Monday, January 5, 2015

Section 12-2: Size Changes Without Coordinates (Day 85)

Welcome to the second semester of Common Core Geometry. There are many important topics for students to learn from Chapters 8 through 15 of the U of Chicago textbook.

But there are a few things to discuss before we begin. First of all, let's take a look again at the Common Core Standards to see what needs to be covered during the second semester. The standards for geometry have been divided into six major topics:

-- Congruence
-- Similarity, Right Triangles, and Trigonometry
-- Circles
-- Expressing Geometric Properties with Equations
-- Geometric Measurement & Dimension
-- Modeling with Geometry

And notice that so far, we've really only covered the first topic, congruence! It appears that we are way behind if we need to cover five topics in one semester. But let's think about it for a moment -- go back to what David Joyce wrote about how a high school geometry course should be organized:

"Done right, the material in chapters 8 [of the Prentice-Hall text, congruence] and 7 [parallel lines] and the theorems in the earlier chapters that depend on it, should form the bulk of the course. Much more emphasis should be placed here."

And this is what we've done. Congruence and parallel lines have formed the bulk of our course -- which is why we spent an entire semester on these topics. Nearly half of the U of Chicago text -- seven out of 15 chapters -- build us up to congruence. This is similar to how traditionalists believe that a third grade course should be organized -- the bulk of the course should be on learning the times tables, with all the remaining topics to be covered only after the tables have been memorized.

The main problem I have is that the PARCC exam will be administered, in the states that will do so, well before the last day of school -- possibly even as early as the start of the fourth quarter. This means that all of the second semester material will have to be taught before the test if students are to do well on that portion of the exam.

The way that I have divided the course into first and second semesters reflects my own philosophy on how a geometry course should be organized. I don't want the students, coming off of a tough Algebra I course, to have to do heavy-duty algebra in the Geometry class. The first seven chapters of the U of Chicago text don't contain much algebra, and so I feel that a student who barely struggled to pass Algebra I with a C- or D- (whichever's the lowest Algebra I grade that the school allows a student to take Geometry) can get a higher grade, maybe even a full letter grade higher, in the first semester of the Geometry course.

On the other hand, suppose that a teacher, seeing the Common Core Standards, feels that at least two of the six topics must be covered in the first semester if there's to be any hope of finishing all six before the PARCC exam. Now imagine our poor struggling student. That student is felling better about him or herself after covering the Congruence topic during the first quarter, and is now getting a grade one letter higher than he or she had received in Algebra I.

But grades after the first quarter, or even the third quaver, don't count -- only the grades at the end of the semester do. And separating our student, with his or her higher grade, from the end of the semester when that grade will count, is the second topic that our teacher is squeezing in. Similarity, Right Triangles, and Trigonometry are all algebra-heavy. Suddenly, our student's grade is going to drop, since someone who struggled in Algebra I will have trouble with these topics. The student feels tantalized by seeing the higher quarter or quaver grades drop at the semester.

And so this is why I divide my course so that only seven chapters fill the first semester. Furthermore, notice that our student may fail the second semester since it covers so many difficult topics. Well, I'd much rather have a student fail the second semester and have to repeat the second semester in the summer, than have him or her fail both semesters and have to repeat them both in the fall.

But the trade-off is that the second semester must be rushed in order to cover all eight chapters by the last day of school -- even more so by the day of the PARCC exam. I'd stated before that a chapter fits comfortably into a three-week period, so we'll have to rush and try to complete each chapter in only two weeks. This would give us 16 weeks for the eight chapters. Even this might not be fast enough to beat the PARCC exam.

I've already fantasized about how the Common Core Standards would be like if I were in charge. So, if I were in charge, I'd try to give the PARCC exam closer to the last day of school -- either that, or have the standards contain some topics that won't be tested on the PARCC and whose sole purpose is for them to be covered between the PARCC exam and the last day of school.

Of course, SBAC doesn't have this problem, since there isn't an end-of-course exam, but instead one exam at the end of junior year, by which the students are hopefully in Algebra II, not Geometry. (Of course, juniors still in Geometry may still suffer.) And of course, all bets are off if the students are in an Integrated Math class.

Now, as I mentioned before winter break, I'm covering the eight chapters in a different order from how they are presented in the U of Chicago. Once again, that order will be:

Chapter 12: Similarity
Chapter 11: Coordinate Geometry
Chapter 13: Logic and Indirect Reasoning
Chapter 14: Trigonometry and Vectors
Chapter 8: Measurement Formulas
Chapter 15: Further Work with Circles
Chapter 9: Three-Dimensional Figures
Chapter 10: Surface Areas and Volumes

So we begin with Similarity, Chapter 12 of the U of Chicago text. But this is the longest chapter in the book, with ten sections, and we have only two weeks in which to cover them. I want to be more careful with our pacing, so let's map out the entire chapter.

Today: Section 12-2, Size Changes Without Coordinates
Tomorrow: Section 12-6 The Fundamental Theorem of Similarity
Wednesday: Section 12-3, Properties of Size Changes
Thursday: Section 12-10, The Side-Splitting Theorem
Friday: Activity
January 12th: Section 12-5, Similar Figures
January 13th: Section 12-9, The AA and SAS Similarity Theorems
January 14th: Section 12-8, The SSS Similarity Theorem
January 15th: Review
January 16th: Chapter 12 Test

This pacing guide reflects the fact that we skipped Chapters 8 through 11. In particular, Lesson 12-1 is on Size Changes on a Coordinate Plane -- but Coordinate Geometry is in Chapter 11, and so we must skip 12-1.

Section 12-7, "Can There Be Giants," discusses how the size changes affect area and volume. The problem with this section is that area and volume occur in the chapters that we've skipped, Chapters 8 through 10! So it will be tough to discuss the areas and volumes of similar figures. I've decided to drop this section altogether.

But this emphasizes why I hate jumping around the text. By skipping Chapters 8 through 11, we've ruined Sections 12-1 and 12-7, which depend on the chapters we missed.

Why did I choose to start with Chapter 12? Well, notice that this matches the Common Core order more closely, since the first topic after Congruence is Similarity, and Geometric Measurement & Dimension (i.e., volume) is given near the end of the year. My hope is that the volume formulas -- which are notoriously difficult to remember -- will be the last topic before the PARCC exam, thereby reducing the time for students to forget them.

Furthermore, by covering Chapter 12 before 11, we will be able to use Similarity to prove the properties of Coordinate Geometry. The U of Chicago does this the other way around -- it uses Coordinate Geometry to prove the properties of Similarity. But the former way is preferred not only by Common Core, but also by Joyce:

In a return to coordinate geometry it is implicitly assumed that a linear equation is the equation of a straight line. A proof would depend on the theory of similar triangles in chapter 10.

This has occurred many times during the first semester -- a traditional geometry course may use Theorem p to prove Theorem q, but our course uses Theorem q to prove Theorem p. This shows us that, while some theorems very obviously follow from others (for example, the Distance Formula obviously derives from the Pythagorean Theorem), for most pairs of theorems it's not so clear, and indeed which one follows from the others depends on which statements we choose as our postulates.

For example, consider AA Similarity, which many pre-Common Core texts take as a postulate. Now most texts won't do the following, but we can consider a hypothetical text in which AA Similarity is used to prove ASA Congruence as a theorem. In fact, the proof is almost trivial -- since the two triangles have two pairs of congruent triangles, they are similar by AA. And since they have one pair of congruent sides, the scale factor must be 1! QED

Most texts, despite having AA Similarity as a postulate, don't do this. But in Birkhoff's axioms -- named after the same G.D. Birkhoff who first proposed the Ruler Postulate -- SAS Similarity is a postulate, and SAS Congruence is proved trivially using SAS Similarity. (Then SSS and ASA can be proved from SAS just as Euclid did over 2000 years ago.)

In most texts, both traditionalist and Common Core, SSS and SAS Similarity are proved using the respective Congruence Theorems -- not vice versa as in Birkhoff. And so in this chapter, we must reverse the order in which statements are proved.

The proofs of Chapter 12 will be based, not on Coordinate Geometry, but on proofs given by Dr. Hung-Hsi Wu. In fact, this explains why, even within the chapter, we will be jumping around from Section 12-2 to 12-6, then back to 12-3, and so on -- the order is completely dependent on Wu. In fact, even Friday's activity comes directly from Wu, and fits right between Thursday's section and next Monday's.

Notice that this week focuses on dilations, while next week is on similarity. Tomorrow's lesson, the Fundamental Theorem of Similarity, contains the longest and most difficult proof of the chapter. I wouldn't mind making this a two-day lesson, today and tomorrow, but I don't want the very first lesson for the students after a two-week vacation to be such a hard proof.  Also, covering all of the Wu theorems means that we don't have time for Section 12-4, on Proportions, even though the students probably need review from Algebra I on how to set up and solve proportions. Instead, I will incorporate proportions throughout the chapter.

But enough about the rest of the chapter -- let's just focus on Section 12-2 for today. Section 12-2 of the U of Chicago text is on Size Changes Without Coordinates. The text describes how to transform a figure -- in the example, it's a person's face -- given a center and a magnitude or scale factor. The U of Chicago writes:

"The transformation described in this section [12-2] has various names. It is called a size change or a size transformation. Some others call it a dilation or dilatation."

And those "some others" include the Common Core Standards:

Verify experimentally the properties of dilations given by a center and a scale factor:

Because of this, even though the U of Chicago prefers the term size change, I will use the term that appears in the Common Core, dilation. Likewise, the text prefers the term magnitude, but I will use the term scale factor. So here is the key definition of the section, rewritten using these terms:

Let O be a point and k be a positive real number. For any point P, let S(P) = P' be the point on ray OP' with OP' = k * OP. Then S is the dilation with center O and scale factor k.

Dilations, along with reflections, rotations, and translations, are the cornerstones of transformation geometry that distinguishes Common Core from traditionalist standards. Because the U of Chicago emphasizes these transformations, it is more in accord with Common Core than most other texts.

This raises an important question -- why is the U of Chicago text, written nearly 20 years before the Common Core, nevertheless more in agreement with the standards than more recent texts? The common denominator is the National Council of Teachers of Mathematics. The NCTM influenced both the U of Chicago text and the Common Core Standards.

Over the holidays I discussed texts that are popular with traditionalists, such as the Saxon and Singapore math texts. It goes without saying that the U of Chicago is unpopular with traditionalists -- but the main reason is not their high school texts, but their elementary texts. Everyday Mathematics is the title of the U of Chicago's text for grade school math.

The following article from 12 years ago -- long before there were any Common Core Standards -- contains a sharp rebuke of Everyday Math:

The author, Matthew Clavel, writes:

"Here’s an example from the updated fourth-grade workbook: 'Homer’s is selling roller blades at 25 percent off the regular price of $52.00. Martin’s is selling them for one-third off the regular price of $60. Which store is offering the better buy?' Now put yourself in the place of kid who hasn’t learned how to multiply quickly, who isn’t sure about what a percentage is, and whose knowledge of fractions is meager. The problem will seem forbidding."

Recall that I agree with the traditionalists regarding the younger grades. Fourth grade, the example given in the text, is on the border between where I'd prefer a purely traditionalist approach and one where I wouldn't mind a mixed approach. A fourth grader may be just beginning to ask questions such as "Why do we have to learn this?" and not simply accept a "sage on the stage" telling him or her "Because I say so." The above question gives an answer to that question -- because you want to buy something and you want to know which is a better buy.

On the other hand, the student in Clavel's example is one who didn't have traditionalist math up to third grade, but instead saw nothing but Everyday Math from kindergarten on. A student who had traditionalist math up to third grade will be ready for this word problem in fourth grade, but a student who had only progressive math won't.

The high school U of Chicago texts aren't as terrible as Everyday Math. But still, "U of Chicago math" isn't a pleasant term to most traditionalists because of its elementary curriculum. This is why I recommended Singapore math for high school. It's possible to compromise here between Singapore and the progressive Common Core math -- namely by using the Singapore grade level text, but in a Common Core order.

Here is Singapore's New Elementary Math for eighth graders, arranged in Common Core order:

1. Indices (Exponents)
2. Algebraic Manipulations
3. Literal and Quadratic Equations
4. Word Problems
11. Motion Geometry (including dilations)
8. Congruent and Similar Triangles
10. Pythagoras' Theorem and Trigonometry
5. Graphs
6. Simultaneous Equations
7. Inequalities
9. Mensuration (volume)
12. Statistics I
13. Statistics II
14. More Algebraic Manipulations

Here I put right triangles just before graphs, even though for U of Chicago, I plan on discussing graphs before right triangles. This is because I wanted graphing to be the first chapter of the second semester rather than the last chapter of the first. Trig is a little tough, but I didn't want the last chapter of the old semester to be the big downer that I know graphing to be. Otherwise, I probably would have placed right triangles after volume.

In today's lesson, the students only have to perform the indicated dilations. Therefore, it's more like an activity than a lesson -- but I don't mind giving an activity for the first day after winter break. And this also frees the teacher for more bureaucratic issues that often occur on the first day of the semester AKA the second first day of school.

In a nod to the first day of school, notice that the one review question I included here is on a traversable network (Question #22 from the text). I also included a preview question -- and it's a proportion, as I mentioned earlier.

I never did get the old printer working again, so this came from the new printer that I purchased the day after Christmas. Hopefully, there won't be so many lessons where I'm forced to write by hand because I don't have access to a printer.

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