Monday, February 5, 2018

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

Lesson 10-3 of the U of Chicago text is called "Fundamental Properties of Volume." In the modern Third Edition of the text, fundamental properties of volume appear in Lesson 10-1.

Meanwhile, you may have noticed it's been some time since my last post labeled "subbing." I haven't been receiving very many substitute teaching assignments from the district whose calendar we've been counting on the blog.

This week I'm doing something about it -- I've added a brand new district. Unlike my old high school only district, this is a unified K-12 district. The new district calendar is a near-Labor Day Start, which is what I call any calendar that starts the week before Labor Day. (Presumably, the calendar was formerly a Labor Day Start calendar, then school started a week earlier to accommodate being closed the entire week of Thanksgiving.)

The second semester began last Tuesday, so that was Day 91. Thus today is Day 95 at the new school, but of course I'll continue to observe the old district calendar on the blog. I'll mention any other differences between the two district calendars on the blog as they arise.

I want to get in the habit of posting a "Day in the Life" for all subbing assignments, even if the subject isn't math. Eventually the goal is to post a Day in the Life for math classes only.

Today I subbed in a seventh grade English class. This district is unusual in that sixth grade is still considered elementary school, and so the middle schools here span only Grades 7-8. (In other states, the trend is the opposite -- to start including fifth grade as middle school!)

8:15 -- I arrive at the middle school. The day begins with a short homeroom period, during which the announcements are made.

8:30 -- Fourth period begins. In this district, middle schools rotate their periods, so that the day doesn't always begin with first period. Today starts with fourth period, while tomorrow will begin with fifth period, Wednesday with sixth, and so on. The rotation is pure in that there is no direct correlation between the schedule and the day of the week.

Why would some schools have a rotating schedule? I suspect it's to fight the correlation between the time of day a class meets and student behavior in that class. On a traditional schedule, students behave best during first period because they are too sleepy to talk or do a lot -- but then again, they might too sleepy to learn as well. On the other hand, students are alert enough in sixth period both to learn and to misbehave.

(Arguably, my middle school last year had a "rotation" of sorts, in that I began the day with seventh graders on Mondays and Thursdays and sixth graders on Tuesdays and Fridays -- and no, don't even get me started with the Wednesday "schedule.")

In all classes today, the seventh graders are preparing to write a persuasive essay. The topic is reality TV and whether it's harmful entertainment or harmless entertainment. Today they are to do research on Chromebooks and come up with a claim and details to support their claim.

9:25 -- Fourth period leaves and fifth period arrives.

10:15 -- The students leave for nutrition.

10:30 -- Sixth period arrives. This class has a TA, an eighth grade girl. I notice that she's wearing a Philadelphia Eagles jersey, and so I ask her about yesterday's Super Bowl LII. Of course, she's delighted with the outcome of the game.

Since the teacher doesn't leave any work for the TA to do, she does work for other classes, including her math class. So this marks the only time I see any math today. The worksheet appears to be on exponents, but she only has two problems to finish before she puts it away. I don't have much chance to look at the worksheet or help her on it.

11:25 -- Sixth period leaves and first period arrives.

12:15 -- The students leave for lunch. As it turns out, this teacher has two conference periods -- namely second and third periods. And so on days beginning with fourth period, this teacher's day ends at lunchtime!

Now when I write these "Day in the Life" post, the emphasis will be on the seven New Year's Resolutions and how well I'm fulfilling them. This is my first opportunity to work on living up to the seven rules of good teaching. Today I look at the first resolution:

1. Implement a classroom management system based on how students actually think.

Today is a grand opportunity to work on this first resolution. Even though the students are preparing to write the first draft of their essay, their teacher is treating this like a test -- so in particular, they are not allowed to talk. This means that I must implement good classroom management in order to enforce the "no talking rule."

Well, the best behaved class is fourth period and the worst behaved class is first period. You might point out that this is easily explained by the schedule -- fourth period starts the day, while first period is just before lunch -- and indeed it explains why schools want to rotate the schedule at all. Perhaps if the classes occur in numerical order, first period would be quiet and fourth period loud -- or maybe not after all. Fourth period is officially an honors class, and so we expect this class to be best behaved no matter when it occurs in the day.

Today I manage the class without any sort of Conjectures/"Who Am I?" game or indeed any sort of points system. In many ways, the game has become a crutch where the only punishment I give students is to deduct points from their group. And besides -- the group aspect of the game doesn't fit with the individual assignment and the "no talking" rule that I'm trying to enforce. Instead, the goal is to write down just the names of the individuals who are breaking that rule by talking.

Still, I think that I could have managed the most talkative class, first period, much better. What keeps happening is that a student talks, and so I remind them not to talk. Then a second person talks, and I give another gentle reminder. It doesn't seem right to give the first person a warning and write down the second student's name -- but what happens is that the first person starts talking again. In other words, the "no talking" rule has become neutered, because no one ever gets punished for talking.

Here's what I should have done -- I should have established a discipline hierarchy. The first time a student talks, I give the student a "teacher look." The second time a student talks, I give the student an official warning by speaking aloud the name which I'm about to write down. Then the third time a student talks, I write down that name. In this case, a seating chart is available, and so I have all of their names available. Looking at the chart and speaking the name aloud also makes it easier for me to remember which students I've already warned, so that I know which names to write down. Even though I do use teacher look today, as well as say some of their names aloud, I didn't follow a strict hierarchy in doing so.

Another problem I've had recently is the issue of students' talking while attendance is being taken. I decide today to call roll directly from the seating chart, which allows me to match names to faces and determine who is following the seating chart and who is disobeying it. I know that when a class is talkative even during attendance, it definitely won't be a well-behaved class in the end. When the first period students are talking during roll, I could have gone back to use teacher look and name aloud the previously called students who where talking. This would have driven home the point that today, this will be a silence-dominated class.

In the end, I write down one name -- the lone male student who doesn't follow the seating chart. Even though several students are talking, he seems to be in the center of the conversation. I don't like singling out students, but in this situation he singles himself out by not sitting in his assigned seat.

Actually, there's a second student who doesn't follow the seating chart -- he tries to sit in the back, but the assistant principal enters the room and takes the second boy with her (presumably for a previous offense in another class). Then the aforementioned boy moves to that same back seat. This should have raised a red flag in my mind -- the sort of student who moves to that back seat is the sort who gets in trouble with assistant principals.

Since first period is the last class of the day in this room, I tell the students to plug the Chromebooks in to be charged. The teacher has labeled all of the charging spots with a number, and so there is a definite place to plug in each laptop. This is a huge difference from IXL time last year, when my failure to label the slots meant that the end of computer time was always chaotic.

So far, I need to continue working on fulfilling the first resolution. In my next post I'll evaluate my ability to keep the second resolution. Yes, I already have a subbing job in my new district for tomorrow -- a far cry from my situation in my old district. I'll be in a different class at a different school -- you'll find out more tomorrow.

This is what I wrote two years ago about today's lesson:

128. One-inch cubes are stacked as shown below. What is the total surface area?

So clearly we have a surface area problem. Notice that in some ways, last week's Dan Meyer problem may prepare the students for this problem:

134. The short stairway shown below is made of solid concrete. The height and width of each step is 10 inches (in.). The length is 20 inches. What is the volume, in cubic inches, of the concrete used to create the stairway?

Once again, notice that these surface area and volume questions are all based on the seventh-grade standards, in theory.

Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

Today I will be doing Lesson 10-3 of the U of Chicago text, on the fundamental properties of volume. Lesson 10-3 naturally flows from last week's Meyer project. Then Wednesday's lesson can cover Lesson 10-5, which then naturally flows from both Lessons 10-1 and 10-3 -- in 10-1 we have surface areas of prisms, and in 10-5 we have their volumes.

(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 lw, 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.

Now that I've returned to the classroom, I'm done posting Euclid's old propositions. The idea was to post Euclid side-by-side with Chapter 9, since David Joyce writes that Geometry students need to learn "the basics of solid geometry." I posted Euclid every day during Chapter 9, as well as the first two sections of Chapter 10 (which is still Chapter 9 in the modern Third Edition). But now that I'm subbing again, my focus needs to be on what's happening in the classroom, not a 2000-year old book.

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