Lesson 12-4: Proportions (Day 124)

I've been thinking lately about how I'm teaching similarity here on this blog. Four years ago -- back when I first started this blog -- I posted Lesson 12-6, "The Fundamental Theorem of Similarity." But the theorem I posted was based on Hung-Hsi Wu, not the U of Chicago text.

Anyway, this morning I received the following question on that old 2015 post:

AJ:
Why did you choose Wu's theorem over University of Chicago's?

And here is my response:

Back when I first posted this, I believed that Wu's method was closer to how the Common Core expects the lesson to be taught.

Since then, I fear that this proof is too difficult for most high school students to understand. Even EngageNY, whose curriculum is based mostly on Wu's method, replaces this proof with a simpler one using the area of a triangle.

Each year I revisit and make changes to the lessons I post, and this week I'm doing Chapter 12. Today I'll blog about our discussion here, and by the end of this week I'll probably end up replacing Wu's proof with the U of Chicago's.

Thanks.

Indeed, I was thinking about these lessons a few weeks ago when the traditionalists were criticizing Common Core Geometry for being "transformation based" instead of "proof based." I had a little to say about how the Core treats similarity, but I wanted to wait until we were actually in the similarity chapter before writing about it.

Well, we're in the similarity chapter. So the time to discuss it is now. I can afford to do so today, since I don't actually have much to say on today's lesson, proportions.

My claim from two weeks ago is that Common Core Geometry is transformation based as well as proof based. We must look at how the Core treats similarity in order to examine this claim.

Let's start with Hung-Hsi Wu. His "Fundamental Theorem of Similarity" actually consists of some of the properties of dilations that appeared in yesterday's Lesson 12-3. If the image of PQ under a dilation with scale factor k is P'Q', then P'Q' | | PQ and P'Q' = k PQ.

Wu proves this in cases based on what sort of number the scale factor k is. For natural number k, Wu proves it using induction on k. This initial case, k = 2, is based on a special version of the Midpoint Connector Theorem of Lesson 11-5. The inductive case from k to k + 1 involves repeating the Midpoint Connector Theorem argument over and over.

For rational number k, if k = 1/q (q natural number), then Wu notes that a dilation of scale factor 1/q is the inverse of a dilation of scale factor q. And if k = p/q (p, q natural numbers), then Wu notes that a dilation of scale factor p/q is the composite of two dilations, with scale factors p and 1/q.

All that's left is to extend the argument to irrational k. Wu hand-waves over this by using what he calls the "Fundamental Assumption of School Mathematics" -- many theorems of pre-college math that apply to all rational numbers also apply to all real numbers. (This assumption also appears in Algebra II when defining what it means to raise a number to an irrational power.)

Even though Wu gives this proof, it's not the sort of proof we expect high school students to figure out easily. In the years since I posted it, I've regretted it. But every year since then, when I return to Lesson 12-6, I keep the Wu proof and change other parts of the worksheet!

As I wrote in the comment above, the EngageNY curriculum is based on the Wu proofs. But there's no way in the world EngageNY would use Wu's "Fundamental Theorem of Similarity" proof.

Instead, they replace it with a simpler proof based on the area of a triangle. I've mentioned the idea before how in proofs, area and similarity are often interchangeable -- this is why the Pythagorean Theorem has both area and similarity proofs. EngageNY's area proof removes the need for Wu's induction on k and subsequent extension to rational and real values of k.

But there's one problem here -- similarity is a "Module 2" topic, but area is a "Module 3" topic. This is the naive order suggested by the Common Core Standards -- since they mention similarity before area, all similarity lessons must appear before any area lessons. Once again, EngageNY justifies this by having students recall the triangle area formula from eighth grade (or earlier).

Notice that in the U of Chicago text, area (Chapter 8) appears before similarity (Chapter 12). Thus the U of Chicago could validly follow the EngageNY sequence of proofs -- except that it doesn't.

Many traditionalists attack the Wu/EngageNY plan from not following Euclid's geometry. But ironically, Wu/EngageNY actually follow Euclid more closely than traditional texts do! His Book VI, which teaches similarity, begins with Proposition 1 (ratios between areas of triangles). This is followed by Proposition 2 (Side-Splitter Theorem), from which both Wu/EngageNY derive their first similarity theorem (Proposition 4, AA~).

Of course, there's also the idea of deriving the similarity theorems "classically" -- that is, by assuming AA~ as a postulate -- and then ultimately deriving the properties of dilations. This is the method used by the old PARCC test, which should be irrelevant since California isn't a PARCC state, and backlash against the Core has led to most PARCC states dropping that test. But I still end up posting old PARCC questions from time to time -- including the Lesson 12-3 worksheet from yesterday.

I've said before that it's better just to follow the U of Chicago text. Therefore I will change this week's worksheets to reflect the U of Chicago lessons -- and finally get rid of the old Wu proof of his "Fundamental Theorem of Similarity."

This has no affect on today's proportions lesson -- but I might go back and change yesterday's lesson.