What a great exam this year... I particularly loved #3, the quantitative-qualitative translation question in which students have to work through a non-constant torque graphically and conceptually. The paragraph question which combines energy and momentum concepts is likely to show up at our year-end conceptual physics tournament in the near future.
My solutions can be found at this TEACHERS ONLY link. Yes, really, teachers only. If you ask for access as a student or parent, Gardner, who runs that site, will take away your birthday, and for extra measure, he'll lay a little spell on you right there. He'll turn you into stone, or a dog, or a chair.
Teachers, if you'd like access, please follow the instructions at the site. Send Gardner a request to join, along with evidence that you are a physics teacher.
The official solutions will be available in the fall on the College Board's AP central website. I'll be grading problem 3 - woo-hoo! (No, I've no clue yet what the rubric will look like, just that I'll be grading thousands upon thousands of problems 3.)
GCJ
I think you have an error in your solution to problem 3c. The reduced friction only starts after 1/2 the time goes past. So the graph for the first half should be identical to the graph in 3a. (unless I am interpreting things incorrectly)
ReplyDeleteOh, you’re right. I started with the correct slope, but the curve shouldn’t start until 1/2 the time has elapsed. Thank you!
ReplyDeleteHonest question about #2: suppose a student graphs "Resistivity as a function of Trial #," comes up with a constant function, does a best fit line, and calculates a pure average as their answer. Legit? Or partial credit? Or total heresy?
ReplyDeleteAlso, just an opinion: #5 is ill-served by being a paragraph question. There are a lot of moving parts to it (pun may or may not be intended), and part b) could easily have been three separate "briefly explain" questions. As is, I predict this problem will be particularly hit-or-miss.
Will,
ReplyDeleteGraphing vs. trial number seems total heresy to me - I can’t see that earning any credit.
I like the paragraph question very much; and I like it phrased as a paragraph. Gardner Friedlander points out that my paragraph could have been shortened substantially by focusing just on the energy argument - since mechanical energy must be dissipated in the inelastic collision, the spring energy at maximum displacement must be less than the kinetic energy at equilibrium. The problem didn’t ask *by what factor* the amplitude changed, just whether the amplitude increased, decreased, or remained the same.
Okay, but hear me out: a student really can compute the resistivity for each cylinder separately, given that data. Filling out one of the columns as resistivity, with appropriate units, is a thing they can reasonably do. When you get right to it, the graph is just not necessary to find the average and accomplish the mission of the lab, but if we're going to have to make one, there are several examples out there in scientific literature where separate trials are in fact compared on one axis to see if they yield the same value. Maybe the title with "as a function of" is problematic, but the basic idea of, "Do four trials and find the average value" is pretty standard practice for finding anything expected to be constant.
DeletePlus, if that's their graph in part a) ii), I don't see how they couldn't get credit for being consistent when they answer part a) iii).
As for question 5, we'll just have to wait and see what the distribution looks like. I'm hoping I'm wrong, and there are a lot of people who get two or three partial credit points out of their paragraph.
Ah, I’m seeing now what you’re proposing... do R/(L/A) in each trial separately, and then graph that. I missed the substance of your proposal the first time. I’m not on that problem this year... but I will ask the table leader the twofold question of (1) did anyone do this, and (2) did they get credit? If I were table leader, depending on how the rubric is worded, I’d try to find a logical way for that approach to earn partial credit.
ReplyDeleteI love the paragraph approach here precisely because it *doesn’t* talk the student through step by step brief justifications - the student has to create the argument out of whole cloth. So, you’re right, of course - I might expect more 4/5 scores and 0/1 scores than middling scores. That’s okay. I’m glad that we’re demanding that students create multi-step logical responses to complex questions without stepwise hints.
I'm okay with multi-step explanations, too, just as long as they're set up and graded so that self-consistent answers don't miss 100% of the points due to one single early mistake in the problem. For example:
Delete"The amplitude will increase, because more mass headed at the same velocity will mean that there is more kinetic energy at the equilibrium point. Greater kinetic energy there means greater elastic potential energy at the point of maximum displacement from equilibrium, which means amplitude would have to increase, because the spring constant is, as the name implies, constant."
The student has only made one mistake: thinking the block will not slow down when the other block is dropped on it. But because of that one mistake, they miss everything, even though the rest of the answer accurately describes what would happen if there were indeed 3m headed at the same v. And to my way of thinking, if a student only makes one mistake, they should only be missing one point out of however many are allotted; i.e. that in a well-designed rubric, a grader who grades additively and a grader who grades subtractively should ideally arrive at the same grade for the student.
*sigh* Someday hopefully I'll be able to join you guys in the grading circle and get to take part in the official discussions. They do sound really interesting in a physics nerd way. Even if it does mean there's a pile of painful paragraphs to peruse.
Will, I just looked at 2017 AP Physics 2 problem 2, which poses a similar experiment. The rubric there pretty much shuts down the idea of calculating resistivity in multiple trials, and graphing vs. trial number. I don't believe that would have earned credit last year; it sounds unlikely that would earn credit this year. I'll discuss this issue in the podcast that posts Monday May 14 2018. And I'll ask that problem's table leaders when I see them this summer.
DeleteWill, the rubrics are usually designed in the way you suggest. We’ll see when the official rubric is published, but the answer you described is likely to get significant partial credit.
ReplyDeleteYeah, crafting and arguing through a rubric is serious intellectual physics nerdery. I hope you get to be part of the process at some point.
Hello Mr Jacobs,
ReplyDeleteRegarding the Free Body Diagram for FRQ number 1, is it OK if the line started at the spacecraft but was drawn all the way to the center of the planet? I do remember on past FRQs how strict the grading was regarding the length of each vector, etc. So I wanted to be as specific as I could since the gravitational force technically should go to the center of the planet..
Also..what are your thoughts regarding how difficult this FRQ section was compared to last year? Personally,I found last year's FRQ a lot easier than this year. I know the curve is different every year to reflect difficulty.
I bought your 5 steps book, and I really thought it prepared me well!
Thank you!
Thank you for the kind words, Anon. As long as your arrow STARTS on the spacecraft and points the right way, it shouldn’t matter how long it is. The reason vector lengths matter is because you can compare several forces, seeing which is larger. Here there’s only one force, so no comparison possible.
ReplyDeleteYeah, difficulty is so subjective, and different for one person than the next. I thought all five problems were exactly the level and style I expected. No surprises. Nothing crazy hard, nothing crazy easy. But, YMMV.
Thanks for your solutions. I felt like this year was a little more straightforward than some questions in the past. All were fair though. The lab (dough) problem seemed easier since they were asked to modify an existing setup with the temp consideration rather than design it all from scratch from the outset. Some of the previous lab problems have required designing the whole experiment initially. Overall, I was pleased.
ReplyDeleteHello Mr Jacobs,
ReplyDeleteOne more question- For the graph in question number 2, is it OK if we started our line at the origin? Conceptually it made sene with the formula...
Thanks
I treated the origin as a data point. Why? Because I “did” the experiment: a piece of dough with length 0 m has 0 resistance.
ReplyDeleteHi, Greg,
DeleteBased on your enlightening explanations elsewhere, I suggest we not include (0,0) as a data point unless it has actually been measured as such using the same equipment that was used to collect the rest of the data. Although (0,0) is true "in theory," we have no way of knowing without actually making measurements at L=0 that the measuring instruments themselves will yield a resistance of 0 ohms there. They certainly might, but it's also possible that there could be a non-negligible resistance in the wire, or even just a miscalibration in the measuring equipment itself, that could result in a nonzero reading for the resistance at L=0. In other words, if the data is our guide, it might be better not to insert data points based only on theory, unless those data points have actually been measured and collected as part of the experiment.
If we include (0,0) as a data point without a measurement at L=0, we run the risk of missing or disguising some nonzero artifact of the measurement process that may be contributing to the resistance calculations at all measured values of L.
Even with a Hooke’s Law spring lab, where it would seem “automatic” that F=0 when x=0, I would suggest that the data point (0,0) should only be included if it is explicitly measured as such. Knowing it should take 0 N to displace a spring 0 m might tempt us into believing that we have already “done” the measurement there simply because we know it “should” be true. But the critical question is, what does the measuring equipment say? We need to make sure that whatever equipment is used to collect the rest of the data is also used to take any measurements at x=0. If we use a force probe to collect force readings at nonzero displacements, we should use the same force probe to measure the force at x=0. If we don’t, and instead insert (0,0) as a data point relying only on our intuition that F=0 at x=0, we would not only miss that perhaps the force probe isn’t tared correctly and needs to be recalibrated, but we would also be “short-circuiting” the point of taking experimental measurements in the first place.
I would agree that following one’s intuition is not an acceptable substitute for taking a measurement. Even with a distance vs. time lab, when the runner begins at d=0, I must still look down at my stopwatch to ensure that the time showing is 0.00 seconds if I want to include (0,0) as a data point. Just knowing the distance should be 0 when the time is 0 doesn’t guarantee that my measuring equipment will agree. If I fail to take an actual measurement at t=0, then I can’t add (0,0) to my graph and still assume that my data is consistent and reliable. If there’s any error in the measurement process with the other data points, it will be obscured if I add “data” that hasn’t been collected via that same measurement process.
In regard to the AP question, I believe we shouldn’t claim to "observe" a resistance of 0 ohms when L=0 unless the student has explicitly performed measurements on the same segment of the circuit without the dough cylinder present (which we generally don't do in physics class because the resulting high current can heat up the wire and damage connected components). To ensure that our data is meaningful, we should be consistent in the way that we collect and record it (relying on measurements that have been explicitly observed instead of on expected results based on theory). Since (0,0) was not measured as an actual data point, it seems best not to include it as “data.” The graph should reflect the measured data itself, not relationships we expect to be true, as you have helped me to understand.
Therefore, in question #2, I believe we should rely only on the data provided in the data table, and not insert additional data points we expect to be true. Sure, we know the resistance of zero-length dough is 0 ohms, but the question is, do we know that the measuring instruments, the same measuring instruments used to collect the rest of the data, would have yielded exactly 0.00 ohms as well?
Hi Greg. For problem 3 on the graphs, I would assume that because none of the graphs had unit scaling provided for the y axis that the lines drawn for the acceleration graphs wouldn't necessarily have to be at a certain distance below the x-axis. For example, a student that drew the acceleration 2 units below the x-axis in the first scenario could get away with drawing the acceleration before t = t1/2 1 unit below the x-axis. Additionally, do you think students will be penalized on part c) ii. if their acceleration hit 0 at t1, so long as the acceleration is clearly not positive at any point?
ReplyDeleteYou’re right... the scaling shouldn’t have to be consistent from graph to graph. Not sure whether acceleration can go to zero.
ReplyDeleteBut to this and the previous unknown respondent, and to all: I love the great physics questions everyone is asking. Let’s be sure we’re not fearing for the lost point, though. If you’re a good enough physicist to be asking these sorts of detailed questions - which you certainly are - then it’s not worth worrying what particular grainy details are in the rubric. The rubric will award points for good physics, and will not award points for bad physics. I haven’t seen the rubric yet; I haven’t seen the papers to which I will be applying the rubric yet, either. Whether you get 10 or 11 points doesn’t matter. Let’s not feed anxiety about peoples’ performance. Let’s just enjoy these outstanding physics problems.
Continue to ask good physics questions, of course - that’s why we’re here! Just let’s avoid the “how many points will I get” or “will I lose credit” questions. I really don’t know the answer to that yet; and when I *do* know the answer for sure, I need to keep my mouth shut until the rubrics are published in the fall. :-)
So, after the questions were released and we could talk about them, I asked my students if anyone graphed the resistivity on the y-axis for FRQ2. Three hands went up. One said he graphed it with "Cylinder #" on the x-axis. Another said he put "Length" on the x-axis. A third said she didn't remember what she put on the x-axis. "Resistivity was constant, so it didn't matter what I put on the x-axis."
ReplyDeleteAlso, while the standard graph may be R as a function of (L/A), there were a significant number--a quarter of my students--who say they solved for resistivity = RA/L, and thus put RA on the y-axis and L on the x-axis, because that was a clear "rise over run" relationship to them. Not bad analysis on the fly for students who, though they'd done electricity labs, didn't do one involving resistivity specifically. (No clue if they got all the units right, though.)
Will,
DeleteAfter having wrestled with this idea for a long time myself (i.e. why can't we just calculate the resistivity using p=R/(L/A) for each trial and then take the average?), I now understand why this practice is not allowed, and why this issue is so fraught with confusion. (Thanks to Greg for the illumination!)
It has to do with understanding the language and context of the question and what we should and shouldn’t assume. When the AP question says the student "determines the resistance R of the cylinder," that wording can mislead students into believing that the values of R in the table are theoretically perfect values for the resistance of the dough cylinder itself. But no. What they mean is, these are the values of R that have been calculated _using the values of V and I that showed up on the measuring tools_, and these values of R might therefore include (in addition to the resistance provided by the dough) some error or some positive or negative contribution unaccounted for in the measuring process. That means if we want to estimate p _of the dough itself_, it is not sufficient to simply calculate p=R/(L/A), because the R values we have in the table do not necessarily indicate the actual resistance provided by the dough cylinders alone.
Simply calculating p in this way for each trial and averaging the results is equivalent to graphing R v. L/A, drawing a line for each trial that connects each data point with the origin, calculating the slope of each such line, and averaging those slopes. This is also similar to forcing a best-fit line through the origin and calculating its slope. These methods can yield ridiculously bad estimates. (Imagine a proper best-fit line to R vs. L/A with a positive y-intercept and a very small but constant slope...The resistivity in this case would be small and constant, but calculating R/(L/A) for each trial would produce resistivities varying wildly anywhere from infinity to 0.)
The proper way to estimate the resistivity of the dough is to graph a line that best fits all the data, regardless of the origin. The slope of that best-fit line _does_ represent the resistivity of the dough, and any non-zero y-intercept will either be statistically “close enough” to 0 for nothing "weird" to be going on (other than typical and expected measurement error), or it will deviate enough from 0 that the y-intercept can be understood to be some unaccounted for contribution by some aspect of the measuring process. In this latter case, drawing a proper best-fit line identifies any constant additive contribution to the resistance by the nature of the measurement (the y-intercept) as well as the resistivity of the dough itself (the slope). Simply "calculating" the resistivity for each trial and then averaging, like forcing a best-fit line through the origin, implicitly assumes that there is no unaccounted for error in the measurement, and therefore this method can produce a bad estimate for the resistivity. If there were no such error (the measurements were perfect) and no additional contribution to the resistance, voltage, or current by some aspect of the measuring process, and the y-intercept was equal to 0, only then would all of the methods mentioned above be equivalent, producing the same answer. But without that assumption, this method ignores real-life aspects of the measurement process and will not provide an accurate estimate.
Students often encounter abstract physics problems outside of an experimental context, such as “For Resistor A, if R=23.6 ohms, A=0.00049 m^2, and L=0.03 m, what is the resistivity?” In this case, they just plug the numbers into the equation and solve for p. No multiple trials, no error, no nothing. But a question such as this is really just asking students to calculate a theoretical value for p based on theoretical values for R, A, and L. When an AP problem involves an experiment simulating real-life measurements, however, we have to switch our brains into “experiment mode” and approach the situation in an entirely different way.
Will - graphing RA vs. L should give the same slope as R vs. L/A, so should be fine. My thinking on graphing calculated resistivity vs Cylinder # does not allow the student to "use the graph" to estimate the value, they're just taking an average. This type of graph question is showing up very commonly, it's important, in the future, to teach your students how to deal with this type of question.
ReplyDelete-Judy A.
Judy,
DeleteNo worries, I do teach graphs, and we see linearized graphs in lots of contexts throughout the year. But who says a graph with a constant function can't be "used" to estimate the value? I mean, heck, the numerical average--which, I may add, will get a better, more accurate and more precise result than eyeballing any hand-made graph's best-fit line--is actually a measurement of the y-intercept of the graph if resistivity is the y-axis value.
It gets to what people view the role of a graph to be in the first place--and, also, my #1 complaint about this problem in the first place: that the graph isn't necessary to accomplish the lab's mission; the graph is just in the problem to make the students show they can do a graph. If it's merely a test of graphing skills, then, a lot of possible answers must be permitted--especially in mere measurement labs like this, where there's no clear independent or dependent variable to go on the x- or y-axis respectively, just a command to "measure the resistivity."
I disagree that the graph serves no purpose or is equivalent to the average of calculated values. It allows you to evaluate trends with what you assumed would be constant more easily than a calculated average does.
Delete