Author: samjshah

I’m so glad I’m not there…

I like my school a whole lot. There’s a ton going for it, from the funny, caring and inviting faculty to the really committed students. My department head has my back, and always has answers to even the glaringly neophyte questions I bring to him. The math faculty make working day in and day out a pleasure. And even though I’ve had my share of griping and complaining (a cathartic activity that is endemic to being a teacher) I couldn’t imagine a better place to have inaugurated my teaching career.

All of that came rushing through my head as I read this article in New York Magazine about a well-known private school in New York in a state of major turmoil.  It’s a fascinating read, about school values, a board of trustees with a strong voice, and teachers and students in battle. An excerpt below:

 The Web page for a Horace Mann Facebook group titled the “Men’s Issues Club” mocked a student organization on campus called the Women’s Issues Club. The 44 members of the parody club included children of both trustees and the legion of prominent names who send their children to Horace Mann, which sits in the top rung of private schools in New York. One club member referred to an English teacher as a “crazy ass bitch” and a French teacher as an “acid casualty.” Another boy boasted that he’s “the only person here who actually beats women when hes [sic] drunk. no joke,” while still another bragged that he had “banged” a teacher “in [the] music dept. bathroom” and “will get great college rec” for the accomplishment. The boys lamented Star Jones’s “fat and wrinkled ass,” “sex in the city,” and “feminism,” proclaiming, “WHERE DO THEY BELONG?!?!????!!! IN THE KITCHEN!! IN THE KITCHEN!!!” The club summed up its mission thus: “For too long men have not had a way to express themselves and their beliefs in society. Men need to have a voice, we aren’t meant to be seen and not heard. Let freedom ring, bitches.”

Spinning on a Globe

This slideshow is the Smartboard I used in this class that I describe below:

The day my students got back from spring break, this week, I started things off with a bang! At least, that was the intention. Instead of doing the whole “Paris was great, where did you go for your vacation?” time waster, I decided to dive right in. They left for break having learned some trigonometry basics, and I had blown their minds with the idea that you can measure angles with something other than degrees (radians). Monday’s lesson was to be on central angles, arcs, and angular velocity.

On Sunday night, I downloaded GoogleEarth onto my school laptop. In an ideal world, I wanted to say:

So kiddos, I flew from Newark to Paris for spring break. Let’s all get out our laptops and find out their longitude and latitude, and the distance between them. [I do it in front of them on GoogleEarth]. Now let’s use this info to the find the radius of the earth.

Unfortunately, after a few minutes of playing around with this problem, it becomes very clear to me that it is not as simplistic and trivial as I had hoped, because the latitude and longitude are different. Picture the following. Draw a line from the center of the earth to Newark, from the center of the earth to Paris, and a curved line (along the globe) from Newark to Paris. You have a sector of a circle. You know the arc length. But what’s the central angle (the angle between the radii)? It’s not a trivial problem for students to calculate that angle.

So instead, I copped out and found two random spots on the equator (one in Brazil, one in the Democratic Republic of Congo), and the distance between them. And then I had students come up with what to do next. It took very little time before they stumbled upon the idea of setting up a proportion: angle/full circle angles = arc/full circle perimeter.

\frac{87.74}{360}=\frac{9610.10 km}{Circumference}

From that, we found the circumference of the globe (about 40,046km). And then we compared it to what Wikipedia said (40,075km), marveled at our own acumen, and talked about the sources of error. And because I truly am a big nerd, I talked about the origin of the meter (see this book).

This warmup took about 10 minutes, and was a perfect segway into a discussion of arc length and central angles.

We set up the same proportion as we did above, but for a general circle of radius r, central angle \theta, and arc length s, to discover the wonderful formula s=r\theta. Then I asked them if it made sense… (The larger the circle’s radius, the larger the arc length… check! The smaller the angle, the smaller the arc length… check!) And when I asked them when they had seen this before, most of my students noticed: “duh Mr. Shah, the Earth problem…”[1]

Finally, I got to the final topic, angular speed. I remember when I was first taught this, I was taught it as a formula. There was no conceptual understanding behind it. And students tend to love to rely on formulas without understanding them. So I introduced the ideas of “linear speed” and “angular speed” — and then asked them to calculate (without showing them any problems) the linear speed and angular speed of someone in Brazil or the Democratic Republic of Congo as the earth spun on it’s axis. And someone’s hand shot in the air, then another, and soon the problem was solved.

In addition, one student noticed that the angular velocity remained the same no matter where on the globe you stood, but your linear velocity decreased the further away you got from the equator.

Which led to the final equation of the day: v=r\omega.

Some more practice problems were done, and that was that. With that concluded one of my better, more cohesive lessons. It all tied together with the globe thing, and they really left with a sense of the concepts, more than the simply memorizing the formulas.

[1] One class noticed we could calculate the radius of the earth with this formula, since in our earlier problem, we knew the distance between the places in Brazil and the Democratic Republic of Congo, and the angle between them… So we did that, and got another wonderfully accurate answer too!

Not funny, people, not funny.

I thought I went through it unscathed. None of my students pulled an April Fool’s Joke on me and I, in return, didn’t do anything to them. It was a merry unspoken reciprocity that we’ve come to love. And so, in my apparently flawed and puny little mind, I thought it was over.

Then I come home and read this and this… I believed the first — and was upset — and then I read the second, which I believed until the end. Then both made sense. Not okay, people, not okay. But the point is: April Fools, I LOATHE YOU, like Tartaglia loathed Cardano. But worse. More like how Cardano loathed Tartaglia.

Update: Dang. I was totally taken in a second time by the same website (see second link above). I’m such a loser. Fool me once, shame on you; fool me twice, shame on me. I was actually, and even I barely believe it, rickrolled. (Look it up on Wikipedia if you don’t know what that is).

Birthday Polynomials

A few days ago, JD2718 wrote a post about “Birthday Triangles” — having students create three coordinates out of their birthdate and then analyzing the triangle that these coordinates make.

Even though it’s just a bit of fun, and you could have students work with any sets of points you give them, there is something great to be said for students creating their own problems that they feel ownership of. As JD2718 writes:

Best evidence, (and mind this, please) almost every class, when they first plot their own birthday triangle, there is one or two sad looking kiddies (it’s not come to tears, but I’ve seen the quivering lip) who thinks their own triangle is ugly. “Nooo” I say “Yours is obtuuuse. Does anyone else have an obtuse triangle that looks as nice as Anna’s?” (it’s usually a girl)

I thought that this idea could work in calculus too, creating “Birthday Polynomials.” My first thought was exactly JD2718’s: take the three birthday coordinates and find a quadratic that would fit them. But that would be a precalc assignment. (With bonus question: With what birthdays could you not create a quadratic?)

But I wanted more. I wanted to come up with something awesome. Something calculus. Something that would knock my students’ socks off. I initially thought something like this… if I was born on April 21, 1978, the birthday polynomial could look like: f(x)=4x^3+21x^2+19x+78 [1].And questions could be: where are the local maxima and minima? where is it concave up and concave down? where is it increasing and decreasing? And of course you could do things with integration too…

But there’s something unsatisfying about that type of question. It’s nice, but I want to wow! my students. I want to knock their socks off. Show them something elegant and unexpected. So I thought…

I want them to create a polynomial using their birthdate which would have an inflection point that was their age.

I was planning on using this amazing property [if there’s a cubic equation that hits the x-axis three times, then there’s a point of inflection, and it will be the average of these three x-intercepts] they would have to discover.

So if my birthday were January 25, 1980 (it is not), and we evaluated this polynomial on March 30, 1980 (after I celebrated my birthday), a birthday polynomial might look like this:

f(x)=(x+1980*3)(x-2008*3)(x)+1x+25.

f(x)=(x+year born*3)(x-2008*3)(x)+month born*x + date born

[Note that the month and date play no role when finding the point of inflection… they are red herrings.]

But there are many annoying problems with this… First of all, that 3 is annoying. Second of all, that 2008 gives some of the fun away. I guess multiplying it by 3 and writing it like 6024 would help disguise it, but not much. Third of all, if I worked the problem on January 5th (or anytime in the year before January 25th), it would get my age wrong by a year. Fourth of all, it’s not elegant.

I’ve spent a little time tweeking it, and thinking of ways to rework it… but I haven’t anything elegant or clever yet. For now, it’s going to have to go on the backburner. Spring break is over and school is starting tomorrow and I have too much on my plate.

I’ll post an actual, good, interesting way to come up with a birthday polynomial with some amazing property (that somehow magically spews out your age, perhaps) when I have time…

[1] Of course I had to do a google search on “birthday polynomial” to make sure I wasn’t reinventing the wheel. One calculus teacher in Texas did something similar.

What do these songs all have in common?

I created an online mixtape, on one of the most graphically sleek sites I’ve seen in a long while [muxtape].

You can hear it by clicking below. For bonus points, try to figure out the common thread which binds these songs together.

cassette_label.jpg

Update: Many people also swear by mixwit, which has a nice interface too. And you can make multiple mix tapes.

Update: Answer to the question in the subject line in the comments.

Generalized coordinates, trajectories, lagrangians, and action

I just spent 3 hours watching Leonard Susskind [Wikipedia page] deliver two lectures on Classical Mechanics via iTunesU. I started this over Winter Break (I watched 4 lectures), but I stopped because I didn’t have the proper time to devote. Once you stop, and enough time passes, you have to start all over [1]. Plus I hadn’t bought a pretty $15 notebook from Paris to take notes in.

But I started over, notebook in hand.

The end result of this is: I’m inspired. I love love love learning about trajectories, generalized coordinates, lagrangians, and principles of least action. Plus, there are some pretty neat digressions or mini-lectures I could use for my multivariable calc class next year.

I highly recommend it. You just need calculus (and some basic multivariable can’t hurt). It’s a bit hard to find on iTunes, but this link might take you to the first lecture. It’s his Stanford PHY 25 course taught in 2007 (October – December).

[1] It reminds me of borrowing the 1st season of LOST from my friend, who had the last seventh disc out on loan to a different friend. Once I finished the first six discs, I was hungry for the last. But still, the other friend refused to return it. And to this day, I have not seen the entire first season of LOST. Now, since so much time has lapsed, I will have to start the season all over again. (Double sigh.)