It is also a multi-day assignment -- my second three-day assignment this month. Therefore I will do "A Day in the Life" today, despite it being a high school non-math class.
But before we begin, let me say something important about today's schedule. Just like my first visit to this classroom nearly a year ago, there is SBAC testing today. But this year, the testing at this same school starts earlier -- last year, I described the test in my April 1st post. (Indeed, I even blogged an April Fool's Day joke about my short day of subbing becoming the regular schedule.) This year, the last day of testing is on St. Patrick's Day.
I wonder whether the move to an earlier testing window has anything to do with the coronavirus and the fear that schools may be closed soon. (In the world, the virus continues to wreak havoc on sports, as one NBA player catches the disease. The rest of the NBA season has now been suspended.)
8:00 -- The standardized test for today is the California Science Test. This is what makes this year's earlier testing really seem strange -- we just barely got the results from last year's CAST a month ago, and now we're already giving this year's CAST! At the high school level, any one of the four grades is given the test -- as is typical, this school chooses to administer it to juniors, since they're the ones taking the other SBAC exams as well.
The other subs and I are asked to spend the two hours of testing in the teacher lounge, where we wait to fill in as a proctor for another teacher who needs to take a short break. None of us are called.
After the testing is over, we return to our rooms. As you might expect, there is a block schedule for the rest of the day, with odd periods today and even periods tomorrow. That's right -- I sub for odd periods two days in a row, yesterday and today, albeit in different districts for different reasons.
But in today's new district, "first period" really means zero period. Yet all seven periods still meet today and tomorrow after the tests. The odd periods today are 1, 3, 5, with 2, 4, 6 tomorrow. As for seventh period, it meets both today and tomorrow but for half as long. (As a young student, my old high school had a similar regular block schedule with seventh period meeting everyday.)
As it turns out, not only does this regular teacher not have first period, but his conference is third -- and so my first class isn't until fifth period, after lunch! I often refer to my teacher's conference period as a "well-deserved" break, but clearly my super-long break today is undeserved.
So what do I do during my long undeserved break? Well, I start looking ahead to Pi Day. As Pi Day is a Saturday this year, teachers who wish to celebrate it will do so on Friday. And since I already know that I'll be in this classroom for three days, I know that this is the class that will get a celebration. I wish that it were a math class getting the Pi Day party instead of English, but oh well.
Last year, I did sub in a middle school math class on Pi Day, but I wasn't called for it until late (after both zero and first periods had already met). I quickly purchased a pie during the next period (which was conference) and then gave out the goodies in the remaining periods.
But this time, I know where I'll be subbing on the day of the party -- right here. And this year, I'm doing something that I may consider if I get my own classroom someday -- having all the classes compete to earn the party.
There will be no testing on Friday, and so all classes will meet in their usual order. As it turns out, second, fourth, and fifth periods are for seniors, and Periods 6-7 are juniors. Therefore I'll have the senior classes compete against each other and likewise with the juniors. I'll award pizza pie to the winning junior class (since both classes meet after lunch -- that is, after pizza parlors open) and fruit pie to the winning senior class.
And so during the long break, I purchase the plates, forks, and utensils. Of course, I won't buy anything edible until Friday, the actual day of the party.
1:20 -- We finally reach my first actual class of coverage -- fifth period, a senior class. There is normally an aide for this class, but she's regularly scheduled only until 1:30. And so today is for the most part a solo day.
Both classes are looking up vocabulary words from the books that they are reading. This class is still studying the same book as last week -- Jon Krakauer's Into the Wild. (One of their vocab words is "egress," and I can't help mentioning the infamous "this way to the egress" prank that PT Barnum played over a century ago.)
And so I introduce the fruit pie incentive. There are ten words for which they must define and write in a sentence. I tell them that the winning class will be determined by the average number of words completed for the entire class. For this class, the average winds up being around 5.7 words -- four students who refuse to do any work are dragging down the average.
The song I choose to perform today is Square One TV's "One Billion Is Big" -- and this is the room with all the guitars, and so I play one of the instruments. This is the first time that I've ever tried "One Billion Is Big" on the guitar. That's because it's a hip hop song/rap (performed by the Fat Boys), and hip hop and guitar aren't really compatible. Three years ago at the old charter school, there was a small drum in the classroom -- and while I normally alternated between drum and guitar on most songs, I always played "Billion" on the drums (which served as a beat box).
But today, I play it on the guitar. There are also ukeleles in this class, but I don't bother. Hip hop would sound even goofier on the uke.
2:25 -- Fifth period leaves and seventh period arrives. This is a junior class.
This class also has a vocab worksheet for the book they've studied since at least last week -- Tim O'Brien's collection The Things They Carried.
As for the pizza incentive, with this class meeting for a shorter time, the average is 4.7 words. Of course, this class will have a chance to build on this score tomorrow as they compete against sixth period, which meets for the full 65 minutes tomorrow. One guy refuses to do any work and puts his head down on the desk nearly the entire period.
3:05 -- Seventh period leaves, thus ending one of my shortest ever days of subbing.
But I will pay for this short day tomorrow. After testing, not only will I not have a conference period, but some of the worst-behaved classes meet tomorrow, especially the sixth period junior class. I'm already guessing that seventh period will wind up winning the pizza party. And deep down, I'm rooting for seventh period to win anyway, only because of my wallet. Sixth period has 19 students while the next period has only seven, so of course I'd rather buy pizza for seven than nineteen.
As for the seniors, fifth period has 18 students while tomorrow's classes have 12 and 13. I can likely get away with buying two pies if one of tomorrow's classes wins, but three will be necessary to feed fifth period. So for the sake of my bank account, I hope one of the other classes will beat fifth period.
Today is Elevenday, so I can focus on any resolution. The fifth rule on seeing the 1955 generation as heroes is still burning in my mind after yesterday's class.
Actually, today the students are allowed to use cell phones since they must look up the definitions of the vocab words. There are no print dictionaries in the classroom, and the Chromebooks have been sent to other rooms for testing (since the California Science Test is completely online). Of course, I still watch for students who use their phones for entertainment instead of their assignment (in particular the quartet of seniors with a blank assignment today).
I also use Elevendays for the millennium resolutions on communication. I do speak a little to the other two subs stuck in the lounge during testing, but for the most part we each do our own thing to pass the time.
As I wrote earlier, I suspect that sixth period will be a problem tomorrow, and that this class will probably lose the pizza to seventh period. But I'm considering giving them a separate incentive -- the students who answer enough vocab questions (say, as many as seventh period) will receive something smaller, such as cookies, at 1:59 -- eleven minutes before that period ends on Friday. Of course, 1:59 is the Pi Day moment -- 3/14 at 1:59 -- which is why I'll hand out the cookies at that time, even though March 13th at 1:59 (3.13159???) has nothing to do with pi.
I once told myself that of all the high schools in this district, this is the one where I'd like to become a regular teacher the most, only because 1:59 on Pi Day is at a convenient time. At the other schools in the district, 1:59 is only 1-3 minutes before the end of sixth period, if not during 6th-to-7th period passing itself. (This is because 1:59 is about an hour before the end of the school day.) But with 11 minutes left in sixth period at 1:59, it makes today's school most suitable for a Pi Day party at the Pi Day moment (especially in 2022, when Pi Day is on a Monday).
Hmm, let me rethink that now that Pi Day now seems to be near the testing window. Actually, 1:59 on a testing day isn't terrible (it would be about halfway through fifth period today), but I'm not sure that having a big test goes well with a party day like Pi Day (even if testing is done well before 1:59, as it is today). I discussed last year whether it's a good idea to give a test on Pi Day.
I wish that California would make it so that no state test is given before March 15th, in order to preserve Pi Day as a test-free day. And besides, I hate the idea of having to give the test as early as today, Day 120 in this district. At least the math test isn't until next week, but still, if there are many juniors in the Geometry class, it means that all of the text up to the Inscribed Angle Theorem must be given before the two-thirds mark of the year, and then there's nothing left to do during the last third of the year after the SBAC.
Interestingly enough, in the copy room, I see a Geometry photocopying a worksheet, which happens to be on -- you guessed it -- the Inscribed Angle Theorem. Thus this teacher is giving this last-minute lesson before next week's SBAC. It's from Lesson 10-4 -- which I assume is the Big Ideas text. If I recall correctly, area and volume appear in Chapters 11-12 of that text. It would be difficult to reach even the pi lesson by Pi Day, much less finish the text in time for the SBAC.
In my old district, yesterday I saw some testing in the library. If this is the SBAC, then it works much differently than in my new district -- apparently, some juniors are pulled to take the test each day, rather than have a two-hour testing block. (I already mentioned that at the lone school that doesn't have a block schedule, testing doesn't begin until April.)
This is what I wrote last year about today's lesson:
Now let's get on with today's lesson. Today is supposed to be Lesson 12-9 of the U of Chicago text, on the AA and SAS Similarity Theorems, since today is Day 129. But there's a problem here.
Chapter 12 is the only chapter of the U of Chicago text with a full ten lessons. This causes a wrinkle in our digit-based pacing plan. Chapter 13, for example, has only eight lessons, so we can cover Lessons 13-1 to 13-8 on Days 131 to 138, then use Day 139 to review for the Chapter 13 Test to be given on Day 140. Then Lesson 14-1 can begin the next day, Day 141. For some shorter chapters, such as Chapter 11, this is even easier. The last lesson of the chapter is 11-6 on Day 116, then this leaves four days to review for and give the Chapter 11 Test.
But with ten lessons in Chapter 12, the pattern would have us cover Lesson 12-9 on Day 129, 12-10 on Day 130, and then 13-1 on Day 131 with no time for the Chapter 12 Test. The best thing to do seems to be to squeeze in the Chapter 12 Test on the same day as Lesson 12-10.
I'll preserve the rest of last year's post, which is a discussion of SAS Similarity -- including a system where SAS~ is assumed as a postulate instead of proved as a theorem:
I've referred to the American mathematician George David Birkhoff several times. Birkhoff was the mathematician who first came up with the Ruler and Protractor Postulates, which are often two of the first postulates to appear in a modern Geometry text. Actually, Birkhoff showed that only four postulates are required to derive all of Euclidean geometry:
- Through any two points, there is exactly one line.
- The Ruler Postulate
- The Protractor Postulate
- SAS~
Let's think about what theorems can be proved from Birkhoff's postulates. One should immediately jump out at us -- from SAS Similarity, we should be able to prove SAS Congruence. In fact, the proof is almost trivial -- two figures are congruent iff they are similar with scale factor 1. Thus if we assume any similarity statement as a postulate, we can immediately prove the corresponding congruence statement.
We can also see how to derive some of Euclid's other postulates. That a line can be extended indefinitely goes back to the Ruler Postulate -- every point on a line corresponds to a real number, so just as the real numbers go on indefinitely, so do the points on a line. That we may draw a circle with any center and radius also comes from the Ruler Postulate, though this is less obvious. But think about it -- starting from the center, we can imagine placing a ruler in any direction. If the center is marked with the real number 0, we consider the point marked with the real number r, the radius. The locus of all points found in this manner is the desired circle. That all right angles are equal obviously comes from the Protractor Postulate, as all right angles measure 90 degrees.
This leaves us with just one of Euclid's postulates -- but it's the one that caused the most trouble for millennia, the Fifth Postulate. It's possible to derive the Fifth Postulate from Birkhoff's axioms, but this is very complicated. But as we try to work it out, we'll learn much about Birkoff's geometry.
But first, let's think about what we've proved so far. We have Euclid's first four postulates and we also have SAS Congruence. This means that we can prove Euclid's first 28 propositions -- the ones that don't require a Parallel Postulate.
In particular, we can prove his Proposition 26, which is ASA. Recall that Dr. Franklin Mason also reproduces Euclid's proof, where he uses SAS to prove ASA.
Now the we have ASA Congruence, we can derive another similarity theorem. We can do it the same way that it's done in many pre-Core Geometry texts -- one similarity result is assumed as a postulate, and we use that postulate and the corresponding triangle congruence statements to derive each similarity theorem. In such texts, AA~ is usually the postulate, and so these texts use SAS to prove SAS~ and SSS to prove SSS~. We are given two triangles satisfying, say, the SSS~ condition and we wish to prove them similar -- the trick is to come up with a third triangle that is both similar to one of the given triangle via the AA~ Postulate and congruent to the other via SSS. This proves that the two given triangles must also be similar.
Birkhoff can use the same trick, except SAS~ is the postulate this time. We can now use SAS~ and ASA to prove a similarity theorem -- but which one? It's the one that corresponds to ASA. A moment's reflection should convince you that the similarity corresponding to ASA is in fact AA~! It can't be ASA~ -- what does it mean for one pair of sides to be proportional? Today's Lesson 12-9 of the U of Chicago text points out that both ASA and AAS correspond to AA~. (Recall that the text proves all three of AA~, SAS~, SSS~ as theorems by using a dilation to produce similar triangles.)
So now we have proved AA~. The next result is often called the Third Angle Theorem (which Dr. M abbreviates as TAT) -- if two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles is also congruent. This easily follows from AA~ as follows -- since the two triangles have two congruent pairs of angles, they are already similar by AA~. And if two polygons are similar, then all of the corresponding pairs are congruent.
Here's a note of warning -- notice that we have not yet proved a Triangle Sum Theorem. If two angles of a triangle have measure x and y, then all we know is that there exists some function f such that the third angle has measure f (x, y). We don't know that f (x, y) = a - x - y for some constant a, much less that a = 180 degrees.
So the Third Angle Theorem is a much weaker result than the full Triangle Sum. But we can use TAT to prove an interesting result:
-- Every Lambert quadrilateral is a rectangle.
And now you're thinking -- what the...? You thought that we were done with all this Lambert and Saccheri nonsense, and here I go talking about Lambert quadrilaterals again!
But notice that we've already proved the first four Euclid postulates thus far. In other words, so far we're in neutral geometry. And so I'm going to use terms that a neutral geometer would use, such as Lambert quadrilateral. And besides, a Lambert quadrilateral only means a quadrilateral with three right angles -- and of course, a rectangle is a quadrilateral with four right angles. We could have said:
-- If three angles of a quadrilateral are right angles, then so is the fourth.
Also, my claim that we can use TAT to prove this theorem is strange. TAT is all about the third angle of a triangle, and now we'll use it to prove something about the fourth angle of a quadrilateral?
Well, here's the proof. Let's call our Lambert quadrilateral ABCD, and declare that A, B, C are the right angles. So our goal is to prove that angle D is also a right angle.
Let's extend sides
Now we look at triangles AEH and BEG. In these triangles, angles A and EBG are congruent as they are both right angles (with EBG forming a linear pair with the given right angle B), and the triangles have angle E in common. Thus AEH and BEG are similar, and by TAT the third angles, AHE and BGE, must be congruent.
Notice we have vertical angles -- AHE and DHF are congruent, as are BGE and CGF. Since AHE and BGE are congruent, we conclude that CGF and DHF are congruent.
Now we consider triangles GFC and HDF. We just proved that CGF and DHF are congruent, and the triangles have angle F in common. Thus CGF and DHF are similar, and by TAT the third angles, C and FDH, must be congruent.
But we are given that C is a right angle. Thus FDH is also a right angle, and (with FDH forming a linear pair with CDH, the same as the given angle D) D is a right angle. Therefore the quadrilateral ABCD is a rectangle. QED
So we just proved that every Lambert quadrilateral is a rectangle. Now as it turns out, it's known in neutral geometry that the statement "every Lambert quadrilateral is a rectangle" is one of many statements equivalent to Euclid's Fifth Postulate. The proof is not simple -- here's a link to a neutral geometry course where this is proved:
http://personal.bgsu.edu/~warrenb/Courses/09STheorems.pdf
The two relevant proofs are listed as 9->6, which takes us from "every Lambert quadrilateral is a rectangle" to Triangle Sum, and then 6->2, which takes us from Triangle Sum to Playfair. This is sufficient, since the most commonly mentioned Parallel Postulate is Playfair. So this concludes how we get from Birkhoff's four axioms to a Parallel Postulate.
Both David Joyce and David Kung say that there should be as few postulates as possible, and you can count me as a third David who agrees. If the fewness of the postulates were all that mattered, Birkhoff's axiomatization would be the winner.
But that's not all that matters. Imagine a high school Geometry course trying to get from Birkhoff's axioms to Playfair. Some of the steps are reasonable for high school students -- SAS to ASA is not commonly done, but the indirect proof does appear on Dr. M's website. The phrase "Lambert quadrilateral" should never be spoken in a high school course, but if we write it as "if three angles of a quadrilateral are right angles, then so is the fourth," the proof isn't terrible.
Of course, the proof that takes us from the rectangle theorem to Triangle Sum is very inappropriate for a high school class. And besides -- this proof sequence is the opposite from what we normally want to do in high school Geometry. We want to use Playfair to prove Triangle Sum, not vice versa, and we want to use Triangle Sum in our proof about the angles of a rectangle, not vice versa.
But actually, this isn't even where the problems with Birkhoff's axioms begin. They actually start with the very first proof -- using SAS Similarity to prove SAS Congruence. Yet of all the proofs, this is by far the simplest, almost trivial: two triangles are congruent iff they are similar with scale factor 1.
An argument can be made that similarity is much more important than congruence. Think about it:
- What can we use to set up scale models and maps? (similarity, not congruence)
- What can we use to prove the Pythagorean Theorem? (similarity, not congruence)
- What can we use to derive the trig ratios? (similarity, not congruence)
- What can we use to derive the slope formula? (similarity, not congruence)
...and so on. Congruence matters very little outside of Geometry class, but similarity matters throughout subsequent high school class, college, and careers.
If we were to introduce an SAS~ Postulate, we instantly have a proof of SAS Congruence. Of course, we already proved SAS Congruence, but we can imagine a class where similarity is taught before congruence is. (I actually once saw a text that teaches similarity before congruence.) Then we can get to the Pythagorean Theorem and the slope formula, and we can still teach congruence as a special case of similarity with scale factor 1. And we can just throw in Playfair as an extra postulate, so we don't run into the same problems as Birkhoff's axioms did.
So what is the problem with this approach? Similarity is a more difficult concept than congruence -- and students don't fully understand similarity until they know what congruence is. This is why I tried so hard to avoid teaching similarity until the second semester, in hopes that students can get a good first semester grade without being confused by similarity. So minimizing the number of axioms is secondary to maximizing student understanding. And so in today's lesson, the first of the second semester, we introduce SAS~, from Lesson 12-9 of the U of Chicago text
There's one more thing I want to say about SAS~. Last week, I implied that I would have to assume SAS~ as a postulate first, then use SAS~ to prove the properties of dilations, and finally use dilations to define similar and prove the other similarity theorems.
But this is hardly logical. To see why, let's look at a statement of SAS~:
SAS Similarity Theorem:
If, in two triangles, the ratios of two pairs of corresponding sides are equal and the included angles are congruent, then the triangles are ______________.
That blank there is intentional, to remind you that we haven't defined "similar" yet! And it's circular to use SAS~ to prove that triangles are "similar," use it to prove the properties of dilations, and only then use dilations to define "similar."
All of this stems from that PARCC question that I mentioned last week. The Common Core Standards tell us that we should use dilations to define similarity:
CCSS.MATH.CONTENT.HSG.SRT.A.2
Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
yet that PARCC question does the opposite -- we have to use SAS~ (and therefore know what "similar" means) in order to prove the properties of dilations.
If all I had was the Common Core standard above, I would have introduced a Dilation Postulate that assumes the properties of dilations, then used dilations to derive SAS~ and the other postulates. If all I had was the PARCC question, I would have followed the classical pre-Core definition of similarity, proved SAS~, and then used it to prove the properties of dilations at the very end of the unit.
I decided to go back to Dr. M's website to see how he teaches similarity, and I see that his method is a compromise of the Common Core method and the PARCC method. He teaches similarity in his Chapter 7, and in his Lesson 7.3 he introduces a postulate:
The Polygon Similarity Postulate:
Given a polygon P and a positive quantity k, we may construct a second polygon Q such that P ~ Q, with scale factor k.
In this case, similarity has already been defined classically, while dilation isn't defined yet. Yet this postulate, while it doesn't say specifically that a dilation (with scale factor k, of course) maps P to Q but only that such a Q exists, serves the same purpose as my Dilation Postulate.
We can then prove all three similarity statements -- SAS~, SSS~, and AA~ -- as theorems by using the corresponding congruence theorem plus the Polygon Similarity Postulate. In particular, we use the postulate to produce a triangle similar to the first given triangle, with the correct scale factor that makes it congruent to the second triangle via the corresponding theorem.
In the past, my worksheets were based on this above method of teaching similarity, rather than the U of Chicago method. Despite my describing the old method here on the blog, today's worksheet is based on the U of Chicago text.
No comments:
Post a Comment