# Suspended Bucket

From a regular physics quiz in late March (the blanked out part is just a small note I made while discussing with the student)—

In the comments—

- What was the student thinking? How did he or she decide to make these particular marks on the paper?
- What question or problem would you pose next to help the student make the next step toward understanding?

Maybe for this one, we should focus on question 2. What questions or problems would help the student move toward a better understanding? What would help him/her reason herself out of this mistake the next time an opportunity for it temptingly appears?

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This is really interesting. I’ve seen students do this and similar things in my class. One in particular, where a student tried to find the the x- and y-components on a v vs. t graph, treating the line as if it were a vector. I think that’s what this student has done with the graph, thinking of the acceleration as the hypotenuse of the line, rather than the slope of the line, and then figuring they can use triangle geometry to find unknowns. From that perspective, what the student does makes sense, they are using the Pythagorean theorem to find the unknown side. Partially the similarity of a line with a vector makes this mistakes more compelling, but I also wonder if the mantra “acceleration is a vector” might also contribute.

I’d probably first focus this student away from the graphical representation for the moment, and more toward the meaning of acceleration. In the problem, the acceleration is 8.91 m/s/s, which means that after one second it’d be going 8.91 m/s, but this student has determined it will be going only 8.71 m/s after 1.59 seconds, slower than it should be after one second! So, I think I’d ask them to verbally interpret the meaning of 8.9 m/s/s, and try to reason through how fast it should be moving after 1s, 2s, etc. Hopefully that would gain us some traction. I value the verbal interpretation over the graphical one (at least as an anchoring idea), and we practice that a lot in my class, so that’s a lot of the reason I’d go there. It’s too easy to know that acceleration is slope of v vs. t without understanding it, so I wouldn’t want to direct our attention their prematurely.

Then, of course, we should return to the graphical representation, and try to figure out what they did, establish that slope is relevant here (and why). We’d probably need to tease apart slope relationships (with graphs) and triangle geometry relationships (with vectors).

This reminds me also of Aaron’s writing about explicitly talking with students about features of graphs that do and DON’T have physical meaning. For example, the length of that line has no physical interpretation.

I think Brian’s questioning along the verbal interpretation of 8.91 m/s/s is a great approach. This is another example where phrasing the acceleraiton as “increasing 8.91 m/s for every second” is so helpful. After asking for velocity at 1 second and 2 seconds and 3 seconds, ask the student to estimate the velocity at 1.59 seconds (another important skill to practice). Then, ask them to reconcile their two answers.

Sorry, I just have to say I love this one. The student is doing just fine and then boom! finds delta v by the pythagorean theorem. I would also follow Brian’s approach. Sorry, nothing new to add there. I have another higher level/meta-comment: I have been changing the way I’m grading. The problem with what the student has written is that there is no real way to tell what (s)he really understands. (S)he nailed the Newton’s law part, so we assume the student understands what (s)he is doing. But it is just as likely that the student has some procedures memorized or well practiced shall we say. (I’ve been reading all the Khan academy brou-ha-ha.)

Okay, long story short. My students have to explain (in writing) what they’re doing and why they’re doing it. If that was a homework question from one of my students I’d return it with a zero and ask him/her to redo it including an explanation of what (s)he is doing and why. Let me just say it works wonders.

I think Brian has the right approach, but I’d like to add one thing. Early in the year, even when we are doing constant velocity, I ask the students the meaning of every possible graphical feature on a position vs. time graph. First of all, although we use the term “graphical feature” a lot, many students don’t really clue into what it means. So spending some time listing graphical features helps: the slope, the area under the curve, the length of the curve, the amount the curve changes vertically, etc. Then I let them talk through what each of the features means… physically. This might not have helped the current student avoid this error, but the she might harken back to “not all graphical features have interesting physical meanings” as she puzzles through Brian’s follow-up questions.

It took me ten minutes the other day (far longer than it should have) to figure out they mixed up slope and length. I’ll repeat my mantra about things not always starting from zero. Ask the student what the graph would look like if the bucket had an initial velocity. That ought to make it clear that this isn’t a triangle, but a degenerate trapezoid with one side of zero length. That might not be the best terminology to use around the student though. Maybe ask them to graph multiple initial velocities, including negative ones, to see how that slides the line vertically.