There are a few things I want to say before we start. First of all, it's the first day of a three-day subbing assignment. Actually, the regular teacher has been out the entire week -- apparently, her daughter living out of state is in the hospital. But my assignment didn't start until today. This marks the second straight year I had a multi-day on the Wednesday-Friday after Thanksgiving (even though last year it was a Spanish class).
This class has a co-teacher for the special ed students. It's actually the same teacher I subbed for recently, in my November 18th post. This time, that co-teacher is present.
In this new district, today is Day 64. As I wrote before, Thanksgiving break divides the trimesters -- and some students switch math teachers at the trimester. This means that these students still haven't met their new math teacher.
Oh, and finally, it's raining in Southern California today. Last year (on the day of our Lesson 7-3 in Geometry, incidentally), I subbed at the same school on a rainy day. That year, there was a special rainy day schedule, but this has been phased out. Instead, the bell schedule is an ordinary (Common Planning) Wednesday.
OK, so without further ado, let's begin:
8:00 -- This teacher has morning duty, and so I must supervise the students in the rain.
8:15 -- The day begins with homeroom, and then all periods rotate at this school. But at this school, homeroom is the same as third period -- and today's rotation just happens to start with third. And so homeroom leads directly into third period.
Today's lesson is on solving one-step equations. This teacher begins each period with writing the agenda, giving a Warm-Up, and then correcting homework. And so my first goal is to make sure that I'm not devoting too much time to opening material and not enough to the main lesson.
There are six problems on the Warm-Up, so I give it six minutes, or one minute per problem. I then decide to devote the same amount of time to the agenda and homework, so that's 18 minutes total.
I indeed finish the homework 18 minutes into the period. But I have the benefit of homeroom and the extra passing period between homeroom and the first class (which wasn't needed because they are the same class, but it's built into the bell schedule). I know I wouldn't have that benefit for the other classes, and so I must work on cutting down the time for opening material -- especially with each class being only 40 minutes on Wednesdays.
This teacher actually grades the homework for accuracy -- she's pre-selected six problems for the students to grade (after exchanging papers, of course). Each problem is worth three points (one point for the work, one point for the answer, and one point for the check), for a total of 18 points. To make the assignment worth 20 points, an additional point is given for writing each name (first/last) -- unless you're a teacher, it's hard to believe how often students forget to write their names on their papers.
The main activity for day is a partner assignment. Partner A and Partner B have eight equations each to solve. The corresponding equations for each partner are the same, so that the kids are able to check each other's answers.
And I do sing a song today while the students are writing agendas. Of course, the song I choose is "Solving Equations," since that's what the students are doing today. Only the first two verses are needed as these are one-step equations.
Third period is the class I fear the most today, since the previous sub had to write the names of two students who had gotten in trouble. This time, I had to write one of their names again, since he only solves three equations after I told the students that they needed to answer four. (His partner solves his fair share of equations.)
Meanwhile, another girl just happens to be wearing a NASA jacket. NASA, hmm, why does that sound familiar? That's right -- we're still in the middle of reading Ottaviani's moon mission book. I actually anticipated having an opportunity to mention this book, so I bring it to class. I show the students a picture of equations on page 89, and inform them that one reason to learn math is that equations are needed to launch a rocket into space. The girl and her partner appear to be struggling on the partner activity, and so I help them out. I tell the girl that maybe if she succeeds in math, she might someday become the first woman to walk on the moon.
9:05 -- Third period ends. Fourth period is the teacher's conference period.
9:50 -- Fifth period arrives.
As usual, I make adjustments to my lesson after the break. This time, I take another idea from Ottaviani's book. Since his book is titled T-Minus, I decide to say "T-Minus" followed by the number of minutes left for each opening task.
This works well, as I'm able to complete all the tasks by the 18-minute limit. Also, I have the homework for tomorrow out on the desk, so that students can take the homework as soon as they finish the partner activity. Of course, the previous subs have written that fifth period was the best class of the day -- and I indeed name them the best class of the day yet again.
10:30 -- Fifth period leaves for snack.
10:40 -- Sixth period arrives. This is the first of two classes with the co-teacher.
This time, I notice that she is stamping -- not the Warm-Up, but the agendas. The previous subs hadn't stamped planners, so I didn't know whether I should have stamped them in the earlier classes.
I must write down the name of one guy who solves the equations, yet shows no work. He's one of the special ed students, so I tell the co-teacher who's there to assist him. She informs me that this is just the way he is -- but I know that failing to show work will ultimately hurt his grade.
There is also a boy in this class wearing a NASA jacket. He appears to be doing fairly well in this class, but his partner does need some extra help.
11:25 -- Sixth period leaves and first period arrives. This is the second class with the co-teacher.
This time, the co-teacher helps me with time management and getting the students on task. I also continue to say "T-Minus" to remind the students how much time they have left.
12:05 -- First period leaves for lunch.
12:50 -- Second period arrives.
During the Warm-Up, one girl just writes "7" as the answer to every question on the Warm-Up. No, seven isn't the correct answer to any of them. She obviously needs extra help, and so I sit with her and her partner.
1:30 -- Second period leaves, thus ending my day.
I'm in this class for two more days, and of course I'll continue to write about it here on the blog. As a multi-day math assignment, this might end up being my most significant assignment of the year. I do know one thing -- I'll keep mentioning the moon landing book this week. Speaking of which --
Let's look at pages 100-108 of Jim Ottaviani's T-Minus: The Race to the Moon. We left off just as we were preparing to take a photo from Apollo 8 -- a lunar orbit:
Frank Borman: Calm down, Lovell.
Well, they finally get their photo, though I can't really show it here on the blog. Instead, we'll skip up to the ninth orbit, 03:14:06:39 GET... after everybody's had a nap:
Jim Lovell: The TV look okay?
Bill Anders: That's very good.
Frank Borman: OK... Welcome from the moon... It's a rather forboding horizon, a rather -- stark, maybe. Stark and unappetizing sort of place. We're now going over our -- approaching one of our future landing sites -- right now... called the Sea of Tranquility.
Bill: And now you can see the long shadows of the lunar sunrise. And so for all the people back on earth... the crew of Apollo 9 has a message that we would like to send to you... In the beginning, God created the heaven and the earth, and the earth was without form and void, and darkness was upon the face of the deep...
Jim: ...And God made the firmament, and divided the waters which were under the firmament from the waters which were above the firmament. And it was so. And God called the firmament heaven...
Frank: ...And God called the dry land earth, and the gathering together of the waters called He seas. And God saw that it was good. And from the crew of Apollo 8, we close with good night, good luck... a Merry Christmas, and God bless all of you.
Now we proceed with the last orbit:
Mission Control: Houston, Apollo 8, over.
Frank: Hello, Apollo 8, loud and clear.
Mission Control: Roger, please be informed there is a Santa Claus.
Frank: That's affirmative. You're the best ones to know.
Mission Control: And burn status report: it burned on time; burn time, 2 minutes, 23 seconds, seven-tenths plus V_GX. Altitude nominal; residuals -- minus five-tenths V_GX; plus four-tenths V_GY; minus 0 V_GZ, delta-V_C, minus 26.4.
Frank: Roger.
(04:06:08:16 GET)
Bill: How was Christmas at your house tday?
Mission Control: Early and busy as usual. I told my son Michael you guys are up there, and he said, "Who's driving?"
Bill: That's a good question. I think Isaac Newton is doing most of the driving now.
Mission Control: Hey -- C.C., c'mon.
(06:02:49:07 GET)
Bill: Quite a ride, huh?
Frank: Yeah... damnedest thing I ever saw.
Ottaviani explains that the Apollo reentry speed was 27,000 miles per hour.
In our final scene for today, we see some information about Apollo 9 and 10, which were also successful lunar orbital missions.
Michael Collins: The CSM's coming in from California on the Super Guppy this week.
Ottaviani tells us to look up the Super Guppy -- what a great plane! He also mentions that the LM came from Bethpage, NY, also travelling by Super Guppy. The space suits are from Dover, DE, and the Saturn V was shipped by barge from Huntsville, AL. Oh, and by the way, Michael is describing all of these parts to the two astronauts who will ride on the next Apollo mission -- namely Neil Armstrong and Buzz Aldrin.
Lesson 7-3 of the U of Chicago text is called "Triangle Congruence Proofs." Let's forget about dilations and get back to the proofs we've been working on.
This is what I wrote last year about today's lesson:
Lesson 7-3 of the U of Chicago text discusses triangle congruence proofs. Finally, this is what most Geometry students and teachers think of when they hear about "proofs."
There's actually not much that I have to say about this lesson. We already know much about how the proofs in this section go -- generally we are given two triangles with some of the corresponding sides and angles given as congruent. In the easiest examples, the given pairs are already enough to conclude that the triangles are congruent. The U of Chicago text points out that often we must work to get one of the needed pairs -- such as the Reflexive Property of Equality when the two triangles have a side in common, or perhaps the Vertical Angles Theorem to find a pair of congruent angles -- then we use SAS, SSS, or ASA to conclude that the two triangles are congruent. Finally, the students are usually asked to prove one more pair of parts to be congruent, which requires the use of CPCTC.
Now technically speaking, CPCTC was covered in Lesson 6-7, which we skipped over. The U of Chicago text uses the abbreviation CPCF, which stands for corresponding parts of congruent figures, for after all, the property applies to all polygons, not just triangles. But the abbreviation CPCTC is so well-known that I prefer to use CPCTC rather than CPCF. Even though we skipped Lesson 6-7, I believe that the students can figure out CPCTC quickly, so I incorporate it into today's lesson.
Last year I didn't create a worksheet, but instead just wrote down ten problems. This year, I will create a worksheet and include some of the problems from last year, but others I had to throw out. In particular, I had to drop the proof of the Converse of the Isosceles Triangle Theorem -- we certainly want to avoid the U of Chicago proof that uses AAS, since we haven't taught AAS yet. I want to cover it at the same time as the forward Isosceles Triangle Theorem -- I'm saving both for next week. I also had to throw out a few "review" questions that once again review lessons that we've skipped.
Notice that Question 5 mentions isosceles triangles, but it doesn't actually require either the Isosceles Triangle Theorem or its converse. We only need to use the definition of "isosceles" to get one pair of congruent sides -- the first S. The definition of "midpoint" gives us the second S, and the Reflexive Property of Congruence gives us the third S, so the triangles are congruent by SSS.
Indeed, this is a good time for me to bring up a point about two-column proofs. I can easily see a student being stuck on this question because none of the three S's are given -- they have to work to find all of them. One thing I like to point out is if the students see a long word like "isosceles" -- or even "midpoint" -- in a question, they will usually need the definition of that word. In particular, a big word in the "Given" usually leads to the students using the meaning half of the definition, and a big word in the "Prove" often needs the sufficient condition half. The meanings of the words "isosceles" and "midpoint" lead to two of the three S's, and then they'll probably see that the Reflexive Property gives the third S.
Of course, soon they'll learn about the Isosceles Triangle Theorem and its converse -- and once those appear, nearly every proof involving isosceles triangles will use one or the other. But before students learn about the theorems pertaining to a key term, it's the definition that appears in the proof. We've already seen a few proofs about parallel lines where it's the definition of "parallel" that matters -- but once we've proved the Parallel Tests and Consequences, we hardly ever use the definition of parallel.
Last thing -- in the U of Chicago text, the "Given" is never included as a separate step. I always include it as this is more common -- so our proofs will have one more step than those in the text.
There's actually not much that I have to say about this lesson. We already know much about how the proofs in this section go -- generally we are given two triangles with some of the corresponding sides and angles given as congruent. In the easiest examples, the given pairs are already enough to conclude that the triangles are congruent. The U of Chicago text points out that often we must work to get one of the needed pairs -- such as the Reflexive Property of Equality when the two triangles have a side in common, or perhaps the Vertical Angles Theorem to find a pair of congruent angles -- then we use SAS, SSS, or ASA to conclude that the two triangles are congruent. Finally, the students are usually asked to prove one more pair of parts to be congruent, which requires the use of CPCTC.
Now technically speaking, CPCTC was covered in Lesson 6-7, which we skipped over. The U of Chicago text uses the abbreviation CPCF, which stands for corresponding parts of congruent figures, for after all, the property applies to all polygons, not just triangles. But the abbreviation CPCTC is so well-known that I prefer to use CPCTC rather than CPCF. Even though we skipped Lesson 6-7, I believe that the students can figure out CPCTC quickly, so I incorporate it into today's lesson.
Last year I didn't create a worksheet, but instead just wrote down ten problems. This year, I will create a worksheet and include some of the problems from last year, but others I had to throw out. In particular, I had to drop the proof of the Converse of the Isosceles Triangle Theorem -- we certainly want to avoid the U of Chicago proof that uses AAS, since we haven't taught AAS yet. I want to cover it at the same time as the forward Isosceles Triangle Theorem -- I'm saving both for next week. I also had to throw out a few "review" questions that once again review lessons that we've skipped.
Notice that Question 5 mentions isosceles triangles, but it doesn't actually require either the Isosceles Triangle Theorem or its converse. We only need to use the definition of "isosceles" to get one pair of congruent sides -- the first S. The definition of "midpoint" gives us the second S, and the Reflexive Property of Congruence gives us the third S, so the triangles are congruent by SSS.
Indeed, this is a good time for me to bring up a point about two-column proofs. I can easily see a student being stuck on this question because none of the three S's are given -- they have to work to find all of them. One thing I like to point out is if the students see a long word like "isosceles" -- or even "midpoint" -- in a question, they will usually need the definition of that word. In particular, a big word in the "Given" usually leads to the students using the meaning half of the definition, and a big word in the "Prove" often needs the sufficient condition half. The meanings of the words "isosceles" and "midpoint" lead to two of the three S's, and then they'll probably see that the Reflexive Property gives the third S.
Of course, soon they'll learn about the Isosceles Triangle Theorem and its converse -- and once those appear, nearly every proof involving isosceles triangles will use one or the other. But before students learn about the theorems pertaining to a key term, it's the definition that appears in the proof. We've already seen a few proofs about parallel lines where it's the definition of "parallel" that matters -- but once we've proved the Parallel Tests and Consequences, we hardly ever use the definition of parallel.
Last thing -- in the U of Chicago text, the "Given" is never included as a separate step. I always include it as this is more common -- so our proofs will have one more step than those in the text.
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