We Will Create A Black Hole Which Will Swallow The Earth

The NYT has recently run two articles (one, two) on the fear that exists (among some) that the Large Hadron Collider — when started — will destroy the earth through the creation of a black hole. Interestingly, this was exactly the topic my undergraduate thesis (in STS) was on, except instead of being about the LHC, it was about the particle accelerator being built at Brookhaven National Laboratories.

But when you delve into the history of it it, it isn’t all that surprising. The guy who is making the black-hole-possibility claim about the LHC is exactly the same guy who made the black-hole-possibility claim at Brookhaven: Walter Wagner. The newspaper stories take the same tenor now that they did back then too.

Although my thesis isn’t awesome (I’ve read some of my friends’ undergraduate theses and I know mine doesn’t stack up) nor well written, I can say it’s timely, so I’m uploading this PDF version of it. I was just trying to find a choice paragraph to lob off and stick up here, but I figured: hey, why not? Unfortunately, in this particular electronic version, some of the websites cited in the footnotes have disappeared (“ERROR!”), and this is the only electronic copy I have. (One of many consequences of the many Great Computer Crashes which have befallen me.)

As an added enticement for those on the fence, I spend a whole “chapter” talking about the fear that the atmosphere would ignite when the atomic bomb was set off for the first time.

Calculus and Chaos Theory

On the same blog, God Plays Dice, I learned two important things today:

(1) The Napkin Ring Problem, which basically says if you take a sphere, and cut out a cylinder from the center (and take off the two caps at the top and the bottom of the sphere), the volume of the remaining solid will not depend on the radius of the sphere at all!

In totally unrelated news, today we wrapped up our initial foray into finding volumes by slicing, by revolution, and by washers in calculus. I wonder what I’ll be doing with them tomorrow. I dearly wish I had an interesting problem for them to work through. Sigh.

(2) Edward Lorenz is dead.

All those years ago (okay, not so long ago) I was at MIT getting my first taste of chaos theory. I took this course with Daniel Rothman, who is an amazing teacher, and who taught from an amazing textbook (Strogatz). I loved the class so much that when I was asked if I was interesting in grading the problem sets for the following year’s class, I jumped at the chance.

It was in that course that I was first introduced to Edward Lorenz and his paper on weather (!) which launched a whole phenomena: chaos. But I also got so into the subject that I tried to fix a broken chaotic waterwheel that was being left unused in the basement of MIT. (I was unsuccessful, though I still wrote my final paper for a class on a modification of the Lorenz equations.) And when I was in graduate school, I was asked to write a “history of chaos theory” article for an encyclopedia that never really got published (to my knowledge). Also when I was in graduate school, I traveled to Beijing for a summer school (4 weeks) on complexity and chaos, thinking that I might want to do my dissertation around the topic of complexity and chaos in the sciences. And now that I’m teaching in high school, I wanted to introduce some of the budding mathematicians to the topic, so in math club, I gave a 3 part mini-lecture on chaos and the logistic equation.

Needless to say, chaos = cool. When I took that initial class with Prof. Rothman, Lorenz came to give a guest lecture. I’m sad that he’s gone. He is partially responsible for a really interesting part of my life that keeps recurring, no matter where I am or what I’m doing.

If you want to read my encyclopedia entry on the history of chaos theory (which is okay, but I’m sure has a number of problems with it), I’ll post it below. It talks about Lorenz’s contribution, if you don’t know.

(more…)

When I’m busy, tired, stressed, I turn to math

This has been a stressful few days. In my school, we have to write comments on each student (about 1/2 page single spaced) and those are due on Thursday. In addition, with the end of the third quarter, there has been a massive grading effort on my part to finish up the video projects I assigned. Plus, that hiring committee I was on took a ton of my time.

Still, even though I have bloodshot eyes and am crashing at random times during the day and evening, I couldn’t help but get addicted to working on Blinkdagger’s Monday Math Madness problem. I think I found a solution, but the great (and cruel) thing about optimization problems is that you can’t check to see if you have the best solution without proving that no other method will beat the method you came up with. Which I’m not going to do. But not knowing if I have the optimal solution is frustrating, but also exciting, because it’s a different experience to not know with certainty if I’ve finished the problem right or not.

I have to say that this problem, more than any other problem I’ve encountered in recent memory, has gotten my brain to work in such a different and crazy way that I have to recommend it to everyone. I think those in computer science (read: not me) will find it easier than I did.

With that said, the hardest part of this wasn’t actually coming up with my solution, but it was writing it in a way that could be understood by someone else. I wanted a student from, say, my high school math club to be able to follow the logic. But whether I succeeded or not, that’s a question that will have to wait until the contest is over and I post my solution.

With that said: check out the problem, test your mathematical mental mettle.

Thinking!

I love it when my students think for themselves.

When learning the law of sines and law of cosines — and when it’s appropriate to use one or the other — our textbook gives pretty prescriptive directions. For example, when you are given SSA (a side, a side, and an angle opposite one of those given sides), you are supposed to use the law of sines. And depending on these values, you can actually get two possible solutions!

Let’s work this out.

If in triangle ABC you’re given that side a=6, side b=8, and angle A is 40 degrees, then let’s solve the triangle.

Using the law of sines, \frac{6}{\sin (40)}=\frac{8}{\sin B}. Rearranging, we get \sin B=0.8571. But we know that sine is positive in quadrants I and II, so we get for B=58.99 or B=121.01. And hence, we have two possible angle values for B, which leads to two different triangles. [For a more detailed explanation, see here.] Another quick application of the law of sines yields that c=9.22 or c=3.04

And looking at the picture below (cribbed from the site above), you can see that both triangles are possible!

Here’s where student thinking is awesome. The book, as I said, says that everytime you have SSA you should use the law of sines. And I agree, it is easier. But it is definitely possible to use the law of cosines too, as one of my students pointed out to me.

Let’s do it:

6^2=8^2+c^2-2(8)(c)\cos(40)

This simplifies, with some rearranging, to the quadratic: c^2-12.26c+28=0. This can be solved to get the two values for c, which are c=9.22 or c=3.04.

I love that it works, and that the student insisted that we could do it. It might be slightly more work, but not that much more, and the exploration aspect is awesome. [1]

[1] An extension for a project for next year might be: how can we use the fact that we can generate a quadratic help us in determining when we have two possible triangles, one possible triangle, or no possible triangles.

Reorganizing Trigonometry

In my trigonometry classes, I decided to deviate from the textbook ordering of concepts. The other teacher, thankfully, was on board. (And next year, I want to tweak things even more.)

Our textbook presentation of trig starts out with reviewing right triangles, SOH CAH TOA, and in the same section, introduces the reciprocal trig functions (csc, sec, cot). It then goes into application problems, involving angle of elevation and depression. Finally, it throws all the hard but juicy stuff into the next section — a section that took me 4+ days to cover. It included introducing the concept of using trig for angles greater than 90 degrees (a VERY hard concept for kids to grasp), reference angles, quadrant analysis, and a variety of different types of problems that students are expected to do.

But then the book starts veering into radians (which I covered already this year, and next year, which I’ll postpone), the graphing of trig functions, and finally goes into the translation and stretching of these graphs.

I skipped this graphing work and an entire other chapter to get to the law of sines and the law of cosines. Um, yeah, hello? Let’s think about this:

Students start by learning that trig helps with right triangles and they do application problems. Then they learn how we can extend this trig work to angles greater than 90 degrees. Me thinks that it would be natural to show them how trig can help them with all triangles at this point – including obtuse triangles. (Importantly, the law of sines also tests a student’s understanding of reference angles.)

My students seem to find doing problems with the law of sines and cosines very tedious, yes, but they also love the grounding and concreteness of it. They told me that. Which makes me think it’s better to put that sort of thing at the beginning of our exploration of trig. And leave the more abstract discussions to later, when the basics are fixed in their minds.

Rethinking the textbook makes me feel good, because it means I’m paying attention to the flow of the subject, to how I’m presenting the topics, and what students are thinking as they learn trig. And it means I’ve started fulfilling one of the goals I set before school started, one that I decried just couldn’t happen my first year.

[1] For giggles, I want to share what the section after the law of sines and cosines is in the textbook: Graphing complex numbers on a complex plane. Can we all say “jarring transition”?

Students teaching students

In the past few weeks in my seventh grade pre-algebra class, we’ve been working on some hard problems involving inscribing circles in squares and squares in circles and, sometimes for good measure, playing around with equilateral triangles too. Radicals abound. And for the most part, I see them getting it.

But recently (for a number of reasons) I’m being a “teacher centered” teacher. I’m at the whiteboard explaining things or doing problems. I call on them from the whiteboard. I let the kids work and I walk around and check to see how they’re doing. But their eyes are always on me or their work [1]. Partly it’s because I know it works. And it doesn’t take a lot of time. (These kids pick up what we’re doing quite quickly… you can actually see their initial struggle, and their breakthrough… sometimes you more than see it… you hear it… “OHHHHHHHHHHH, Mr. Shah! It’s sooooo simple!”) [2]

But today I had an extra 10 minutes and so I had a student come up and take over the classroom. She was presenting the solution to a problem, and I gave her total control. She could call on students, or be at the board, or do whatever she needed to to explain the solution to the class. And with great poise, she strode up there and started asking good questions (“How do we know what that side length is?” “What’s the area of a circle?”) and adroitly led her classmates through the solution. The other students were into it: their hands waved in the air, eager to answer her, as they are eager to answer me. [3]

The cherry on top of the sundae? Once she finished, about 75% of the kids had their hands up wanting to present the next problem. They want to be teacher. Which made me happy to be their teacher. I’m sad I’m not going to be teaching seventh grade next year.

[1] Okay, that’s not entirely true. They are a collaborative class; they work with each other answering questions, running ideas of each other, and comparing answers to see if they’re on the right track.

[2] Of course, on the other — less happy — side, you’ll get to a topic where a few kids will proclaim their hatred for the subject at hand, at which case I feign (do I feign? or is it real?) pain that someone could say something so awful about something I’ve devoted my life to, and how I want to curl up in a corner and cry.

[3] I noticed that her teaching style was a lot like mine, which means that I’m rubbing off on them.