Well, fortunately today I subbed in a middle school math class. And so I can finally say, let the Pi Day party begin!
And I definitely want to do "A Day in the Life" today. But first, let me describe the events that occurred before the subbing even began. Last night, I accepted a subbing job at this middle school for a math teacher. It was a part-time job, where I only needed to cover the last three classes. (I assume the regular teacher has to leave early today for some appointment.)
Anyway, this morning I received a call from the school. Another math teacher called in sick, and suddenly my part-time assignment turned into a full-day job.
As it turns out, it's the same Algebra I class that I covered the week before Thanksgiving. You may read about this class in my mid-November posts. I also provided period coverage for this teacher for one period -- you may read about this in my January 11th post. (You might want to read the January 11th post anyway today, since I wrote about Lesson 8-9 and the number pi that day.)
This teacher has a zero period class, but with the late change, I didn't arrive in time for that class.
8:30 -- Instead, I arrive near the start of first period proper. This is an eighth grade Algebra I class.
This class is beginning trinomial factoring. I tell the students that back when I was a young Algebra I student, my teacher referred to factoring as "the F-word" of Algebra I, since it's so tricky.
Teaching factoring is also tricky. Many teachers teach it in different ways, and these students haven't seen a factoring lesson from their regular teacher yet. Today I show them how to draw the big X and write in the desired product and sum -- but there's another method with a big X where the desired product and sum are not written down. Fortunately, today's lesson in all about monic trinomials (that is, those with a leading coefficient of 1). I hope the regular teacher will return to teach the students how to factor trinomials that are not monic.
Pi Day should be all about food and giveaways, but since I'm running late, I don't have much. I give two pencils away to two students on the good list. (They're green pencils, so I try passing them off as early St. Patrick's Day pencils.) But at least I play a few Pi Day songs during the last few minutes of the period, when the students are beginning the factoring homework.
9:20 -- This is the middle school where all classes after the first rotate based on day of the week. As today is Thursday, the rotation goes 1-5-6-2-3-4.
Fifth period is the teacher's conference period, which leads into snack. Naturally, I use the time to go out and purchase some Pi Day treats for the remaining classes.
10:30 -- Sixth period arrives. This is another eighth grade Algebra I class.
This time, I open the class with a simple Pi Day song, to the tune of Jingle Bells:
Pi Day songs,
All day long,
Oh what fun it is,
To sing a jolly Pi Day song,
In a fun math class like this!
I follow this up with Bizzie Lizzie's Pi Day song "Digit Connection," which I sing while the students are checking Tuesday night's homework. Also, by now I have some treats to pass out -- some cookies for Pi Day. (I bought cookies for the sixth graders at the old charter school as well.) Then I begin the factoring lesson.
11:25 -- Sixth period leaves and second period arrives. This is the last eighth grade Algebra I class.
Second period goes just as smoothly as sixth period, with more cookies. I have some extra cookies to give away, and one guy who solves a tricky word problem (area of a field) earns these cookies.
12:20 -- Second period leaves for lunch.
1:05 -- Third period begins. This is an ASB class. The students are setting up tickets to sell for the upcoming fair for International Week.
Back when I subbed in this class on November 15th, there just happened to be some leftover pumpkin pie from a pre-Thanksgiving eating contest. And so I decide to purchase some more pie for my grand return to this class! The local Ralphs even sells some pumpkin pies for $3.14 today. (Note: I personally prefer cherry pie on Pi Day.)
Notice that in my November 15th post, I even mused about what I'd do if I were to return to this class on Pi Day -- and that actually happens today!
1:32 -- Around the midpoint of class is when I serve the pie (so that the students aren't too distracted by pie to work on the tickets). No, this isn't 1:59 -- but hey, 1, 3, 2 are digits somewhere in pi. (Look at this year's "Vi Day" video below, where Vi Hart also skips around the digits of pi.)
1:58 -- Third period leaves...
1:59 -- ...one minute before the big pi moment. A few third period students linger around, and so I announce the official Pi Day moment to them. I meanwhile am trying to gather all the leftover plates, forks, and knives quickly, since third and fourth periods are actually in adjacent classrooms for some strange reason.
2:03 -- Fourth period arrives. This is the math support class. It's a new trimester (Day 121 in this district) and so there are different students from second trimester -- although some first trimester students are returning to the class for third tri. They remember the November pie -- and when I present them with more pie today, they declare me to be their favorite sub. After all, every time I come to this class I have pie!
Since 1:59 has passed, I serve them pie right away. As I wrote back on November 15th, pie at the start of class means the students may be too distracted to learn. Some of them are supposed to be working on homework for their main math class -- but two other math teachers are also out with subs today, and so I'm unable to verify the other assignments. (Yes, three math teachers are all out on Pi Day for some reason. One of them is the teacher I'm originally supposed to be subbing for.)
The few students who do work either do problems out of a textbook (on the Stats/Prob middle school standards for Common Core) or do ALEKS on Chromebooks.
2:55 -- Fourth period leaves, and we all go home to finish celebrating Pi Day.
As usual, here are some pi-related videos. This time, I wish to link to the videos that I actually showed the classes today:
1. Parody of "Happy" by Pharrell Williams (first period)
This was first posted in 2015, but somehow I never noticed it until this year.
2. Parody of "American Pie" by Don McLean (first period)
Because "pi" and "pie" sound alike, this is one of the most commonly parodied songs for Pi Day. I also sing along with the video in third period ASB. (I was originally hoping to sing Bizzie Lizzie's version of "American Pi" along with her "Digit Connection," but it's so much easier just to sing along with this version.)
3. "Pi" by Hard and Phirm (sixth period)
This "song" is just a simple reading of the digits. After I play in sixth period, I repeat the two videos that I played for first period.
4. "Pi Day Song" by Michael Bautista (second period)
This is a brand new song for 2019. Yes, new Pi Day songs are being created every year. And of course, its length is 3 minutes and 14 seconds.
5. "Pi Day Anthem" by John Sims and Vi Hart (second period)
This duet is also a reciting of digits, but is slightly more interesting than the digit video above.
6. "6 Digits of Pi" by Vi Hart
I don't play this in any class, but I list it here just below the other Vi Hart video. This time, the famous mathemusician is trying to make 3/14/19 into another "Pi Day of the Century." She points out that the date contains the first six digits of pi -- that is, if you skip the 5.
7. "Pi Day Music Video" by musicnotes online (third period)
This video is also a few years old, but it's another one that I don't notice until this year.
8. Parody of "Dynamite" by Taio Cruz (third period)
I've watched this video several times before. The students who create this video are fifth graders -- two or three years younger than the students I see today.
9. "Pi Song: Mathematical Pi" by Mr. R's Songs for Teaching (third period)
I have more time to play songs during third period ASB, since I don't have to teach these students how to factor.
10. "Pi Day 2015 Song" by Singing Nerd (fourth period)
Yes, there were many Pi Day videos created during the "Pi Day of the Century" hype of 2015.
11. "A Song About Pi," Lucy Kaplansky (fourth period)
This is another new one for me. I notice that when the singer recites "3, 1, 4, 1, 5, 9," the notes sound like "E, C, F, C, G, D" -- as if this were inspired by the note digit songs.
12. "How Pi Sounds as Music," Saher Galt (fourth period)
This one actually is a note digit song, even though it's the first time I've noticed it. Notice that low B is used for digit 0.
13. "Song from Pi," aSongScout (fourth period)
I've played this note digit video several time before. It's based on A minor scale, not C major. Since this class meets after 1:59, the entire period is party-like and so I play the most songs. Thus there's extra time to spend on these note digit songs.
14. "Pi Day," X-Phaze
The fourth period students really enjoy this one.
15. Parody of "Lose Yourself (in the Digits of Pi)" by Eminem
During the previous video, someone mentions Eminem. I inform him that Eminem has nothing to do with the previous video -- but here is the real Eminem-inspired Pi Day song.
On the way home I hear Meghan Trainor's "All About the Bass" playing on the radio, so of course I had to sing my parody "All About the Bass and Height" on top of it. I don't sing this song in class, since it's not directly related to pi (although pi is implied for cylinders). For that matter, I could have tried looking for a factoring song to play during the Algebra I classes, but I didn't make the effort.
Oh, and let's not leave Mocha out of the fun:
http://www.haplessgenius.com/mocha/
70 N=1
80 FOR X=1 TO 32
90 READ A
100 SOUND 261-N*(18-A),4
110 NEXT X
120 DATA 3,1,4,1,5,9,2,6,5,3
130 DATA 5,8,9,7,9,3,2,3,8,4
140 DATA 6,2,6,4,3,3,8,3,2,7
150 DATA 9,5
This song is based on 18EDL. As I've said before with pi, 0 doesn't first appear until relatively late, which is way some musicians cut off the song just before the first 0
For 18EDL, we use 0 and 9 as the tonic -- the beginning and ending of the scale. This might sound strange when there's no 0 to represent the root note. We might consider changing the song to 16EDL and make 1 and 9 the tonic notes. (This entails changing 18 to 17, not 16, in line 100.) I don't write about 16EDL much since it contains only one third -- a neutral 16/13 third. This sounds unfamiliar to modern music ears.
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
Given: P' on
Prove: OP'/OP = OQ'/OQ = P'Q'/PQ
Statements Reasons
1. P' on
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.
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 proof, 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.)
Another idea to avoid giving this test on Pi Day is to do what I did two years ago -- combine the test with yesterday's Lesson 12-9 rather than 12-10. (That year, I did it because Day 130 happened to land on a Monday following a four-day weekend.)
This is a traditionalist post -- the first such post scheduled for Pi Day. Of course, the last thing traditionalists would want to see is a Pi Day party!
I don't really want to tie this post up with the traditionalists' debate. But I did read something lately on Concrete, Pictorial, Abstract -- another paradigm that many math teachers might wish to use, but traditionalists probably won't approve.
It's worth discussing in a separate post, though. Concrete, Pictorial, Abstract reminds me a little of the Illinois State text -- and I wonder whether I would have been more successful with Illinois State at the old charter school if I'd kept this in mind.
Then maybe I wouldn't have had to leave my old charter just before Pi Day. That was a very bittersweet Pi Day -- I bought my sixth graders a pizza and delivered it to the school even though I was no longer a teacher there. In many ways, today's Pi Day shows me what the party might have looked like two years ago at the old charter if only I had stayed.
(By the way, I walked past a seventh grade classroom today. The lesson appeared to be on angle measures, not pi. Last year I saw an honors 7th grade class give a pi lesson near Pi Day.)
OK, that's all I have to say about traditionalists. Here are today's worksheets -- one for Lesson 12-10 and the other for the Chapter 12 Test.
I like to post on Pi Day at 1:59 PM Pacific Time, but I couldn't, since I was moving around between classrooms. Instead, this is being posted at 1:59 AM Greenwich Time (albeit it's the morning of the 15th in that time zone).
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