried about these two troublemakers. But let's revisit the STEM (not science) project that I gave two years ago on Halloween:
On the October holiday two years ago, I tried to do another seventh grade Illinois State project, "Orienteering." I made the activity fit the holiday by dressing up as a pirate and having the students search for my treasure. I wrote on the blog that the activity didn't work as well as I wanted, because the public school (co-located with our charter) restricted us to a small area of the playground. It might have worked better this year now that we're no longer co-located -- if only I were still a teacher there.
Meanwhile, normally when I sub for science, I compare this class to what I should have taught two years ago at the old charter school -- but it's difficult to think about science because I'm so worried about these two troublemakers. But let's revisit the STEM (not science) project that I gave two years ago on Halloween:
On the October holiday two years ago, I tried to do another seventh grade Illinois State project, "Orienteering." I made the activity fit the holiday by dressing up as a pirate and having the students search for my treasure. I wrote on the blog that the activity didn't work as well as I wanted, because the public school (co-located with our charter) restricted us to a small area of the playground. It might have worked better this year now that we're no longer co-located -- if only I were still a teacher there.
Today on her Mathematics Calendar 2018, Theoni Pappas writes:
The perimeter of this regular octagon is 20.4 cm. To the nearest [square] cm, what is its area?
(Yes, Pappas forgot to write square centimeters for the area.)
Notice that each side of the octagon has length 2.55 cm. The octagon can be divided into eight isosceles triangles with base 2.55 cm and vertex angle 360/8 = 45 degrees. Each triangle can be divided into two right triangles with leg 1.275 cm and opposite angle 22.5 degrees. These means that the other leg is 1.275/tan(22.5) = 3.078 cm, which is the apothem of the octagon. So the area is:
A = (1/2)ap = (1/2)(3.078)(20.4) = 31.3968 cm^2
To the nearest square cm the area is 31 cm^2 -- and of course, today's date is the 31st, Halloween.
(Notice that tan(22.5) is exactly equal to sqrt(2) - 1, but a proof requires half-angle formulas not often taught in general trig, much less Geometry.)
Lesson 5-4 of the U of Chicago text is called "Properties of Kites." In the modern Third Edition of the text, kites appear in Lesson 6-5.
This is what I wrote last year about today's topic:
This is what I wrote last year about today's topic:
Section 5-4 of the U of Chicago text covers kites. The kite is a relatively new quadrilateral classification. Not only did Euclid never define kite, but many texts made no mention of kites -- including my class geometry textbook from 20 years ago. Nowadays most texts define kite, but some include kites only in bonus questions, not in the main text.
Here's what John Conway wrote about the kite -- over 20 years ago, right around the time that I was taking my geometry class:
In fact it's not quite true, either, because "kite" is not
a very traditional name - it was obviously inserted because
this was a type of quadrilateral that SHOULD have received a
traditional name, but didn't, until recently.
Why do we include the kite - plainly because it represents
the one type of symmetry not otherwise mentioned. But this
reason suggests we should also EXCLUDE the non-isosceles
trapezoid.
a very traditional name - it was obviously inserted because
this was a type of quadrilateral that SHOULD have received a
traditional name, but didn't, until recently.
Why do we include the kite - plainly because it represents
the one type of symmetry not otherwise mentioned. But this
reason suggests we should also EXCLUDE the non-isosceles
trapezoid.
David Joyce, meanwhile, doesn't find kites to be necessary at all:
Too much is included in this chapter. The sections on rhombuses, trapezoids, and kites are not important and should be omitted.
Notice that both Conway and Joyce want to exclude trapezoids -- but Joyce is the only writer I know who wants to omit rhombuses. Of course, for Joyce, the emphasis should be on triangles and parallel lines, not quadrilaterals like rhombuses. Also, notice that kites are not specifically mentioned in the Common Core Geometry Standards. But I find that students can identify a kite more readily than a rhombus or trapezoid.
Now every kite contains two special vertices, known as its "ends." As defined by the U of Chicago:
"The common vertices of the equal sides of a kite are the ends of the kite."
Notice that Michael Serra doesn't define ends, but does give a name to the two angles whose vertices are the ends of the kite -- the vertex angles, in analogy with the vertex angle of an isosceles triangle. I see that from a proof standpoint, this makes sense, since the first thing that we do in the proof of our main theorem is divide the kite into two isosceles triangles.
And now here is our main theorem: the Kite Symmetry Theorem. As I mentioned back in the lesson on isosceles triangles, we use symmetry to determine the properties of kites. (A pre-Common Core proof might divide the kite into two triangles to be proved congruent by SSS.) I will post a proof here of the Kite Symmetry Theorem. It is taken directly from the U of Chicago, except that I, as always, add a Given step to the beginning of the proof. Since the U of Chicago's proof has eight steps, mine has nine:
Kite Symmetry Theorem:
The line containing the ends of a kite is a symmetry line for the kite.
Given: ABCD is a kite with ends B and D.
Prove: Line BD is a symmetry line for ABCD.
Proof:
Statements Reasons
1. ABCD is a kite with ends B and D 1. Given
2. AB = BC, AD = DC 2. Definition of ends of kite (meaning)
3. Tri. ABC and ADC are isosceles 3. Definition of isosceles triangle (sufficient)
4. Let m be the perp. bis. of AC 4. A segment has exactly one perp. bisector
5. A' = C, C' = A 5. Definition of reflection (sufficient)
6. m contains B and D 6. The perp. bis. of the base = angle bis. of the vertex angle
(so it contains the vertex)
7. B' = B, D' = D 7. Definition of reflection (sufficient)
8. ABCD reflected over m is CBAD 8. Figure Reflection Theorem
9. m (Line BD) is a symmetry line 9. Definition of symmetry line (sufficient)
Notice that more than half of the reasons in this proof are definitions. This underlines how important definitions are to the study of quadrilaterals.
As for the other theorems in this lesson, the Kite Diagonal Theorem follows directly from lines 4 and 6 of the above proof. It makes the symmetry diagonal the perpendicular bisector of the other diagonal and so the diagonals of a kite are perpendicular, and the symmetry diagonal bisects the other one.
Finally, we have the Rhombus Symmetry Theorem. It states that a rhombus has two symmetry lines, as both of the lines containing its diagonals are symmetry lines. This follows directly from the classification of a rhombus as a kite. In texts that define kite exclusively, the theorems "The diagonals of a kite are perpendicular" and "the diagonals of a rhombus are perpendicular" are two separate theorems, often in two separate sections. But here we can easily see why the diagonals of both the kite and the rhombus are perpendicular -- because the rhombus is a kite! Also, since we are defining kite inclusively, we don't need any extra steps in the proof to ensure that our kite isn't a rhombus.
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