Last time I described an idea about how to use matrices to study simple weather models. Really simple weather models; in fact, the model we used was a two-state Markov system. And like all good simple models, it was interesting enough and at the same time inaccurate enough to give us some meat to chew on.
I used it as one session in a teacher institute I just helped present (October 2019), where “matrices” was the topic we were given for the five-day, 40-contact-hour event. Neither my (excellent!) co-presenter Paola Castillo nor I would normally have subjected teachers to that amount of time, and we would never have spent that much time on that topic. But we were at the mercy of people at a higher pay grade, and the teachers, whom we adore, were great and gamely stuck with us.
One purpose I had in doing this session was to show a cool use for matrices that had nothing to do with solving systems of linear equations (which is the main use they have in their textbook).
Just running the model and recording data was fun and very important. Teachers were unfamiliar with the underlying idea, and although a few immediately “got it,” others needed time just to experience it.
Making the connection between the randomness in the Markov model and thinking about natural frequencies did not appear to cause any problem. I suspect that this was not an indication of understanding, but rather a symptom of their not having had enough time with it to realize that they had a right to be confused.
The diagram of the model was confusing.
Let’s take the last bullet first. The model looked like this:
We’re careening towards to the end of the semester in calculus, and I know I’m mostly posting about stats, but this just happened in calc and it applies everywhere.
We’ve been doing related rate problems, and had one of those classic calculus-book problems that involves a cone. Sand is being added to a pile, and we’re given that the radius of the pile is increasing at 3 inches per minute. The current radius is 3 feet; the height is 4/3 the radius; at what rate is sand being added to the pile?
Never mind that no pile of sand is shaped like that—on Earth, anyway. I gave them a sheet of questions about the pile to introduce the angle of repose, etc. I think it’s interesting and useful to be explicitly critical of problems and use that to provoke additional calculation and figuring stuff out. But I digress.
Reflecting on the continuing, unexpected, and frustrating malaise that is Math 102, Probability and Statistics, one of my ongoing problems has been the deterioration of Fathom. It shouldn’t matter that much that we can’t get Census data any more, but I find that I miss it a great deal; and I think that it was a big part of what made stats so engaging at Lick.
So I’ve tried to make it accessible in kinda the same way I did the NHANES data years ago.
This time we have Census data instead of health. At this page here, you specify what variables you want to download, then you see a 10-case preview of the data to see if it’s what you want, and then you can get up to 1000 cases. I’m drawing them from a 21,000 case extract from the 2013 American Community Survey, all from California. (There are a lot more cases in the file I downloaded; I just took the first 21,000 or so so we could get an idea what’s going on.)
I don’t quite know how Beth does it! We’re using Beth Chance and Allan Rossman’s ISCAM text, and on Thursday we got to Investigation 1.6, which is a cool introduction to power. (You were a .250 hitter last season; but after working hard all winter, you’re now a .333 hitter. A huge improvement. You go to the GM asking for more money, but the GM says, I need proof. They offer you 20 at-bats to convince them you’ve improved beyond .250. You discover, though the applets, that you have only a 20% chance of rejecting their null, namely, that you’re still a .250 hitter.)
I even went to SLO to watch Beth Herself run this very activity. It seemed to go fine.
But for my class, it was not a happy experience for the students. There was a great deal of confusion about what exactly was going on, coupled with some disgruntlement that we were moving so slowly.
Slinkies are great. You can demonstrate waves. You can make them go down stairs. They are super-dynamic physics toys. They make a great sound.
But they are also pretty great when static. Consider, for example, a hanging slinky. How far down does it hang?
Well. It depends.
For this post, I’ll skip the question-posing part of this and go directly to what it mostly depends on: the number of coils (slinks) that are hanging down.
Let’s skip all the way to the data. Here is a graph of the length (in cm) of a hanging slinky as a function of the number of slinks. You should, of course, record your own data, if for no other reason than to experience the glorious difficulty of measuring the distance.
We can pause here and make sure the graph makes sense. What do you see in the slinky itself? How would you describe the spacing of the coils in the hanging slinky? How does that pattern get reflected in the data and in the graph? Continue reading The Hanging Slinky
You should already have read or skimmed part 1 and part 2 of this saga. In part 2, we showed the picture of the globe (reproduced here) that purported to show how the aspect ratio of a 10° by 10° box at 60° was 2:1.
Of course, it’s only approximately 2:1, I told the students, because (small pause to get attention) the scale on the globe is changing with every degree. In fact, it’s changing continuously. At this point, the students muttered, “mmm, calculus.” So at least they smelt it coming.
I’d like to be able to say that at this point I turned it over to them: “Indeed! Calculus! We’re trying to find , the function that gives us the total y-distance on the map as a function of the latitude. Work in your groups to figure out just what calculus you need to do to find that function, and be ready to present in about five minutes.”
But I can’t. Being realistic, it was good that they sensed calculus, but in this unusual context—the sphere and everything—it would have been excruciating. So I scaffolded like my life depended on it. I reproduce the chart we developed last time that shows the dimensions of the 1° boxes.
“Indeed! Calculus! After all, we want , the y-distance as a function of latitude. So to find the total distance, to, say, 60°, I’m going to start at 0° and use this number we have in the table, sec(0°), times 1° because that’s the height per 1°, right?. Then for the next degree, I have to add sec(1°) times 1°, then for the next one, sec(2°) times 1°, all the way up to 60.”
I write the sum on the board, (still ignoring the constant factor, by the way, but that was OK):
“And of course, within each of these 1° sections, it’s changing continuously too. So what should we do?”
Now, almost in unison, “integrate secant,” and a student bravely writes,
Doing the Integral—the way cool way
“And of course, you all know the integral of the secant, right?”
They squirm, but I take them off the hook. “Of course you don’t. Nobody should remember that. But what’s the most practical way to find this integral?”
“A substitution?” they ask. “Do we, like, do it by parts somehow?”
“All good suggestions, but it’s really not obvious what to substitute. And for parts, usually there’s more stuff, right? So you can have a v and a du? What I’m really asking is, what’s a really practical way to find this integral?”
“Look it up?”
“That will work, but I want to show you another way.” I pull out my iPhone and hook it up to the projector. “Siri, integrate secant of x.”
“You’re kidding me,” says one kid. Class chatter rises.
“Hmm, let me think…,” says Siri. She shows me an approximation and a link to Wolfram Alpha, which I tap. And this appears:
At this point, they returned to desmos and their data to see if this function actually worked. And it did, to enormous satisfaction throughout the class. That this obscure function, a log of a combination of trig functions, fit these data, was a bit of a miracle, a triumph of actually using calculus.
Actually Finding the Integral
We did not find the integral in class, but in case you care, here is a derivation, which shows just how arcane these can get. This is why you look these up. And why it’s so great that Wolfram Alpha exists, because they give you the answer and the derivation.
It turns out that there is a great substitution! First you multiply top and bottom by :
Now we do a u-substitution, choosing
When you take its derivative, you discover that by some miracle,
Which means that
Which turns out (after even more mind-numbing algebra than it took to find du above) to be equivalent to the horrible formula with the half-angles that Wolfram gave us at first.
Perhaps there is a way to anticipate that that particular substitution would work out so well, but I sure don’t have that kind of insight.
You really don’t want to read all the details. But among the crucial steps in getting the lesson outlined last time to work right, the crucialest might be the place where the students figure out that the y-scale goes as the secant.
How do we help them figure that out? (And what do we mean by that?) That’s what we’ll talk about today.
First of all: that y-scale is the number of centimeters per degree in the y-direction, and that depends on latitude. And when you look at a Mercator-projected Earth (like the one in the figure) you can see that this scale increases with latitude. How do we know? Because the lines of latitude get farther apart. So more centimeters per degree.
Last week, Zoya let me into her calculus class to do a data-rich activity of my choosing. Ideally it would involve calculus, appropriate for these kids who had already taken the AB exam. Most of my activities that use elementary functions to model data we get from the world (The Model Shop) or measure ourselves (EGADs) don’t involve calculus, although I think they suit a wide range of students.
For some weeks I thought about freeways, and the optimization problem of figuring out at what speed freeways have the greatest flow of traffic. It’s yummy because of the optimization, of course (and that gives us calculus, or at least smells that way) and also because you have to wrap your mind around what you mean by flow. I also found public data from CalTrans—but that’s all a story for another time.
Wisely, I think, I backed off that idea and instead went with a problem I faced long ago when I tried to write a program to draw a Mercator-projection map of the world. Namely: what’s the function in the y-direction that relates distance on the map to latitude?
A million thanks to Zoya Voskoboynikov and her two sections of “regular” calculus at Lick for letting me come litter their otherwise pure math class with actual data. Of course, it was after the AP exam, and these are last-quarter seniors, so my being there didn’t interfere with any learning they needed to get through.
It worked great. It had what I most wanted: the aha experience of arriving at the destination by another route. Fortunately (and unsurprisingly), none of these successful math students remembered the theorem from the geometry class they took as frosh.
What we did
I set up the problem and had them predict, informally, what the function would look like. The main purpose of this is to orient students to what we’ll be measuring and to the idea that if you measure two quantities, you can see their relationship in a graph. Continue reading Chord Star in the Classroom
Last month, in Falmouth High School in Maine, some Honors Physics students were estimating the period of a mass hanging from a spring. They used InquirySpace/Data Games software and Vernier motion sensors, and got data that looks like this (Reading is in meters; Time in seconds):
To do their investigations, they needed the period of this wave.
Some students found the peak of one wave, and subtracted its time from the peak of the next wave. This is the most straightforward and obvious. But if you do that, your period will always be a multiple of the time between points, in this case, 0.05 seconds. (This is part of what must have happened in the previous post.)
Some students—sometimes with prodding—would take the time difference across several waves, and divide by the number of periods. It’s not obvious to students that this technique gives a more precise measurement for the period. It’s interesting to think about how we know that this is so; for example, if you use five periods, it’s now possible to get any multiple of 0.01 seconds; but does that mean it’s actually more precise? (Yes.) This technique also gives students a chance to be off by one: do you count the peaks? No. You have to count the spaces between the peaks. (Getting students to explain why is illuminating.)
We could imagine trying to fit a sinusoid (and some students would, but it’s hard) or using a Fourier Transform (which is a black box for most students).
But this post is about an alternative to all of these techniques—one that uses all the data and gives a much more precise result than the first two.