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:

But most physics students first see it in integral form:

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 ().

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.

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?

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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:

- The current doesn’t change because all of it still needs to flow through R1
- The current increases because there are now more paths for the current to flow through
- 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:

- Give them the answer
- Leave them hanging
- 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:

- It can’t stay the same because V=IR and R is clearly changing.
- It can’t increase because the equivalent resistance goes up.
- 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.

- It stays the same because of the psychic argument (which I now called interestingly wrong).
- It increases because the equivalent resistance goes down (and here’s a chalkboard with a numeric example).
- 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:

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:

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What works:

- I think students are putting much more time into learning how to do problems than last year.
- Grading is pretty straight forward, with everyone getting a single standard score using my Frank-Noschese-stolen rubric.
- Someone usually groans after the die rolls. Sometimes it’s me.

What doesn’t:

- The “turning inside out” takes up some time, usually 3-4 minutes. That means when it’s all done and they’ve turned it in and they’ve gotten into groups we only have about 45 minutes left. It’s not the end of the world, especially as I see value in the conversation about turning things inside out, but I’m nervous about it.
- My friend Joss had a really interesting comment on my last post about this, and I do see some evidence that this is happening. Basically he talked about students finding the governing equation for a problem and just plugging in the knowns to solve for the unknowns. That seems like it could be what we want our students to do, but I think I’d rather have them thinking about physics than memorizing governing equations.
- The fact that Monday’s quizzes overwrite one of the scores from the previous week still takes some getting used to by my students.

What’s surprising:

- I didn’t realize how much I would like the fact that I can have a much larger impact on what they’re practicing by picking particular types of problems. I tend to assign one “Question” type problem that the chapters have, 1-2 “exercises,” and 1-2 “problems.” Last year every reassessment was an “exercise.” I can have them attack problems of all sorts. For example, all the symmetry types are showing up for the Gauss’ law problems they’re doing for an upcoming class. Last year students just submitted re-assessments on whatever symmetry they understood the best.
- I really like the forced reassessments on Monday. On Fridays we have a review session (the nb.mit.edu usage hasn’t really taken off for that, but we’ll see) and then over the weekend they have 2 problems on Monday’s info and 2 on Wednesday’s info. It’s great to have them really reviewing stuff before we move on.
- There was a problem this past week that involved figuring out the force on a charge based on the location of 2 other charges. All I did to “turn it inside out” was change the sign of one of those charges. It fundamentally changed all the directions in the problem, and the class had an average of about 2 out of 4 on that one. I thought it was really interesting that such a seemingly minor change caused such problems for them. It makes me a little nervous that that type of change, ie the type that I’m excited about making them really think, is too deep/subtle/hard for them to swallow in 10 minutes.

Overall I’m pretty happy about it. We’ll see if any of them chime in on this post below in the comments section to see if they like it.

Your thoughts? Here’s some starters for you:

- I’m in this class and I really like the quizzes. What I like best about this approach is . . .
- I’m in this class and I really don’t like the quizzes. What I think would be better is . . .
- I like to assign problems that take a while to do. Isn’t 10 minutes too short?
- I’ve been wrestling with whether Joss’ concern is something to be worried about or not. I think that . . .
- I wouldn’t waste time having the students help you turn problems inside out (also, I think it’s bourgeois of you to constantly put that in quotes). I’d just figure out how I’d want them changed ahead of time and just tell them right after the die roll what the quiz is.
- I think that the “turning inside out” discussion is probably one of the most valuable things you seem to be doing with your students. I think you should do way more of “that,” even at the cost of not even doing the quiz.
- This is all a waste of time. You should use class time getting them to talk with each other and really learn, not regurgitate the work they’re doing outside of class. Don’t you trust them to do what’s best for their learning?
- I stumbled onto this blog after I heard that you did a gig with Bill Nye once. I’m really disappointed that there’s not image of him in this post.

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It then led to a conversation where Alice talked about how she’s challenged herself to never address the whole class, instead focusing on small groups or even individual conversations with students during class. She makes sure the instructions for what to work on are available (she uses Google Classroom quite a bit, I know) and creates a culture where the students come in and get to work right away.

This got me thinking about my general physics class coming up this semester (first day is this Wednesday!). We spend quite a bit of class time working in groups, using whiteboards, trying to figure out approaches, apply ideas, predict what’ll happen in demos etc. I spend a lot of my time walking around engaging with groups and individuals. So far, so good, and I can see how Alice’s ideas could help me get to that faster. I could make sure that the questions/issues/content/demos are laid out on some web site or projected handout or whatever and have them get right to the groups right away. I like thinking about how I could save time by not re-explaining the instructions to the whole class even when only one group or individual has the question. It would mean having some pretty explicit instructions but I don’t think that bothers me.

But I see another way that would save time and make the learning better. Here’s the scenario that I think could be improved:

Student (or group): … but when we do that, we get this?!

me: ooh, cool! HANG ON EVERYBODY, THIS GROUP HAS SOMETHING YOU SHOULD SEE

Of course I’m happy to use live group work to help others learn. However, the big problem with that scenario is the “HANG ON” part. I’m forcing everyone else to freeze their thought process and try to focus on something that’s not hitting them in stride. There’s all kinds of variations to that scenario, ranging from “we don’t get why this matters” through “We googled this and . . .” to “ours seems to be different than everyone else’s.” All of those can often lead to learning for everyone if everyone could give some attention to it. But forcing *when *that focusing happens causes problems.

So what if I used technology to allow those moments to be captured (probably a photo of their work along with a caption from either me or them (assuming I encouraged them to do it)) and added to a streaming slide show that they can all access and look at. Alice says it well:

In other words, when people are ready to focus on something that another group has done, they can access the slide show, pause it on the appropriate slide, and learn!

I think I’d still like something of a board meeting (borrowing from the Modeling community) in there somewhere so that different groups could interact, but this idea could really foster some great conversation.

What I would need:

- A website (sounds easiest) that could display the instructions and the evolving slideshow
- The ability for anyone in class (including me) to add to the slide show with both photos and captions
- Repeat with a fresh (empty) slide show for each class

I’m sure I could build that into my homebuilt LMS but I’ll likely look around for something first. Maybe Flickr could do it? Maybe a google folder? I’m happy to take suggestions below.

Questions/thoughts/ideas/anecdotes/complaints? Here are some starters for you:

- This is Alice and I like how you’ve captured this idea. Here’s a bunch of other resources along these lines …
- This is Alice and you’ve totally screwed up this idea. I really hope no one’s reading this comment because that means they’ve scrolled through this whole stupid post.
- Why do you think that students reading instructions is better than you giving a (nuanced?) explanation at the beginning of class?
- I hate it when I interrupt students to give them something new to think about. I like this approach, though I worry they won’t attend to the slide show. Here’s how I’d fix that . . .
- I love it when I interrupt students to give them something new to think about. It usually brings back the groups that have gone off the rails and later they can discuss with other groups better because they’ve all thought about the same things. I think you should do this to fix this idea . . .
- I’m in your general physics class this semester and I’m excited about some of the ideas in this post. Mostly I’m excited to . . .
- I’m in this class this semester. Where can I get a drop card?

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I really like letting the students weigh in, and to really debate those choices in the previous paragraph. It works really well in an upper division course when there can be some great debate over what’s interesting. In such a course we spend the day grappling with weird/deep ideas, and we end the class recognizing that we’ve only used broad strokes in class and that the details will come from further resources (the text, my screencasts, etc). The students have taken a bunch of physics courses so they get the subtleties involved in when they should study a derivation vs studying applications.

However, I don’t think this works as well in an intro course. Physics majors are a minority in my general physics 2 course, with the majority being students who really feel like they’re supposed to learn calculations and applications. I don’t actually fully agree with this, but it’s what they think they should do and so that’s what they always vote on at the end of each class period.

What’s even worse, though, is that we’ll spend the day doing the broad strokes and students will vote that the standard should only cover those. I get a lot less of that in the advanced courses, but I found myself quite frustrated with that last year in this intro course.

I don’t think setting up the standards ahead of time is really that bad, even in the advanced courses. I think I’ll do it for intro and play it by ear in the advanced courses as I move forward. One really nice thing about doing it ahead of time is that the day spent in class is very focussed. The students know what they’re going to be assessed on so they can help nominate further resources that they think they’ll need.

Thoughts? Here’s some starters for you:

- I’m going to be in this class and I think this is a cool idea. Here’s why . . .
- I’m going to be in this class and I think this is a dumb idea. Here’s why . . .
- What are some examples of standards that made you frustrated in the past?
- How has this worked in your non-science-major courses?
- If you set all the standards up ahead of time, what happens if you fall behind?
- I think having intro students doing derivations is a waste of time. I hope you don’t ask them to do any for your standards.
- I think having students do derivations is the only important thing. I hope that’s all you ask them to do.
- All students should be in the class because they love physics. If they don’t you should force them out.

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The typical problem to be solved with FEM involves trying to find a function on a 2D surface that obeys a (typically) second order differential equation. A few examples include

- The
~~voltage~~electric potential inside a region where the~~voltage~~potential on the walls is known - The shape of a rubber sheet hung from a frame
- The possible standing waves on a surface
- In quantum mechanics this could be possible eigenfunctions
- In acoustics this represents the possible resonance modes

Normally you divide the surface into lots of triangles and then turn the continuous differential equation into a finite number of linear equations, solving for the value of the unknown function on the nodes. Here’s a great (though long) description of how it works.

For years I’ve been jealous of my friends who are facile with software that can do this sort of computation, because if you try any of those examples above with Mathematica 9 or below it laughs at you and gives up. But now, in Mathematica 10, it’s built into my favorite command for doing differential equations: NDSolve.

The first thing you have to do is define the surface you’re interested in. There are a ton of built in shapes like Disk[] that you can use, but you can also define them using commands like ImplicitRegion and RegionUnion and RegionDifference. You can also make regions out of graphics and a lot of the built-in data in Mathematica like the shape of countries, states, counties, and cities.

Once you have the region specified, you can do a command like this:

`NDSolve[{Laplacian[u[x,y], {x,y}]==0, Boundary conditions ...}, u, {x,y}\Element region]`

where I’ll discuss the boundary conditions below (and “region” has already been defined). The Laplacian command is equivalent to:

Alright, here are some examples:

For this example I set the Lagrangian equal to a nonzero value and set the frame (borders) to be interesting shapes (every frame in the gif is a different solution). You set the border using a Dirichlet boundary condition like this: DirichletCondition[u[x,y]==some cool function, border you’re interested in]. In this example I did a different condition for the inner and outer borders.

Here I solved the following equation on the region in the shape of Minnesota (obtained using DiscretizeGraphics[EntityValue[ctrl-=Minnesota, “Polygon”]])

Another way to do the same thing is to find the resonant modes of Minnesota and combining them to start as a thump:

By playing with Neumann boundary conditions (where you can set the value of the slope at the boundary) I tried to simulate flapping flags:

I didn’t like the edges very much, so I tried again with a square US flag, adjusting the boundary conditions a little more:

Note that you can get the flag image using flag=EntityValue[ctrl=United States, “Flag”] and you can put it on 3D graphics using PlotStyle->Texture[flag].

Finally I’ve played quite a bit with various drum heads and the sounds they’d make. You solve this equation:

But you do it with an awesome brand new command in 10.2: NDEigensystem. It takes just the left hand side of the equation (with an odd negative sign that took me a while to get), along with appropriate boundary conditions. It returns the values of and interpolation functions for the resonant functions. Here’s the 25th resonance of a Texas-shaped drum:

You can use the ‘s to figure out what a drum head will sound like, and you can figure out how much of each resonance you’ll hear based on where you thump the drum by expanding your “thump” function into the resonant functions. I cheat and use a delta function thump so I just need to know the value of the resonant function at the location that I want the thump. By the way, the functions returned by NDEigensystem are orthonormal, which makes my last sentence somewhat more palatable to my mathematician friends.

Here’s a movie I made combining Mathematica’s ability to do all the calculations, image generation, Minnesota data look up, Traveling Salesman solutions, and sound generation with FFMpeg’s ability to combine images and sound into a movie. It shows what it would sound like if you thumped a Minnesota-shaped drum head in the center of all 89 counties in an order determined by the shortest tour through those points.

Fun times, huh? I’ve certainly enjoyed learning a ton about FEM and what you can do with it, and I’ve gotten decently facile with Mathematica’s implementation.

Thoughts? Here are some starters for you:

- Why did you strike out voltage in the early examples?
- You said it was typically second order problems. What about the beam equation, huh? That’s fourth order, can you handle that? I bet you’d have to use something like COMSOL to do that, not wimpy Mathematica (you’re right).
- Hey, I’ve done some of this stuff with Mathematica (x<10), you’re a liar.
- What’s up with the negative sign you have to use in NDEigensystem?
- Your description of a delta function thump has tons of problems, not least of which is that you have to calculate all the resonances, and I have a sneaky suspicion you only do, oh, I don’t know, 20.
- How did you decide to put dampening in the sound? It’s clear you’re not just playing those frequencies combined.
- Why didn’t you do this in python, that seems the obvious choice for anything computer related?
- How could you utilize this when teaching Laplacian solutions in a electricity and magnetism class?
- I tried your flapping flag simulation and I got some negative eigenvalues. What the heck? (Yeah, I get that too when I set the Neumann boundary condition equal to the function itself, which is what I thought a flapping flag would look like. My mathematician friend tells me that then I can’t trust any of this.)
- Why do you only do Minnesota examples? What, you think you’re too good for the rest of the country?

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Back when I graded homework/practice I felt that I was bad at holding the line in my office hours. What I mean is that students seemed to be pretty good at tricking me into doing their homework for them. I was also getting pretty crabby about trying to figure out if the students were doing their own work. So my solution was to have daily 10 minute quizzes. I would assign four problems per class with the promise that in the next class I would roll a 4-sided die to pick one of the four to use as the quiz. This had a lot of benefits/features/odd-side-effects:

- I was perfectly happy to help students in my office hours
- I would “turn the problem inside out” in the quiz by asking for a different unknown, so the students understood that to study they should solve the problem and make sure they understood all the connections among the various variables.
- I would randomly change the numbers involved by asking the class to shout some out. Before letting them take the quiz I would let them discuss whether they expected a weird answer based on the random numbers they’d thrown out. For example, if they said that a capacitor had 100 Farads, they guessed that the rest of the problem would have answers very different than other problems they’d worked.
- Nearly every class day someone would groan after the die was rolled. Sometimes it was me if I felt it had rolled to one of the easy ones. But I liked that it was truly random so they didn’t spend time the night before trying to guess which problem I was going to pick.
- I liked how they had to work on a problem long enough to have the confidence to pull it off in class in 10 minutes. Often their first pass at the problem the night/day before took much longer than that. Combined with point 2 above, this led to some decent problem solving skills

I felt it worked quite well. Plus it dramatically reduced my grading from 4 problems every day from each student to one.

So I was wondering if this would work in my new SBG implementation where I’m trying to hold the line against giving points for homework. I think it might. Here’s what I’m thinking:

- Assign 4 problems per day just as above. Design them to measure students understanding/mastery of recent standards, not necessarily just the one covered on that day.
- Do the 10 minutes on Monday, Wednesday, and Friday even though Friday will still be a review day (there will still only be two standards per week).
- Grade the quiz with standard scores (my 1-4 rubric)
- Continue to allow students to do video reassessments

Content days (Mondays and Wednesdays) would now have 10-15 minutes less per day. I handled that in the past pretty well, but I’ll definitely have to tighten up the activities that we’ll do. I had several sloppy days last year where directions weren’t clear and students spun their wheels quite a bit. I think much better preparation on my part for things like clicker (ok colored cards) questions and white board/group work will pay off huge for this.

Review day will now only have a 10 minute quiz so the rest of the day I can use the awesome nb site from MIT. That’s like google moderator on steroids, as students can ask questions about daily notes and vote each others questions up and down.

I really like point 5 above about students needing to practice enough to get fast at the inside-out problems. That seems to be a good indicator of mastery.

I’m worried that, due to my reassessment policy, students won’t take these quizzes seriously. I’d appreciate any thoughts you might have about that.

Thoughts? Here are some potential comments I’d appreciate:

- I’m going to be in this class and this sounds intriguing. How will ______ work?
- I’m going to be in this class and this sounds terrible. Instead why don’t you . . .
- Why 4? Is it just because you have a 4-sided die?
- Do you like this better than the ticket idea from the last post? What would you miss from that if you didn’t do it?
- Can you give some examples of turning a problem inside out?
- Why do you let them suggest numbers that clearly wouldn’t be physically reasonable?
- Can you say more about how your office hours changed when you switched to this in the past?
- How exactly do you plan to use the nb MIT site?

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**Practice is your ticket**

It’s a Monday, Wednesday, Friday course and I still really like the idea of only 2 standards per week and so therefore a review/assessment day on Fridays. Last year we spent ~30 minutes reviewing material based on their submissions to Google Moderator and then ~30 minutes on a quiz that tended to assess the two new standards of the week (though often hitting earlier standards as well). My thought for this year is that their work on practice problems would drive the discussion/work during the first 30 minutes, but that they could only take the quiz if they submitted some practice work.

Because it’s standards-based grading, missing the quiz isn’t the end of the world. My students are (nearly) always allowed to resubmit video assessments for any active standard. Last year the quiz was always the first assessment on that week’s standards and some students turned in reassessments later. So if they demonstrated that they’ve practiced, they get to take the quiz. If they do well, the only reason they’d ever have to do some reassessment is if I force them to during one of the two oral exams.

I’d actually like to say that submitting evidence of practice is the ticket for even the first 30 minutes (when the review happens) but I feel a little weird about barring students from class.

**What does evidence mean?**

At first I was thinking that students could come with some sheets of paper where they’ve worked on some problems. They’d bring them and turn them in so that they could get the quiz. They’d also use their successes and failures of their practice to focus our conversation during the 30 minute review part. But then I had a different idea:

What if the students did some practice and submitted photographic evidence. This does a number of nice things for me:

- The students tackle the logistical problem of photographing and submitting their work, both of which are necessary (though not sufficient) for figuring out how to do the video reassessments.
- Often students would not do any video assessments because they said they couldn’t figure out all the logistics.

- It gives me an organized way to present students (anonymized) work to the class during the review session.
- I like it being anonymous because we can all just look at work done to find common errors or cool tricks/solutions.

- Let’s me take a look at some of the work prior to the 30 minute review time.

**Other possible options**

I could of course just give points for homework. I tend to not like that option due mostly to cheating but also because homework should be practice, not a demonstration of mastery. I realize, though, that I’m likely being too stubborn about this, so we’ll see.

What do you think? Here are some starters:

- I was in the class last fall and I loved not doing homework. I don’t think it affected my learning at all.
- I was in this class last fall and I wish I had done more homework. Getting points for it would have motivated me.
- Not letting a student take a quiz seems harsh. Why not make it so that if they don’t do homework they tend to fail the quiz?
- Just grade the homework! Yes some might cheat, but it’ll ensure most will get the practice they need. Stop clinging so hard to your misguided principles!
- Why not do what Evan does, having a system of credits that students can earn to have a chance at a reassessment?

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For this first pass, I thought I’d look at a bouncing ellipse (in 2D). I figured it would sometimes bounce low (and spin fast) and sometimes high (and spin slow). I thought I could look to see if it “bounces high” in any particular pattern.

Here’s an animation of what I did:

You can see that the height of the bounce does change. Here’s a plot of the height of the center as a function of time:

There’s definitely some variability, but not as much as you can see in some rugby matches:

Here’s a plot showing several height plots for several different initial orientations

It’s clear that there can be quite a bit of variability of heights. Here’s an animation showing that:

So I don’t think this really shows a “every third bounce” pattern, but I think it’s a good start.

**How I did it**

I tried three different approaches to get these calculations. Only the last one worked, but I learned a lot.

First I thought I’d use the new support in Mathematica for regions. You can define a region (the ellipse) and the region for the ground (half plane) and then look for the intersection to see if they’ve hit each other. Then you can calculate the area of the intersection to use in calculating a potential energy. Good in principle, but super slow and the derivatives needed to do on the potential energy didn’t work. Oh well.

Next I thought I’d try to learn how to use the new mesh/Finite Element procedures. I thought I could find the best triangular mesh for the ellipse and treat all the lines between nodes as springs. Here’s the (very slow!) result:

As you can see, the “ball” gets severely distorted upon the bounce. Also, I think treating all the mesh lines as springs doesn’t correctly represent the stiffness of the internal structure of the ball.

Finally I just decided to say that the ground was a giant one-way spring (it only pushes, it doesn’t pull) and then identify when (and where) the ellipse pokes into the ground. It took me a while to figure out the lowest point of an ellipse, but it was really just differentiating the parametric equation to find a minimum in y. So then I could say that my potential energy is just proportional to y^2 where y is the distance that the lowest point gets below the ground plane. That’s the form of a spring and lets the ground push the ellipse back up. In my experience, this is a useful way to do bounces because it tends to conserve energy and correctly model the time it takes to collide.

What’s cool about doing it using a Lagrange approach (with a potential energy) is that I don’t have to do things like “when it hits, reverse the momentum” and “when it hits, figure out the impulse torque to figure out the change to angular momentum.” Instead I just put the potential energy in as a function of y and theta and the Euler-Lagrange equation takes care of all the rest. Specifically it gets the angular momentum just right without every calculating an impulse torque.

So, the upshot is, I think that high bounces are possible using this mechanism, though a predictable “once every three bounces” doesn’t seem to be happening. This was fun modeling though, and I look forward to adding other approaches based on your comments below (hint hint).

Your thoughts? Here are some starters for you:

- I’m in this class and, wait, never mind, it’s summer
- I think this is cool, but I think you can make it better by . . .
- I think this is dumb, what you forgot about was . . .
- Why don’t you add friction?
- Why don’t you add some x velocity?
- Here’s a better way to deal with that mesh . . .
- Mathematica does Finite Element Analysis now? Cool, can it do . . .
- What is Finite Element analysis?
- Why was finding the bottom of the ellipse so hard? Seems easy to me!
- Please let me know when you redo this in Python, then I’ll read it.

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I came up with three ways (four if you count straight to the center of the earth) and all of them give a different shape for the well:

- Slowly reel out a heavy plumb bob on a line. When you stop reeling, it’ll point out away from the axis of rotation of the earth due to the centrifugal force. As long as you do it slow enough, you can ignore the coriolis force. Have the well shape match the path the plumb bob takes during its (slow) descent.
- Slowly reel out a chain or rope that doesn’t have negligible weight but also doesn’t have a heavy plumb bob at the bottom. Have the well shape match the shape of the rope when you’ve got it as deep as you like it.
- Drop a rock and see which way it goes. Have the shape of the well match the trajectory. This one has the benefit that future rocks won’t hit the side!

So I set about trying to figure out each of those shapes. For those that don’t want the details, here’s a plot showing all three. Well 1 is in blue, well 2 is in red, well 3 is in green.

The easiest one is the green one. I just do a pretty straightforward integration of the equations of motion where:

Where the first term is gravity (note that g changes like a spring as you go deeper), the second is the centrifugal force, and the third is the coriolis force. Once you know the form of the force, doing an NDSolve command in Mathematica is pretty easy.

The other two wells were a little harder to calculate. For well 2 I modeled several (19) masses connected by inextensible strings. I started with them all anchored at the top and hanging straight toward the center of the earth. Then I turn on time with a lot of friction to see where they all settle down to. Here’s an animated gif of that process:

I actually used Lagrange multipliers for all the strings and simply solved for x,y, and z of every particle along with the tension forces necessary to keep their spacing correct.

For well 1 I only used one particle but I changed the constraint to be time dependent. Basically I just had the length between the ball and the anchor grow linearly.

Cool, huh? I wonder which is the better way to do it. What do you think? Here’s some starters for you:

- I’m in this class and I love this. I’ll redo that standard using all of these just so you’ll brag about me.
- I’m in this class and I’m pissed. Why couldn’t you post this a long time ago?
- This is cool. What happens with well 3 if you add air resistance?
- I’m confused. What do you mean when you say “g” changes like a spring as you go deeper?
- This is cool, but I don’t think you did _____ correctly. Here’s a post of mine that does it correctly.
- This is dumb. I build wells for a living and we don’t do this at all. Instead we . . .
- This is cool, but as usual you haven’t bothered to post your code, jerk.
- Having just redone all your work, I’m now convinced that you used a huge omega value (0.5) and a tiny earth (10 m radius) with g=9.8 at the surface, all at 45 degrees N latitude. You could have just put that in your post, you know. Jerk.

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