(nothing)
That's because this is the opening page of Chapter 10, "Mathematics & Architecture." There is nothing on this page except a photo captioned, "The Oracle office building, Redwood City, CA."
Now that we've finished the science chapter in Pappas, there's no need for me to write daily about my science class from last year. I know that to the readers, writing about my science class is like crying over spilled milk -- I didn't teach the class the way I wanted, and it's over. Still, I continue to keep that class in mind. It's the reason that I'm not teaching in a class right now -- and if I ever want to teach again, I must avoid the mistakes I made in that class. So I'll be continuing to write about my old class from time to time (but not in today's post, at least).
I'll post the table of contents for Pappas Chapter 10 tomorrow. Notice that we discussed a little on architecture in Chapter 0 of Serra. But once again, the Pappas and Serra topics miss each other on the blogging schedule.
Instead, our focus is now the U of Chicago text. Just like the Serra text, it's an old Second Edition (1991), and there are newer editions in which the chapters are ordered differently. Since my plan this year -- unlike past years -- is to follow the order strictly, let's revisit the chapter order in my text:
Table of Contents
1. Points and Lines
2. Definitions and If-then Statements
3. Angles and Lines
4. Reflections
5. Polygons
6. Transformations and Congruence
7. Triangle Congruence
8. Measurement Formulas
9. Three-Dimensional Figures
10. Surface Areas and Volume
11. Coordinate Geometry
12. Similarity
13. Logic and Indirect Reasoning
14. Trigonometry and Vectors
15. Further Work With Circles
Let's compare this to the modern Third Edition of the U of Chicago text. The first thing we notice is that the new text has only 14 chapters, not 15. We observe that the first twelve chapters are more or less the same in each text, and so it's Chapter 13 that is omitted in the new version. Instead, the material from the old Chapter 13 has been distributed among several different chapters.
You might recall that in the past when I used to juggle the lessons around, it was Chapter 13 that I moved around the most. So you could argue that when I was breaking up Chapter 13, I was actually adhering to the order in the new Third Edition -- unwittingly, of course!
Let's look at Chapter 13 in the old text, and I'll give the lesson in the new text to which the old Chapter 13 material has been moved:
-- Lessons 13-1 through 13-4 (on indirect proof) are now the first three lessons of Chapter 11, just before coordinate proofs. (Lesson 13-2, "Negations," is no longer a separate lesson in the new text.)
-- Lesson 13-5, "Tangents to Circles and Spheres," is now Lesson 14-4, in the circles chapter.
-- Lesson 13-6 through 13-8 (on exterior angles of polygons) have been incorporated into Lessons 5-6 and 5-7 (on Triangle Sum).
Some of these changes are those I once made by myself -- for example, including tangents to circles with the other circle lessons.
Besides the breakup of Chapter 13, here are the other major changes made in the Third Edition:
-- Chapters 4 through 6 exhibit many changes. In my old version, reflections appear in Chapter 4, while the other isometries don't appear until Chapter 6. In the new version, all isometries are defined in Chapter 4. With this, the definition of congruence (and some of its basic properties) have now moved up from Chapter 6 to Chapter 5. Only Triangle Sum remains in Chapter 5 -- the properties of isosceles triangles and quadrilaterals have been pushed back to Chapter 6.
-- With this, Chapter 3 has a few new sections. Two transformations are actually introduced in this chapter, namely rotations and dilations. This may seem strange, since rotations are still defined as Chapter 4 as a composite of reflections in intersecting lines -- and reflections themselves don't appear until Chapter 4. It appears that the purpose of rotations in the new Lesson 3-2 is to introduce rotations informally, as well as tie them more strongly to the angles of Lesson 3-1. (Rotations appear before reflections in Hung-Hsi Wu, but Wu does for different reasons.) Arcs also now appear in Lesson 3-1 instead of having to wait until 8-8. Meanwhile, the new Lesson 3-7 on dilations (which are still called "size transformations") is essentially the old Lesson 12-1 and 12-2. Again this is only an intro -- dilations are still studied in earnest only in Chapter 12.
-- Chapter 7 is basically the same as the old text, especially the first five sections (except that SsA in Lesson 7-5 now has an actual proof). The new Lesson 7-6 is the old Lesson 8-2 on tessellations. I see two new lessons in this chapter, Lesson 7-9 on diagonals of quadrilaterals and Lesson 7-10 on the validity of constructions. (David Joyce would approve of this -- but he'd take it a step forward and not even introduce the constructions until this lesson.) Meanwhile, the old Lesson 7-8 on the SAS Inequality (or "Hinge Theorem") no longer appears in the new text.
-- Chapter 8 has only one new section -- Lesson 8-7, "Special Right Triangles," is the old 14-1. This is so that special right triangles are closely connected to the Pythagorean Theorem.
-- Chapter 9 was always a flimsy chapter in the old book -- it's on 3D figures, yet most of the important info on 3D figures (surface area and volume) don't appear until Chapter 10. Now surface area has moved up to Chapter 9, reserving Chapter 10 for volume (except for the surface area of a sphere, which remains in Chapter 10). The old Lesson 9-8 on the Four-Color Theorem has been dropped, but that was always a lesson that was "just for fun."
-- The last section of the old Chapter 12 (side-splitter) is now the first section of Chapter 13, which is the new trig chapter. Lesson 13-2 is a new lesson on the Angle Bisector Theorem, and Lesson 13-4 is a new lesson on the golden ratio. I've actually seen these ideas used before -- including on the Pappas Mathematical Calendar -- but this is the first time I've seen them in a text as separate lessons. This is followed by lessons on the three trig ratios. Vectors, meanwhile, have moved up to Lesson 4-6, so that they can be closely connected to translations.
-- Chapter 14 should be like the old Chapter 15, but there are a few changes here as well. Ironically, I, like the text, moved tangents to circles to this chapter (Lesson 14-4) so that it would be closer to the other important circle theorem, the Inscribed Angle Theorem. But inscribed angles have been moved up in the new text to Lesson 6-3. This places that lesson closer to the Isosceles Triangle Theorem, which is used in the proof of the theorem. Meanwhile, Lesson 14-6 technically corresponds to 15-4 ("Locating the Center of a Circle") of the old text, but it has been beefed up. Instead of just the circumcenter, it discusses the other three concurrency theorems (important for Common Core) as well as the nine-point circle of a triangle.
Meanwhile, of immediate concern are Chapters 1 and 2 of the new text. Unlike the others, these chapters haven't changed much from the old text. The only difference in Chapter 2 is that Lesson 2-3, on if-then statements in BASIC, has been dropped. (After all, who uses BASIC anymore, except on computer emulators, as in yesterday's post?) In its place is a new lesson on making conjectures.
Two of the lessons of Chapter 1 have been dropped. One of them is actually today's Lesson 1-1, as its material has been combined with the old Lesson 1-4. Meanwhile, Lesson 1-5, on perspective, has been delayed to Chapter 9 (which makes sense as perspective is definitely related to 3D). The last lesson in Chapter 1 is on technology -- a "dynamic geometry system," or DGS. (That's right -- goodbye BASIC, hello DGS!) Officially, it still corresponds to the last lesson of the old Chapter 1, since this lesson still introduces the Triangle Inequality Postulate (but now students can test out this postulate for themselves on the DGS).
On the blog, I'll continue to follow the old Second Edition of the U of Chicago text. But if I ever get to sub in a classroom again, the classroom has priority over the U of Chicago order. In this case, if an important lesson is skipped, I could sneak the lesson in by following the Third Edition order instead.
Okay, without further ado, let's finally start the U of Chicago text!
Lesson 1-1 of the U of Chicago text is called "Dots as Points." This isn't a lesson that I covered on the blog before, since for the first three years of this blog, we always began with Lesson 1-4. But I did mention one important idea from this chapter -- the first description of a point:
First description of a point:
A point is a dot.
This is the start of a new school year. Many students enter Geometry having struggled throughout their Algebra I class. Now they come to us in Geometry, and after all the frustration they experienced last year, the first question they ask is, "Why do we have to study Geometry?" Well, the answer is:
A point is a dot.
The old U of Chicago text writes about dot-matrix printers. This isn't relevant to the 21st century, and indeed they don't appear in the modern edition. But here's another question to ask students -- if you didn't have to take math, what would you do at home instead of math homework? And if the answer is "play video games," then guess what -- video game graphics consists of millions of dots. Or, more accurately, they consist of millions of points, since:
A point is a dot.
Images on video games don't come out of nowhere -- someone had to program in the millions of dots, treating them as points -- therefore using Geometry. So without Geometry, video games don't exist. If you want to answer that question -- "What would you do if there was no math?" -- then next time choose something that doesn't require math to build.
In the modern version of the text, there is a brief mention of pixels as part of both computer images and digital camera images. Again, it's not emphasized as much, since "dots as points" must share the new Lesson 1-3 with "network nodes as points."
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