Today I subbed in a high school science class. It's in my LA County district. Since it's a high school class that isn't math, there's no need for "A Day in the Life" today.
Two of the classes are Anatomy and the other is Biology. The two Anatomy classes are mostly juniors, with a few seniors as well. Due to an email error, I never see the actual lesson, but I do know that their assignment is on the Gizmo website -- a site with which I'm not familiar. There is one Gizmo for classwork and another for homework.
The Biology class has an aide, and so she sets up the lesson for these freshmen. They begin by watching a video on genetics. Then they also have a Gizmo for classwork. But their homework is on Edpuzzle -- and that's a site we know all too well.
As for the song for today, I do have a few standbys for science classes. I choose "Earth, Moon, and Sun," since today is Earth Day and the lyrics do mention our planet's name. But I could have selected "Meet Me in Pomona," since that's my usual go-to song for Bio classes. Believe it or not, there's another, more appropriate song I could have performed for Anatomy as well. As one of the classes is leaving, I ask a student to tell me what the lesson is on, and her reply is "the heart." This means that I could have sung "Count the Ways" from Square One TV -- a song that's all about heartrates.
Today is Nineday on the Eleven Calendar:
Resolution #9: We pay attention to math as long as possible.
Let's replacing "math" with "science" here. We recall that this is the district where students only have to stay on Zoom for a half hour. In all classes, I set up breakout rooms, one of which is for collaborating with the other students and the other is for working independently. No one chooses collaboration in either of the Anatomy classes, while some Bio kids do collaborate with the aide.
In second period, I send the students off right at the end of the half-hour, but some students stay in the fourth period Bio aide's breakout room for most of the period. Because of this, I only get to sing "Earth, Moon, and Sun" to the students who are in-person.
The fact that I'm in a science class reminds me of science at the old charter school -- and the fact that today's Earth Day reminds me of charter science even more. Recall that there was a Green Team project that was supposed to reach its climax on Earth Day. I left the school just barely before the Green Team was to get started, and so there's no way for me to know how exactly the projects went that year.
Once again, we can't exactly compare high school to middle school science. But I do know that genetics is part of the middle school life sciences curriculum that I should've taught my seventh graders. I'm not quite sure about the human body -- I keep thinking that in middle school, genetics is followed by evolution and then the human body. But things may have changed.
Today is Thursday, and so it's time for my weekly series on COVID-97 and high school Track. This week, my alma mater participated in the second dual meet of the season -- the first league meet. It was held on the opponents' home track.
This time, we have both boys and girls participating in the 1600. The girls contest the 1600 at both the Varsity and Junior Varsity, while the boys run it at the Frosh Soph level in addition to Varsity and JV. As we're comparing this season to my own senior season, we can ignore the Frosh Soph race. Although I'm hoping to make it to Varsity, it's more realistic that I'll start my season at the JV level.
At Varsity, both our team and our opponents have two runners. (Unlike Cross Country where a Varsity team should have seven runners, I don't think there's a fixed number of Varsity runners for Track.) Our top two milers (a sophomore and a junior) finish the race in the low 4:40's. In the JV race, the top two for our school run just over five minutes.
I don't think it's realistic that I would've dropped more than five seconds from last week's race. So let me set up my COVID-97 What If with an educated guess of what my 1600 times would've been:
On April 22nd, 1999 (second dual meet, first league meet), my 1600 time would have been 5:17.
Since my goal is to make it past League Finals, a comparison to the other league runners is highly significant here. Interestingly enough, my time of 5:17 would place me very close to the opponents' second runner in both the Varsity and JV races. If I fail to beat the second opponent, then that would make me last place in the Varsity race, but not JV. For this COVID-97 What If, let's assume that I'm still running in the JV race. Perhaps I might have run it slightly faster than 5:17, in an effort to beat the opposing runner.
Interestingly enough, my time of 5:17 would beat our top senior in the 1600, who ends up running in the high 5:20's. At our school, the Class of 1999 had several strong runners, so it's possible that there would have been faster seniors in the COVID-97 race. The only other senior distance runner this week contested the 3200 instead -- his time was in the high 12:40's, a time that I probably could have beaten.
I also notice that some of our school's juniors pulled off some interesting doubles -- that is, they ran in two races, but not the two that you'd expect. Two runners completed the 800/3200 double, and another ran a 200/800 double. All three of these juniors ran 1600 last week. One sophomore pulled off an even rarer double -- 400/3200. For this What If, it's important to pay attention to these doubles -- some runners who might have beaten my 1600 time might choose to contest a different distance instead at League Finals. (Indeed, this happened on the original timeline in 1999 as well.)
Lesson 15-1 of the U of Chicago text is on Chord Length and Arc Measure. The key theorem of this lesson is:
Chord-Center Theorem:
a. The line containing the center of the circle perpendicular to a chord bisects the chord.
b. The line containing the center of the circle and the midpoint of a chord bisects the central angle determined by the chord.
c. The bisector of the central angle of a chord is perpendicular to the chord and bisects the chord.
d. The perpendicular bisector of a chord of a circle contains the center of the circle.
Proof:
Each part is only a restatement of a property of isosceles triangles.
a. This says the altitude to the base is also a median.
b. This says the median to the base is also an angle bisector.
c. This says the angle bisector is also an altitude and a median.
d. This says the median to the base is also an altitude. QED
Why didn't the text just say "diameter" instead of "the line containing the center of a circle"? I assume it's because a diameter is a segment, but bisectors are lines. Here is the other key theorem:
Arc-Chord Congruence Theorem:
In a circle or in congruent circles:
a. If two arcs have the same measure, they are congruent and their chords are congruent.
b. If two chords have the same length, their minor arcs have the same measure.
The U of Chicago text points out that we can't use the terms "are congruent" and "have the same measure" interchangeably. Two angles are congruent if and only if they have the same measure, but two arcs with the same measure aren't necessary congruent. A 50-degree arc of a tiny circle is nowhere near congruent to a 50-degree arc of a large circle. The theorem tells us that an additional condition is needed -- the circles must be congruent also.
But what does it mean for two circles to be congruent? The U of Chicago text proves that two circles are congruent if and only if their radii are equal. Recall that in Common Core Geometry, we can only show two figures congruent by showing that some isometry maps one to the other:
Lemma:
Two circles are congruent if and only if they have equal radii.
Proof of Lemma:
If two circles X and Y have equal radii, then one can be mapped onto the other by the translation mapping X to Y. So they are congruent. Of course, if they do not have equal radii, since isometries preserve distance, no isometry will map one to the other. QED
Proof of Part a of Arc-Chord Congruence:
In circle O, you can rotate Arc AB about O by the measure of Angle AOC to the position of CD. Then the chord
Part b is left in the text as an exercise. A hint is given -- the measure of an arc equals the measure of its central angle. This suggests that we could use a traditional two-column proof via SSS:
Given: AB = CD in Circle O
Prove: measure Arc AB = measure Arc CD
Proof:
1. AB = CD 1. Given
2. AO = CO, BO = DO 2. All radii of a circle are congruent.
3. Triangle AOB = COD 3. SSS Congruence Theorem
4. Angle AOB = COD 4. CPCTC
5. Arc AB = CD 5. Definition of arc measure
The text warns us that in circles with different radii, arcs of the same measure are not congruent -- they are similar. This isn't proved in the text, but notice that one of the Common Core Standards directs students to "prove that all circles are similar." So let's do so right here:
Theorem:
All circles are similar.
Proof:
If the two circles are concentric, then their common center is also the center of a dilation, with the scale factor obviously R/r, with r the smaller radius and R the larger radius. If the two circles have different centers, then we can translate one of the circles so that its center matches the other, then perform the dilation. QED
Actually, a single dilation will work if the centers aren't the same, but it's difficult to locate the center of this dilation, so it's easier just to translate first.
Here is my first worksheet for Chapter 15:
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