8:00 -- You already know what I do outside at this time. (It does rain, but not until much later.)
8:15 -- You already know what the day begins with.
8:20 -- The rotation for today begins with fifth period.
Today, the main activity is the Week #1 Quiz. It's labeled "Week #1" because it's the first week of the second trimester. (Recall that Thanksgiving divided the trimesters in this district.)
I begin with the usual opening activities, and then I sing a song. This time, I finally perform the song that I started to prepare for yesterday:
WHODUNNIT?
First Verse:
Was Mrs. fixing snacks in the kitchen?
Was Dr. hearing songs in atrium?
Professor's iPad by the pool now?
Did Miss kill the victim?
Second Verse:
A cougar's mauling was the method?
Or had a green snake's bite begun it?
Or object's falling was the weapon?
Just tell me... Whodunnit!
I used Mocha to compose parts of this song in the 12EDL scale -- the scale which, as I've mentioned before, is the main EDL scale I want to compose in. I'll write more about the tune later.
On the quiz, several students fail to show work before turning it in. When I spot a few quizzes without work, I hand them back and tell them to add it in.
9:15 -- Fifth period leaves and sixth period arrives. This is the first class with the co-teacher.
This time, I end up stamping the agendas, but then the co-teacher helps out with the rest of the opening activities and the quiz. She makes sure that the students remain silent for the quiz.
I also pass out some candy as a reward for anyone who solved yesterday's Whodunnit -- indeed, the idea is to hand out the candy and then sing the Whodunnit song as they eat it. (Actually, I wanted to do the same in fifth, but recall that only one student in that class solved yesterday's mystery. So instead, in fifth I had three students work on my replacement Clue #3 and then give those kids candy.)
This time, I make sure that nobody turns in the quiz who hasn't shown the work. I remind them that their teacher deducts points for failing to show the work.
10:10 -- Sixth period leaves for snack.
10:20 -- First period arrives. This is the second of two classes with the co-teacher.
This class goes more or less like sixth period. In fact, I declare that the best class of the day is a tie between sixth and first periods.
11:10 -- First period leaves and second period arrives.
This class is slightly louder than yesterday, but they do get quiet at quiz time. When there are a few minutes left after finishing the quiz, I read the next page (118) of Ottaviani's book just to show them that the Apollo 11 does maintain enough fuel to land softly on the moon. Today, second period isn't in the silent reading "slot" after lunch, but I at least want to resolve the cliffhanger.
12:05 -- Second period leaves for lunch.
12:50 -- Third period arrives.
This class is now in the silent reading "spot." And so I read them pages 118-124 of Ottaviani's book -- the conclusion.
Everyday this week, names must be written down for this class, and today is no exception. First, the teacher requests that I leave her the name of anyone without a silent reading book. I must write down one guy's name -- the same boy who was on yesterday's bad list.
But then more problems arise. First, there is continuous talking during the opening activities -- including my reading of Ottaviani's book and the Whodunnit song. (Recall that second period is also talkative, but they at least are quiet when I'm reading or singing.) While other classes get quiet when it's quiz time, I assume that third period won't get silent -- based on their track record this week.
So I explain to them the importance of being quiet on quiz days even when there's a sub, as well as the need to show work on the quiz. One guy (a boy who made the bad list with the previous subs) complains that he shouldn't have to show his work. This turns into an argument, and I come very close to yelling. In fact, I need to err on the side of being hard on myself, so let me correct that last sentence -- I actually yell.
Lately, I've been ending arguments/yelling by singing, so I sing "Whodunnit" at this point. Then I pass out the quiz.
The same guy who complains about showing work now asks whether he can use the Photomath app on the quiz. I tell him that calculators aren't even allowed, much less Photomath. Then another girl asks whether notebooks are allowed. I respond that yes, the teacher does allow notebooks. (These do appear to be the interactive notebooks which are becoming increasingly common.) Then the first guy complains again, claiming that Photomath is equivalent to a notebook.
The talking continues during the quiz. I'm sitting down to make sure that students are showing work, which makes it more difficult for me to see who's talking. I end up writing the same guy's name on the bad list, since I clearly see him disturbing another student.
2:00 -- Third period leaves.
But just as class ends, two girls inform me that some students have indeed been cheating. (These two are new transfers to this class for the second trimester -- but I notice that, in this and at least one other class, the transfer kids are the top students.) They hand me a list of two boys and four girls who were using Photomath during the quiz. I believe them easily, since one of them is the guy who complained about the app earlier. So apparently, he indeed used the app and encouraged others to cheat. And so I must strongly recommend that the regular teacher give all six students zeros on the quiz.
My day ends here since fourth period is conference. But after spending so much time in this class during my three-day assignment, I can't help but want to compare this class to the old charter school, in both academics and classroom management.
Let's start with academics. I never mentioned what this class did on Monday and Tuesday -- it appears that they took notes (in the interactive notebook) both days. Tuesday, like the rest of the week, began with a Warm-Up. On Monday (the first day after Thanksgiving break), however, the class began with a multiplication worksheet.
You already know about the Wednesday and Thursday lessons. Both days were activities -- the partner activity on Wednesday and Whodunnit yesterday.
Then the class ends Friday with a weekly quiz. Today's quiz contains ten questions -- four of which review previous standards (integer arithmetic and order of operations) and the other six assess this week's standard (one-step equations).
I like how this week's lesson plan goes, but it wouldn't have worked at the old charter. First of all, Mondays were lost to coding. And if we're looking at the seventh grade, then I also lost Wednesdays to music. In addition, I wouldn't have been able to give activities such as the partner activity or Whodunnit, because I was bound to give projects from the Illinois State text.
So the closest I could get to a week like this is a traditional lesson on Tuesdays, an Illinois State project on Thursdays, and a weekly assessment on Fridays. (I've proposed something similar to this in previous posts -- but the project might need to be science instead of math.) Also, notice that for someone who makes a big deal about drens, I only gave Dren Quizzes once a month. It's possible that this teacher gives the multiplication worksheet every Monday. (It's also possible that the worksheet is only given right after long holidays since there's nothing to review in the Warm-Up -- these would then be about as frequent as my old Dren Quizzes.) I could have given my Dren Quiz every Monday, as something to do in the few minutes remaining in class after the coding teacher left.
And once again, interactive journals are something that I could have implemented back then, if only I was willing to deal with students who forgot them all the time. Allowing students to use them on quizzes (though not tests) might have been an incentive for them to bring them regularly.
That year, I didn't really reach one-step equations in my seventh grade class until February, just before I left the school. I had used the Illinois State STEM Project text to decide in what order to teach the standards, but the intended order was the follow the naive order given in the Common Core Standards -- first RP, then NS, EE, G, and SP.
But the Big Ideas text used in middle schools in this district presents the seventh grade standards in a different order. As listed in the Table of Contents, the order is NS, EE, RP, G, and SP. This allows the most important strands, NS (Number Sense) and EE (Expressions and Equations) to be taught earlier in the year. Also, the Big Ideas order allows proportions (RP) to be solved using equations (EE). On the other hand, the Illinois State text presents all RP standards before introducing any EE standards -- which is why we had to use line diagrams and tape diagrams to solve proportions in Grades 6-7.
It's possible that I could have taught all NS/EE standards before RP that year -- but I was stuck on the idea that the STEM projects should determine the order of the standards. (Either order may allow us to reach G -- including standard on the circumference/area of a circle -- by Pi Day.)
Now let's think about classroom management. I thought that I was taking a few small steps forward throughout the week, but then I took a huge step back during third period today.
I can't help but think about the day when some of my eighth graders cheated on the volume test -- especially since it also happened the first week in December. If you prefer a seventh grade example, consider the November quiz on positive and negative numbers, during which the seventh graders talked throughout.
In both classes, I had trouble identifying the talkers. In eighth grade I correctly identified two cheaters, but one girl I gave a zero to turned out to be innocent. Seventh grade was even worse -- at least half the students were talking, but not necessarily the same students I ended up punishing.
One problem today with identifying the cheaters today instead of looking at the class to see who's talking, I'm checking quizzes to see who's written down enough work. Perhaps it's better to check for work in only well-behaved classes (here any class except third period). But in misbehaving classes, I only check for behavior -- just let those kids lose points for showing no work. (It's better to lose a few points than get a zero for cheating.)
Today, the students complain when I make them show work, or remain silent even if they're finished with the test if others are still working. But in this case, there's real cheating (rather than a mere suspicion of cheating). Making the students show work helps to stop cheaters (since Photomath doesn't show the work), as does keeping them quiet after they finish the quiz. Yet the students think that I'm a dictator for making them show work and remain quiet -- and thus refuse to do either. (It's possible that "Because I said so" is an effective answer to "Why must I show work?" or "Why can't I talk if I'm done?")
In all classes, it's tough to stop talkers when at least half the class is speaking during a quiz. The best cure for this situation is prevention. The class is talking so much because they think they can get away with it -- in each case, I must have shown such weak classroom management earlier that the students try to do whatever they want.
Today's class has had subs all week, including me on Wednesday. During the week, third period must have seen me fail to punish someone who misbehaved. And of course, my failure to punish students at the charter (including the ineffective participation points system) is well-documented on the blog.
Notice that third period, the bad class, is the same as homeroom. I wonder whether there's something I could have done during the first five minutes of the day that might have helped control them.
By the way, what disappoints me the most today about third period is that one of the girls who cheats is the one who wore a NASA jacket. I was really hoping that this girl, who clearly respects NASA and its work, would resonate with my reading of Ottaviani's book -- and strive to be more successful in her math and science classes. Instead, all she wants to do on today's quiz is cheat. It almost makes me want to skip reading the last seven pages of the moon book -- almost.
Let's look at pages 118-124 of Jim Ottaviani's T-Minus: The Race to the Moon. As I mentioned earlier, we left off with Apollo 11 possibly running out of fuel:
Gene: Well?
Steve: N-not a problem. Go.
Charlie: Nnnno abort, we're g... go flight.
Gene: We got -- we're go on that alarm. Eagle, Houston, you're go for landing. Over. Roger. Understand. Go for landing. Over.
Michael: Okay...
Buzz: 3000 fe -- Program alarm. It's a 1201.
Charlie/Steve: Same type. Go flight.
Gene: Roger that, 1201 alarm. We're go
Ottaviani explains, "The 1202 and 1201 alarms happened because Buzz had the landing and rendezvous computers both running (just in case) so the AGC's memory -- barely big enough to hold four seconds of an MP3 -- struggled to keep up.
Gene: AGC, Control, Fuel. How much left?
Jack: 60 seconds, Flight.
Charlie: Okay... Flight. 45 seconds.
Michael: OK. 75 feet. Down a half, six forward.
Charlie: 30 seconds.
Michael: Kicking up some dust. 30 feet, 2 1/2 down.
Charlie: 25 seconds. 20... 19.
Michael: Mode control -- both auto. Descend engine override -- off. Engine arm-off.
Of course, the Eagle lands, and Neil takes his one small step. And now, back to C.C., on the good earth, for the final scene.
Ivy: Did you see the landing?
C.C.: Oh... um, yeah. It was great, Ivy. Really great.
Ivy: Did you even watch it, C.C.?
C.C.: Well, sure. I mean, some of it. But there's this stuff we're working on, and... you know.
Ivy: What stuff?
C.C.: Apollo-Soyuz. We gotta modify the Command Module Docking mechanism if we're going to do an EOR with the Russians.
Ottaviani explains, "Remember EOR? It's Earth Orbit Rendezvous."
Ivy: That's the next thing? With the Russians? Wow!
C.C.: No, Skylab is the next thing. Apollo-Soyuz isn't until 1975.
Ottaviani explains, "Alexei Leonov flew the Soviet half of Apollo-Soyuz, so the USA + USSR scene you saw C.C. draw on page 62 came true... sort of."
Ivy: Well, who's working on Skylab then? I've been so focused on the space shuttle that --
C.C.: Oh, we're already done with that. The drawings are over there. So, uh... if you don't mind...
Ivy: Hey -- speaking of docking -- I noticed something the other day. What if you changed the hatch t something like... this (indicating a certain drawing)? Maybe we'll get a better connection with the Russian spacecraft?
C.C. Yeah? Let me see...
The final image is labeled "T-plus..." -- this means time after the moon landing, just as "T-minus" means time before landing. Our students are living at T-plus 50 years and counting -- and maybe some of them might contribute to future space exploration at NASA, if only they would concentrate on solving equations today.
I definitely enjoyed reading this book. And I'll be reading another one of Ottaviani's science comics one of these days.
Lesson 7-5 of the U of Chicago text is called "The SSA Condition and HL Congruence." This lesson introduces the final congruence theorems.
This is what I wrote last year about today's lesson:
Lesson 7-5 of the U of Chicago text is on SSA and HL. I've already mentioned how I'll be able to prove HL without using AAS, since we have to wait before I can give an AAS proof.
Meanwhile, we know that SSA is invalid, but the U of Chicago text provides us with an SsA Congruence Theorem, where the size of the S's implies that it must be the longer of the congruent sides that is opposite the congruent angle. The text doesn't provide a proof of SsA because the proof is quite difficult -- certainly too difficult for high school Geometry students.
But you know how I am on this blog. I'm still curious as to what a proof of SsA entails, even if we don't ask high school students to prove it. Last year, I mentioned how SsA leads to the ambiguous case of the Law of Sines.
I've noticed that when using the Law of Sines to solve "both" triangles for the SsA case, the "second" triangle ends up having an angle sum of greater than 180 degrees. We can use the Unequal Sides Theorem to see why this always occurs -- that theorem tells us that the angle opposite the "s" must be smaller than the angle opposite the "S" (the known angle). We know that if two distinct angles between 0 and 180 have the same sine, then they are supplementary. So the second triangle would have angle sum of at least 180 minus the smaller angle plus the larger angle -- which always must be greater than 180. (This is often called the Saccheri-Legendre Theorem, named for two mathematicians with whom we're already familiar and associate with non-Euclidean geometry. Of course spherical geometry is not neutral, and Saccheri-Legendre fails in spherical geometry.)
That the sum of the angles of a triangle can never be greater than 180 (but we don't necessarily know that it's exactly 180) is neutral, but the Law of Sines is not neutral. Nonetheless, it is known that SsA is a neutral theorem. So the Law of Sines can't be behind the secret proof that U of Chicago text doesn't print in its text.
But one of the questions from the U of Chicago text hints at how to prove SsA -- and it's one that I included on my HL/SsA worksheet last year. Here is the question:
Follow the steps to make a single drawing of a triangle given the SSA condition:
a. Draw a ray XY.
b. Draw angle ZXY with measure 50 and XZ = 11 cm.
c. Draw circle Z with radius 9 cm. Let W be a point where circle Z and ray XY intersect.
d. Consider triangle XZW. Will everyone else's triangle be congruent to yours?
The answer is that they most likely won't. In step (c) we are to let W be a point -- not the point -- where the circle and the ray intersect. This implies that there could be more than one point where they intersect -- and in fact, there are two such points. But there can never be a third such point, because a circle and a ray (or line) intersect in at most two points. We can prove this indirectly using the Converse of the Perpendicular Bisector Theorem:
Lemma:
A line and a circle intersect in at most two points.
Indirect Proof:
Assume towards a contradiction that there exists a circle O that intersects line l in at least three points A, B, and C, and without loss of generality, let's say that B is between A and C. By the definition of circle, O is equidistant from A, B, and C. From the Converse of the Perpendicular Bisector Theorem, since O is equidistant from A and B, O lies on m, the perpendicular bisector of
Now both m and n are said to be perpendicular to l, since each is the perpendicular bisector of a segment of l. So by the Two Perpendiculars Theorem, m and n must be parallel -- and yet O is known to lie on both lines, a blatant contradiction. Therefore a line and a circle can't intersect in three points, so the most number of points of intersection is two. QED
So let's prove SsA now using this lemma and the construction from the problem above. Once again, the proof must be indirect. Even though most of our congruence theorem proofs call the two triangles ABC and DEF, I will continue to use the letters XZW so that it matches the above question.
Given: AB = XZ < BC = ZW, Angle A = Angle X
Prove: Triangles ABC and XZW are congruent.
Proof:
We begin by performing the usual isometry that maps AB to XZ. As usual, we wish to show that the final reflection over line XZ must map C to W -- that is, C' must be W.
So assume towards a contradiction that C' is not W. As usual, the given pair of congruent angles allows us to use the Flip-Flop Theorem to map ray AC to ray XY. Just as in the problem from the text, we know that C' must be a point on ray XY (since ray AC maps to ray XY), and it must be the correct distance from Z (since reflections preserve distance). Now the set of all points that are the correct distance from Z is the circle mentioned in the above problem. We know that the intersection of a circle and a ray is at most two points, and so the assumption that C' is not W implies that W must be one of the two points of intersection, and C' must be the other point.
We must show that this leads to a contradiction in the SsA case -- that is, when AB and XZ are longer than the sides BC and ZW. We see that if C' and W are distinct points equidistant from Z, then the triangle ZC'W must be isosceles, and so its base angles ZC'W and ZWC' must be congruent.
Now we look at triangle ZC'X. It contains an angle, ZC'X, which forms a linear pair with ZC'W, so its measure must be 180 - m/ZC'W -- that is, m/ZC'X = 180 - m/ZWC' by substitution. It also contains an angle ZXC' -- which (renamed as ZXW), we see must be larger than ZWC' (renamed as ZWX) by the Unequal Sides Theorem -- ZXW is opposite the longer side ZW (in triangle ZXW), so it must be the bigger angle.
So now we add up the measures of two of the three angles in triangle ZC'W -- the angle ZC'X has measure 180 - m/ZWC', and angle ZXC' is known to be greater than ZWC'. So the sum of the two angles is greater than 180 - m/ZWC' + m/ZWC' -- that is, it is greater than 180. That is, the sum of the angles of triangle ZC'W is greater than 180 -- which is a contradiction, since by the Triangle-Sum (actually Saccheri-Legendre) Theorem, the sum must be at most 180.
Therefore the assumption that C' and W are different points is false. So C' must be exactly W. QED
Of course we wouldn't want to torture our students with this proof. It depends on three theorems -- Isosceles Triangle, Unequal Sides, and Triangle-Sum -- that we have yet to prove. On my worksheet, I added an extra note to this problem explain how it leads to SsA.
Again, I retain references to non-Euclidean geometry from the old post. The new Third Edition of the U of Chicago text includes an actual SsA proof. It's similar, but not quite like, the proof given here. I dropped the mention of SsA on the worksheet since today's the second day of the SSASS activity. Oh, and since SsA works, so does SsAsS. Two short sides are adjacent to the angle, which implies a convex quadrilateral.
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