#1

Both focused adequately on how the velocity would decrease over time. Neither accounted for (perhaps by displaying double arrows for acceleration that changes velocity or by giving examples of how the velocities were calculated) why the velocity changes, but it probably made sense in their heads. Their work differs in that Student #1 attended to the average velocity, and Student #2 added time information. Each thought that they could “add” the velocities in appropriate ways to get the displacements. Whereas Student #1 added the average velocities on each time interval displayed, Student #2 added the instantaneous velocities (or perhaps just the initial velocities) at the endpoints of each interval. Student #2’s writing of the velocity in the middle of the interval might indicate forgetting that the written velocities were instantaneous. Both students neglected to consider that the intervals did not all have the same time interval. I like the idea of including heterogeneous time intervals to probe understanding. Did any students attempt to make the diagram more Motion Map-like by showing the changes every 0.5 seconds? Were they more successful at calculating displacement?

#2

First, “How did you calculate the displacement?”, to anchor the conversation.

Then, ask about the first interval. “How long did it take? How far did it go? How do you know?”

Then, move on to the other intervals or perhaps jump to the last one. “How long did it take to slow from 5 m/s to 0 m/s?”

#2

“When and where was the stone/ball traveling up at 15 m/s?”

“I noticed that you didn’t draw the arrow all the way to the maximum height in the last time interval. Can you tell me why?”

I’m sure there are more questions to ask, but that’s all that comes immediately to mind. For both students, once they have demonstrated their understandings, one might ask, “It seems like a lot of work to calculate the displacement by adding up all the smaller displacements. Is there an easier way to calculate the displacement here?”

]]>But the real student’s real error is another thing entirely. I smell a whiff of Zeno’s paradox: at the infinitesimally small instant that the car is at the marker, there can be no motion. The student fails to do a “common sense” check or “ballpark estimate”. Maybe the teacher should ask what the speed is between 2 and 3 pm.

]]>I wonder if the student answered there is “no speed” because (s)he’s thinking about the singular instant that the car is at mile marker 72- and thus is thinking that the change in position at that instant is zero and/or the change in time at that instant is zero. Thus the speed is zero at mile marker 72.

Perhaps having the student draw a position-time graph would help, since that would come out as a nice straight line showing that the speed of the car is constant, and then have them figure out what the slope of that long straight line is?

FWIW- as someone who will be teaching physics for the first time in quite awhile next school year I’m very much enjoying this blog and the analysis in the comments. It’s a bit intimidating since I clearly don’t have the analysis & support skills the contributors have, but I’ve already found it very helpful just as an exemplification of how to use student mistakes to drive the next step in instruction. That’s something that often gets talked about as a good practice but is rarely demonstrated as clearly as it is on these pages. This is Good Stuff.

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