Today I subbed in a high school Spanish class. It's in my new district. Since it's a high school class and not math, there's no need for "A Day in the Life" today. As usual, I will say a few things about the class.
First of all, it's a Spanish for Spanish speakers class. Most of the students are freshmen, though there are a few sophomores as well. The assignment is to complete a ten-minute warmup, work on some vocabulary, and prepare for an upcoming test. It's my last day of handing out Easter pencils and candy.
For my song today, I revert to my favorite in any Spanish class -- "Sign of the Times" from Square One TV, since it contains the line "Equis es el simbolo de los tiempos."
When I arrive at the school, I thought that it would be a minimum day, but it isn't. (Originally yesterday was supposed to be virtual Open House -- many districts have minimum days on the day after Parent Night rather than Parent Night itself. A minimum day was printed on the school calendar in January -- most likely, that Open House was cancelled.) Thus I have an unexpected tutorial today -- each of the three blocks ends with the 20-minute extra time. Of course, I fill it with another song, "Mathematics of Love," also from Square One TV -- except instead of Roman numerals, I sing Spanish numbers.
During first period tutorial, one girl asks for help on her math assignment -- she has an Algebra I test on quadratic functions coming up the following period. She shows me her notes, and I notice that there's mention of some quadratic song, which I assume to be the "Pop Goes the Weasel" parody. And so I sing my version of the song, including my new verse on the vertex formula. One of her questions is on finding the vertex of a parabola, and so my song helps her.
I continue to sing all three songs the rest of the day -- "Sign of the Times" during the proper period, and then both "Mathematics of Love" and "Quadratic Weasel" during tutorial. (I was tempted to add both my "U-N-I-T Rate! Rate! Rate!" to celebrate UCLA in the Final Four, and "Big March" to mark the end of the toughest stretch of the year. In the end, I decide to keep it a mini-concert.)
In third period (at this school, odd periods meet on Wednesdays/Fridays), the class is Zoombombed -- someone tried to get into the meeting under the name "Jesus Christ." Yes, someone claims to be Jesus on Good Friday -- and it's during the class that ends at noon (the traditional time that Jesus was crucified according to the Bible). Of course, I don't let "Jesus" into the meet.
Today is Elevenday on the Eleven Calendar, my day to focus on communication skills. Well, today I'm in a language class that's all about communication. I ask the students to help me translate some of the Spanish words the regular teacher wrote in her lesson plans, and I speak to them using (very basic, of course) Spanish. I also often use Elevendays to communicate with my fellow teachers, but today I buy lunch off-campus on a day when I don't have a conference period, and so I rush out to get the meal and just barely get back before class resumes.
Meanwhile, there's no Miller wager today. I take attendance on paper rosters that don't indicate cohorts, so I can't be sure exactly how many students have opted out of hybrid. Third period has the most students in the room -- but it's also the largest class by far, so we can't conclude that I would have lost the wager in third period and won it in the other classes.
But speaking of Miller, one girl in third period stays for fifth period. She tells me that even though she's a freshman, she's not scheduled for fifth period due to having P.E. zero period. The actual Darren Miller tells us that many of his students would rather go to school zero period and then leave early (at least they did before the pandemic). Then again, this girl doesn't go home early -- she decided to delay her spring break by two hours, presumably to walk home with her friend in fifth period. (There's also another girl who's enrolled in fifth period and has an open third period -- today she attends third period and not fifth period!)
Lesson 14-2 of the U of Chicago text is called "Lengths in Right Triangles." In the modern Third Edition of the text, lengths in right triangles appear in Lesson 13-3.
Recall that Chapter 13 of the new edition corresponds to Chapter 14 of the old edition (since the old Chapter 13 has been split up into different chapters). The new Lesson 13-1 is the last lesson of the old Chapter 12, while the new Lesson 13-2 is on the Angle Bisector Theorem -- a theorem that doesn't appear in the old text (but occasionally appears on the Pappas calendar).
This is what I wrote two years ago about today's lesson:
Lesson 14-2 of the U of Chicago text is on lengths in right triangles -- specifically, those lengths that are related to the altitude and involve the geometric mean.
Geometric Mean Theorem:
The geometric mean of the positive numbers a and b is sqrt(ab).
(Note: This may sound like a definition, but actually the U of Chicago defines geometric mean to be the number x such that a/x = x/b, so we need a theorem to get the geometric mean as sqrt(ab).)
Right Triangle Altitude Theorem:
In a right triangle:
a. The altitude of the hypotenuse is the geometric mean of the segments dividing the hypotenuse.
b. Each leg is the geometric mean of the hypotenuse and the segment adjacent to the leg.
In this lesson, I give the proof of the Pythagorean Theorem based on similarity, but this time I gave the proof in the book, which mentions the geometric mean. Let's look at the proof -- as usual, with an extra step for the Given:
Given: Right triangle
Prove: a^2 + b^2 = c^2
Proof:
Statements Reasons
1. Right triangle 1. Given
2. a geometric mean of c & x, 2. Right Triangle Altitude Theorem
b geometric mean of c & y
3. a = sqrt(cx), b = sqrt(cy) 3. Geometric Mean Theorem
4. a^2 = cx, b^2 = cy 4. Multiplication Property of Equality
5. a^2 + b^2 = cx + cy 5. Addition Property of Equality
6. a^2 + b^2 = c(x + y) 6. Distributive Property
7. x + y = c 7. Betweenness Theorem (Segment Addition)
8. a^2 + b^2 = c^2 8. Substitution (step 6 into step 7)
It is uncertain whether this is the proof that Common Core intends the students to learn, or whether my earlier proof that avoids geometric means suffices.
Actually, since posting this last year, I've decided to check both the PARCC and SBAC released test questions for those related to the proof of the Pythagorean Theorem. There were a few questions that required use of Pythagoras, but none directly related to the proof. Of course, some people lament that there aren't very many proofs on the Common Core tests.
Today is an activity day. I found a Desmos activity on geometric means, created by Stacie Bender:
https://teacher.desmos.com/activitybuilder/custom/6055f7c649ff4a449e91a1d9
As usual, I include the first worksheet from two years ago, to be paired with the Desmos activity.
It is now spring break. Expect a special spring break post or two in the next few days.
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