Thursday, March 3, 2016

Lesson 8-3: Fundamental Properties of Area (Day 115)

Yes, today I'm having yet another traditionalist post. There are two things that I wish to discuss in today's post.

First, I'm still thinking about my grandmother, who lived in Kansas City. Now Kansas City lies right on the Kansas River (for which the city is named). On one side of the river is Kansas City, Missouri, and on the other side is Kansas City, Kansas. My grandmother had lived in both states at different times in her life.

Now regarding Common Core, Missouri hasn't been in the news much lately. The Show-Me State recently dropped the SBAC, but they are still working on replacement standards and a new test, the last I've heard. Kansas, on the other hand, is in the recent news:

Here's the thing -- not only do lawmakers want to abolish the Common Core, but they seek to get rid of any test that's not produced in the Sunflower State -- including the AP and IB Tests!

Surely this is not what the anti-Core traditionalists want. Traditionalists oppose Common Core math because it doesn't make it easy to reach AP Calculus in senior year -- so the last thing they'd want is to abolish AP Calculus!

Still, it does show that for many, Common Core has poisoned the well for national standards. Even if one comes up with some replacement national standards that are excellent and genuinely improve education for students, teachers, and parents, Kansas would oppose them simply because they'd originate from outside the state.

The other story I want to mention comes from my own state of California. This article is titled, "Gov. Brown proposes competition to create new high school math course":

Anyone familiar with this blog knows that I've written long lists of proposed curricular units several times before, so this "competition" is a natural fit for me. The only thing that's surprising for the readers is that it took me two full weeks to notice and say something about the article -- but of course I was distracted with my grandmother's passing.

Of course, most of my proposed math units were for Geometry courses, since this is, after all, a Geometry blog. But unfortunately, Geometry isn't exactly what the governor is looking for:

Brown is proposing spending $3 million for a competition to develop a year-long math course that is closely aligned with the California State University’s expectations for incoming freshmen, and will help students avoid having to take remedial classes when they get there.

The additional high school course would mesh with the Common Core State Standards, which envisage a three-year sequence of math courses but also strongly recommend a fourth year, without specifying what that fourth-year course should be.

Similarly, the new math course would target students with comparable ability in math who have passed Algebra 2, which is required to gain admission to CSU and UC, and who scored “conditionally ready” on the Smarter Balanced assessment.

OK, so the new math course is for seniors who passed Algebra II but struggled with it -- perhaps earning the lowest possible grade acceptable to college admissions (C, or even C-), and scored only 2 out of 4 on the SBAC. The usual class to take after Algebra II is Precalculus, but these are students who probably wouldn't succeed in a standard Precalc class.

If I had it my way, the class would be Geometry II -- if there's Algebra I, why not Geometry II? I'd mentioned several times on the blog that one thing I like about starting Geometry is that it gives a break to students who just finished struggling through Algebra I -- the last thing they want to see is more Algebra. (This is why I'm posting the topic of area so late on this blog.) So likewise, Geometry II would be a break for those who just finished struggling through the Algebra II class.

But I know this isn't what Gov. Brown is seeking. The point of the new class is to be equivalent to college entrance exams without needing remedial classes -- and I know that these entrance exams are heavy in Algebra as much as Geometry, if not more so.

Here is a link to the Entry-Level Mathematics (ELM) exam -- the test for which this class is supposed to prepare the students:

According to this link, the breakdown is 35% Numbers and Data, 35% Algebra, and 30% Geometry, so this should be the focus of the class.

Recently, I spent so much time on discussing my "Presidential Birthday" and "Gubernatorial Christmas" plans. Of course, these were just fantasies about what I'd do if I were president or governor -- which of course isn't going to happen.

But this is the opportunity to design a real high school math course. I have no idea who will be eligible to participate in the competition. You'd think that at the least, teachers should be eligible -- then again, many people criticize the Common Core for lack of teacher input. I wonder whether I, a mere substitute teacher, would be eligible. Well, if I can participate, you bet I will participate!

I haven't come up with my new senior math course yet, but even if I had, I would certainly not post it here on the blog -- after all, what part of "competition" don't you understand? I'm trying to win the competition, not have others steal my idea and win it themselves!

But I will tell you about the skeleton of my plan, as much of it will be based on information that's already posted here on the blog. You may be asking, will this be my Presidential Birthday or Gubernational Christmas plan? Well, it should be "Gubernatorial" something -- since after all, it's the governor who's proposing the competition in the first place. Notice that even though this class will be based on the national Common Core Standards, developing this course entails dividing the high school standards into courses, which is a state-level decision.

Here on the blog, as part of my Gubernatorial Christmas plan, I described a class that is for students who finish Algebra II but aren't ready for Precalc -- which is exactly what Brown is seeking. This is what I wrote back on Christmas Eve about this new class, which I called Algebra III:

The West Virginia standards do contain some form of tracking. In particular, we see that both West Virginia and Sidwell Friends have an Algebra III course for students who barely pass Algebra II and so aren't ready for Pre-calculus. By contrast, Algebra III classes aren't common here in California.

So we see that these low-track students cover most of Algebra I as freshmen. Then as sophomores, they finish Algebra I and start Geometry. In junior year, the first semester is devoted to Geometry and the second semester to Algebra II. Finally, as seniors they finish Algebra II and begin just a little bit of Pre-calculus -- a course that's approximately equivalent to the Algebra III courses that we can find on the East Coast. So in a way, the low-track students are following a sort of Integrated Math plan, since they see both Algebra and Geometry during their sophomore and junior years. This justifies Integrated Math in a way that the Common Core doesn't.

My Gubernatorial Christmas plan actually slows down the college-track curriculum for all four years, not just senior year. But we can still see how Algebra III fits after a standard Algebra II. It would also fit after Integrated Math III at the integrated schools.

In the Edsource link about, the commenter Seth Rogers praises the idea:

I think that this is an excellent idea. Math needs to be accessible to all students. We need to stop promoting the belief that “some people are just good at math and others aren’t”. We don’t allow people to say “Some people can read and write and others just can’t do it”, so we shouldn’t make those allowances for math either. It isn’t an either/or scenario. People are perfectly capable of being proficient in both math and language arts. If you leave High School without a solid grasp on mathematics, then your educational system has failed you.
I was a product of that thinking. I spent most of my life believing that I wasn’t good at math because I just didn’t “have the math gene” as my mother would put it. In all of actuality, I wasn’t good at math because I was told to memorize formula after formula and when I asked where the formula came from, I was told that it didn’t matter (probably because the teachers I had were either unable to tell me or uninterested in trying to explain it to me.) I would go homer confused, and my mother would tell me that it was her fault, that she wasn’t good at math and she had “passed it on” to me. What was really happening, was that she was unable to help me with my homework, so I was on my own if I didn’t understand what was taught in class the first time. Looking back, what I really needed was a tutor, what I got was a fear of (and later disdain for) math. It wasn’t until college that I realized that not onIy could I do math, but that I actually quite enjoyed it. The reason that I had missing out on math all of those years was because I am not the type of learner who does well at learning or memorizing something without understanding the “why” of it first.
Of course, what Rogers writes here is the opposite of what traditionalists want. Traditionalists feel that understanding the "why" only slows down the learning of the math. Of course, I don't expect traditionalists to support Algebra III for seniors at all. The only class they want to see seniors take is, of course, AP Calculus.

By the way, if I propose this idea, I'm not sure whether I like the name "Algebra III." Once again, these are students who just scraped by Algebra II. The class will contain Algebra comment, but I'm not sure whether I like the word "Algebra" in the name of the course. As JK Rowling, the author of Harry Potter, once wrote, fear of a name increases fear of the thing itself.

Don't forget that this Algebra III course is only for California. Then again, I expect that Kansas has a need for a new senior year course, since they're dropping AP Calculus.

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

Lesson 8-3 of the U of Chicago is on the fundamental properties of area. That's right -- we have finally reached the notion of area on the blog.

Let's see what other authors have to say about this very important concept. We begin, as usual, with David Joyce:

Chapter 5 is about areas, including the Pythagorean theorem. It begins with postulates about area: the area of a square is the square of the length of its side, congruent figures have equal area, and the area of a region is the sum of the areas of its nonoverlapping parts. A theorem follows: the area of a rectangle is the product of its base and height. There is no proof given, not even a "work together" piecing together squares to make the rectangle. An actual proof is difficult. It would depend either on limiting processes (which are inappropriate at this level), or the construction of a square equal to a rectangle (which could be done much later in the text). It would be just as well to make this theorem a postulate and drop the first postulate about a square.

So far, Chapter 5 of the Prentice-Hall text sounds very similar to Chapter 8 of the U of Chicago text, as the Pythagorean Theorem is included in the area chapters of both texts. But I've already discussed my treatment of the Pythagorean Theorem.

According to Joyce, the Prentice-Hall text assumes as a postulate the area of a square, and claims without proof the area of a rectangle. Joyce states that there are two ways to prove the theorem. The first is by "limiting processes." You may wonder, why are "limiting processes needed" -- why can't we just divide the rectangle into squares? After all, isn't that what "square units" are -- a rectangle with area 32 square units (the first rectangle shown in the U of Chicago text) can be divided into 32 unit squares.

The problem is, what if the sides of the rectangle have irrational lengths? Dividing the rectangle into unit squares only works for the rational numbers, not the real numbers. If we had a rectangle with dimensions 1 and sqrt(2), we wouldn't be able to divide it into squares so easily. The well-known paper size A4 is supposed to be a rectangle whose ratio of length to width is about sqrt(2). Another famous irrational rectangle is called a golden rectangle. This rectangle defies division into squares -- cut off the largest possible square, and the remaining rectangle is similar to the original, so that cutting off another square leaves a third rectangle similar to the original, and so on. The ratio of its length to width is (1 + sqrt(5)) / 2 -- a well-known irrational number known by the Greek letter phi.

So limiting processes are needed to show that this works for irrational side lengths. We'll revisit the idea of limiting processes later on when we discuss another famous irrational number -- pi.

Joyce also mentions that one can construct a square that has the same area as a rectangle. We can imagine how to accomplish this construction. Think about it -- if the length and width of a rectangle are x and y, then the square must have side length sqrt(xy). Notice that this is the geometric mean of x and y, so it implies that the geometric means of Section 14-1 are relevant. Once we have the square, we must then prove that it and the rectangle really have the same area. Most likely, what will happen is that both the square and the rectangle are divisible into triangles -- and we don't need to know how to find the triangles' area, as the triangles making up both figures will be congruent. So the proof would be similar to the area proof of the Pythagorean Theorem.

We move on to Dr. Franklin Mason. As it turns out, Dr. M provides a clever way to derive the rectangle formula from that of the square -- and no limiting processes are needed! He notes that a square with side length x + y must have area (x + y)^2, which is x^2 + 2xy +y^2 according to the polynomial manipulation of Algebra I. This square can be divided into four rectangles -- two of them squares of sides x and y, the other two rectangles x by y. Subtracting x^2 and y^2 leaves 2xy as the combined area of the two rectangles, so that xy must be the area of the each of these rectangles with dimensions x and y. QED

Dr. Hung-Hsi Wu, meanwhile, discusses the area of rectangles with rational sides in the context of multiplying fractions. He points out that, for example, the area of a rectangle with dimensions 1/2 by 1/3 must be 1/6, because clearly six congruent copies of them fill the unit square. Actually, Dr. M does the same with squares with fractional side lengths before giving his clever derivation of the rectangle formula. Wu then points out that by a form of hand-waving used to avoid limits, The Fundamental Assumption of School Mathematics -- any formula that works for rational numbers also works for irrational numbers, so we have the rectangle formula.

After all of this, what does the U of Chicago do? It provides us with an Area Postulate:

Area Postulate:
a. Uniqueness Property: Given a unit region, every polygonal region has a unique area.
b. Rectangle Formula: The area of a rectangle with dimensions l and w is lw.
c. Congruence Property: Congruent figures have the same area.
d. Additive Property: The area of the union of two nonoverlapping regions is the sum of the areas of the regions.

Therefore the U of Chicago, in part b, does exactly what Joyce suggests:

It would be just as well to make this theorem [the rectangle formula -- dw] a postulate and drop the first postulate about a square.

All sources provide the Congruence and Additive Properties, albeit with different names. For example, Dr M. calls the Additive Property the Area Addition Postulate -- to parallel other postulates such as "Segment Addition Postulate" and "Angle Addition Postulate."

Notice that the U of Chicago text states, "To find the area of a region, cover it with congruent copies of a fundamental region [emphasis mine]." The phrase "fundamental region" appeared in yesterday's lesson -- the fundamental region is the region that is repeated in a tessellation. So we can see another reason for including yesterday's lesson on tessellations -- one can find the area of a region by tessellating it with squares.

No comments:

Post a Comment