Part XI of Benoit B. Mandelbrot's The Fractal Geometry of Nature is on miscellaneous topics, and consists of Chapters 36 through 39. In Chapters 36 and 37, he applies fractals to such far-flung fields as physics and economics.
In Chapter 38 Mandelbrot discusses scaling and power laws without using geometry. A well-known example of such a power law is the hyperbolic distribution, also called Zipf's law. Zipf's law is used to predict the frequencies of words, the distribution of income, and -- an example that I've read about outside of Mandelbrot's book -- the populations of cities. For example, New York City has the largest population in the U.S., then the second largest city Los Angeles has about half of NYC's population, then the third largest city Chicago has about one-third NYC's population, and so on. Mandelbrot ties this to fractals in that the distribution isn't necessarily F/u but Fu^-D, where D could be interpreted as a fractal dimension.
Mandelbrot devotes Chapter 39 to various mathematical ideas about fractals. He mentions that just as there exist self-similar fractals, there can also be self-affine fractals. I once mentioned the concept of an affine transformation here on the blog -- affine transformations include all of the Common Core transformations as well as the transvection, or "shear." Unfortunately, Mandelbrot doesn't give any specific examples of self-affine fractals (other than the self-similar ones), but one could image a fractal generated by curdling a figure made out of parallelograms that aren't squares.
In this chapter, Mandelbrot also mentions the famous Weierstrauss function -- a function that is continuous everywhere but differentiable nowhere. This reminds me of another math teacher blog I've been wanting to link to -- its name is "Continuous Everywhere But Differentiable Nowhere":
The blog author is Sam Shah, a high school teacher from New York City (yes, the city that has about twice LA's population and thrice Chicago's population). One Geometry post that he posted last year (indeed, exactly one year ago today) is related to the lesson that I posted yesterday -- finding the center of a rotation:
Now Shah does this lesson differently from the EngageNY site where I found yesterday's lesson -- he has his students do it on graph paper. Notice that the graph paper is there just to make it easier to keep track of the points, not to use some formula for the rotations -- after all, we're trying to find the center of rotation, so we can't use formulas that only work when the center is the origin. It's also apparent that his class devoted several days -- possibly the entire week -- to this activity, not just one.
The lesson originally scheduled for today was the first theorem of Dr. Hung-Hsi Wu. And so today's the day that I'm jumping out to Chapter 7 and triangle congruence.
I'm stated numerous times why I'm making these two changes -- first to drop Dr. Wu's lessons, and second to start congruence now. In short, it's because I want to emphasize parallel lines less and congruent triangles more for the PARCC. In particular, Dr. Wu's lessons give priority to alternate interior angles while the PARCC emphasizes corresponding angles. And we've seen a few questions on the PARCC where it's better to give a proof based on SSS, SAS, and ASA rather than attempt other proofs that are given in the U of Chicago text.
For those who don't like all this jumping around in the text, recall that triangle congruence appears in Chapter 4 in most pre-Core texts, and Chapter 3 in Dr. Franklin Mason's text -- and we've already covered most of Dr. M's first two chapters, so it's time for his Chapter 3. On the other hand, it does not appear until Chapter 8 in the much-aligned Prentice-Hall text (Dr. David Joyce's review) -- and the U of Chicago's 7 is much closer to 8 than to the usual 3 or 4. And so I move up triangle congruence to the time that it's normally taught.
I've spent the last few days discussing how much proof I want students to do in my class. Last year, I introduced the concept of low-, medium-, and high-level proofs. These categories aren't rigid, but here's an approximate division:
Low-level: Prove SAS Congruence from first principles (i.e. transformations, if it's Common Core)
Mid-level: Prove the Isosceles Triangle Theorem from SAS Congruence
High-level: Prove the Equilateral Triangle Theorem (i.e. that an equilateral triangle is equiangular) from the Isosceles Triangle Theorem
So we can somewhat see the difference among these levels -- in particular, we may use the lower-level theorems in the proofs of the higher-level theorems.
But there's a more important distinction among these levels in the Geometry classroom. Teachers are more likely to ask students to prove higher-level than lower-level theorems. Many Geometry texts, especially pre-Core, don't expect students to prove our low-level theorems, such as SAS Congruence from first principles. Indeed, they absolve students from the responsibility of proving SAS completely by making it a postulate!
And now we see where the opponents of Common Core come in. They point out that geometry based on transformations is too experimental to appear in the classroom. Instead, they favor the pre-Core status quo -- just declare SAS a postulate and throw out transformations altogether,
Now here's the problem with this thinking -- low-level is to mid-level as mid-level is to high-level. I can now imagine a hypothetical class where not only do we avoid the low-level derivation of SAS from transformations, but we can avoid the mid-level derivation of the Isosceles Triangle Theorem from SAS as well. Instead, just declare the Isosceles Triangle Property to be a postulate and throw out SAS Congruence altogether! Students can still prove interesting theorems from this Isosceles Triangle "Postulate," including the Equilateral Triangle Theorem -- even the first problem from that Weeks and Adkins page from two weeks ago can be proved using only the Isosceles Triangle "Postulate" (and its converse, which could be declared yet another postulate).
One may argue that no Geometry text actually does this -- but au contraire, there really is a text that does something like the above. In Lesson 5-1 (old version) of Michael Serra's Discovering Geometry, Conjecture 27 is the Isosceles Triangle Conjecture (and Conjecture 28 is its converse), while Conjecture 29 is the Equilateral Triangle Conjecture (stated as a biconditional), with a paragraph proof provided to show how 29 follows from 28. So even though all three of these statements are labeled as "conjectures," the net effect is that 27 and 28 are postulates (as no proof is given), while 29 is actually a theorem proved using postulate 28.
Of course, this may seem silly -- Serra doesn't avoid SAS altogether, but instead gives it later on in the same chapter (Lesson 5-4, old version). And ultimately when we reach the end of the book when two-column proofs are taught, students are asked to prove the Isosceles Triangle Theorem using SAS, just as in most other texts.
But it does show that the pre-Core status quo is attacked on two fronts. If you argue that students should learn SAS so that the Isosceles Triangle Property can be a theorem rather than a postulate, then why not take it further in the direction of more rigor, and teach the students about reflections so that SAS can be a theorem rather than a postulate? Or going the other way, if you argue that students shouldn't have to learn how SAS follows from reflections, then why not take it further and say that students shouldn't have to learn how the Isosceles Triangle Property follows from SAS? It's not at all obvious why the exact status quo (SAS a postulate, Isosceles Triangle Property a theorem) is neither excessively nor insufficiently rigorous.
Now there is an argument that, if true, would vindicate the status quo defenders. It could be that the mid-level derivation of the Isosceles Triangle Theorem from SAS is easy for the students to understand, thereby preparing them well for the rest of Geometry and subsequent courses, but the low-level derivation of SAS from reflections is too hard for them and makes them cry after the test, thereby discouraging them from taking subsequent math courses. If this can be demonstrated, then the status quo is exactly right. Then again, until such a demonstration can be made, it's just as likely that the Common Core transformation approach, or even Serra's conjecture approach, could be correct.
There's one more thing that we must take into consideration -- the distinction between the Common Core Standards and the Common Core tests. Much of all my chapter juggling has occurred because I'd originally set up my lessons to match the standards, only to see something else on the tests. The standards state that students should learn how SAS and the other congruence and similarity theorems follow from the properties of transformations -- but such low-level proofs don't appear on the PARCC or any other Common Core test. It's actually easier to test for medium- and high-level proofs on a test, and the PARCC is no exception.
The PARCC question where students have to derive the Alternate Interior Angles Consequence from the Corresponding Angles Consequence is a mid-level proof. The PARCC question where students have to prove that the sum of the exterior angles of a triangle is 360 is definitely a high-level proof (after all, the triangle sum ultimately goes back to alternate interior angles). A low-level proof in this tree would be to show how the Corresponding Angles Consequence goes back to transformations.
Mid-level proofs on the PARCC are problematic -- and it's these questions that drive most of the changes in my curriculum. We saw last week how although the Common Core Standards ask students to derive SAS Similarity from the properties of dilations, a PARCC question asks them to derive a mid-level property of dilations from SAS~ instead.
But high-level proofs cause the fewest curriculum problems. I consider the classic two-column proofs of U of Chicago's Lesson 7-3 -- where students use SAS to prove two triangles congruent, but the "S" comes from the Reflexive Property or the definition of midpoint and the "A" comes from the Vertical Angles Theorem or some other result -- to be high-level proofs. This is because they appear at the top of the proof tree, rather than branch out to be used in other theorems. Such proofs don't require the students to derive SAS as a theorem.
On this blog, I will present low-level proofs in worksheets in order to meet the Common Core Standards, but I don't expect students to reproduce them in the exercises or on a quiz or test. On the other hand, students will have to know and understand the mid- and high-level proofs.
Let's get to today's worksheet. Now as it turns out, not only did I begin Chapter 7 last year near the Thanksgiving break, but it was also when I was purchasing a new computer and working hard to get it installed and connected to the Internet. These two facts combined mean that I don't necessary have a great Lesson 7-1 worksheet from last year.
Last year I gave some sort of an activity, where students were given parts of a triangle such as SS, AA, SSS, AAA, and so on, to determine whether they are sufficient to determine a triangle. Then I would follow this with a discussion of the results followed by some "review" problems. Once again, juggling the lessons around means that the students would be "reviewing" concepts that I haven't taught this year yet, such as Triangle Inequality and Triangle Sum. (Notice that this are related to the SSS and AAA conditions, respectively.)
So I decided to keep the activity-like part of the lesson and replace the Triangle Inequality and Triangle Sum questions with some more information about isometries and polygon congruence -- that this year's students haven't seen yet since we haven't covered Lesson 6-7 yet. This is what the resulting worksheet looks like: