Wednesday, March 29, 2017

Lesson 12-9: The AA and SAS Similarity Theorems and Chapter 12 Test (Day 129)

This is what Theoni Pappas writes on page 88 of her Magic of Mathematics:

"Albrecht Durer (1471-1528) was an artist of many talents. He felt the study of mathematics enhanced art, in particular geometry, perspective and ideas of projective geometry."

On this first page of the Durer section, Pappas shows a woodcut from Durer's Treatise on Geometry. I post another Cut the Knot link on some of Durer's geometric ideas:

http://www.cut-the-knot.org/pythagoras/DurerPentagon.shtml

This is a link to the construction of a "regular" pentagon with straightedge and compass. The regular pentagon construction goes back to Euclid, but it's a bit complicated -- which explains why we have high school students construct squares and regular hexagons but not regular pentagons.

As it turns out, Durer doesn't construct a regular pentagon either. His much simpler construction does produce a pentagon that is equilateral, but not quite equiangular. It may be interesting to have high school students try to construct Durer's pentagon.

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 are two problems here.

First, 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 -- at least it would be were it not for the second problem.

And that problem is -- Cesar Chavez Day. I first wrote about Chavez Day two years ago on the blog as a day celebrated in the LAUSD, but not most other school districts. Well, not only does the blog calendar observe all LAUSD holidays, but as I mentioned in February, the blog calendar also includes an extra PD day. The PD day is tomorrow to create a four-day weekend for the students.

This is what I wrote two years ago about Chavez Day:

Some schools -- and most notably universities -- here in California observe Chavez Day. As I mentioned earlier, there's a trend to sever the link between spring break and Easter, in order to avoid having the school holidays be tied to a holiday that can vary by over a month like Easter. In the University of California and California State University systems, spring break is tied not to Easter, but to Chavez Day.

Notice that for schools whose the first day of school is after Labor Day, that end-of-summer holiday is the most important holiday on the calendar. The first day of school is either early or late depending on whether Labor Day is early or late, which means that the last day of every quarter and trimester, including the last day of school, is early or late depending on the previous Labor Day date. In 2014, Labor Day fell on its earliest possible date, September 1st, while [in 2015], Labor Day will fall on its latest possible date, September 7th. This means that dates for the end of each semester will fall later during the 2015-6 school year than in the 2014-5 school year. Schools on an Early Start calendar may choose a different holiday to be the most important holiday on the calendar. For example, some Early Start schools are set up so that the last day of school is before Memorial Day, so that holiday is the most important. Others set up the year so that a fixed number of weeks occur before winter break, thereby making Christmas the most important holiday. The LAUSD Early Start calendar appears to have a fixed number of weeks before winter break -- but winter break itself is defined to begin three weeks after Thanksgiving. I won't know for sure until the holiday reaches its earliest possible date again on November 22nd, 2019, but it appears that Thanksgiving is the most important holiday on the LAUSD academic calendar.

Well, the most important holiday for the UC and Cal State college is Chavez Day. For example, at my alma mater, UCLA, the school year is divided into three quarters -- fall, winter, and spring. Spring break contains the observed Chavez Day and separates winter quarter from spring quarter. Since each quarter has a fixed length (ten weeks plus a finals week), and the length of the breaks is also fixed, all the dates in the UCLA academic calendar can be determined starting from Chavez Day.

There is a slight difference between the Cal State and UC calculations of the Chavez Day date. In the Cal State system, Chavez Day is on the labor leader's actual birthday, March 31st, and so spring break is the week containing the last day of March. Since most Cal States operate on a semester system, the spring break week is actually halfway during the spring semester, so that midterms can be given the week before the holiday. But at UC's -- well, at least at UCLA -- Chavez Day is defined to be the last Friday in March -- so that [in 2015], March 27th was Chavez Day Observed. This means that at the Cal States, this week is spring break, while at the UC's, spring break was last week (just as it was on the blog) and spring quarter has already begun. Notice this means that there are often classes held at UC on the labor leader's actual birthday on the 31st, while there is never school this day at Cal State. I point out that in either case, it's more convenient, especially at UC, to tie spring break to Chavez Day rather than to Easter, in order to avoid quarters of differing lengths in years when the Christian holiday is exceptionally early or late.

LAUSD also observes a Chavez Day holiday. It is observed on the actual date, March 31st, in some years, while in others it's moved to the nearest Monday or Friday. [Of course in 2017, March 31st is a Friday, so the LAUSD observes Chavez Day on the actual date this year -- dw]

[Four] years ago, Easter actually fell on Chavez Day. There was a Google Doodle that day celebrating Cesar Chavez -- which angered some Christians expecting to see a Google Doodle for Easter. I point out that Chavez was a devout Catholic, and so one could argue that Chavez himself would have rather seen Google celebrate Easter than himself.

Returning to 2017, here are a few more things I want to say about Chavez Day. The UCLA calendar is relevant to my classroom this year due to the presence of the Bruin Corps students. We see that the university observes Chavez Day on the last Friday in March, which is also the 31st. Thus this entire week is the UCLA spring break, and so the Bruin Corps students weren't in my classroom. It also means that last week was winter quarter finals -- and with so much study time needed, Bruin Corps wasn't in my room last week either. One of them should be back in the classroom next week for the first week of spring quarter -- but for the other student, winter quarter was his final quarter at UCLA, and he's now graduating. I wish him congratulations.

As I wrote two years ago, it's strange that UCLA would select the last Friday in March to observe as Chavez Day -- this means that the holiday could occur as early as the 25th (as it did last year). The students always return to school the following Monday -- which means that students usually return on or before the 31st. To me, this seems to be dishonoring the labor leader -- ironically, students are more likely to attend school on the 31st now than before Chavez Day was implemented (which was my final year at UCLA), when a different formula was used to determine spring break.

It would have been more logical for UCLA to observe the last Monday in March as Chavez Day rather than the last Friday. Then the entire following week would be spring break -- and this week always contains the 31st. Students start the spring quarter the following Monday -- the first Monday in April, which would always be after the labor leader's actual birthday. The county library also closed Monday for the holiday -- when I was at the book sale on Saturday, I saw signs announcing the holiday closure.

I was curious to see whether any other California K-12 districts close for Chavez Day. Here in Southern California, Lynwood Unified also observes the holiday on Friday. I also found a few Northern California districts that close Friday as well.

So this means that with the PD day tomorrow, the Big March actually ends today -- by definition, the Big March extends from President's Day to the next school holiday, which for me is Chavez Day, not Easter or spring break. It does mean that there is only a short time left until our spring break. Notice that the Big March is thus almost never the longest stretch without a break. Even when Easter is late, the existence of Chavez Day means that the stretch from President's Day to spring break is broken up by Chavez Day, while the stretch from Easter to Memorial Day never is. Still, I consider the Big March to be the toughest time of the year. At least in April and May we can see summer just beyond the horizon, while in March summer is still far away.

But now that there's a four-day weekend, you can see why we can't have the Chapter 12 Test on Day 130 as the digit pattern suggests. Day 130 falls on a Monday after a four-day weekend, and that's the last day we'd ever want to give a test.

So my idea is to give the test today, just before the long weekend. We squeeze in Lesson 12-9, which is on AA~ and SAS~, before the test. Fortunately, I'm using last year's worksheets, and on the worksheets I created last year, I actually combined AA~ with SSS~ on the 12-8 worksheet, and so today's worksheet has only SAS~. This means that the students had an extra day to study AA~ before the test. (The reason I did this last year has nothing to do with Chavez Day --  it wasn't even March when I posted Chapter 12 last year!)

Here I will cut-and-paste what I wrote last year on SAS~ followed by the test. Notice that I'm using last year's Chapter 12 test -- but which I actually posted last year was a Chapter 11-12 Test! (It would be more logical to wait to have a Chapter 12-13 Test, as that solves the Day 129-130 problem.) So I only post the second part of the test, where the questions are numbered 10-20. Of these eleven questions, only one would have to be thrown out, #15, as it strongly references Lesson 12-10 -- this leaves ten questions, which is convenient for grading out of 100. Of the remaining questions, three questions use AA~ (including two proofs), which the students learned yesterday. One question mentions SAS~, which is today's lesson -- but that question only requires students to identify two triangles that are similar by SAS~, not give a proof.

As for the SAS~ material from last year, it's rather long -- but hey, it's a four-day weekend and besides, it's all cut-and-paste anyway. Scroll down past the SAS~ material for test answers!

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~

This is astounding -- only four postulates are required? Even Euclid himself had five postulates, and now Birkhoff claimed that he can derive Euclid's geometry in only four? And as Birkhoff's first postulate is the same as Euclid's, the American's other three postulates should somehow be equivalent to the Greek sage's other four. But how can this be?

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 (xy). We don't know that f (xy) = 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 ABC are the right angles. So our goal is to prove that angle D is also a right angle.

Let's extend sides AB and CD a little. That is, we choose point E on ray AB beyond point B, and point F on ray CD beyond point D. Notice that rays AB and CD point in opposite directions. So when we connect E and F, we have that EF intersects both BC and AD, and we'll call the points of intersection G and H respectively.

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 DD 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 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.

This solves most of our problems. Today we use SAS and Polygon Similarity to prove SAS~, with the idea that SAS~ can be used to prove the properties of dilations as in the PARCC question. Yet it still has the same idea expressed in the Common Core Standards -- we don't just assume any of the three major similarity statements, but prove all of them by producing similar triangles.

Test Answers:

10. b.

11. Yes, by AA Similarity. (The angles of a triangle add up to 180 degrees.)

12. Yes, by SAS Similarity. (The two sides of length 4 don't correspond to each other.)

13. Hint: Use Corresponding Angles Consequence and AA Similarity.

14. Hint: Use Reflexive Angles Property and AA Similarity.

15. [deleted in 2017]

16. 9 in. (No, not 4 in. 6 in. is the shorter dimension, not the longer.)

17. 2.6 m, to the nearest tenth. (No, not 1.5 m. 2 m is the height, not the length.)

18. 10 m. (No, not 40 m. 20 m is the height, not the length.)

19. $3.60. (No, not $2.50. $3 is for five pounds, not six.)

20. 32 in. (No, not 24.5 in. 28 in. is the width, not the diagonal. I had to change this question because HD TV's didn't exist when the U of Chicago text was written. My own TV is a 32 in. model!)

Chavez Day is on Friday, so the next post will be Monday, April 3rd. On that day we'll finish Lesson 12-10 and then begin Chapter 13, Logic and Indirect Reasoning.



Tuesday, March 28, 2017

Lesson 12-8: The SSS Similarity Theorem (Day 128)

This is what Theoni Pappas writes on page 87 of her Magic of Mathematics:

"Having been influenced by perspective in the art of the Renaissance and desiring to help artists, 17th century architect/ engineer/ mathematician Girard Desargues initiated the study of projective geometry. He wrote a number of theorems (Desargues' Theorem) that are still essential in the study of projective geometry."

Of course, I had to check out Desargues' Theorem. Here's a link to Cut the Knot:

http://www.cut-the-knot.org/Curriculum/Geometry/Desargues.shtml

Let A1B1C1 and A2B2C2 be two triangles. Consider two conditions:
  1. Lines A1A2, B2B2, C1C2 joining the corresponding vertices are concurrent.
  2. Points ab, bc, ca of intersection of the (extended) sides A1B1 and A2B2, B1C1 and B2C2, C1A1 and C2A2, respectively, are collinear.
Desargues' Theorem claims that 1. implies 2. It's dual asserts that 1. follows from 2. In particular, the dual to Desargues' theorem coincides with its converse.
Curiously, Desargues' theorem admits an intuitive proof if considered as a statement in the 3-dimensional space, but is not as easy in the 2-dimensional case, where it is often taken as an axiom [that is, a postulate -- dw].

In researching Desargues' Theorem, I also stumbled upon the concept of homography. All the transformations mentioned in this section of Pappas as well as Lesson 1-5 of the U of Chicago. Of course, we're supposed to imagine a homography as mapping (or projecting) a figure from one plane to another, but we can more easily consider a homography mapping figures in the same plane. In this case, homography has a simple definition -- it's a 1-1 correspondence that preserves B-C, betweenness and collinearity.

Clearly all isometries are homographies. All similarity transformations are homographies. Last year, I wrote about the concept of an affine transformation -- Lesson 6-1 of the U of Chicago text gives a simple affine transformation, S(x, y) = (2x, x + y). Well, all affine transformations are homographies.

Isometries and similarity transformations map squares to squares. Affine transformations map squares to parallelograms. As Pappas writes, homographies map squares to any quadrilateral -- indeed, given any quadrilateral, there exists a homography mapping a square to it.

Homographies are often understood using matrices, just as isometries, similarity, and affine transformations often are, but this is tricky. I mentioned how McDougal-Littell Integrated Math II text (which I didn't buy -- I have only the Math I text) uses transformation matrices. Mostly likely, the text uses matrix multiplication for reflections and rotations, but matrix addition for translations. This is unsatisfying as it seems that we should be able to use multiplication for all isometries. But the problem is that multiplying any 2x2 matrix by the vector (0, 0) gives (0, 0), yet translations clearly don't map the origin to itself.

The trick is to pretend that the xy-coordinate plane is actually the plane z = 1 in 3D. Then a particular matrix multiplication can actually map (0, 0, 1) to (h, k, 1) and represent a translation. It turns out that all homographies in the plane can be represented by 3x3 matrices -- the only restriction on the matrix is that its determinant be anything but zero. For the homography to be an isometry, though, the condition on the matrix is complicated -- it's necessary that the determinant of the matrix be 1 or -1, but that's not sufficient. For example, trigonometry is required to find the matrix of a rotation whose magnitude isn't a multiple of 90 degrees.

OK, that's enough about homographies. I just write about them in order to point out that there's a whole world of transformations beyond isometries and similarity transformations -- affine transformations and homographies are the next two steps up the ladder. So in particular, isometries and similarity transformations aren't things invented by the Common Core to torture traditionalists.

This is what I wrote last year about today's lesson:

Let's get to our Geometry lesson. We are now working on the AA~ and SSS~ theorems, which complete our study of similarity. There are several ways we can prove these at this point. We can use the original dilation proofs given in the U of Chicago text (Lessons 12-8 and 12-9), or we can use the one similarity theorem we already have (SAS~) plus the corresponding congruence theorems (ASA and SSS, respectively). [2017 update: last year I reversed Lessons 12-8 and 12-9 but this year I'm preserving the original order, so no, we can't use SAS~ to prove SSS~.]



Monday, March 27, 2017

Lesson 12-7: Can There Be Giants? (Day 127)

Last weekend was the biannual book sale at our local library. Ordinarily, the spring book sale is held the first Saturday in April -- the only reason it would be the last Saturday in March instead would be to avoid Easter weekend. Yet the book sale was last Saturday, even though Easter is still several weeks away. Who knows what the librarians were thinking with the strange date here?

As usual, I purchased several math texts for one dollar each. For the first time, I actually found two Integrated Math I texts. The first was published by McDougal Littell in 1998 -- that's right, well before the Common Core. We know that some traditionalists oppose both Common Core and Integrated Math, but not only is it possible to have Common Core without Integrated Math, but vice versa as well. I see that the text comes from a school in Olympia (the capital of Washington state) -- and so I wonder what the book is doing here in California! But anyway, other states apparently had Integrated Math years before there was ever a Common Core.

The McDougal Littell text is divided into ten units:

McDougal Littell Integrated Mathematics 1
1. Exploring and Communicating Mathematics
2. Using Measures and Equations
3. Representing Data
4. Coordinates and Functions
5. Equations for Problem Solving
6. Ratios, Probability, and Similarity
7. Direct Variation
8. Linear Equations as Models
9. Reasoning and Measurement
10. Quadratic Equations as Models

The other text is published by Pearson -- so it's obviously a Common Core text. Two years ago, I wrote about the Pearson texts I found near the back of a classroom where I was subbing, but only the Integrated Math II and III texts were in the classroom that day. Well, since that day I remember actually seeing the Math I text in another classroom where I was subbing -- and yet I didn't mention Pearson Integrated Math I anywhere on the blog. I think what happened was that I saw the texts unused near the back of the room, but the students were actually working in another text. So I blogged only about the books the kids were actually using, not the Pearson texts.

The first difference between McDougal Littell and Pearson is obvious -- the latter is divided into two volumes, yet the former is only a single volume. I actually found only Volume 2 at the book sale.

Here is the table of contents for Pearson Math I Volume 2, which covers Chapters 8 to 14:

Pearson Mathematics I
Volume 2
8. Transformations
9. Connecting Algebra and Geometry
10. Reasoning and Proof
11. Proving Theorems About Lines and Angles
12. Congruent Triangles
13. Proving Theorems About Triangles
14. Proving Theorems About Quadrilaterals

Let's compare this to the contents of Pearson Math II Volume 1, which I wrote about two years ago:

Pearson Mathematics II
Volume 1
Chapter 1: Reasoning and Proof
Chapter 2: Proving Theorems About Lines and Angles
Chapter 3: Congruent Triangles
Chapter 4: Proving Theorems About Triangles
Chapter 5: Proving Theorems About Quadrilaterals
Chapter 6: Similarity
Chapter 7: Right Triangles and Trigonometry
Chapter 8: Circles

That's right -- the last five chapters of Math I and the first five of Math II are identical! I remember noticing that oddity two years ago -- and yet I never said a single word about it on the blog! Actually, the day I wrote about Math II and III, I noticed that these texts had one chapter in common, but of course I wrote nothing about Math I.

Let me repeat some of what I wrote two years ago about Pearson Math II Volume 1 text, since it applies to Pearson Math I Volume 2 as well. Note that we must add nine to every chapter number in Math II to obtain the corresponding Math I chapter.

Pearson's Chapter 2 is on Lines and Angles. Section 2-2 introduces a postulate on parallels -- and it happens to be the Same-Side Interior Angles Postulate. This is highly unusual, as most texts choose either Corresponding Angles (U of Chicago) or Alternate Interior Angles as a postulate. Section 2-3 contains the Parallel Tests -- and all of the are listed as theorems. The first test is the Converse of the Corresponding Angles Theorem, and its proof is said to be given in Section 13-5 -- but that chapter is on Probability. I once saw another text that does give the proof -- it's an indirect proof, and so Pearson probably intended to wait until Chapter 4 to give the proof, but it mistakenly omits it.

Now that we see the Math I text, this all makes sense. Chapter 4 of Math II is the same as Chapter 13 of Volume I, and this is indeed where the indirect proof of the Corresponding Angles Consequence is actually given. When renumbering the chapters, Pearson mistakenly kept "Lesson 13-5" instead of changing it to "Lesson 4-5" for Math II.

Looking at Lesson 13-5 in my Math I text, I see that in the exercise where the Corresponding Angles Consequence is to be proved, a hint is given: use the Triangle Angle-Sum Theorem. We've had several discussions regarding this proof and how other authors prefer geometry to be taught, such as David Joyce and Hung-Hsi Wu.

We know that Joyce wants there to be only one postulate for parallels. Pearson has two -- the Same-Side Interior Test and the Parallel Postulate of Playfair. Pearson uses Playfair as the first step of Triangle Angle-Sum, which is then used to prove Corresponding Angles Consequence. But as we've seen in the U of Chicago, we can use the Angle Measure (or Protractor) postulate in the first step of Triangle Angle-Sum instead of Playfair. Then Playfair becomes superfluous, and so there is only one parallel postulate, as Joyce suggests.

On the other hand, we ultimately use that one postulate in the proof of Corresponding Angles Consequence, but Wu shows us that the parallel consequences (as opposed to the tests) are provable in neutral geometry, without any parallel postulate at all. The proof requires replacing Triangle Sum with the weaker TEAI (Triangle Exterior Angle Inequality), which is provable in neutral geometry. I have tried something like this in the past on the blog, but it ended up being awkward. Trying to avoid a parallel postulate just because a result holds in neutral geometry turns out to be an unnatural restriction that only confuses high school students.

Let's get back to two years ago:

Pearson's Chapter 3 is on Congruent Triangles. We notice that in Sections 3-2 and 3-3, SSS, SAS, and ASA are given as postulates. But then in true Common Core fashion, congruence transformations (isometries) are used to verify these postulates in Section 3-8. I notice that as important as the isometries are to Common Core, translations, reflections, and rotations aren't defined anywhere in this section. I suspect that since these isometries form the foundation of Common Core Geometry, Pearson actually defines them in its Math I text. There's no way for me to know for sure since I don't have access to the Math I text.

Now I have the Math I text, so now we know -- transformations appear in Math I Chapter 8, while Math II starts with the same chapter as Math I Chapter 10.

Pearson's Chapter 5 is on Proving Theorems About Quadrilaterals. Most of this material appears in the same-numbered chapter of the U of Chicago, except some of the theorems on parallelograms don't appear until Chapter 7. Of course I had to check to see that Section 5-7 of Pearson gives the exclusive definition of trapezoid. As I said earlier, the inclusive definition was spotted in a PARCC question, but I have yet to see it. Then the rest of the chapter gets into coordinate geometry -- which I suppose makes sense as many coordinate proofs involve quadrilaterals. The harder concurrency theorems from Chapter 4 are given coordinate proofs here as well. Many of these coordinate proofs require the Distance or Midpoint Formulas, which are never introduced. Once again, I suspect that these formulas are taught in the Math I text and students are expected to remember them.

Apparently, looking at the Math I index, Distance and Midpoint appear in Chapter 7 of Math I -- the last chapter of Volume 1. I still don't have a copy of Volume 1, but considering what I wrote two years ago:

Notice that the four constructions in the Pearson text are exactly the same as the four in Prentice-Hall, even numbered in the same order! I checked the list of authors for the Pearson text and saw that there are several, since both algebra and geometry writers are needed. But I notice that one of the geometry authors is Laurie Bass, who is also, according to Joyce, the author for Prentice-Hall -- small wonder, then, that the same four constructions are here.

...I wouldn't be surprised that Math I Chapter 7 is similar to Bass Chapter 1 -- the Prentice-Hall text that Joyce criticizes. Part of the criticism is that the Distance Formula appears well before the Pythagorean Theorem on which it depends.

Okay, that's enough from two years ago. What I want to do now is compare the Pearson text to the McDougal-Littell text that I purchased this weekend, so we can compare Math I to Math I.

First of all, we see that, for an "Integrated" Math text, algebra and geometry are separated! In fact, we could teach a complete Geometry course by starting with the last chapter of Math I Volume 1 and going up to the first chapter of Math II Volume 2, with the intervening volumes (Math I Volume 2 and Math II Volume 1) both being pure geometry texts.

On the other hand, McDougall Littell doesn't have separate algebra and geometry chapters. Many of the units include both topics. For example, Unit 6 is "Ratios, Probability, and Similarity." This chapter includes both the algebra of solving proportions and their obvious application to similarity, which is geometry. The following Unit 7, "Direct Variation," also includes connections, as its first lesson is "Direct Variation, Slope, and Tangent." This lesson connects similarity to slope via the trig ratio of tangent. Common Core tells us to tie similarity to slope, and here we see the connection made in this pre-Core text.

As it turns out, the transformations of translation, rotation, and so on also appear in this text -- again, these transformations existed long before the Common Core (as we've seen with the U of Chicago text, which is also pre-Core). Translations and rotations are in Unit 4, "Coordinates and Functions," while dilations obviously appear with similarity in Unit 6. Reflections don't appear until the final unit, "Quadratic Equations as Models" -- which in a way makes sense when we consider reflections along with the line of symmetry of a parabola. By contrast, the only truly integrated chapter of Pearson is Chapter 9, "Connecting Algebra and Geometry" (which covers perimeter, area, and slope).

McDougall Littell provides a complete listing of the topics that appear in each of the Integrated Math I, II, and III texts:

Course 1
Algebra: linear equations, linear inequalities, multiplying binomials, factoring expressions
Geometry: angles, polygons, circles, perimeter, circumference, area, surface area, volume, trig ratios
Stats/Prob: analyzing and displaying data, experimental, theoretical, and geometric probability
Logical Reasoning: conjectures, counterexamples, if-then statements
Discrete Math: discrete quantities, matrices to display data, lattices

Course 2
Algebra: quadratic equations, linear systems, rational equations, complex numbers
Geometry: similar and congruent figures, coordinate and transformational geometry, right triangles
Stats/Prob: sampling methods, simulation, binomial distributions
Logical Reasoning: inductive, deductive, valid, and invalid reasoning, postulates and proof
Discrete Math: matrix operations, transformation matrices, counting techniques

Course 3
Algebra: polynomial functions, exponential functions, logarithmic functions, parametric equations
Geometry: Inscribed figures, transforming graphs, vectors, triangle trig, circular trig
Stats/Prob: variability, standard deviation, z-scores
Logical Reasoning: identities, contrapositive and inverse, comparing proof methods
Discrete Math: sequences and series, recursion, limits

As we've seen above, some of the topics listed under "Course 2" are introduced in Course 1.

I've written before about the relationship between Common Core 8 and Integrated Math I. In many ways, Integrated Math I can double as a Math 8 course. But there are a few differences, depending on whether we use Pearson or McDougal-Littell as our Math I course.

I suspect that Pearson Math I Volume 1 (and I can only guess its contents since I don't have that text) matches the algebra content of Math 8. But the geometry content of Math I Volume 2 is a bit too advanced for eighth graders, especially with proof being introduced in Chapter 10.

With McDougal-Littell, it's the other way around. Most of its geometry content appears in Common Core Math 8 (or even Math 7), except for trig ratios. But the algebra content differs greatly from the Math 8 course. Multiplying binomials appears in McDougal-Littell Course 1 but not Math 8. Notice that some factoring actually appears in Math 8 (and even Math 7), but it's just GCF factoring, not full trinomial factoring.

I sort of like the way topics are introduced in the McDougal Littell text. Algebra and geometry really are integrated throughout the text. I would recommend the text for both eighth grade and Integrated Math I classes. The last chapter, on quadratic equations, is a little advanced for Common Core 8, but the appearance of the Quadratic Formula in the final lesson (Lesson 10-8) reminds me of the old Dolciani text that I used as a young Algebra I student. That text also teaches the Quadratic Formula in the very last lesson.

If I were at a school that uses the Pearson text, I would note that with the last five chapters of Math I repeating as the first chapter of Math II, I'd only cover each chapter once. That is, I'd teach Chapters 1 to 12 of Math I the first year, and then Chapter 4 to 15 of Math II the second year. This means that exactly 12 chapters is given each year -- six chapters per semester, one chapter per three weeks. I've stated before that I like the idea of covering each chapter in three weeks.

This means that the switch from Volume I to Volume II each year doesn't happen exactly at the semester mark. But it does mean that the transition from algebra to geometry in Math I, as well as the transition from geometry back to algebra in Math II, does occur right at the semester, which is convenient for semester finals.

The Pearson text also provides a loophole for traditionalist teachers and principals who oppose both Integrated Math and the delaying of Algebra I until ninth grade, rather than their preferred eighth grade Algebra I and ninth grade Geometry. Traditionalists who are forced by their districts to use Pearson can start with Math I Volume 1 in eighth grade, which is all Algebra I. Then the freshmen can use Math I Volume 2 and Math II Volume 1 to make up their Geometry course. Sophomores take Math II Volume 2 and Math III Volume 1, which comprise a credible Algebra II class. Juniors wrap up Math III Volume 2 and then any missing topics they need to prepare for senior year Calculus.

The trickiest part of this idea is the eighth grade Algebra I course. Eighth graders go to middle school while ninth graders attend high school, so high school teachers would need the middle schools to cooperate, plus the Math I Volume 1 texts must be delivered to middle school campuses. The volume plan is more convenient with semesters, so it may be tricky at trimester middle schools. Finally, there's one problem that any eighth grade Algebra I class has in the Common Core era -- the students still have to take the PARCC or SBAC in May of the eighth grade year, and those tests include geometry and other topics not included in Algebra I. My preference is to include the other PARCC or SBAC topics in the first semester and then Math I Volume I algebra the second semester. Some topics like quadratic equations are missing from Algebra I, but of they will appear in Algebra II. (Here's an idea that fits into trimesters -- first trimester: Geometry and Stats/Prob, second trimester: Pearson Math I Chapters 1 to 4, third trimester before PARCC/SBAC: Chapters 5 to 7, after the test: end with the Quadratic Formula.)

The Pearson text is typical of many texts in the Common Core era. Both the Illinois State and Pearson texts are divided into two softcover volumes rather than one hardback volume. And it's telling that both Illinois State and Pearson appear to have connections to England -- I wonder why texts based on our national standards need to be developed in another nation.

There are a few other books that I purchased at the book sale. One is Guiding Children's Learning of Mathematics by Leonard Kennedy and Steve Tipps (2000). The book seems to focus mainly on elementary school math, but there are a few middle school topics mentioned there (such as decimals, Stats/Prob) and besides, the book only costs a quarter. Another book is Lee Canter's Succeeding With Difficult Students (1993). In my quest to become the ideal classroom manager, the instructional aide at my school recommends that I read Lee Canter. This book appears to be based on an "inservice video package" -- that is, Canter and associates would visit schools on their PD days and show the teachers some videos. Since I don't have access to these videos (but who knows, maybe they're on YouTube), this book won't be as effective -- again, but this one also sets me back only a quarter.

The last book I'll mention will also be my newest side-along reading book. And now you're asking -- what the heck is going on? I stopped writing about my middle school classes and even took two weeks off from posting due to lack of time during the Big March, and now suddenly I have time to go back to posting about side-along reading books on the blog! (I know that this post is long, but then again, today is coding Monday.)

Well, my side-along reading book is The Magic of Mathematics by Theoni Pappas. I already planned on making up some "Pappas questions" that I would cover in my classes and on the blog. That fell apart when the administration started cracking down on questions from sources other than Illinois State and forced me to change my Warm-Up from Pappas to the Illinois State Daily Assessment. My so-called "Pappas questions" didn't come from Pappas herself, but were just modified versions of her questions with the same idea that the answer was the date.

So now I'm replacing Pappas questions with an actual Pappas book, which she wrote in 1994. Here is the table of contents:

0. Preface
1. Mathematics in Everday Things
2. Magical Mathematical Worlds
3. Mathematics & Art
4. The Magic of Numbers
5. Mathematical Magic in Nature
6. Mathematical Magic from the Past
7. Mathematics Plays its Music
8. The Revolution of Computers
9. Mathematics & The Mysteries of Life
10. Mathematics and Architecture
11. The Spell of Logic, Recreation & Games

I'll still going to follow the Pappas tradition and tie in the date somehow. And here's how I'll do it -- today is March 27th, the 86th day of the year. So I'll read page 86 today. Surely I should have time to read one page per day. This will take us to page 311 (the solution page) sometime in November. Of course, since I just bought the book I could just label this Day 1 and start from there. But it's not as if I won't be skipping pages anyway, since I don't post here everyday.

Page 86 is in the middle of Chapter 3, "Mathematics & Art." The current section is called "Projective Geometry & Art," which begins on Page 85. Pappas writes:

"Projective geometry is a field of mathematics that deals with properties and spatial relations of figures as they are projected -- and therefore with problems of perspective [emphasis hers]. Just as topology studies the properties of objects that remain unchanged after they have undergone a transformation, projective geometry studies properties of plane figures that do not change when they undergo projections."

She adds that a circle, when viewed in perspective, becomes an ellipse, and a square is projected into a different quadrilateral -- but she doesn't specific which one. Perspective is mentioned in Lesson 1-5 of the U of Chicago text, so maybe we can read it to figure out what happens to our square.

In the U of Chicago text, we read:

"In perspective drawings, horizontal (or vertical) lines remain horizontal (or vertical) and parallel. But oblique parallel lines will intersect if extended. The box below has two vanishing points, P and Q. Vertical lines on the box remain vertical and parallel, but the box has been tilted to have no horizontal edges."

So assuming that this box is a cube, each face is a square. The vertical sides of the square are still vertical, so we have at least one pair of parallel sides -- a trapezoid. But this trapezoid is definitely not a parallelogram, since the other sides ultimately intersect at the vanishing point. So the answer is that a square is projected into a trapezoid.

The U of Chicago also gives a nonperspective drawing of a cube. In this drawing, the sides of the square do not meet at a vanishing point, so the image of the square is a parallelogram. But this is definitely not a projection.

Now Lesson 1-5 isn't today's lesson -- today is actually Lesson 12-7, so let's hurry up and begin.

Lesson 12-7 of the U of Chicago text is called "Can There Be Giants?" Last year we skipped over this lesson, so I don't have any old worksheets for it. This is one of those "fun lessons" that we can cover if there's time, but in the past we bypassed it to get to 12-8 and the all-important SSS Similarity.

As I wrote above, Lesson 12-7 naturally lends itself to an activity. The whole idea behind it is that while dilations preserve shape, they don't preserve stability. This is because of the Fundamental Theorem of Similarity -- a dilation of scale factor k changes lengths by a factor of k, areas by a factor of k^2, and volumes by a factor of k^3. Weight varies as the volume, or k^3, while strength varies only as the area (as in surface or cross-sectional area), or k^2. Therefore, the answer to the question in the title of the lesson is no, there can't be giants because their k^2 strength, couldn't be strong enough to carry their own k^3 weight.

Here are the worksheets that I've created for this lesson: