Today is the third and last day of finals week. It has been my tradition on the blog on the third finals day to post a review of the old semester and preview of the new semester.
Yesterday, when I was searching for MTBoS blogs, I did find the following blog:
http://cheesemonkeysf.blogspot.com/
The blog belongs to a Northern California high school math teacher. (Apparently the letters "sf" in the URL stand for "San Francisco.") Even though the blog is anonymous, a commenter I quoted yesterday referred to this author as "Elizabeth." Thus I'll refer to cheese monkey by the name Elizabeth (and use feminine pronouns to her).
Anyway, Elizabeth is a Geometry teacher, and she refers to Geometry proofs in recent posts:
And so for our review, let's compare our U of Chicago course to Elizabeth's to look for any key similarities and differences.
Let's start with the first semester plan from the U of Chicago text:
1. Points and Lines
2. Definitions and If-then Statements
3. Angles and Lines
4. Reflections
5. Polygons
6. Transformations and Congruence
7. Triangle Congruence
Let's start with the first semester plan from the U of Chicago text:
1. Points and Lines
2. Definitions and If-then Statements
3. Angles and Lines
4. Reflections
5. Polygons
6. Transformations and Congruence
7. Triangle Congruence
In her November 6th post, Elizabeth writes:
So this year, when I had to be out of school for a few days, I designed a Proof Portfolio project for them to do in my absence.
Each day had four small, reasonable proofs students had to do — and they could collaborate on these. But then... they had to write a number of short-answer reflections to analysis questions based on their own proofs in the day's set.
And by "a few days" she means "four days," since she links to four worksheets -- a grand total of sixteen proofs. The first worksheet includes proofs about parallel lines (Chapter 3 of the U of Chicago text), while the other three are the more common triangle congruence proofs (that would have to wait until Chapter 7).
Did Elizabeth's students enjoy the project? Well, that's not exactly the case:
When I returned, there was a great deal of wailing and moaning and gnashing of teeth about How Hard This Project Was and How Hard They All Worked.
Yet she declares the project a success:
But as I'm reading their work, I am blown away by how much they seem to have learned!
The following week, Elizabeth blogs about this project some more:
The complaints and lamentations were filled with drama. "OH MY GOD, DR. S — THAT ASSIGNMENT WAS HARD." But they could tell that they had accomplished something.
(OK, so apparently Elizabeth's last initial is S.) She explains how she grades the assignment:
My assessment strategy was to be rigorous about completion but merciful with points. It was only worth a quiz grade (100 points), and my default score for students who completed every section was a 95. There are rewards for following instructions. Missing sections or components left blank cost more points.
She concludes the post as follows:
I am excited to see what happens on the next major test that includes a proof. Photos of student work to follow.
But unfortunately, she hasn't posted anything on Geometry since then, so we don't know yet how her students fare on the next major test.
Notice that the cornerstone of this project is the peer review. Elizabeth feels that feedback from peers is more effective than that from teachers. Traditionalists would disagree -- they believe that feedback from the teacher (someone who knows math) is better than that from another student (someone who doesn't know as much math). But traditionalists make the assumption that students will actually listen to the teacher just because she says so. Elizabeth's project is based on the possibility that the traditionalists' assumption is false -- that students are more willing to listen to each other than to the "sage on the stage."
Our first semester plan contains some activities, but nothing as ambitious as Elizabeth's four-day proof project.
Here is our second semester plan. We'll begin with Chapter 8:
8. Measurement Formulas (January 8th-14th)
9. Three-Dimensional Figures (January 15th-29th)
10. Surface Areas and Volumes (January 30th-February 13th)
11. Coordinate Geometry (February 14th-28th)
12. Similarity (March 1st-14th)
13. Logic and Indirect Reasoning (March 15th-April 4th)
14. Trigonometry and Vectors (April 5th-18th)
15. Further Work With Circles (April 23rd-May 6th)
Since the new semester begins on Day 86, we start with Lesson 8-6, "Areas of Trapezoids." It means that Lessons 8-3 (rectangles), 8-4 (irregular figures), and 8-5 (triangles) are omitted. But I'll find a way to squeeze in the missing lessons.
Is there anything on Elizabeth's blog about second semester Geometry topics? Well, I did find a link to the following post from over a year ago:
http://cheesemonkeysf.blogspot.com/2017/07/things-that-work-1-regular-vocab.html
This post is all about Geometry vocabulary. Her example is on circles, This appears to be more like the basic circle lessons of Chapter 13 than the more advanced work in Chapter 15 (where we have to deal with inscribed angles, power of a point, and so on).
Our worksheets refer to vocabulary. But Elizabeth takes the extra step of actually giving her students a vocabulary quiz. She writes:
At some level, I recognize that this sounds stultifying. But at another level, it was incredibly empowering for the students. Everybody understood exactly what was being asked and expected. And everybody saw it as an opportunity to earn free points. Students gave each other encouraging written comments and cheered each other on. They saw their scores as information—not as judgment. They used what they knew to make flash cards or Quizlet stacks. They quizzed each other. They helped each other.
And nobody ever complained about the regularly scheduled vocab quiz. It was a ritual of our course.
And apparently, she gives these quizzes every week:
It also ensured that everybody spent a little quality time on the focus task of preparing for the vocab quiz on Thursday or Friday. And this, in turn, meant that everybody was a little more ready to use the correct and appropriate mathematical vocabulary in our work. They noticed more because the owned more.
On the circle answer, Elizabeth gives a term, "2. concyclic points," that doesn't appear anywhere in the U of Chicago text. I believe the correct answer is "N. points that lie on the same circle" -- that is, "concyclic" is to circles as "collinear" is to lines. Also, the answer to "7. party hat situation" is "J. the situation where two tangent segments are drawn to a circle from a point external to the circle." This situation (which does indeed sort of look like a party hat) occurs often on the Pappas calendar, yet the U of Chicago text never gives it a formal name.
Elizabeth writes:
- There should be many more definitions in your right-hand list than there are terms in your left-hand list. Also definitions can be re-used. This way there isn't a zero-sum outcome if someone misses an answer.
Yet in her given circle unit example, there are exactly 16 terms and 16 definitions. But perhaps she follows her own advice on the other vocab quizzes.
I know the importance of vocabulary to any math class, especially Geometry -- particular these days of the Common Core when students must explain their answers. But I've never considered giving students a weekly Geometry quiz.
There are many other examples of "things that work" on Elizabeth's blog. It definitely gives me something to consider if I ever return to my own classroom someday.
Officially, this is my last post before the winter break. But since I didn't actually sub today, it's as if winter break has already started -- and so this is almost like a winter break post. This is especially since I don't plan on making as many winter break posts this year as last year.
I want to establish the tradition of posting more Mocha computer music every year at the holidays. So I'll be converting two more songs to the Mocha EDL scales this year.
Once again I emphasize that new scales are for writing new songs, not converting old ones. I hope that someday I'll write more original songs in the EDL scales -- and back in October I did compose some original Halloween songs in the new scales. Still, converting songs is a great exercise to learn what the new scales sound like before trying to compose new music.
Let's start with "O Holy Night." Last year, we converted this song to the Bohlen-Pierce scale, which isn't truly one of our new EDL scales -- but still, the BP scale is easier to approximate in Mocha than on a standard 12EDO keyboard.
Halloween songs are often played in minor scales, which are easier to play in Mocha. But most Christmas songs are played in major scales. As we've seen before, none of our simple EDL scales (from 12EDL to 20EDL) contains a full major chord (root-3rd-5th-octave) -- the simplest EDL with such a chord is 30EDL. (On the other hand, the minor chord first appears as early as 6EDL.)
We've found out that the best EDL's for converting a major song are 18EDL and 20EDL. In 20EDL we have a major 3rd but no perfect 5th, while 18EDL has a perfect fifth but no just major 3rd. (It does have a just minor 3rd, which is why 18EDL is also suitable for minor music.)
Let's look at the 20EDL scale once more:
The 20EDL scale:
Degree Ratio Note
20 1/1 green C
19 20/19 19u C#
18 10/9 white D
17 20/17 17u D#
16 5/4 white E
15 4/3 green F
14 10/7 red F#
13 20/13 thu G
12 5/3 white A
11 20/11 lavender B
10 2/1 green C
First of all, I'll point out that neither 17 nor 19 will be needed for today's songs. After all, if this were used to replace a C major scale, then we don't need the notes C# or D#.
On the other hand, both 11 and 13 are needed, since a C major scale needs both G and B. The G at 13 is very sharp (more like G# than G), while the B at 11 is flat (between Bb and B).
Our trick will be to minimize the need for 11 and 13 so that the rest of the song fits on the other notes that sound more like standard notes. "O Holy Night" is known for having many long notes that span more than a full measure. If any of these long notes were 11 or 13, then the song will sound off.
My music score has the song "O Holy Night" in C major, and so I was tempted to use 20EDL. But then I noticed that all of the notes of the C major scale appear long except for F (4th) and A (6th). In particular, both G and B appear as long notes. In 20EDL we'd be forced to play these as 13 and 11, which will make the song sound out of tune.
But if we use 18EDL instead of 20EDL, 13 and 11 now appear as the 4th and 6th -- exactly the notes that don't appear long in "O Holy Night"! So 18EDL appears more suitable than 20EDL.
Let's look at 18EDL again:
The 18EDL scale:
Degree Ratio Note
18 1/1 white D
17 18/17 17u 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
Yes, 18EDL lacks the usual 5/4 major 3rd. Instead it contains 9/7, a supermajor 3rd. This still sounds recognizably as a major 3rd, so it's not terrible to use it in our song. On the other hand, the seventh now plays as 9/5, a minor 7th. Neither 18EDL nor 20EDL has much of a leading tone, but perhaps the 20/11 seventh of 20EDL sounds slightly more like a leading tone. (This is a rare time when an 11 actually fits better in the song than a 7-limit note.)
Here's our 18EDL version of "O Holy Night" in Mocha:
http://www.haplessgenius.com/mocha/
70 N=9
80 FOR X=1 TO 21
90 READ A,T
100 SOUND 261-N*A,T
110 NEXT X
120 FOR I=1 TO 400:NEXT I
130 FOR X=1 TO 75
140 READ A,T
150 SOUND 261-N*A,T
160 NEXT X
170 DATA 14,4,14,2,14,2,12,6,12,2,11,2,11,2
180 DATA 13,2,11,2,9,8,12,2,12,2,14,2,18,2
190 DATA 18,4,14,2,13,2,12,4,13,2,16,2,18,12
200 DATA 14,4,14,2,14,2,12,6,12,2,11,2,11,2
210 DATA 13,2,11,2,9,8,12,2,12,2,13,2,14,2
220 DATA 10,4,12,2,11,2,10,4,9,2,10,2,14,12
230 DATA 12,4,12,4,11,4,16,4,12,4,11,2,12,2,9,2,14,2,12,4
240 DATA 12,4,12,4,11,4,16,4,12,4,11,2,12,2,9,2,14,2
250 DATA 12,8,9,12,10,2,11,2,10,12,10,4
260 DATA 8,8,11,4,11,2,11,2,9,8,9,6,9,2
270 DATA 7,8,8,6,12,2,9,12,10,2,11,2
280 DATA 12,8,12,4,11,2,12,2,12,12,9,4
290 DATA 8,12,12,4,6,12,8,12
300 DATA 9,8,10,2,10,2,9,2,8,2,9,8
Don't forget to click the "Sound" box before you RUN the program!
This song starts with line 70 rather than 10. That's because I cut-and-pasted the version of this song from last year, where we need to set up the Bohlen-Pierce scale. But EDL scales are more natural in Mocha, so we don't need to set up the scale.
Instead, we only need the line 70 N=9. This sets the root note to middle white C, which matches my own musical score. Any value of N from 1 to 14 is playable in Mocha.
Our other song will be "Happy Holiday," made famous in Irving Berlin's Holiday Inn. In my score, this song is written in the key of Eb major. So how should we convert it?
Well, the only long note that spans more than a measure in this song is the note C, which is the sixth note of the Eb major scale. A just major 6th appears in 20EDL, so we should use 20EDL rather than 18EDL for this song.
Also, in my score, the song repeats in a different key (that is, it modulates). It drops a perfect 4th, from Eb major down to Bb major. Since a perfect fourth is 4/3, we do this in Mocha by having a FOR V loop that ranges from 3 to 4, and then multiply the Degree by V. We must RESTORE the data lines before the song repeats. In fact, this is one difference between SOUND and PLAY -- we can't modulate as easily using PLAY (even though PLAY is based on the standard 12EDL scale).
Here's our 20EDL version of "Happy Holiday" in Mocha:
NEW
10 N=2
20 FOR V=3 TO 4
30 FOR X=1 TO 50
40 READ A,T
50 SOUND 261-V*N*A,T
60 NEXT X
70 RESTORE
80 NEXT V
100 DATA 13,3,13,1,12,3,12,1,12,24
110 DATA 13,3,13,1,12,3,12,1,12,24
120 DATA 13,3,13,1,11,2,11,2,12,8,11,4,10,4,13,8
130 DATA 16,3,16,1,13,2,13,2,15,8,18,4,16,12
140 DATA 13,3,13,1,12,3,12,1,12,24
150 DATA 13,3,13,1,12,3,12,1,12,24
160 DATA 13,3,13,1,11,2,11,2,12,8,11,4,10,4,13,8
170 DATA 16,3,16,1,13,2,13,2,15,8,18,4,20,12
Here I intentionally skip Line 90, so that Lines 100-110 are the same length. In fact, Lines 140-170 are a repeat of Lines 100-130 except for the last note in Line 170.
By using N=2, the key modulates from green F major to middle green C major. We can also use N=1 which is an octave higher. Also, N=3 modulates from green Bb major down to green F major. The Bb section matches the Bb in my score, except that it goes down to F rather than up to Eb.
Well, that does it -- this is the end of the first semester, and tomorrow starts winter break. I plan to post just twice during winter break. Both such posts will be updates of some of the winter break posts from last year.
The first day of the second semester will be Tuesday, January 8th, 2019. As I wrote earlier, this will be Day 86 and Lesson 8-6.
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