And so Mandelbrot devotes a chapter to each of the types of fractals discussed earlier in the book, but this time giving a randomized example. For example, in Chapter 23, he describes random "curdling" similar to the formation of the Cantor set -- instead of always choosing the middle thirds interval, one chooses a random interval, then a random subinterval after each step. If one begins with a cube of three dimensions, the resulting fractal has approximately 1 dimension, and this is a more accurate model of the universe than can be given using nonrandom curdling, as the locations of the galaxies appear more or less at random.
In Chapter 24, Mandelbrot moves on to randomized fractals that are analogous to the Koch coastlines and snowflakes. Afterward, he moves on to Brownian motion, which describes how subatomic particles move. Recall that these fractals are random, and so they aren't self-similar -- no dilation maps the fractal to any part of the fractal.
Today is a quiz day, and so it's time for another post about the traditionalists and their objections to Common Core. And I go right back to the traditionalist Dr. Katharine Beals, because she has written another post criticizing Common Core Geometry.
We see that Dr. Beals mentions PARCC Practice Test Question 16 in this post, but if we compare it to Question 16 posted on this blog, it's not the same question (though both are about dilations). This is because Beals got this question from the Practice Performance Based Assessment (PBA), whereas my questions come from the End-of-Year exam (EOY). I've heard rumors that next year, the PBA and EOY will be combined into a single test, a move that I certainly like.
Question 16 of the PARCC Practice PBA exam is, as I said above, about dilations:
A dilation centered at point C with a scale factor of k, where k > 0, can be defined as follows:
1. The image of point C is itself. That is, C' = C.
2. For any point P other than C, the point P' is on Ray CP, and CP' = k * CP.
Use this definition and the following diagram shown to prove the following theorem:
I've discussed this theorem several times here on the blog. In the U of Chicago text, it appears in Lesson 12-3 as part of the Size Change Distance Theorem. (Recall that a "size change" is simply what the U of Chicago text calls a "dilation.") Dr. Hung-Hsi Wu also mentions this theorem, but he gives it a more impressive sounding name: The Fundamental Theorem of Similarity.
Now the U of Chicago text proves the theorem by appealing to the previous Lesson 12-1, which is about dilations on the coordinate plane. Thus it's basically a coordinate proof -- and the Distance Formula is ultimately used to prove that the image distance is k times the preimage distance.
It may also appear that this is just a straightforward application of SAS Similarity. But there's a problem with both the coordinate proof and the SAS~ proof -- both the properties of coordinates and SAS~ are themselves supposed to be proved using dilations (including the Size Change Theorem)! So either proof ultimately becomes circular.
Now, Dr. Wu goes through the trouble of writing a non-circular proof of his Fundamental Theorem of Similarity from first principles. But this proof is very complicated -- it depends on induction to prove it for all natural k, then inverses are used to extend this to all rational k, and then finally he appeals to a "Fundamental Assumption of School Mathematics" to extend this to all real k.
In January, I posted the Wu proof as one of the first worksheets of the second semester. But in hindsight, I decided that this proof is excessively difficult for high school students. In fact, this year I considered simply making A'B' = k * AB into a new postulate, to be called the "Dilation Postulate," so that students wouldn't have to prove it.
And so how does the PARCC expect students to prove Question 16? Obviously, we can't just call it a postulate, since PARCC is expecting a proof. The Wu proof is too long and difficult for the PARCC to expect students to reproduce it as an answer. And I doubt that PARCC is expecting a coordinate proof -- otherwise the test would have provided a coordinate plane. (Indeed, this is exactly what happens in Question 26 of the practice EOY -- Part D requires a coordinate proof, and so the figure is actually placed on the coordinate plane.)
So all that's left is the proof based on SAS Similarity. It's the only proof that's reasonable enough to be included as a PARCC question.
And so we see that once again, what appears in the Common Core Standards differs from what actually appears on the PARCC exam. In the standards, we see:
Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
According to this standard, we must prove the properties of similarity transformations first, and then we use those properties to prove AA Similarity. We can also use those properties directly to prove SAS~, or we can derive SAS~ from AA~ as is common in most pre-Core textbooks. In either case, the corresponding congruence theorem, SAS, is used. Yet on the PARCC, we appear to be using SAS~ to prove the properties of dilations! This is blatantly circular.
Recall that it's the contents of the Common Core -- meaning the Common Core tests -- that take priority over all other considerations. And even though the PARCC no longer has separate PBA and EOY tests, the fact that it appears on a practice test implies that it could appear on the combined test.
It may be best simply to declare AA~ a postulate, as in most traditionalist texts. Dr. Franklin Mason actually declares SAS~ to be the postulate -- a tradition that goes back to George Birkhoff, the creator of the Ruler Postulate. I myself might be tempted to let the SSS Similarity be the postulate, only because SSS~ actually comes before AA~ and SAS~ in the U of Chicago text.
Once again, Beals compares this problem to one from her 1970 Weeks & Adkins text:
P is a point on the side
I don't always agree with Beals, but this time she has a point. A concern about Common Core is that it could dictate a national curriculum. Lower- and mid-level proofs that appear on Common Core tests might force a national curriculum -- for example, that PARCC question about deriving the Alternate Interior Angles Consequence from the Corresponding Angles Consequence forces one to teach the latter before the former, although some texts do the opposite and derive the latter from the former. On the other hand, higher-level proofs such as those from Weeks & Adkins avoid this problem -- one can safely use SSS~ and other theorems to prove it without having to worry about circularity or how exactly SSS~ was derived in a particular test.
This is what I wrote last year about today's quiz, provided that any teacher wishes to use it at this point in class:
1. C' is the same as C, but D' goes up diagonally to the left. This is tricky because the line of reflection is not perfectly vertical.
2. I' goes up diagonally to the right.
3. There are two symmetry lines -- the segment joining the two points and its perpendicular bisector.
4. The angle measures 62 degrees.
5. The angle measures 2x degrees.
6. The reflection image over line AD of ray AB is ray AC. This is tricky because it's been a while since we've seen the Side-Switching Theorem.
7. This is officially the Figure Reflection Theorem -- just make the right vertices correspond.
8. Reflections preserve distance.
9. The orientation is clockwise.
10. The orientation is counterclockwise, because reflections switch orientation.
11. There are three pairs: angles B and C, angles BAD and CAD, angles ADB and CDB.
12. There is one line of symmetry -- the line containing the angle bisector. This follows from the Angle Symmetry Theorem.
13. F' = E follows from the Flip-Flop Theorem. FG = EH is because reflections preserve distance.
1. MO = MN 1. Given
2. M' = N 2. Given
3. MO = NO 3. Reflections preserve distance
4. MNO is equil. 4. Definition of equilateral
It's possible to add more details, such as O' = O, Transitive Property, etc.
15. The rectangle has two lines of symmetry, one horizontal, one vertical.
16. The isosceles triangle has one line of symmetry, and it's horizontal.
17. The images of the vertices are (1,3), (7,1), and (6,-2).
18. The image is (c, -d).
19. The angle measures 140 degrees.
20. The shortest distance is the perpendicular distance.