Thursday, January 28, 2016

Lesson 12-10: The Side-Splitting Theorem (Day 92)

Lecture 18 of David Kung's Mind-Bending Math is called "Filling the Gap Between Dimensions." In this lecture, Dave Kung continues his description of paradoxes related to Geometry -- obviously a topic dear to my heart considering the title of this blog.

Kung begins by discussing a three-dimensional solid called Gabriel's horn. As it turns out, Gabriel's horn has a finite volume, yet an infinite surface area. He shows this using Calculus -- this is a bit beyond AP Calculus, but it might be understandable by an AP student.

The Quick Conundrum involves balancing a yardstick on two fingers, one on each hand. If the hands approach each other quickly, they meet at the midpoint (the 18" mark), but if he moves them slowly, the ruler slides first one way, then the other.

Then Kung moves on to figures which, as the title goes, fill the gap between the dimensions. Of course, I'm talking about fractals. Kung calculates the similarity dimension of three fractals -- all of which I discussed on the blog back in October when I read the fractal book written by the Polish mathematician Benoit Mandelbrot:

Cantor Set (mentioned in the other two links below)
Sierpinski Triangle (Sierpinski was also Polish, born in the same city as Mandelbrot.)
Menger Sponge

In describing how to find the similarity dimension of a fractal, Kung mentions what the U of Chicago calls the Fundamental Theorem of Similarity in Chapter 12 (as these fractals are self-similar). Since we are currently working in Chapter 12, we ought to be discussing this right now on the blog -- except that we aren't.

Notice that we're skipping from Lesson 12-3 (yesterday's lesson) to 12-10 (today's lesson). Well, actually we'll be covering some of the skipped lessons next week. But the material mentioned in Kung's lecture comes from Lessons 12-6 and 12-7, which we aren't fully covering on the blog.

So what's going on here? The problem is that when I first wrote these similarity lessons last year, I based them on the writings of Dr. Hung-Hsi Wu. He used the name "Fundamental Theorem of Similarity" to refer to a different theorem from the U of Chicago -- the text uses the name FTS to refer to the idea that areas of similar figures vary as the square of the length and volumes as the cube of the length, while Wu uses the name FTS to refer to what I called the Dilation Distance Theorem in yesterday's post.

I had labeled Wu's FTS proof as "Lesson 12-6" even though it referred to a different theorem. But Wu's proof was highly complicated -- and, as I later decided, too difficult for high school. And so I ended up dropping Wu's FTS proof and using the PARCC proof (using SAS~) instead. But Kung points out that it's the U of Chicago's FTS that leads to the similarity dimension of fractals -- the Cantor set doubles in content as its length triples, so its dimension is log_3(2).

As for Lesson 12-7, I never included it as a lesson on the blog. But this lesson is actually fun -- and Kung refers to it in his lecture. The question is, "Can There Be Giants?" The answer, as it turns out, is no, for the weight of a giant increases as the volume (the cube of the height) while its strength varies as the area of a bone's cross-section (the square of the height). So a giant isn't strong enough to stand up. On the other hand, as Kung explains, a tiny person can't exist because the amount of blood its heart can pump drops fast, like the volume (the cube of the height) -- so not enough blood can make it all the way to its feet and head (the first power of the height). Large and small animals must have completely different shapes -- they can't be geometrically similar to humans.

Well, here is the section of the U of Chicago text for which we do have a worksheet all ready and made -- Lesson 12-10, the Side-Splitting Theorem. This is what I wrote last year about the theorem:

The U of Chicago version of the theorem is:

Side-Splitting Theorem:
If a line is parallel to a side of a triangle and intersects the other two sides in distinct points, it "splits" these sides into proportional segments.

And here's Dr. Wu's version of the theorem:

Theorem 24. Let triangle OPQ be given, and let P' be a point on the ray OP not equal toO. Suppose a line parallel to PQ and passing through P' intersects OQ at Q'. ThenOP'/OP = OQ'/OQ = P'Q'/PQ.

Notice that while the U of Chicago theorem only states that the two sides are split proportionally, Wu's version states that all three corresponding sides of both sides are proportional.

Moreover, the two proofs are very different. The U of Chicago proof appears to be a straightforward application of the Corresponding Angles Parallel Consequence and AA Similarity. But on this blog, we have yet to give the AA Similarity Theorem. So how can Wu prove his theorem?

We've seen several examples during the first semester -- a theorem may be proved in a traditionalist text using SSS, SAS, or ASA Congruence, but these three in Common Core Geometry are theorems whose proofs go back to reflections, rotations, and translations. Instead, here we skip the middle man and prove the original high-level theorem directly from the transformations. We saw this both with the Isosceles Triangle Theorem (proved from reflections in the U of Chicago) and the Parallelogram Consequences (proved from rotations in Wu).

So we shouldn't be surprised that Wu proves his version of the Side-Splitting Theorem using transformations as well. Naturally, Wu uses dilations. In fact, the names that Wu gives the points gives the game away -- O will be the center of the dilation, and P and Q are the preimage points, while P' and Q' are the images.

Here is Wu's proof: He considers the case where point P' lies on OP -- that is, the ratio OP'/OP, which he labels r, is less than one. This is mainly because this case is the easiest to draw, but the proof works even if r is greater than unity. Let's write what follows as a two-column proof:

Given: P' on OPQ' on OQPQ | | P'Q'r = OP'/OP
Prove: OP'/OP = OQ'/OQ = P'Q'/PQ

Statements                                           Reasons
1. P' on OPQ' on OQPQ | | P'Q'      1. Given
2. OP' = r * OP                                    2. Multiplication Property of Equality
3. Exists Q0 such that OQ0 = r * OQ  3. Point-Line/Ruler Postulate
4. For D dilation with scale factor r,    4. Definition of dilation
    D(Q) = Q0, D(P) = P'
5. P'Q0 | | PQP'Q0 = r * PQ               5. Fundamental Theorem of Similarity
6. Lines P'Q0 and P'Q' are identical      6. Uniqueness of Parallels Theorem (Playfair)
7. Points Q0 and Q' are identical          7. Line Intersection Theorem
8. OQ' = r * OQOP' = r * OP,            8. Substitution (Q' for Q0)
    P'Q' = r * PQ
9. OP'/OP = OQ'/OQ = P'Q'/PQ = r    9. Division Property of Equality

Now the U of Chicago text also provides a converse to its Side-Splitting Theorem:

Side-Splitting Converse:
If a line intersects rays OP and OQ in distinct points X and Y so that OX/XP = OY/YQ, then XY | | PQ.

The Side-Splitting Converse isn't used that often, but it can be used to prove yet another possible construction for parallel lines:

To draw a line through P parallel to line l:
1. Let XY be any two points on line l.
2. Draw line XP.
3. Use compass to locate O on line XP such that OX = XP.
4. Draw line OY.
5. Use compass to locate Q on line OY such that OY = YQ.
6. Draw line PQ, the line through P parallel to line l.

This works because OX = XP and OY = YQ, so OX/XP = OY/YQ = 1.

The U of Chicago uses SAS Similarity to prove the Side-Splitting Converse, but Wu doesn't prove any sort of converse to his Theorem 24 at all. Notice that many of our previous theorems for which we used transformations to skip the middle-man, yet the proofs of their converses revert to the traditionalist proof -- once again, the Parallelogram Consequences.

Another difference between U of Chicago and Wu is that the former focuses on the two segments into which the side of the larger triangle is split, while Wu looks at the entire sides of the larger and smaller triangles. This is often tricky for students solving similarity problems!

Let me review the three MTBoS links to which I referred in yesterday's post:

Julie Morgan is a Scottish high school math teacher. Here's what I wrote about her questions, which are related to a lesson on the area of a sector:

Yes, I heard of the idea of an “Anticipatory Set,” a introductory question that motivates the students to start thinking. I’m glad that your Anticipatory Set question worked well!
I’ve tutored students in the past, and when they reached this lesson, I motivated them by bringing them an actual pizza slice and make them calculate the area before I let them eat it!
As an American, I find it interesting to read about education in other countries. So it’s nice to read your blog about teaching in Scotland!

This is the second time that I referred to fellow California math teacher Marisa Aoki. Her question often shows up in the traditionalist debates. This is what I wrote:

Don't worry about being unable just to "follow the prompt" -- some of my MTBoS posts end up going off on a tangent as well.

I suppose this was the thinking behind the Common Core Standards -- idea is to increase math comprehension and make sure that the students do understand what the numbers mean. But so many people are afraid that students are spending too much time on comprehension and not enough on learning the addition and multiplication tables.

And this is Aoki's response:

I don't think fluency is bad - but when kids tell me that 9x8 is impossible because they can't remember it, that is an issue. If a kid really knows what multiplication means, even when facts are forgotten, there is a strategy to figure it out (instead of just giving up or guessing). I wish there weren't such a pervasive idea that comprehension and fluency are necessarily at odds. If anything, I feel like they should compliment each other.

Massachusetts teacher writes about her Advanced Algebra I class:

Yes, this must be an Accelerated class if they're already using complex numbers in Algebra I!

That being said, now that the students have learned complex numbers, they should know that ALL quadratic equations have a solution -- which means that the answer "no solution" that they tried to write is NEVER correct. (This, of course, is the Fundamental Theorem of Algebra.)

But yes, I know that many students often wonder why they must learn more than one method. I was once a long-term sub in an Algebra I class and when we were learning about factoring, this very bright girl had her father show her the Quadratic Formula, so when asked to factor x^2 + 5x + 6 (which is not an equation, just an expression to factor), she'd just write -2 and -3 and didn't want to learn how to factor.

Here are the worksheets. After watching Kung's lecture, I decided to include an extra worksheet on Lessons 12-6 and 12-7 -- the only problem is that since we switched the order of the chapters, students following the blog haven't truly learned about area or volume yet. A disclaimer is included at the top of that page.

Kung will continue with Geometry paradoxes in his next lecture.

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