Today I subbed in a high school math class. It's in my new district. Since it's a math class, I definitely want to do "A Day in the Life" today.
6:40 -- The day starts with zero period. (Yes -- here "zero period" actually means zero period.) This is an AP Calculus AB class.
Unfortunately, zero also means zero -- as in zero seniors. Today, apparently, is "senior ditch day." There are certainly no in-person students today in this class.
As for online students, I have problems accessing the class. Normally, this teacher uses Zoom, but in order to avoid changing the Zoom host, he sets up a temporary Google Meet for today. So I go to this Meet and see that, sure enough, there are no students here either. But I make one huge mistake with the Meet today.
Here's what the error is -- Google shows that there's no one in the Meet, but then I never actually join the Meet myself. (If you've used Google Meet, you know what I mean -- just before you click on the Meet, it shows you who else is there.) This mistake is unacceptable -- after all, I used Google Meet before during the long-term assignment, so I should know how to use it. What threw me off, of course, is there being no students already in the Meet -- there's nothing like senior ditch day at a middle school. When there's someone else in the Meet, of course I know that I need to join the Meet for that student to see me. (Also, lately most teachers have been letting me use Zoom, so the sudden switch to Google also threw me off.)
By the way, this senior ditch day raises the question, what will I do if I ever become a regular teacher with seniors in the class? Some traditionalists have discussed this before -- one trick would be to assign some sort of "test" today. No, it wouldn't be like today's actual assignment -- to practice the free response question from the 2015 AP Calc exam. Instead, it would be more like a Hero Quiz -- every question would be very easy so that anyone who attends the class is almost guaranteed an A. For example, the test could have 10-15 derivatives, where every function is a polynomial or something very simple like e^x.
7:55 -- Second period begins. This is the first of two Geometry classes.
Since it's Geometry, there are no seniors in this class. A handful of in-person students arrive, but there still are no online students -- and that's when I figure out my Google Meet error. I immediately enter the meet, and I finally see all the online students.
This explains why what I do in zero period is a mistake. When I first enter the password and it shows no one in the call, that's only true for the moment that I type in the password. If someone logged in five minutes before me, I don't see it. And if someone logs in five minutes after me, I still don't see it -- the message "no one else is in this call" remains on the screen, even if a dozen students show up. That's why I should have joined the meet, not keep the "no one else is in this call" message up. Instead, some students might have tried the enter the meet, see the "no one else is in this call" message themselves, and then just quit, figuring that they did something wrong. Since I have the password, I'm the one who needs to have taken the first step and join the call myself.
By the way, there are a few juniors enrolled in this AP Calc class. Only one of these juniors is in Cohort B and hasn't opted out of hybrid, but as I said, no student is in-person zero period. (This happens -- often when there is only one in-person student scheduled, that student doesn't appear. Indeed, this occurred with some of my Grade 10-11 classes yesterday and the day before. Oh, and I already declared the wager over, so don't try to calculate whether I win or lose with today's classes.)
I can't keep moping on about the Calc class that I ruined -- Geometry is in the room now, and that's our favorite math class. These students have a worksheet on finding surface areas and volumes of prisms.
The song for today is a no-brainer -- "All About That Base and Height" (3D) is the obvious choice. I perform this right at the start of class, and then repeat the relevant lines for each problem. At the end of class during tutorial (last 20 minutes), I also sing "U-N-I-T Rate! Rate! Rate!"
This worksheet, called "Surface Area & Volume of Prisms Plug & Chug," is an interesting one. The answer to each of the fifteen problems is used in the next problem. For example, the first question is:
A. Find the lateral area of a regular hexagonal prism that has a side length of 3 and a height of 12.
The formula to use here is L.A. = ph. Since the base is a regular hexagon with side length 3, its perimeter is 18. Then the lateral area is (18)(12) = 216 square units. Then the next problem is:
B. Find the height of the cube if the volume is ______ cm^3.
So we plug in the answer for A into the question for B -- the volume of the cube is 216 cm^3. (The fact that we switched from square to cubic units is irrelevant -- the number is all that matters.) The height, of course, is cbrt(216) = 6 cm. Then the next problem is:
C. Find the surface area of a rectangular prism that has a base edge of ______ cm and 10 cm and a height of 3 cm.
So we plug in the answer for B into the question for C -- the box has dimensions 6 * 10 * 3 and we need its surface area, and so on.
Some students ask me about Question H. Of course, we can't skip any questions due to the chain of questions and answers here -- we need to know the answer to G before we can look at H. But notice what Question G is:
G. What is the side length of a cube if its lateral area is ______ units^2 and the height is 2 units?
Notice that we don't need to know the answer to Question F to fill in the box. I'll give you a hint -- rewrite the question without the part with the blank:
What is the side length of a cube if its height is 2 units?
Ahh -- the dimensions of a cube are all equal, so if the height is 2 units, so is the the side length! Now we can plug in the answer 2 into Question H:
H. Find the surface area of an equilateral triangular prism with a side length of _____ and a height of 6.
The formula we need here is S.A. = L.A. + 2B = ph + 2B. We know that h = 6, and since the side length of the equilateral triangle is 2, we also have p = 6 as its perimeter. This leaves us to find B, which we can do by dividing it into two 30-60-90 triangles. The side opposite 30 is 1, and so the side opposite 60 is sqrt(3), thus giving (1/2)(1)sqrt(3) = sqrt(3) as the area.
Substituting this into the formula, we obtain S.A. = (6)(6) + 2sqrt(3) = 36 + 2sqrt(3). This is the answer to H that we must plug into Question I. But there's a problem with Question I. I'll save it for the end of this post, since I want to finish "A Day in the Life" first.
During tutorial, I send out an email via Canvas to the entire zero period class, explaining how I've messed up the Google Meet. I tell them that if they have time, they should log into the Meet during either second or fourth period tutorial. Two girls do this -- one senior and one junior. And so I mark them as attending online on the roll sheet.
9:50 -- Second period leaves for snack break.
10:15 -- Fourth period arrives. This is the second of two Geometry classes.
During this period, I think that Question J also contains an error. But what I don't realize is that while my key contains the error, the student version has been corrected.
Unfortunately, I don't realize this until I come home to type this blog entry. And I believe that I confuse some of the students here -- I write one of the incorrect numbers from my key, and an online student asks where I get those numbers from.
12:05 -- Fourth period leaves for lunch. A third Calc student -- a senior guy -- logs in for attendance. I also have another guy remain from fourth period to get extra help on one of the problems.
In fact, I stay in the room for lunch in case any more zero period students appear. During this time, I show my notes for Question I on the board so that the teacher sees that the "book" made errors with these problems.
12:55 -- For sixth period, this teacher is a Track coach. In fact, he coaches girls Cross Country and distance Track. The regular teacher tells me that I don't have to stay since assistant coaches will deal with attendance. But since this is my sport, I decide to stay -- and so this is a continuation of our Track discussion from yesterday's post.
Today's practice is a "premeet" -- that is, a short practice on the day before a Track meet. In fact, the athletes will be participating in two difference races tomorrow. To my surprise, these races are more like invitationals where many schools participate. Recall that there were no XC invites, and my alma mater has only dual Track meets until League Prelims.
I notice that there appears to be many girls in this school's distance program. Recall that at the dual meet I attended earlier this week, no girls from my alma mater participated in the 1600. I wonder how many girls from today's school will contest the 1600 tomorrow.
I'm also paying attention to the number of seniors participating in Track, which is especially relevant to my COVID-97 What If story. One coach tells me that most of the athletes he sees today are freshmen and sophomores -- but then again, it's still senior ditch day. He's hoping that some of the seniors who ditched their academic classes will come to school in time for the workout.
1:10 -- I decide that I've seen enough of the premeet workout. Since I don't really need to stay for this period, I leave at this time.
Meanwhile, right at this moment, the regular teacher sends out an email to the Calc students. They are to work on a complete practice test over the weekend. He addresses today's obvious senioritis in his email -- he reminds them that no matter how far they're looking ahead, they shouldn't look past the AP exam coming up in 2 1/2 weeks.
Today is Sunday, the third day of the week on the Eleven Calendar:
Resolution #3: We remember math like riding a bicycle.
Obviously, the Geometry students need to remember their surface area and volume formulas. Recall that there's an entire lesson of the U of Chicago text -- Lesson 10-6, Remembering Formulas. (The formulas themselves appear in Lessons 10-1 and 10-5.) And of course, today's "All About That Base and Height" is there to help students remember the formulas. Indeed, I tell them that it's often easier to remember what is sung that what is said.
This is what I wrote two years ago about today's lesson:
Lesson 14-7 of the U of Chicago text is on adding vectors using trigonometry -- and we can't skip it because it appears in the following Common Core standard:
CCSS.MATH.CONTENT.HSN.VM.B.4.B
Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
Notice that we are essentially converting the vectors from polar to rectangular form, adding their components, and then converting the sum back to polar form. And all of this is done without the students' even knowing what polar coordinates are.
In an actual trig course (as part of precalculus), we find out that we can avoid thinking about angles larger than 90 degrees by considering the reference angle -- the angle formed by the x-axis and the terminal side of the original angle. We see that even though the U of Chicago's Geometry text doesn't teach reference angles, the angles shown in the text are always formed using the x-axis -- that is, the west-east axis -- and never using the y-axis. So the U of Chicago, while not explicitly teaching reference angles, clearly has these reference angles in mind when writing this chapter.
The text states that one of the two components is found using the sine of the angle, while the other is using the cosine. But because of the way the angles are drawn, the horizontal component will always use the cosine and the vertical component will always use the sine -- just as they would be for polar coordinates in a trig class.
To convert from rectangular back to polar coordinates, we use the distance formula and the inverse tangent function. This is the only time that an inverse trigonometric function appears in the U of Chicago text -- although many other geometry texts discuss inverse trig in more detail.
Today is an activity day -- and perhaps I should search for a Desmos lesson on vectors. But I like to priority to things that happen in an actual classroom over all other considerations. And so I'd rather post today's Plug & Chug worksheet instead. (Is it pandemic-friendly? Presumably, yes -- it was given in a real classroom during a real pandemic, so it counts.)
I'm justified in giving this activity now, even though we're long past Chapter 10, the U of Chicago chapter where surface area and volume are taught. That's because questions like H above require knowledge of 30-60-90 triangles -- and such triangles are taught right here in Chapter 14. In fact, some might consider this a weakness of the U of Chicago text -- since measurement is taught before special right triangles, we can't use the special right triangles in our volume formulas. In most other texts, special right triangles are taught first, and so students are able to find the surface area of an equilateral triangular prism.
The Plug & Chug question chain is admittedly tricky -- if students get one question wrong, then they'll get the rest of the worksheet wrong. Of course, other areas of math are like this -- including the free response question that the AP Calc students are (supposed to be) working on today. But today's questions are more contrived than those on the AP -- the Calc free response parts are at least related, while Geometry has a 216 cm^3-volume cube just because the previous problem has a 216-unit^2 surface area. Today I go over Questions A, B, and then G through J. This means that they don't need to go more than five questions (from B to G, and from J to O) without knowing whether they're correct or on a path of consecutive wrong answers.
Of course, before we can really commit to using today's worksheet as an activity, we need to figure out what's wrong with Question I. Let's look at these now:
I. A regular hexagonal prism has a surface area of _____. What is the new surface area if its dimensions are changed by three and two-thirds?
First of all, this is a poorly worded question. What does it mean to "change" the dimensions of a prism "by three and two-thirds"? Here, "three and two thirds" sounds like the mixed number 3 2/3 = 11/3 -- perhaps it means that we're performing a dilation with a scale factor of 11/3. So the surface area would need to be increased by a factor of (11/3)^2.
Well, here's what the teacher wrote in the key -- recalling that the original surface area is 36 + 2sqrt(3), which we get from Question H:
(36 + 2sqrt(3))(3)(2/3) = (36 + 2sqrt(3))2 = 72 + 4sqrt(3)
Oh, so the "three" and "two-thirds" are intended to be two separate factors, not a mixed number.
But then there's another, deeper concern here. This problem is assuming that if we multiply a dimension of a prism by a constant, then we automatically multiply the surface area by that constant. As it turns out, this is definitely true for volume (and Question D asks this for volume), but it's not necessarily true for surface area.
Think about it -- the surface area of a prism is S.A. = LA + 2B = ph + 2B. Suppose we triple the height of this prism. Then the lateral area ph also triples -- but the base B remains unchanged. Thus the entire surface area does not triple (even though the volume does). Claiming that the surface area triples is like saying that if each you and I each have $100, and I triple my money while you keep the same amount, then somehow our money has tripled to $600.
Consider a unit cube -- let's triple one dimension and take 2/3 of another dimension. The resulting box does indeed have volume 2, but what is its new surface area? Taking 3 to be the new height:
S.A. = ph + 2B = (1 + 2/3 + 1 + 2/3)(3) + 2(1)(2/3) = (10/3)3 + 4/3 = 30/3 + 4/3 = 34/3
which is not the same as double the surface area of the unit cube (which would be 12).
And this is starting with a unit cube. The actual Question I is even worse, since here we're beginning with a regular hexagonal prism. Notice that by stretching one dimension by three and shrinking another by 2/3, the base is no longer a regular hexagon! Also, as it turns out, the answer depends on which of the dimensions we triple and which one we multiply by 2/3.
Why is surface area so different from volume in this regard? Notice that ordinary area seems to follow the volume pattern -- if we take a plane figure and multiply one dimension by m and another by n, the area is indeed multiplied by mn. So why doesn't it work for surface area?
If you think about it, the Fundamental Theorem of Similarity in Lesson 12-6 is proved by estimating a plane figure with unit squares (cubes), and then replacing them with larger or smaller squares (cubes) of the new dimension. This works even when we scale by different factors in different dimensions -- this time, we tile the image with rectangles (boxes) with the correct dimensions.
But this doesn't work for surface area. Suppose we take a unit cube and triple it in the x dimension, and then we unfold it into a net. We might be careful enough to keep the x sides in the x direction and the y sides in the y direction -- but what about z. When we unfold, some of the z sides are in the x direction (specifically those on the yz faces) and some in the y direction (those on the xz faces). If we then tried to stretch this net by a factor of three in the x direction, the z sides in the x direction get tripled, while those in the y direction don't.
This explains why the Fundamental Theorem of Similarity doesn't work for surface area unless we scale by the same factor in all dimensions -- only then will all the z sides (in both directions) will also get multiplied by the same factor.
Let's try solving Question I for a hexagonal prism. We ought to do it for a prism of surface area of exactly 36 + 2sqrt(3) as specified in the original question. But I'll make a slight change -- the surface area will be 36 + 12sqrt(3) instead. Then each base can have area 6sqrt(3) -- obtained by gluing six of the equilateral triangles in Question H to make a hexagon. Each side has length 2, and so the perimeter must be 12. The lateral area is 36, implying that the height is 3.
Now we perform the transformation, Let's do the easy one first -- we multiply the height dimension by 2/3, so the new height is 3(2/3) = 2. The harder task is to stretch the hexagonal base by a factor of 3 -- we do know that the area will also triple to 18sqrt(3), but we need to see what happens to the perimeter.
Two of the sides of the original hexagon are already in the direction of the stretch, so their new length will triple to 6. But the other four sides won't. Instead, we can think of them as vectors (yes, taking it back to today's original topic) -- the magnitude is 2, but we must resolve them into their components that are perpendicular.
If we draw it out, we see that the original hexagon can be divided into a rectangle with side lengths 2 and 2sqrt(3), and then four 30-60-90 triangles with legs 1 and sqrt(3), with the 2 and 1 sides in the direction of the stretch. Then the new triangle has legs 3 and sqrt(3) -- which, as you might notice, is still a 30-60-90 triangle since one leg is exactly sqrt(3) times the other. The new hypotenuses, forming four sides of the image hexagon, are 2sqrt(3). (Notice that the new hexagon has four 150-degree angles and two 60-degree angles.)
So the new perimeter is 12 + 8sqrt(3). Multiplying this by the height 2 gives new the lateral area, which works out to 24 + 16sqrt(3). Adding this to the two bases, each with area 18sqrt(3), finally gives us the new area, 24 + 52sqrt(3). Obviously, going from 36 + 12sqrt(3) to 24 + 52sqrt(3) isn't a doubling.
OK, so that's what's wrong with Question I. I can't really post this worksheet because it does contain a copyright symbol. And the owner of the copyright is not a publisher -- it's Katrina Newell.
Newell has both a blog and a Twitter account, although she hasn't blogged since before the pandemic and hasn't tweeted since calendar year 2020. Here's a link to her blog:
https://newellssecondarymath.blogspot.com/
I don't want to criticize Newell too much. After all, I enjoy her worksheet -- it supplements what's missing from the U of Chicago text very well. And apparently, she graciously fixed the error that someone must have spotted in Question J. Indeed, I look back at her old post from June 2019 and notice the following:
https://newellssecondarymath.blogspot.com/2019/06/2018-2019-school-year-review.html
Singing in math class -- that's a silly idea! (Note: Pay no attention to any references to "All About That Base and Height" and "U-N-I-T Rate! Rate! Rate!" from earlier in this post.)
Hmm, perhaps I might refer to Question I on Twitter in order to draw attention to it.
Here's a link to where Newell discusses today's worksheet:
https://newellssecondarymath.blogspot.com/2016/04/surface-area-volume-of-prisms-unit.html
Of this "Plug & Chug" activity, she writes:
OK, so here's a link to today's old lesson. The intention is to follow it with something like Newell's "Plug & Chug" activity. Notice that Newell's lessons are copyrighted, and they do cost money. I will not violate that copyright by posting this lesson, or even writing my own version (with, say, a new version of Question I).
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