Lesson 10-3: Fundamental Properties of Volume (Day 103)

Today I subbed in a high school English class. Since it's not a math class, I won't do a full "Day in the Life" post today.

I will say that there are three freshman classes and two senior classes. Let's focus on the freshman classes, since that's obviously when the behavior problems occur.

The students are currently reading Harper Lee's To Kill a Mockingbird. As we've seen, this is the time of year for persuasive essays in English classes. So today, the freshman are to do a pre-write for such an essay, based on a specific claim about the novel (its characters and their motives). They complete the pre-write on Google Classroom in groups of no more than five.

The first freshman class is second period. In that class, some students take out a deck of cards and claim that they're "multi-tasking" -- a common student trick. I decide that too many students aren't paying attention to the assignment (which they have only today to complete), and so my plan is to try harder to keep the students on task in subsequent classes.

The second freshman class is sixth period. For some strange reason, this class has an aide, even though all classes are gen ed. Thus she's the one who keeps the students on task.

The last freshman class is seventh period. So this is when I make sure to watch the students to make sure they don't stray off task. I begin by taking the printed roster and writing down which students are in which groups. This will help me later on when I write down the good and bad lists.

But there's a problem -- one of the groups has six students, even though I clearly tell them (and it's stated on Google Classroom) that there are to be no more than five students. I inform the students that on my roster, I'll only list five of them. And indeed, five of the students tell me that the sixth student doesn't belong in their group. It's an easy to decision to write this guy on the bad list, especially since I catch him breaking another class rule -- no food in class. He's eating some Girl Scout cookies and passing the rest of them to other students. (Yes, it's that time of year again -- and in fact, I buy three boxes myself from a Girl Scout in one of the senior classes.) At the end of class, the five students inform me that this guy hasn't done any of the work.

Then after school, this sixth guy makes a counterclaim -- in reality, he's done all of the work and the other five are the freeloaders! He then tells me that his group has forgotten to click "Submit." I warn him that the regular teacher might be grading the assignments over the weekend, and so by the time they see her again, she'll have already assumed that his group has done no work.

But unfortunately, he interprets this as an argument on my part -- that I'm saying that he personally did no work, that I'm believing the other five over him. He storms out of the room, takes out his cell phone, and texts the regular teacher to give his side of the story.

These days, I wish to solve problems without arguing. But here's the thing -- if the student interprets what I'm saying as an argument, then it's an argument, period. This sixth guy thinks that I'm arguing with him, but all I want to do is warn him. Regardless of who's really doing the work, my fear is that he'll assume that he can just submit the assignment when he sees his teacher on Monday. My fear is that the teacher will grade the assignments on Saturday and enter them into the grade book on Sunday, so that by the time they see each other on Monday, all six of them will already have a zero. I want to know how I can communicate those fears without making it sound like an argument. In other words, somehow I needed to separate "Did you submit the work?" from "Who did the work?"

Whose side should I have taken -- the sixth guy or the other five? On one hand, I write the sixth guy's name because he's eating Thin Mints in class, which is forbidden. (Instead of Girl Scouts, he should have been thinking about the girl named Scout -- the protagonist of the novel.) Yet as wrong as he is to eat the cookies, that has nothing to do with whether he or the others are doing the work. On the other hand, I do spot one of the other five looking at pictures of cars on his Chromebook, which is indeed evidence that he's not fully on task. (Then again, the other four might still be on task.)

This also goes back to the idea of group assignments -- how many students there should be in a group, or whether there should even be group assignments at all. Many traditionalists oppose group assignments for this reason -- only some of the group members will actually work. And larger groups are more likely to have freeloaders than smaller groups. Of course, I'm just a sub who must follow the lesson plan, so whether I personally agree with the traditionalists is a moot point. I'm required to give the group assignment, with the group sizes as specified by the regular teacher. And the regular teacher writes that the students are to choose their own groups.

Two years ago at the old charter school, I had a similar argument with my sixth graders. The Illinois State text contained a partner assignment (groups of two). I figure that some of the students will try to form groups of three anyway, so I tell them that they can make groups of 2-3. The argument occurs when four students want to form a group.

Teachers know that students will complain that they don't want to be separated from their friends, and so we often let them choose their own groups. But chances are, the groups they form won't all be of the correct size. Today, while one group has six members, another group has only two. And we even see this on the first day at the old charter, where I let them choose their own seats (not groups). The seats are clustered in groups of four, but then five eighth grade girls wish to sit together (with the "special scholar" being the one left out).

How to handle the seating chart is a bit more complicated. But as for a group project, we can assume that no matter what size of group we specify, someone will try to make a group one larger. So if we want groups of, say, four, then five students will inevitably try to form a group. Thus, instead of telling them that we want groups of four, we tell them to form groups of three. That way, we can show leniency when the inevitable foursome tries to form (since four is the maximum size that we're willing to have anyway).

I could have tried this today -- tell them to form groups of four instead of five. But this trick likely wouldn't have worked -- they would have logged into Google Classroom, read the direction to form groups of at most five, and then figure out that I'm trying to trick them. But this trick might work if I'm in my own classroom instead of subbing.

By the way, when I was a young student, I never read To Kill a Mockingbird. The reason I missed this novel was a bit complex. You see, Mockingbird wasn't always a freshman novel. A few decades ago, our school taught it to juniors. My own class, the Class of 1999, was the last class to read this novel as juniors -- the Class of 2000 was the first to read it as freshmen. (This means that for two years, both juniors and freshmen were assigned the novel.)

And so my junior year came. Most likely, our class would have read the novel right around this time of year (the first of February). The problem was that I never made it to February! At the start of November, I was transferred from the regular program to the magnet program. (I explained this transition in several old posts where I wrote about my old science classes.)

The magnet had its own curriculum. One major difference was that both U.S. History and American Literature were taught to magnet sophomores, instead of juniors as in the regular program (and in the rest of California). Since I was a year behind the other magnet students, I was placed as a junior into magnet sophomore English. I was the only '99'er, surrounded by members of the Class of 2000.

Freshman English was more or less the same inside and outside the magnet. This included To Kill a Mockingbird -- 2000 was the first class to read it as freshmen inside and outside the magnet. Thus in transitioning from regular to magnet, I was moving from a class that was three months away from reading the novel to one that had already completed it nine months earlier.

And that's the real reason why I never read To Kill a Mockingbird.

Today I will be doing Lesson 10-3 of the U of Chicago text, on the fundamental properties of volume.

(Also, I might add that Lessons 10-1 and 10-3 also flow naturally from last month's 8-8 and 8-9. Both the formulas for a circle appear in the surface area formula of a cylinder -- the circumference of a circle leads to the lateral area of a cylinder and the area of a circle leads to the full surface area including the bases.) But some people might point out that this would confuse the students even more. Instead of doing all of the surface area formulas at once (as the U of Chicago does) and all of the volume formulas at once, we'd keep going back and forth between surface area and volume. But another argument is that it's better to do all of the prism formulas at once, then all of the pyramid formulas, and finally all of the sphere formulas.

The cornerstone of Lesson 10-3 is a Volume Postulate. The text even points out the resemblance of the Volume Postulate of 10-3 to the Area Postulate of 8-3:

Volume Postulate:
a. Uniqueness Property: Given a unit cube, every polyhedral solid has a unique volume.
b. Box Volume Formula: The volume of a box with dimensions l, w, and h is lwh.
c. Congruence Property: Congruent figures have the same volume.
d. Additive Property: The volume of the union of two nonoverlapping solids is the sum of the volumes of the solids.

Just as we derived the area of a square from part b of the Area Postulate, we derive the volume of a cube from part b of the Volume Postulate:

Cube Volume Formula:
The volume of a cube with edge s is s^3.

And just as we can derive the area part of the Fundamental Theorem of Similarity from the Square Area Formula, we derive the volume part of the Fundamental Theorem of Similarity from the Cube Volume Formula:

Fundamental Theorem of Similarity:
If G ~ G' and k is the scale factor, then
(c) Volume(G') = k^3 * Volume(G) or Volume(G') / Volume(G) = k^3.

Last year I stopped posting Euclid's propositions at this point, and so I do likewise this year.

Today is an activity day, the only such day this week. Last year, I connected the worksheet to a special activity I did for Lesson 10-1, which I didn't post with the worksheet on Wednesday. Since my old Lesson 10-3 worksheet refers to the 10-1 activity, it makes sense to restore that project for today's activity day.

This is what I wrote last year about that project:

Today's lesson is on surface areas. But recall that back at the end of the first semester, I mentioned Dan Meyer, the King of the MTBoS (Math Twitter Blogosphere), and his famous 3-act lessons. I pointed out how one of his lessons was based on surface area, and so I would wait until we reached surface area before doing his lesson. 

Well, we've reached surface area. And so I present Dan Meyer's 3-act activity, "Dandy Candies," a lesson on surface area.

Meyer includes some additional questions for teachers to ask the students, but I only included what fits onto a single student page. Those who want the extra information can get it directly from Meyer:

http://www.101qs.com/3038