Lesson 1-1: Dots as Points (Day 11)

Today on her Mathematics Calendar 2019, Theoni Pappas writes:

A 112-meter rope is used to mark off a rectangular planter box. What width should the rectangle be to maximize the area?

As it turns out, of all rectangles with the same perimeter, the square has the most area. Thus we are asking for the side length of a square whose perimeter is 112.

Clearly, the answer must be 112/4 = 28. Therefore the desired width is 28 meters -- and of course, today's date is the 28th.

Let's locate this lesson in the text. The area of a rectangle appears in Lesson 8-3. Meanwhile, the Isoperimetric Inequality (i.e., that a circle maximizes area for a given perimeter) appears near the end of the text, in Lesson 15-8. If it's not known that a square maximizes area for a given perimeter, then we must use algebra (or even calculus) to prove this fact.

These lesson numbers refer to the U of Chicago text, of course. And anyway, speaking of that text...

This is what I wrote last year about today's lesson:

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 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 the Mocha computer emulator for music?) 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 lesson has 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."

Here is the Blaugust prompt for today:

Observe yourself!  Record your lesson using your phone in your pocket and use it to reflect

Well, you already know that I have very few photos or videos of my classroom to post. Anyway, today's a great day to sit out Blaugust anyway. I already spent much of this post discussing the structure of the U of Chicago text.

Meanwhile, today we return to the blog of Jenna Laib:

https://jennalaib.wordpress.com/
https://jennalaib.wordpress.com/2019/08/28/choosing-a-story-to-tell-examining-a-lesson-close-in-1st-grade/

I had planned to spend my first time in Room 12 observing the classroom teacher and getting to know her students. Instead, the teacher texted me the day before to let me know that she had come down with pneumonia. I immediately offered to teach the class.

…not that I had any ideas for a lesson. First Grade. I didn’t know the students. The classroom teacher, Natalie, would probably be out all week.

Once again, Laib is an elementary math coach. Well, apparently she's a substitute teacher this week -- just as I am. Then again, I don't sub in first grade classrooms.

So as usual, the only Blaugust poster is unrepresentative of what I hope to teach someday. But it's interesting to view Laib's post in light of today's Blaugust topic -- photos and videos of lessons.

No, Laib doesn't post any videos, but she does post photos. And I notice that all photos in both of her Blaugust posts are of student work. And so even if I never take photos with a phone in class, there's nothing preventing me from scanning and posting student work.

Well, there's one thing -- copyright. I usually avoid posting anything with a copyright symbol -- usually worksheets published by a textbook company. So if I teach or even sub in a class and wish to post student work, it's likely that I'd be violating the copyright of some company.

Laib's photos contains copyrighted worksheets -- but then again, it's a photo, not a copy. It seems so much worse to scan a worksheet than it is to take a picture of it.

What I could do someday is take a worksheet that I created here on the blog, have students work on it, and then post their student work here. This is tricky -- it would require a multi-day assignment (which Laib is in fact doing this week). The first day is for finding out what lesson the students are on (to determine which worksheet to use), the second for actually assigning the worksheet, and the third is for taking the worksheet home, posting it, and returning it to the student. (The third day is not needed if I have access to a photocopier.) Even then, it only works if it fits into the regular teacher's lesson plan (for example, if we happen to exhaust everything on the lesson plan).

I may never be able to have a video of one of my lessons. But classroom photos are common -- and indeed, most teacher blogs or Twitter accounts contain many such photos. I believe that my cellphone (low-level, not a smartphone) has a camera, but I almost never use it.

If I ever get my own classroom, I could adopt the following phone policy -- cellphones are almost always forbidden in my class. Here's one exception for the "almost" -- if the student offers to take a bloggable (or tweetable) photo of the classroom for me. In other words, if I see a phone out, the student must take such a photo and send it to me to avoid confiscation. Here "bloggable"/"tweetable" means that it contains no student faces, since these can't be posted without parental permission. For example, Laib's photo of the girl P. is bloggable because it contains only her hand and her work, not her face.

That's all I really have to say about Laib's post. Oh, I guess I will point out that Laib's first graders, in the process of learning addition, already know their "doubles" (4 + 4 = 8, 5 + 5 = 10, and so on). In learning the arithmetic tables, some facts are known before others.

Here is the Lesson 1-1 worksheet: