Can a pendulum save you?

I’m so thankful to my friend Chija for pointing out this video for me:

Here’s her tweet

When I saw it I started to wonder if angular momentum was enough to explain it. So I set about trying to model it. Here’s my first try:

Green ball is 20x the mass of the red. No contact or air friction.

It does a pretty good job showing how the fast rotation of the red ball produces enough tension in the line to slow and then later raise the green ball. Here’s a plot of the tension in the line as a function of time:

Tension in the line as a function of time. The green line is the strength of gravity. The reason everything is negative is a consequence of how I modeled the constraint (a Lagrange multiplier)

So how did I model it? I decided to use a Lagrange multiplier approach where the length of the rope needs to be held constant. Here’s a screenshot of the code:

“ms” is a list of the masses. “cons” is the constraint.

You define the constraint, the kinetic and potential energies, and then just do a lagrangian differential equation for x and y of both particles:

$\frac{\partial L}{\partial x}-\frac{d}{dt}\frac{\partial L}{\partial x'}+\lambda(t)\frac{\partial \text{cons}}{\partial x}=0$

(note that in the screen shot above there’s actually some air resistance added as an extra term on the left hand side of the “el” command).

Very cool. But what about the notion that the rope wraps around the bar, effectively shortening the string? I thought about it for a while and realized I could approach the problem a little differently if I used radial coordinates. First here’s a code example of a particle tied to a string whose other side is tied to the post:

“rad” is the radius of the bar. Note how the initial “velocities” of the variables need to be related through the constraint.

I’ve changed the constraint so that some of the rope is wrapped around the bar according to the angle of the particle. Here’s what that yields:

Ok, so then I wanted to feature wrapping in the code with both masses. Here’s that code:

Note the negative sign before “l[2][t]” and the “$\theta[2][t]$” in the constraint.

And here’s the result, purposely starting the more massive object a little off from vertical:

Fun times! Your thoughts? Here are some starters for you:

• Why do you insist on using Mathematica for this? It would be much easier in python, here’s how . . .
• Some of the animations don’t look quite right to me. Are you sure that . . .?
• This is cool, do you plan to do this for your students soon?
• What about contact friction between the rope and the bar? I would think that would be a major part.
• In the video he just comes to a rest instead of bouncing up. Clearly you’ve done this all wrong.

Portfolio SBG

My last post talked about a way to have daily quizzes in my Standards-Based Grading (SBG) optics course. It (and the comments) got me thinking about how to do it even better and I think I’m closing in on a better plan.

The main idea is to have daily quizzes that are problems randomly selected from the previous day’s work. It reduces the amount of homework I have to grade, and tackles the cheating problem since it’s now a no-notes quiz. I liked it a lot in my fall class and I definitely want to keep those strengths. My suggestion was six problems per day that would act as the only contexts for any future assessments (quizzes, screencasts, oral exams, and office visits). One commenter noted that might be too much to ask the students to absorb from Tuesday to Thursday. Also, I wasn’t too happy about the double quiz I suggested on Tuesdays (one for the previous Thursday material and one to act as a re-assessment of week-old information). So, here’s my new thinking:

1. Assign 3 problems per night
1. Have them be substantial, covering various aspects of what we talk about in class.
2. Each day do a quiz on a randomly selected problem from the previous 6 problems (three each from the last two days of new material).
3. Have the students maintain a portfolio of all the problems so that they can act as context for all future assessments

• Finding 3 solid problems sounds much more fruitful (and easy for me) than finding six every day.
• I really like the portfolio idea. Want to come improve your standard score? Bring in your portfolio and I’ll randomly ask about one of those three problems. For each of the standards the students will (hopefully) be encouraged to really comprehend the issues around the three problems, especially given that they and I will be encouraged to “turn them inside out” for every assessment.
• Before every quiz they should be touching up six problems in their portfolio. Admittedly if the quiz is on one they’re not ready for, they get a crappy grade but they can redo it via screencast, office visit, . . .
• Something we’ll go over today might show up next time or the time after that, allowing for some cycling (we will likely discuss the context of the quiz beforehand and often the details of the quiz afterwards, especially if it seems people are unsure how to approach the problem).
• Three problems times ~25 standards is a workable number of problems that the students need to master (especially considering that they are in groups of three with common ideas). Certainly it’s easier than six times 25.

• The students “only” have to know how to do three problems per day. Master those, and they’re guaranteed an A. I get student evals sometimes that say I need to do some sort of high stakes exam to make sure they really know it. I’ve tended not to heed such advice, but this has me thinking about that again.
• There’s a chance that a standard might not ever be quizzed (25% chance, I guess). That means that they’ll need to submit something on their own. I guess I could use my old “one week rule” (here’s a post back when I called it the two week rule) or something. I could also weight the random selections differently to reduce that 25% to, I don’t know, 10% or something.
• Hopefully the notion of keeping up a solid portfolio will lower the barrier to having them submit something.
• If I had the quiz be on the last 9 problems, there’s an even greater chance that a standard doesn’t ever get quizzed (29.6%)
• The days could devolve into “how do we do these three problems” instead of active learning around the content.
• Students might want to do their own problems for the oral exams (that’s how I’ve tended to do it) instead of just coming with their portfolio ready.
• A compromise could be that I’ll tell them which standard they’re going to be reassessed on and they can polish up those three problems, of which I’ll randomly select one to grill them on.
• Another approach could be “bring your whole portfolio to the oral exam and I’ll randomly select anything in there.” I think that would really drive home the notion of keeping up a good portfolio but they might rebel.

So that’s where I’m at (for today :) Your thoughts? Here’s some starters for you:

1. I think 3 is too many/few and that instead you should subtract/add x and here’s why…
2. I’ve taught with a portfolio approach before and here’s where I think your system is going to fail . . . (this is a cue for my friend Bret to weigh in)
3. You definitely should also have assessments that do completely different problems and here’s why . . .
4. How would you teach the students to “turn a problem inside out?”
5. Here’s how I’d solve the 25%-that-won’t-get-quizzed problem . . .
6. I think for the oral exams you should limit what they’ll need to bone up on and here’s why . . .
7. I think for the oral exams you should make everything on the table and here’s why…
8. Why not have every quiz be a random selection from anything in the portfolio?
1. Below is a histogram of running 1000 semesters and finding how many problems would never get quizzed using this approach. The average is just a little over the 25% that I get with my approach above
Posted in syllabus creation, teaching, Uncategorized | 7 Comments

Daily quiz help

I’m preparing my syllabus for my upcoming Physical Optics course and I’d love some feedback on a policy I’m polishing regarding daily quizzes. Here’s a post from last summer laying out what I did in a recent class (general physics 2). For this upcoming class I don’t have 3 days per week, allowing for Mondays to be a reassessment day, so I was thinking of just doing a longer quiz on Tuesdays.

Here’s what I was thinking

• every day assign 6 problems
• randomly select one for the quiz on the next day
• on Tuesdays additionally select a problem from two weeks prior

In addition I’m thinking that the assigned problems could be the context for both oral exams and office visits. In other words, it’s the only problems they’ll work on. Note, of course, that on all quizzes and exams the problems will be “turned inside out” so that they really represent a type of problem, instead of a specific problem.

Ok, first I realize that I have to be super careful selecting the six problems each day. There really can’t be any fillers in there or super hard ones with fancy tricks that’ll only work in weird situations. I’m up for that challenge.

Here’s one question I have: In the past class I assigned all new problems for the review day so they really had 6 problems for every standard (4 on the day we “covered” the material and 2 for the review homework). Should I assign 6 every night for this Tuesday-Thursday class? Or should I go with 4 since it doesn’t seem too hard to tackle them from Tuesday to Thursday (admittedly Thursday to Tuesday is easier)?

Second question: If a problem is randomly selected, can it be selected again? If so, maybe I should never provide solution sets. I guess I’m leaning toward that already so that they’ll know to just really have a good handle on all the problems (since they could show up anywhere: quiz, oral exam, office visit, etc).

I guess I’m right now circling around 6 problems per class and repeats are fine with no solution sets. What are the downsides I’m not seeing?

• I’m going to be in this class and I’m really excited about this. Here’s why . . .
• I’m going to be in this class, where can I get a drop card?
• I think x problems per class is the perfect number, here’s why
• Why do you put “covered” in quotes?
• If you’re just giving them the problems they have to do, they’re not going to learn since there’s never a surprising question on an exam. You need to assess their understanding, not their ability to refine a fixed set of problems.
• Can you give some examples of “turning a problem inside out?”
Posted in syllabus creation, Uncategorized | 4 Comments

Unstable rotation (spinning handle in space)

First, watch this:

Cool, huh? My students found this last year when we were studying rigid body rotation. One of the things we did a lot was try to spin a tennis racquet about an axis in the plane of the head and perpendicular to the handle without it rotating about the handle. It turns out it’s pretty hard and the reason is the same as the explanation for the video above.

My friend Will posted that vid recently again and I sent him the an animation I made showing a similar result.

He asked for a blog post, so here you go. To make it a fun challenge, I wanted to see if I could do it “off the top of my head,” in other words I wanted to see if I could put together the calculation without checking my notes from last spring when I was teaching this stuff (and hence it was all at my fingertips).

I knew I couldn’t do all the inertia tensor stuff off the top of my head, so I thought I’d see if I could do it with a small number of masses so that the inertia tensor benefit wasn’t huge.

First, I laid out a few point masses to model the handle in the video. I put one at the screw, two at the handle ends, and one at the crossing point. I knew I needed to calculate the location of those points for any arbitrary Euler rotation, so I had to think about Euler rotations first. Basically these are the rotations you can do to an object to put it in any orientation (without changing the center of mass which I put at the origin). It reminded me of the discussions my students and I had about how to do that (before we’d read about Euler rotations) and I decided that sounded easiest:

1. Rotate about the z-axis by $\psi$.
2. Rotate about the y-axis by $\theta$.
3. Rotate about the z-axis by $\phi$.

What that does is the usual theta and phi orientation for a direction from the origin and an additional phi rotation of the body around that direction. It’s not how Euler rotations are sometimes presented:

1. Rotate about the z-axis by $\phi$.
2. Rotate about the new y-axis by $\theta$.
3. Rotate about the really new z-axis by $\psi$.

It just turns out that’s harder to do numerically since you have to find the new and really new axes. In Mathematica you can do my recipe by:

RotationMatrix[$\psi$, {0,0,1}].RotationMatrix[$\theta$, {0,1,0}].RotationMatrix[$\phi$, {0,0,1}].(points you care about)

The period is how Mathematica does matrix multiplication (including dot products).

Ok, so now I need to find the locations of my 4 points and then take time derivatives recognizing that my time-dependent variables are theta, phi, and psi. The time derivatives produce the velocities that I can use to calculate the kinetic energy as a function of the variables and their time derivatives. Then I’m in business because I can just do the euler-lagrange approach at that point. Here’s a screenshot of the code:

The locs are the locations of the dots as described before with the handle screw part being 1 unit long and the handle width being that crazy square root of 3/2 + 0.01 which will make sense below. The m function is the rotation matrix described above. The newlocs function determines where all the points are at some arbitrary theta, phi, and psi and the ke is the kinetic energy (note the D used for derivative). The el function is the Euler-Lagrange operator and the sol command puts it all together, including some initial conditions set to rotate the handle as similar to the video as I could do (note that if you don’t set the psi variable to a little off zero you don’t see the instability). Here’s the result:

And here’s an animation looking at the path the screw takes (it’s animated just so the camera can sweep around)

I remembered from the intertia tensor analysis that the stable axes of rotation (among the 3 eigenaxes) are the ones with the highest and lowest eigenvalues. So I calculated those and found that when the length is sqrt(3/2) there is not one in the middle. Here’s a comparison with the length both 0.1 above and below that magic length:

Cool, huh? I hope Will’s happy.

1. I was in that class and this really helped me understand . . .
2. I was in that class and this was a complete waste of time because . . .
3. I love this! What should I do with my antiquated vpython scripts that couldn’t possibly do this?
4. I hate this! When I flip my tennis racquet it never rotates.
5. What other initial conditions show (or don’t show) that instability?
6. How did you calculate the eigenvalues off the top of your head. What you just happened to know what the eigenaxes were or something?
Posted in mathematica, physics, Uncategorized | 4 Comments

Best bingo board

My son is in the third grade and his math homework is to play games. The other night we played one that really got me thinking. Each player makes a 4×4 board and puts in any even number between 8 and 48 in every box (note there are more then 16 to choose from and that you can have repeats if you like). I just used the first 16 numbers (8-38) randomly on my grid. Then you roll 4 6-sided dice, add up the total, and then double it (so it’s testing low integer adding and doubling for the homework). You play until someone gets four in a row. As we played we both noticed that 28 kept coming up. I had it once on my board and he didn’t have it at all, so it really just kept extending the game. I told him that 28 would be expected to be the most common (avg roll is 3.5 and 3.5 x 4 x 2 = 28) so we got talking about whether next time we should try a board with all 28s. This post is all about what I learned when trying just that.

I decided to code up the game in Mathematica (this is the century of the decade of the year of the week of the hour of code after all). The low hanging fruit was to match an all-28 board against a board with random numbers on it without any repeats. It’s low-hanging because not having repeats means I don’t have to teach Mathematica how to make a choice when a repeated number is rolled (see below for my try at that). To simulate a roll I just produce 4 random integers between 1 and 6, add them, and double them. Here’s a plot of the probability of each roll:

To check if a bingo (four in a row) happens, I just check the board after each roll for any possible bingos.

Instead of playing matches, I just calculated how many rolls it would take to get a bingo for each type of board. Here’s a histogram of 1000 runs for each type (each bin is the count of the runs that took that many rolls to get a bingo for both types of boards).

Yellow is for the board with all 28s, blue if a random, non-repeat board. Gray is where they overlap.

I was a little surprised by this result. The random boards beat the all 28s board by a fair margin (on average). Did it surprise you?

So then I started wondering about better boards. I realized that if I wanted to do boards with some repeats on them, I’d have to teach Mathematica an effective strategy for making decisions. For example, say you rolled a 22 and you had 3 22s on your board. How do you decide where to put your bingo marker?

What I decided to go with was to go for the spots that help out as many potential bingos as possible. That means corners and the inner square are worth more than non-corner edges. What I mean is that a corner spot could be a part of 3 potential bingos (left-right, down-up, and diagonal). The same is true for the inner square. But the non-corner edge spots only have left-right and up-down. So, if given a choice, it’ll go with one of the better ones. If all choices are in the same sort of spot (either all good or all slightly-less-good) then just do it randomly. However, if any of the choices gives you a bingo, I go with that one.

First I tried boards with randomly selected possibilities on each space. This allowed for repeats, since each space re-ran the random selection. Then I made boards where the randomness just mentioned was weighted by the probability expectation seen above. Here’s a comparison of all 4 types of boards:

It’s really interesting to see that the all 28s board is the worst, on average, even though we expected it to be better based on our (very limited) experience. It’s also interesting to see that the average number of rolls for a bingo is half as much for the weighted random (with repeats) board.

So what’s the best board? I don’t know, but what I did was generate 100 weighted-random boards and play 100 games with each. I then looked for the one with the lowest average. Here’s the winning board:

26 40 24 22
18 36 34 18
38 30 36 26
32 20  26 34

And here’s a histogram of running that board 1000 times:

Note that once it got a bingo in four consecutive rolls! Also note that the board doesn’t have any 28s in it!

Ok, that’s my fun for the week/day/hour of code. I hope you enjoyed it. Thoughts? Here are some starters for you:

• I’m in your son’s class, thanks! But I tried your best board and my friend beat me once. Therefore this is all wrong.
• I’m your son’s teacher and I really wish you hadn’t posted this. Now every single time my students play they tie since they always use the same board.
• I’m a lawyer at a Bingo ™ board manufacturer. I need your mailing address to send a cease and desist letter.
• Here’s a better idea for an algorithm to deal with the choices that need to be made when you have a repeat board, because the one you used is dumb.
• Thanks for this! Now I can quit school and stick it to the casinos!
• Why did you only run 100 boards at the end. What, you didn’t want to stay up even later on a Friday night to let it run longer? Wimp.
• I don’t believe this. The all 28s board should have trumped everything. You must have a mistake in your code.
Posted in fun, math, mathematica, parenting, Uncategorized | 9 Comments

First let me say that I thought for sure I’d written this post last year, but it appears not. That could also mean that maybe I didn’t write it because someone else had a good breakdown of my argument, but a quick check of google didn’t really find anything. So, here goes.

Faraday’s law is sometimes called the law of induction. It’s really one of the four Maxwell equations that (supposedly) explain all of electromagnetism.  In differential form it’s:

$\vec{\nabla}\times\vec{E}=-\frac{\partial \vec{B}}{\partial t}$

But most physics students first see it in integral form:

$\text{EMF}\left(=\oint \vec{E}\cdot\vec{dl}\right)=-\frac{d}{dt}\int \vec{B}\cdot\vec{dA}\equiv -\frac{d}{dt}\Phi_B$

Basically it says that you’d get current flowing around a circuit if the magnetic field flowing through the circuit changes.

However, I claim it’s way over used.

What I mean is that at least half the homework problems you’ll find in typical physics texts about Faraday’s law could be done (and often easily) using the concept of motional EMF, a concept that grows right out of the “magnetostatics” the students have been studying up to this point. Motional EMF simply has you calculate the actual magnetic forces on the charges in the wires of the circuit and has you figure out which way they’ll move. There’s no need for some strange “induced electric field” and the direction of current flow is attainable from a straightforward right-hand-rule that they’ve already learned ($\vec{F}_B=q\vec{v}\times\vec{B}$).

Any time the magnetic field is fixed in space and the wire moves (generators, rail guns, etc) you don’t need Faraday. The motional EMF gives you a perfectly fine answer, and it doesn’t require a deeper understanding of the connections between the magnetic and electric fields.

Don’t get me wrong, I think we should teach the crap out of Faraday. It’s incredible! Apparently if you have a time changing magnetic field, you get a new type of electric field produced (very much unlike the field that students have seen before!). But too often the homework doesn’t actually involve a time-varying magnetic field. It’s just a uniform, constant field that shines on a cool circuit that moves. If that’s the case, you don’t need Faraday.

Certainly I agree it’s cool that the motional EMF calculation can be rejiggered to show that it’s equal to the time rate of change of the magnetic flux. But so what? If it’s not what’s actually happening in the physical situation, why confuse our students with it?

Here’s an example. Consider a rectangular circuit where one side can be grabbed and moved further down the rails of the neighboring sides (case 1 in the image below). If there’s an unchanging magnetic field directed down through the plane of the circuit, if you move the bar, you’ll get current flowing around the loop. Cool! But it’s “just” because the free/mobile charges in the bar are responding to the magnetic force and moving along. No weird new electric field. Now consider the usual next case in a textbook (case 2 in the image below). The bar remains the same but a “magnetic field flashlight” sweeps in from the other direction. Whoa, it seems pretty similar to the first case, it’s just that the relative motion has been switched.

2 cases for a typical Faraday’s law discussion

But now the wire (and all the charges inside it) aren’t moving, so it can’t be the magnetic force getting them to go around. Instead we have something new! The field inside the circuit is not constant (nor uniform). So now it’s creating a circulating electric field which pushes the charges around. Awesome! Do you get the same answer? Amazingly yes. Though students don’t seem too surprised since they’re pretty convinced by this time of Galilean relativity.

Interestingly there’s a great footnote in chapter 7 of Griffiths’ famous electrodynamics book for undergraduates. He says that perhaps we should call Faraday’s new field the G field and point out that it pushes charges around just like an E field. However, it has a curl and no divergence!

So what do I propose? Just keep the homework problems a little more separate. Or at least ask your students if the current is due to (boring) magnetic field forces or (interesting) weirdo Faraday’s electric field forces.

Your thoughts? Here are some starters for you:

• Why do you say supposedly when talking about Maxwell’s equations?
• Why do you put vector symbols over your vector entities instead of bolding them?
• What do you mean when you say that Faraday’s electric field is different than Coulomb’s electric field?
• I’m in your class right now and I like this distinction. It helps me . . .
• I’m in your class right now and I don’t really care about this distinction. As long as I can . . .
• Why should we introduce a new field if it acts just like an E field? (it doesn’t)
• I love having students figure out whether the amount of B field is growing or falling in generator problems. Leave me alone!
• It’s all relative, students should use whatever calculation approach helps them get the right answer.
• I love this post because I like to have my students focus on what’s really happening microscopically.
• I hate this post because I don’t want to have to explain what a curl is to my introductory students.
• In motional EMF you have to hand waive the idea that the field is constant in a wire. How do you approach that?
Posted in electromagnetism, general physics, physics | 6 Comments

Circuits: find out now or leave it hanging?

I had a really fun class today. It was a review day, like every Friday, and I started out with a pretty innocuous clicker question (though I don’t use clickers). Go ahead, give it a try. The correct answer changes to bold when you hover over it.

The initial vote turned out to have a pretty even mix of the 3 realistic answers (rarely do batteries explode). I was a little surprised because 1) it’s not that hard of a problem compared with some of the complicated circuits we’ve been doing, and 2) I knew it wasn’t as hard as a similar question about R2. But because they were showing their confusion with their vote, I decided to really have all three answers be fleshed out.

For each answer, I asked for someone willing to be a public advocate. I got these three arguments:

1. The current doesn’t change because all of it still needs to flow through R1
2. The current increases because there are now more paths for the current to flow through
3. The current decreases because the total resistance is now larger so the battery doesn’t have to work as hard.

For each a student first articulated their thoughts and I said it back to them with some clarifications to make sure I had the argument right. After the three arguments had been laid out, I told them that 2 of them were wrong. So I had them share with their neighbors some more and this time I wanted them to vote an argument “off the island.” In other words, don’t vote for the correct answer, instead vote for the argument that you are most sure is wrong.

The vote was basically even once again!

Ok, here was the cool part. I looked at the clock and realized that we had already spent 10 minutes on a question that I thought would only take us 2 minutes. So I asked them which of the following would help their learning the most:

2. Leave them hanging
3. Let them talk it out some more with their classmates

Interestingly, as I gave the options for the vote, right away a vocal minority shouted for a combo of “tell us now” and “definitely don’t leave us hanging.” So I asked this question:

Are you voting for what you want now or what will help you learn this for next week’s oral exam?

I thought it was interesting that a few of them really seemed to get that they would vote two different ways based on the answer to that question. So, after a brief talk about what I meant by that question, we voted again and it was resounding for “let us talk some more about it.”

So I told them that I’d put them in three groups to hash it out. I asked if they wanted the groups to be made up of people who liked the three different possibilities OR if they wanted the groups to be made up of people who really disliked one of the possibilities. They voted overwhelmingly for the latter, which surprised me. But I went with it.

So they got into their groups and I told them to make sure everyone in their group understood their main arguments and that I would randomly select a person from each group to present the argument. It’s interesting that I made the mistake of asking “are you ready” before making it clear that I would randomly pick someone. Before that clarification they said they were ready. After the clarification there was a strong sentiment that they needed more time.

Here were the three counter arguments:

1. It can’t stay the same because V=IR and R is clearly changing.
2. It can’t increase because the equivalent resistance goes up.
3. It can’t decrease because the charges aren’t psychic (they have no idea that there’s a change ahead when they’re in R1).

Ok, I thought, let’s get into positive groups (groups that are separated by the original answers). Again I told them to have an argument ready.

1. It stays the same because of the psychic argument (which I now called interestingly wrong).
2. It increases because the equivalent resistance goes down (and here’s a chalkboard with a numeric example).
3. It decreases because the equivalent resistance goes up (and here’s a whiteboard with a symbolic approach).

I thought the juxtaposition of the last two was really interesting.

Alright, at this point we had been working on this for 30 minutes! But the energy in the room was great! The next thing I did was follow up with what one student (by himself) had started to do on another whiteboard:

$R_\text{equivalent}=R_1+\frac{R_2 R_3}{R_2+R_3}=R_1+R_2\left(\frac{R_3}{R_2+R_3}\right)

There were a few “aha’s” at that point so I did an oral vote and now everyone was pretty convinced of the correct answer (the current increases).

So I thought it was fun. I liked how they asked to spend more time on it, neither wanting to be left hanging nor just being given the answer. I also really liked the different modes of argument (arguing for an answer versus arguing against an answer).

What do you think? Here’s some starters for you:

• I’m in this class and I thought it was great. I was confused at first but . . .
• I’m in this class and I just wish you would have given us the answer because . . .
• I’m in this class and you’ve totally misrepresented what happened. Instead this is what happened . . .
• I don’t see the difference between what I want now and what helps me with next week’s oral exam.
• I would have fired up PhET and shown them after the initial prediction (Note that one twitter buddy basically said just that:

• I would have wired up the circuit and shown them.
• I wouldn’t have spent more than your original plan of 2 minutes. This is too easy of a problem to spend 30 minutes on.
• What do you have against clickers? They’re the single greatest technological invention in the last 100 years!
• Do you have students really advocating for themselves when you say you might pick them randomly?
• This class sounds great, how do your students like all these crazy things you do?
• It’s too bad you weren’t asking about R2. That’s a much more interesting problem. Note that another twitter bud of mine characterized that as the pizza problem:

Posted in physics, teaching | 4 Comments