Today is Pi Day Adam -- the day before Pi Day Eve. Of course, Pi Day Adam becomes significant in years when the actual math holiday falls on a Sunday. In school we frequently celebrate the Adams of holidays that fall on Sundays (such as Mother's Day every year, and Halloween this calendar year).
I was hoping to sub in a math class on this special day, but I didn't. Instead, I subbed in a special ed class in my new district. It's the same class that I recently covered in my February 26th-March 1st posts.
This is one of the reasons that I started giving away pies on Wednesday -- at the time, I already knew that I'd be in the special ed class today, so Wednesday was my last math class before Pi Day. Something similar happened last year -- the last math class before the math holiday was on March 9th. But unfortunately, I taught a zero period that day and didn't have time to get any snacks.
And of course, we all remember what else happened last year. I covered that special ed English class on the last three days -- including Pi Day Eve -- before the coronavirus closure.
Last year, I played Pi Day music on the guitars and ukeleles that teacher had in his classroom. This year, one of the aides is a ukelele player -- but she doesn't bring her instrument to school today. Thus I'm the one who supplies all the music today. I will describe how I attempt to celebrate Pi Day Adam in a special ed class.
For starters, I sing a different Pi Day song each hour on the hour. Sometimes I choose the songs, and at other times I have one of the in-person students select it. Here's what I performed:
OK, today's not supposed to be "A Day in the Life," but this post turns into just that as I mention all the Pi Day songs I perform. In that case, I might as well as the "subbing" label.
In addition, just as I did on Wednesday, I give the students the opportunity to earn individual pies to eat from 7-Eleven. Unfortunately, these students are very low-level and haven't learned about pi yet, so how can I ask them questions about that constant for Pi Day?
(That's right -- the two classes to which I sing most of the Pi Day songs are a sixth grade class and a special ed class, neither of which has officially learned about that constant yet. Again, that's just the luck of the draw, as these are the two classes I spend the most time with leading up to the math holiday.)
So here's how I do it -- last night, I searched for the simplest possible pi worksheet. Here's what I found:
https://www.superteacherworksheets.com/geometry/circ-circle-1_TWDBF.pdf
On this worksheet, every question gives the diameter and asks for the circumference. There are no trick questions where the radius is given instead of the diameter, or where the circumference is given, or any of the other questions that we might ask a gen ed class. Nor do I ask for the area (the problem here isn't with the r, but with the "squared"). So all the student has to do is enter the number in the question on a calculator and multiply it by 3.14.
One guy (all the in-person students in this class are guys -- the ones opting out of hybrid are all girls) is already working on a rectangle perimeter worksheet, so this worksheet is not that much of a stretch from his current level. Another guy is doing mostly addition worksheet, but near the end of the day, he needs one last worksheet to earn the pie, and so I give him the pi worksheet. As soon as I tell him to press the times key on the calculator and not the plus key, he figures it out.
In addition to pies, I hand out pencils -- St. Patrick's Day pencils. I usually try to keep my Pi Day and St. Patrick's Day rewards separate, but I usually hand out pencils in special ed classes and since we're within a week of St. Paddy, I might as well pass out the green pencils.
By the way, the movie the students watch at the end of the day is Pixar's Brave. At first, I think to myself how appropriate this film is for St. Paddy, until I remember that Merida is a Scottish princess, not an Irish lass. (Still, I can see how one could get away with passing this off for St. Patrick's Day, as the Scots and Irish share a Celtic heritage.)
Today is Friday, the first day of the week on both the Eleven and Gregorian Calendars. It's the second straight Eleven week that I begin in this particular special ed class.
Resolution #1: We are good at math. We just need to improve at other things.
And the students do see that they are better at math than they think -- at least the two guys who complete the pi worksheets. They learn a brand new concept today.
A long-term sub is supposed to fill this teaching position starting next week. Therefore this should be the last time that I cover this class, at least for now.
Today on her Daily Epsilon on Math 2021, Rebecca Rapoport writes:
In the close-packing of equal spheres, the number of neighboring spheres for each sphere.
As it turns out, this number has a special name -- the kissing number:
https://mathworld.wolfram.com/KissingNumber.html
In 3D, the kissing number is 12 -- and of course, today's date is the twelfth. This is not very easy to prove at all. No, we can't just use the surface area formulas involving pi to find the answer, not even on Pi Day Adam (even though all the spheres are touching, or "kissing," on their surfaces).
Lesson 12-7 of the U of Chicago text is called "Can There Be Giants?" This is one of those "fun lessons" that we can cover if there's time, but in the past we bypassed it to get to 12-8 and the all-important SSS Similarity.
As I wrote above, Lesson 12-7 naturally lends itself to an activity. The whole idea behind it is that while dilations preserve shape, they don't preserve stability. This is because of the Fundamental Theorem of Similarity -- a dilation of scale factor k changes lengths by a factor of k, areas by a factor of k^2, and volumes by a factor of k^3. Weight varies as the volume, or k^3, while strength varies only as the area (as in surface or cross-sectional area), or k^2. Therefore, the answer to the question in the title of the lesson is no, there can't be giants because their k^2 strength, couldn't be strong enough to carry their own k^3 weight.
This year, not only does Lesson 12-7 land on our weekly activity day anyway, but it's the day that we observe Pi Day in the classroom. If we can't teach an actual pi lesson on Pi Day (and we can't, since we're always in the middle of Chapter 12 on Pi Day), the next best thing is a "fun lesson" like this one.
All that remains then is a way to make this lesson pandemic-friendly. (Yes, I do give a printed pi worksheet in class today, but this is a special ed class where the students regularly get worksheets.)
There does exist a Desmos activity about a giant. It's called "Marcellus the Giant":
https://teacher.desmos.com/activitybuilder/custom/58093e7b37d6769f0b7fde92
Unfortunately, this doesn't quite capture the full idea of Lesson 12-7 -- Marcellus the Giant can't exist because his surface area-to-volume ratio is too small. Indeed, this activity mentions neither surface area nor volume. And when I force a Google search for biological mentions of surface area-to-volume ratios, the results are mostly about why cells can't be too big, not about why organisms can't be too large.
I'll keep the second worksheet from last year (since there are a few questions about pizza on this Pi Day Adam) -- perhaps we can make an activity out of the Marcellus activity on Desmos and then discuss why he can't really exist. We can even combine it with the "Can There Be Humans?" riddle that I asked last year to make a complete activity. But it would be much more convenient if the entire activity (from the dimensions of Marcellus to the "Can There Be Humans?" riddle) were on Desmos.
Once again, a Brobdingnagian is a giant, while a Lilliputian is a tiny being:
Prove: Brobdingnagians don't exist.
Indirect Proof:
Assume that Brobdingnagians exist. In each of the three dimensions, a Brobdingnagian is 12 times as large as a human. So each bone of a Brobdingnagian would have to carry 12 times as much weight as a human bone. Thus a Brobdingnagian standing is like a human carrying 12 times its own weight. But no human can carry 12 times its own weight, a contradiction. Therefore Brobdingnagians can't possibly exist. QED
But there's a problem with this proof -- it can be used to prove that humans don't exist:
Prove: Humans don't exist.
Indirect Proof:
Assume that humans exist. In each of the three dimensions, a human is 12 times as large as a Lilliputian. So each bone of a human would have to carry 12 times as much weight as a Lilliputian bone. Thus a human standing is like a Lilliputian carrying 12 times its own weight. But no Lilliputian can carry 12 times its own weight, a contradiction. Therefore humans can't possibly exist. QED
The proof is clearly invalid, since humans do exist. My question is, which step is invalid?
Answer: "But no Lilliputian can carry 12 times its own weight, a contradiction."
This step is invalid, because in the real world, Lilliputians can carry 12 times their own weight -- indeed, much more than 12 times their weight. (Real-world Lilliputians are often called "ants.")
END
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