Tuesday, October 6, 2015

Representing and Describing Transformations (Day 29)

Today I subbed in a math class at a continuation school. The students were working out of the same Integrated Math I text as yesterday, but this time I noticed that there were Integrated Math II texts in the classroom as well. Here's the table of contents for the Houghton Mifflin Harcourt Math II text:

1. Analyzing Functions
2. Absolute Value Functions, Equations, and Inequalities
3. Rational Exponents and Radicals
4. Adding and Subtracting Polynomials
5. Multiplying Polynomials
6. Graphing Quadratic Functions
7. Connecting Intercepts, Zeros, and Factors
8. Using Factors to Solve Quadratic Equations
9. Using Square Roots to Solve Quadratic Equations
10. Linear, Exponential, and Quadratic Models
11. Quadratic Equations and Complex Numbers
12. Quadratic Relations and Systems of Equations
13. Functions and Inverses
14. Proofs with Lines and Angles
15. Proofs with Triangles and Quadrilaterals
16. Similarity and Transformations
17. Using Similar Triangles
18. Trigonometry with Right Triangles
19. Angles and Segments in Circles
20. Arc Length and Sector Area
21. Volume Formulas
22. Introduction to Probability
23. Conditional Probability and Independence of Events
24. Probability and Decision Making

The volume divide is between Modules 13 and 14.

But as I said earlier, the students were working out of the Math I text, not the Math II text. Three of the periods were working on Module 3, just as the freshmen were yesterday -- they were on Lesson 3.2, Understanding Relations and Functions. With these classes, I actually mentioned the mnemonic DIX-ROY -- which means "domain-input-x-value, range-output-y-value." It's another one of those mnemonics I found at the website of Oklahoman math teacher Sarah Hagan. And no, it's not the same as that HOY-VUX line I mentioned earlier on my blog, though I also found it on this website:


I'm not sure how many students will actually remember DIX-ROY, especially since I subbed in that class for one day only and they'll never hear that line again. Then again, Hagan herself writes how she's not sure how many of her students actually use her DIX-ROY and HOY-VUX mnemonics.

The other two periods were actually in the geometry volume -- they were on Lesson 16.3, Representing and Describing Transformations. And so not only will I focus on Lesson 16.3, but I've changed today's lesson -- originally scheduled for Lesson 4-6 of the U of Chicago text -- to what the students did today. After all, Lesson 16.3 of the Houghton Mifflin Harcourt Integrated Math I text is about transformations on a coordinate plane. We know that there are several questions on the PARCC Practice Exam about transformations on the coordinate plane. On the other hand, there are no questions on the PARCC Exam about clockwise or counterclockwise orientation -- the main topic of the U of Chicago's 4-6.

And besides, we actually are transforming polygons in Harcourt's 16.3. In some ways, Harcourt's 16.3 fits better with Chapter 4 of the newer editions of the U of Chicago text, since reflections, rotations, and translations are all covered in the new Chapter 4.

The regular teacher assigned example questions 5 and 6 for the students. They are given a triangle and its image under some transformation, with the coordinates of the vertices of both the preimage and image listed along with a graph of both triangles. Students were directed to use coordinate notation to write the rule that maps each preimage to its image, then identify the transformation and confirm that it preserves length and angle measure (in other words, that it's really an isometry).

Question 5 happened to be a 180-degree rotation about the origin, while Question 6 was a translation whose vector is <1, -2>. Notice that students were asked to identify the transformations as a rotation or translation, even though they don't formally learn about these until Chapter 17. So my placing this lesson on the blog today -- before our students have seen rotations or translations -- isn't any different from what they had to do today. A common error was to call the 180-degree rotation a "flip" -- which usually indicates a reflection. Of course, in a way a 180-degree rotation really is a type of reflection -- a point reflection rather than a line reflection. But in a Common Core class, a 180-degree rotation is definitely not considered to be a reflection. (After all -- going back to the lesson we skipped over -- a 180-degree rotation preserves orientation, while a line reflection reverses it.)

To confirm that these transformations are isometries, the students had to use the Distance Formula (that unfortunately appears earlier in Chapter 16) to find the lengths, but to find the angle measures, they could use a protractor (since that would otherwise require trigonometry). Unfortunately, many of the students had trouble using the protractor correctly.

Meanwhile, yesterday I mentioned how I decided to read Benoit B. Mandelbrot's The Fractal Geometry of Nature, and I'll read it one "part" (labeled by a Roman numeral). Each of the twelve parts contains three or four chapters, giving 42 chapters in all.

Part I of Mandelbrot's The Fractal Geometry of Nature is simply an Introduction. Mandelbrot begins the first chapter, "Theme," as follows:

Why is geometry often described as "cold" and "dry?" One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline, or a tree. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.

Mandelbrot writes that the shapes of many real-life objects are better described as fractals. Indeed, it was Mandelbrot himself who first came up with the name "fractal":

I coined fractal from the Latin adjective fractus. The corresponding Latin verb frangere means "to break:" to create irregular fragments. It is therefore sensible -- and how appropriate for our needs!-- that, in addition to "fragmented" (as in fraction or refraction) fractus should also mean "irregular," both meanings being preserved in "fragment."

The concept of fractal is intimately tied up with the concept of dimension. Indeed, Mandelbrot ultimately defines fractal in terms of its dimension in Chapter 3:

A fractal is by definition a set for which the Hausdorff Besicovitch dimension strictly exceeds the topological dimension.

I believe that Mandelbrot will define Hausdorff dimension later in the text. But right now, let me define topological dimension. Let's recall what topology is -- as Mandelbrot writes, "Topology ... considers that all pots with two handles are of the same form ...," or as is more popularly stated, it considers a doughnut to be of the same form as a coffee cup.

Now both geometry and topology agree that a point has dimension 0, a line has dimension 1, and a plane has dimension 2. (Yes, this is the Dimension Assumption, part of the Point-Line-Plane Postulate in Lesson 1-7 of the U of Chicago text.) The difference is that the geometrical dimension of a circle is 2, while its topological dimension is 1. This is because even though a circle fits on a plane, on the circle itself one can only travel one way or the other, just like on a line. Likewise, a sphere may have three dimensions in geometry, but it's a two-dimensional manifold in topology.

Throughout Part I of the book, Mandelbrot provides many pictures of fractals. One picture shows us two artificial fractal "flakes." One of these has Hausdorff dimension 5/2, the other dimension 8/3. As we can see, these lie between two and three dimensions. These computer-generated fractals remind me of a recent episode of Futurama. In "2-D Blacktop," the main characters suddenly find themselves smashed into two dimensions. Fortunately, the Professor has previously installed "Dimensional Drift" on his spaceship, which allows the crew to return to the third dimension. On the way from dimension 2 to dimension 3, several fractals appear in the background. These fractals, which look just like the fractal flakes in Part I of the book, have dimension between 2 and 3. Recall that many of the writers on Futurama were professional mathematicians, and they made sure that the dimensions of the fractals shown on the episode gradually increased from 2 to 3 as the crew made its slow return back to its home dimension.

I look forward to reading more from Mandelbrot's book soon.

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