Today is Day 130. It's halfway between Day 80 (the end of the first semester) and Day 180 (the end of the second semester), and so we expect it to be near the end of the third quarter. And indeed it is -- third quarter grades are due today. (This explains why yesterday's students were frantically making up assignments -- the grade deadline is today.) And some students are starting fourth quarter classes today.
Today I subbed in a middle school music class. It's in my new district -- and in fact, it's my second visit to this class. I describe my first visit in my January 21st post.
Since it's a middle school class -- and one classroom management issue does come up today -- I will do "A Day in the Life" today.
8:45 -- First period arrives. This is a Grade 7-8 Band class.
Just like in January, the regular teacher sets it up so that students don't have to play instruments on sub days -- instead, they have another EdPuzzle to work on. It involves watching a video, so the assignment spans most of the period.
Notice that January 21st was a Thursday, while today is Wednesday. And recall that in this district, middle schools are on Hybrid Plan #2, so that Cohort A meets Tuesday/Thursday. Thus the kids I see in the classroom today are different from the ones I saw in January.
I always sing songs in music class, and today is no exception. Today I perform two songs. First, since this class hasn't heard "The Big March" yet, I sing this one first. Then I bring my keyboard music book again, hoping that I could play on the same keyboard I found in this room in January -- but it's not there today. I had to sing a song from this book instead. I choose "Has Anybody Here Seen Kelly?" That's Kelly from the Emerald Isle -- as in Ireland -- as in St. Patrick's Day -- as in today's date.
9:40 -- First period leaves and second period arrives. This is a sixth grade Band class.
These students get an Edpuzzle as well. I'm not sure whether it's the same as the first period Edpuzzle, but (as you'll see later on) I have reason to suspect that it is.
10:35 -- Second period leaves for snack break.
10:45 -- Third period arrives. This is a sixth grade Exploratory Wheel class.
As you might recall from January, this is a one-quarter class -- and as you might recall from this top of this post, students are starting fourth quarter classes today. It's awkward for the very first day the students have in a class to be with a sub, and yet that's exactly what happens today.
This class also gets an Edpuzzle, but it's definitely different from the Period 1-2 assignment. This Edpuzzle is more like an introduction to the class. Some of the students ask me whether they'll get to play real instruments in this class -- but of course, I don't know, since I'm just the sub.
11:40 -- It is time for tutorial.
At this school, students remain in their third period class, unless they tell the teacher in advance that they are seeing another teacher. There are seven in-person students and two leave to go to their P.E. teacher (P.E. is independent study, so this is the only they get to see their P.E. teachers). But only four students remain -- apparently a third guy leaves for P.E. without telling me.
Meanwhile, there are nine online students, but only eight are in tutorial. I call out names, and figure out who's missing -- one girl asked to go to the restroom (at her house) 14 minutes before the end of third period, but then she logs out and never returns for tutorial.
But later on, I realize that I made an attendance blunder. One girl went to my tutorial from another class, so there were only seven students from third period, not eight. I never figure out until the students are leaving for lunch, and so I don't know who the other missing student is.
As I usually do for tutorial, I sing extra songs. I have the students choose extra songs from my math songbook, and they select "That's Math" and "Wanna Be" -- these are two inspirational "Why should I learn math?" songs from Square One TV.
I said that there's no keyboard for me to play in this class, but there is another instrument that I am able to play here -- a banjo. This is the second banjo I've seen in a music class this month -- but this banjo doesn't have six strings tuned to standard guitar tuning (EADGBE). Instead, it has five strings -- and I suspect that with five strings, this is a real banjo, not a fake banjo that's really just a six-string guitar.
Even though I do wish to discuss what I learn about the banjo today, this is also a traditionalists' post, so I don't want to tie it up with a musical discussion. This deserves its own post.
Here's what I will say about the banjo for now -- the five strings are tuned to GDGBD, which is a G major chord. If you recall, "That's Math" was the first song I played on my guitar (during my long-term assignment) when it was stuck on EACGAE tuning -- that day I played the entire song using only one chord, a G chord (since much of the song was just talking, not singing). This makes it the perfect song for me to play on the banjo. I also play the "Row Row Row Your Boat" parodies ("Mode Mode Mode" and "Same Sign"), as these are also relatively simple songs.
Oh, and I will tie this to the traditionalists' debates -- the traditionalists (and yes, they do bring up music in their arguments) are laughing every time I mention the banjo or ukelele in my posts. They'd say, if I really wanted to learn those instruments, why don't I just Google them -- I'd probably find some sort of instruction manual or lessons online. Then I won't have to guess how to play chords and songs on them.
Well, my response is that I enjoy playing music by ear rather than by rote. I've told the story on the blog of how I "discovered" the G major scale for myself on my keyboard. Yes, I know I could have Googled banjo and ukelele strings and learned about those instruments in minutes. But you can't just use Google to create new music -- and besides, learning about the banjo and uke are helping me learn more about how to create EDL music for Mocha (which can't be Googled or learned by rote).
But that's enough about traditionalists for now -- let's just get through the rest of the day.
12:05 -- The sixth graders leave for lunch. Much to my delight, there is a potluck St. Patrick's Day meal of baked potatoes and other green foods being served in the Home Ec room and teacher's lounge. Since it's still a pandemic, we must put a glove on the serving hand before we put anything on our plate.
12:45 -- Fourth period arrives. This is a Strings 7-8 class.
Like the Band classes, these students get an Edpuzzle. But unlike the Band classes, here the teacher has accidentally posted the Edpuzzle in Spanish. And so the students can't understand the lesson.
Since the kids have no lesson to do, I decided to -- you guessed it! -- sing the entire period. I begin with the songs that either I or the tutorial sixth graders have chosen earlier, and then let these kids add a few songs of their own. Here's what my entire concert looks like:
And that takes us through the rest of the period.
1:40 -- Fourth period leaves and fifth period arrives. This is a sixth grade Strings class.
Once again, the students see only a Spanish Edpuzzle, so I start up my concert again. Unfortunately, this time the five in-person students keep talking during the song and fooling around. A few students take out their violins -- and then start hitting each other with the bows.
This is a tricky issue that goes all the way back to the old charter school -- when I'm singing, what should the students be doing? It's not easy to answer, especially in this case where the students don't actually have an assignment due to the language error.
Perhaps I should have insisted that the students stay seated and either listen to my concert or do other work for this/any other class. If they don't, then I should stop singing and start writing down names for the bad list to leave for the teacher.
I also could have had an incentive, such as the St. Patrick's Day pencils that I was giving away earlier this week. I didn't do it today, St. Paddy's Day itself, since pencils in music class are awkward. And even if I did, I might have given them all away in the morning classes -- there's no way for me to know that the assignment on Canvas wouldn't work in the afternoon.
This should be the last time I mention Pi Day and St. Patrick's Day in the classroom this season. I must avoid temptation -- tomorrow I might go to another classroom and said, hey, I've subbed these kids before but didn't get to celebrate pi or Patrick with them and there's nothing else to do, so keep the holiday going unnecessarily. After all, that's what happened today with Pi Day.
I ultimately sing the same songs in this class as fourth period, except that I replace the pi songs (after all, sixth graders haven't learned pi yet) with "All About That Base," 2D version (after one girl mentions that she's learning about triangle area in her math class).
2:35 -- Fifth period leaves, thus completing my day.
Today is Sixday on the Eleven Calendar:
Resolution #6: We ask, what would our heroes do?
The math songs "That's Math" and "Wanna Be" show us how to emulate our heroes. Perhaps I could have found a way to discuss this with the sixth graders in fifth period -- and such a discussion might have kept them out of trouble.
It's time to get back to the traditionalists. Our main traditionalist, Barry Garelick, celebrated Pi Day in his own way, with a blogpost:
https://traditionalmath.wordpress.com/2021/03/14/math-zombies-and-conceptual-understanding-in-math/
Conceptual understanding in math has served as a dividing line between those who teach in a conventional or traditional manner (like myself), and those who advocate for progressive techniques. The progressives/reformers argue that understanding of a procedure or algorithm must precede the procedure/algorithm itself; failure to do this results in what some call “math zombies”.
Garelick quotes a certain high school football coach:
“WORRYING ABOUT MATH ZOMBIES IS LIKE WORRYING THAT YOUR FOOTBALL PLAYERS ARE TOO GOOD AT PASSING THE BALL — ON THE BASIS THAT THEIR POSITIONAL PLAY IS NO BETTER THAN THE REST OF THE TEAM, AND THEREFORE THEY OBVIOUSLY DON’T UNDERSTAND WHAT THEY ARE DOING WHEN THEY PASS BEAUTIFULLY.”
As usual, here's the difference -- when a football player has trouble passing, he doesn't complain that he doesn't understand it. When a math student has trouble doing math, he complains that he doesn't understand it and leaves it blank. This is why understanding is more relevant to math than to football.
Our regular commenter SteveH isn't here in the comment thread. Let's see who's here instead:
First of all, notice EB's sleight-of-hand here -- both traditionalists and progressives say that we should focus on the "hands-on" part in math. But here, EB uses "hands-on" to mean (standard) algorithms and practice (sets, traditionalist of course). That's definitely not what, say, advocates of project-based learning mean by "hands-on."
Then EB follows this up with a musical example -- appropriate for today's post. Once again, we notice EB's sleight-of-hand -- progressives/constructivists would let students (like young me) discover the G major scale while traditionalists directly teach the music theory of what the G major scale is.
Once again, in both music and math, of course we should at some point teach the basics like what a major scale is and what addition is. But I think that the discovery method has its place in both subjects.
The other main commenter here is eanelson2014:
Cognitive experts tell is that to be able to do that, they need to be taught new concepts, and practice applying them to simple problems they can solve by mental math (memorized math facts). Recallable mental math leaves room in the brain’s working memory to process what the concept is about. Processing moves the context cues of the concept into long-term memory (LTM). That’s what learning is: Storing information in LTM so it can be recalled and applied to solving problems.
This means, the cognitive experts tell us, once the student is taught the concept (which does not take long), to apply the concept to solve a problem of any complexity, the student must memorize by practice and apply an algorithm.
An algorithm is simply a stepwise procedure which it has been proven the brain can manage to solve a problem without overloading working memory. Without an algorithm to manage the data of a problem, working memory overloads quickly, and confusion results.
I agree with Nelson only up to a point. I agree with storing basic math facts in "LTM" (or what I've called "the bicycle" in many posts) and as much higher math as possible. I encourage students to memorize as much as possible ("keeping the bike inflated"). But at some point, we recognize that students can't remember everything, and so traditionalism has its limits.
As usual, "algorithm" here means standard algorithm. Many students are able to keep the standard addition/subtraction algorithms in their bike/LTM but struggle with the standard algorithms for multiplication and division. They may prefer other algorithms besides the standard (such as the maligned "lattice method").
Lesson 12-10 of the U of Chicago text is called "The Side-Splitting Theorem." This is what I wrote last year about today's lesson:
The U of Chicago version of the theorem is:
Side-Splitting Theorem:
If a line is parallel to a side of a triangle and intersects the other two sides in distinct points, it "splits" these sides into proportional segments.
And here's Dr. Wu's version of the theorem:
Theorem 24. Let triangle OPQ be given, and let P' be a point on the ray OP not equal toO. Suppose a line parallel to PQ and passing through P' intersects OQ at Q'. ThenOP'/OP = OQ'/OQ = P'Q'/PQ.
Notice that while the U of Chicago theorem only states that the two sides are split proportionally, Wu's version states that all three corresponding sides of both sides are proportional.
Moreover, the two proofs are very different. The U of Chicago proof appears to be a straightforward application of the Corresponding Angles Parallel Consequence and AA Similarity. But on this blog, we have yet to give the AA Similarity Theorem. So how can Wu prove his theorem?
We've seen several examples during the first semester -- a theorem may be proved in a traditionalist text using SSS, SAS, or ASA Congruence, but these three in Common Core Geometry are theorems whose proofs go back to reflections, rotations, and translations. Instead, here we skip the middle man and prove the original high-level theorem directly from the transformations. We saw this both with the Isosceles Triangle Theorem (proved from reflections in the U of Chicago) and the Parallelogram Consequences (proved from rotations in Wu).
So we shouldn't be surprised that Wu proves his version of the Side-Splitting Theorem using transformations as well. Naturally, Wu uses dilations. In fact, the names that Wu gives the points gives the game away -- O will be the center of the dilation, and P and Q are the preimage points, while P' and Q' are the images.
Here is Wu's proof: He considers the case where point P' lies on OP -- that is, the ratio OP'/OP, which he labels r, is less than one. This is mainly because this case is the easiest to draw, but the proof works even if r is greater than unity. Let's write what follows as a two-column proof:
Given: P' on OP, Q' on OQ, PQ | | P'Q', r = OP'/OP
Prove: OP'/OP = OQ'/OQ = P'Q'/PQ
Statements Reasons
1. P' on OP, Q' on OQ, PQ | | P'Q' 1. Given
2. OP' = r * OP 2. Multiplication Property of Equality
3. Exists Q0 such that OQ0 = r * OQ 3. Point-Line/Ruler Postulate
4. For D dilation with scale factor r, 4. Definition of dilation
D(Q) = Q0, D(P) = P'
5. P'Q0 | | PQ, P'Q0 = r * PQ 5. Fundamental Theorem of Similarity
6. Lines P'Q0 and P'Q' are identical 6. Uniqueness of Parallels Theorem (Playfair)
7. Points Q0 and Q' are identical 7. Line Intersection Theorem
8. OQ' = r * OQ, OP' = r * OP, 8. Substitution (Q' for Q0)
P'Q' = r * PQ
9. OP'/OP = OQ'/OQ = P'Q'/PQ = r 9. Division Property of Equality
Now the U of Chicago text also provides a converse to its Side-Splitting Theorem:
Side-Splitting Converse:
If a line intersects rays OP and OQ in distinct points X and Y so that OX/XP = OY/YQ, then XY | | PQ.
The Side-Splitting Converse isn't used that often, but it can be used to prove yet another possible construction for parallel lines:
To draw a line through P parallel to line l:
1. Let X, Y be any two points on line l.
2. Draw line XP.
3. Use compass to locate O on line XP such that OX = XP.
4. Draw line OY.
5. Use compass to locate Q on line OY such that OY = YQ.
6. Draw line PQ, the line through P parallel to line l.
This works because OX = XP and OY = YQ, so OX/XP = OY/YQ = 1.
The U of Chicago uses SAS Similarity to prove the Side-Splitting Converse, but Wu doesn't prove any sort of converse to his Theorem 24 at all. Notice that many of our previous theorems for which we used transformations to skip the middle-man, yet the proofs of their converses revert to the traditionalist proof -- once again, the Parallelogram Consequences.
Another difference between U of Chicago and Wu is that the former focuses on the two segments into which the side of the larger triangle is split, while Wu looks at the entire sides of the larger and smaller triangles. This is often tricky for students solving similarity problems!
Now it's time to give the Chapter 12 Test. Again, it's awkward to combine Lesson 12-10 with the test, but that's the way it goes. I can argue that by doing so, we're actually following the modern Third Edition of the text. Yesterday's lesson on SAS~ and AA~ is the final lesson of Chapter 12 in the newer version (Lesson 12-7). In the Third Edition of the text, the Side-Splitting Theorem is the first lesson of the next chapter, Lesson 13-1.
Test Answers:
10. b.
11. Yes, by AA Similarity. (The angles of a triangle add up to 180 degrees.)
12. Yes, by SAS Similarity. (The two sides of length 4 don't correspond to each other.)
13. Hint: Use Corresponding Angles Consequence and AA Similarity.
14. Hint: Use Reflexive Angles Property and AA Similarity.
15. 3000 ft., if you choose to include this question. It's based on today's Lesson 12-10.
16. 9 in. (No, not 4 in. 6 in. is the shorter dimension, not the longer.)
17. 2.6 m, to the nearest tenth. (No, not 1.5 m. 2 m is the height, not the length.)
18. 10 m. (No, not 40 m. 20 m is the height, not the length.)
19. $3.60. (No, not $2.50. $3 is for five pounds, not six.)
20. 32 in. (No, not 24.5 in. 28 in. is the width, not the diagonal. I had to change this question because HD TV's didn't exist when the U of Chicago text was written. My own TV is a 32 in. model!)
By the way, I changed the first worksheet below in order to give the U of Chicago version of the Side-Splitting Theorem proof, rather than the Wu/EngageNY version. When I did so, I notice that the old worksheet had a nine-step proof. The U of Chicago proof also lists nine steps (actually eight since that text never provides the "Given" step).
The test, meanwhile, starts with Question #10. This numbering has nothing to do with nine-step proof, since I used to post Questions #1-9 on an old defunct version of my test. But it suddenly gave me the bright idea to treat this lesson as part of the test, since we're giving Lesson 12-10 on the same day as the Chapter 12 test. (But this might be awkward for the students, especially Steps #5-9 of the proof which is all algebraic manipulation. We wouldn't want the students to miss 5/20 or 25% of the test due to confusion with the algebra.)
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