M and I: Chapters 1 and 2

I'm not a big fan of how Matter and Interactions introduces some algorithms upfront, well before their application, like looking at calculating relativistic momentum and breaking changes in momentum into parallel and perpendicular components before even broaching the momentum principle, but I can reorder a bit with no harm; my first set of assessments will group chapters 1 and 2 into a single unit, anyway.

They can also undercut my affinity for symbolic algebra, but I think that we'll be able to press ahead ok symbolically, for the most part.

Here's a progression of whiteboard problems that they did on the second day of school; they read about both relativistic momentum and the momentum principle the night before. They were familiar (though not by name) with the momentum principle already, having done a lot with both Newton's 2nd and impulse last year.

• Projectile motion: given the initial velocity (speed and angle), use the momentum principle (not kinematic equations!) to find the velocity of the particle after some amount of time.
• Projectile motion: use the position update formula to determine the location of the projectile at that time (quickie review of average velocity, including vector form now - hey, we're actually using average v for something!). This is great - two dimensions, no waiting (and no toolkit equations)!
• Space shuttle: if it used its maneuvering thrusters to fly towards Proxima Centauri, what would the issues be? Fuel consumption's a huge one, but let's pretend that it's not, and that the shuttle's mass stays constant (removing this assumption would be a great place to start a capstone!), and that it can get away from Earth easily enough. Using some nominal mass value and the real thrust of the maneuvering thrusters, how fast would it be going after, say, 15 years (using the momentum principle)?  It's going to take a while to get there, after all! Answer: more than the speed of light.  That's a problem. Now we look at the model that we use, and how the p and v grow with time (use program referenced in earlier post). What really happens, though is this (use relativistic def'n with the program). The momentum principle's always true - it's that def'n of p that was wrong.
• OK: how fast will it be going, then after 15 years?
• How far will have gone? Oops - that's too far. 
• Unfortunately, we can't solve analytically for how long it'll take, because of p's complex dependence on v. Time for a program! We sketched it out in pseudocode together, and they finished it in class or for HW.

This Year's Marshmallow Challenge

I always start off the year by doing (never by talking about the syllabus!).  We start with the Marshmallow Challenge, where the students try to build the tallest structure made of 20 sticks of spaghetti, one meter of tape, and one meter of string, to support a marshmallow as high off the ground as possible.

The TED talk accompanying it is great, and it's a great intro to standards-based grading.  What happened to those creative kindergarteners that were unafraid to fail and benefitted from their own mistakes to turn them into disaffected failure-shy business school grads? Well, school, mostly. The combination of high stakes (grades that are indelible) and low skills (what learner doesn't have low skills at the beginning?) creates all kinds of damaging behavior in the long-term (poor retention, little conceptual understanding, focus on "the answer," cheating, cramming) in the service of short-term gains, because those short-term gains are incentivized and they don't yet have the tools to earn them legitimately.

Anyway, this year I had 6 of 16 teams end with intact structures. The two tallest, both at 60 cm are below.

Last year's post

PS: I do allow them to use guy wires, but only connected to the table. Is that cheating? I dunno, but that's what I do.

Assessment Strategy for AP C

I'm running my AP C: Mechanics course a bit differently this year. Most notably, I'm using Matter and Interactions as the curriculum, which takes some major head-wrapping. I've done as much of that as I can before actually seeing students in the course. Frank Noschese, Mark Hammond, and I came up with standards and aligned old AP problems to chapters in M and I this summer, but the grainier nature of those standards presented me an assessment scheme challenge. Usually, my standards are much bigger, translating to fewer per term, and I only allow one standard to be reassessed per day (and only on M, W, and F, after registering via Google doc). Those standards, for fall and winter, look like this.

The AP standards look like this, with chapters 1-5 in the fall term, 6-10 in the winter, 11 in the spring (with some mechanism for including previous term standards that I haven't devised yet).

I was toying with some more project/portfolio-based assessment for this course, especially after I saw success with limited screencast reassessment at the end of last year).

What I'm going with is a combination of in-class and out-of-class assessment, which I'll put under the umbrella of 'portfolios.' Basically, there are a lot of ways that you can prove to me that you have nailed a standard (and it's your responsibility to make that case to me for each standard):

• Show me how well you did on in-class assessments that covered that standard and/or reassessments (I'm only preparing one or maybe two per unit)
• Show me how you applied this understanding to an in-depth analysis ('capstones')
• Show me a lot of problem-solving from the text (M and I's problems are generally pretty robust and most are not the kind of sterilized problem that you see in Giancoli, Walker, etc.)
• Ideally: all three.

I need a mechanism to make sure that not everybody's just doing problems, with all of the potential issues (ethics and others) that that entails. Maybe I'll make a tally sheet for them, so that they have to color-code the methods that they used, so they (and I) can see at a glance how they met the standard. I'd like to make sure that they do at least three capstones per term, as well.

I'm thinking about a binary scale (Yes, Not Yet) and a 50 + 50*(% of standards met) algorithm.

Thoguht?

Standards for (most of) the year

Apart from being deep in accreditation report-writing, I'm, like everyone else, deep in class prep. In case anyone's interested, here are my standards for the first two terms for Physics and Honors Physics and the whole year for AP Physics/Mechanics.

The links are to PDFs of the standards; there's a brief synopsis of the course content for each course so that you know what you're getting.

Physics (my own text - motion, then oscillations (not much force analysis), waves, sound, shadows, astro, optics)
Honors Physics (Mechanics for these terms, pretty modelly, using Giancoli as the text (read: problem bank), but probably Knight next year)
AP Physics (First half of Matter and Interactions)

M and I: Young's Modulus

After looking at the relatively involved apparatus used by the authors and the $1100 Pasco apparatus, I was pretty worried about measuring the Young's modulus for materials. After a very fun meet-up dinner with M and I adopters (seriously - when's the last time that you, Halliday, Resnick, and 20 excited adopters had dinner?) last night, I got three different methods to measure it from folks there:

•  First, from Mark Hammond, the idea of just hanging a wire from an I beam in the ceiling, putting a pointer on it, and measuring.  I hadn't gotten around to trying this, but I was thinking that it'd be a good last-ditch substitute. 
• Second, from Aaron Titus (do you have a blog Aaron?), the idea of doing that but with a motion detector underneath the mass hanger.  If the wire stretches enough, this will measure the distance more accurately than the first method (which is apparently possible - I thought that the stretch wouldn't be much relative to the motion detector's resolution, but I guess that I'm underestimating one of the two).
• Third, also from Aaron, attach one end of the wire to a clamp, run it horizontally (I ran it a little more than 6 meters to maximize the stretch amount) over a Pasco rotary motion sensor.  Have a short length of wire hanging over the pulley and wrapped around a mass hanger.  It only took about 500 g of mass to get a nice range of stretch.  Even a 50 gram difference caused an easily measurable stretch for the sensor (around .05 radians/50 grams).

For the third method:

• No knots: wrap the wire around both connection points - you don't want kinks and knots slipping.
• I measured the tension force with a force probe as well as calculating it from mg.  They were pretty equivalent in practice (as well as in theory, of course), so I'll just use mg.
• I used thin enameled magnet wire (.17 mm diameter - don't know the gauge); I'd like to get aluminum wire as well, so that I can predict resonant frequencies for the singing rods.
• Edit: put a catch box underneath.  As I was writing this, the wire gave way with 500 g hanging from it.
• Here's the data - 118.9 GPa (117 on Wikipedia). I'll surely take that. I'll use this to predict v for a transverse wave, too, and see if I can get agreement there, but this is seriously encouraging.

M and I: Speed of Sound from First Principles

I'm excited about using the Matter and Interactions curriculum to connect the micro-world to the macro-world. Two of the big candidates are speed of sound and specific heat. Right now, I'm looking at the speed of sound.

Using the interatomic bond length and stiffness to determine an expression for the speed of sound (longitudinal) in a material, the authors arrive at:

 

Here, k is the interatomic bond stiffness and d is the interatomic bond length (modeling the solid as a cubic lattice of atoms connected by Hooke springs), and m is the mass of one atom.

What I really want is to be able to put the physics students (who cover sound and know the standing wave stuff and can experimentally determine the frequency of a standing wave and calculate the speed of sound) together with the AP Physics students (who can calculate the speed of sound from experimental measurements, if I can get a Young's Modulus apparatus) to see if the approaches agree.

While I don't teach it, this made me think of the speed of sound in a wire expression:

Here, T is the tension and mu is the mass per unit length.  This has the obvious dependence of speed on tension explicitly and made me think about the M and I expression - tension's not (obviously) there.  Is this an approximation or will the approach only work under no tension (as the value was calculated)?

Actually, the two (pretty much) reduce to each other. Starting at the speed in a wire expression:

Comparing the tension in the wire to the tension in an individual "strand" of atoms:

Substituting, simplifying, rearranging:

The approximation was that one fourth equalled one sixth, from the volume/area substitution. The problem is that this really only works if we call the tension force between two atoms kd, which isn't really what it should be.  I'm not sure if we're dealing with a limitation of the M and I model, some unstated assumptions in the wire model, or both.  ...or a longitudinal/transverse mismatch. Can anybody help?

I'm encouraged about the possible agreement, though it's not really necessary for the expression to agree for me to test the prediction made by the M and I model.  I tried to test it out a bit using a 17 cm long 2 cm thick copper cylinder.  I dropped it on its end on the floor and recorded the high-pitched ringing while it was in the air.  The frequency spectrum:

The fundamental frequency for that bar (using all of the available simplifications) should be around 15 kHz, but it's quite a bit lower (less than 3 kHz).  The spacing's not even here, so I'm thinking that I have an issue with the cylinder having some odd vibrational modes, but I can't find a reference on what to expect.  In any event, I'd like something with well-behaved harmonics that my physics students can analyze successfully. A copper wire seems great, but the waves are the wrong mode, and definitely wouldn't agree.

Another idea that I had: a long aluminum rod (one of those Arbor Scientific dealies).  It's 74.5 cm long, and I recorded this spectrum from the demo video on the Arbor site (I can't find the rosin):

These peaks are nice and harmonic, with a fundamental right around 3346 Hz. This means a speed of sound of 4986 m/s, which should agree nicely with the value that we'll get from using the bond strength of aluminum (OK - I need to get some aluminum wire.  Hopefully that's a thing.).  It looks like this is the way to go.

Other suggestions, anyone?  I'm going to try thumping a marshmallow, too, since the modulus is easy to determine, but I'd have to do some voodoo with the calculations, since it's not an elemental substance.

M and I: chapter 1 - Relativistic vs. Classical Momentum

In my last post, I promised to share thoughts and materials.  Here's the first!

Chapter 1 is a lot of review for my second-year students. The new bit is relativistic momentum, and the revelation that our definition of momentum from last year was really just a special-case limit.

One of my themes for the year is 'pulling back the veils' on all of those more complicated bits that we approximated and hand-waved away last year. I want this to be a major selling point of the course. Unfortunately, as presented, I think that the momentum definition isn't accessible or real to the kids.  We can't really discover it, and it's just an abstraction at first.

I wrote a VPython program to give this some context. You can run it twice for the students - in each case, a 60 g block is pushed from rest by a constant 1 million Newton force. The momentum and velocity of the block are graphed as time goes on.

Here's where another pillar of the course is going to come in to play: fundamental laws of the universe. Really, we didn't deal with too many last year, at least not on a really visceral and important level. Here, though, the universal speed limit becomes important. Look at the v graph pass right by it without blinking, when the program uses the classical definition of p (and finds v from there):

That's a problem. The problem's with our model, though: momentum just doesn't really equal mass times velocity.  Uncomment the gamma definition in the program to use the real relationship now:

Here's a great way to distinguish which is the fundamental principle here: the momentum principle still holds just fine, even in the relativistic case, but our definition of v (as determined from the momentum) was broken. With the relativistic relationship, the velocity approaches the universal speed limit.  Two days in and kids not only know a fundamental law of the universe, but they know how it really does fit in with what they knew about momentum and what the missing piece in their model was.

Script:

 from __future__ import division, print_function  
 from visual import *  
 from visual.graph import *  
 scene.background = color.white  
 scene.height = 50  
 scene.width = 50  
 scene.x = scene.y =0  
 ## Josh Gates 2012, started with kernel of Ruth Chabay program  
 print ("""  
 Click anywhere in display window to start.  
 Click to stop.  
 Momentum and velocity are shown as .6 kg block is pushed with a constant 1,000,000 N force.  
 Uncomment gamma line to remove classical p def'n/assumption""")  
   
 ## CONSTANTS  
 delta_t = 0.01  ## for 100 steps  
 mblock = 0.06  
 c=3e8  
 Fnet = vector(1e6,0,0)  
   
 ## OBJECTS & INITIAL VALUES  
 pblock = mblock*vector(0,0,0)  
 gamma = 1  
   
 # start time at 0  
 t = 0  
 scene.center = (0,.1,0)  # move camera up  
 scene.range = 0.15  
   
 ## GRAPH STUFF  
 gp = gdisplay(background=color.white, foreground=color.black, y=0, x=250, height=300,  
        title='Momentum vs. time: block with constant F', xtitle='Time (s)', ytitle='Position (m)',  
        ymax=3e7,xmax=30)  
 blockp = gcurve(color=color.magenta)  
 blockp.plot(pos=(t,pblock.x)) ## initial pos.x of block  
   
 gv = gdisplay(background=color.white, foreground=color.black, y=300, x=250, height=300,  
        title='Velocity vs. time: block with constant F', xtitle='Time (s)', ytitle='Velocity (m/s)',  
        ymax = 3e8, ymin=0, xmax=30)  
 blockv = gcurve(color=color.blue)  
 blockv.plot(pos=(t,pblock.x/(gamma*mblock))) #initial velocity  
 asymptote=gcurve(color=color.red)  
   
   
 scene.mouse.getclick()  
 stopper='go'  
   
 a=.1  
 while a<1000:  
   asymptote.plot(pos=(30/a,3e8))  
   a=a+1  
   
 ## CALCULATIONS  
 while stopper=='go':  
   rate(500)   
   
   pblock = pblock + Fnet*delta_t  
   
   # comment to make classical approximation  
   #gamma = (1-(pblock.x)**2/(mblock**2 * c**2+(pblock.x)**2))**-.5  
     
   # update time  
   t = t + delta_t  
   # plot pos.x, velocity.x of block  
   blockp.plot(pos=(t,pblock.x))  
   blockv.plot(pos=(t,pblock.x/(gamma*mblock)))  
 ##  print t, block.y  
   if scene.mouse.clicked:  
     stopper='stop'