Graphing Interpreting Functions

Impossible Graphs

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How would you mark these problems? What do the kids understand? What don’t they?

5 replies on “Impossible Graphs”

The first students has a point – in some sense you “can’t go back in distance”. If a car has 50,000 miles on the odometer there is no way to undo some of that distance. Maybe the student is thinking of that sort of distance (miles on a car) as opposed to something like the distance from a wall or distance from home.

Student #2 might be wondering why the graph stopped, and whether that would imply that the object also stopped. It does seem kind of inviting to extend that graph so that the distance would be 0 when the time/domain ends. I am guessing the notion of teleporting was used in class to describe what would be needed for the vertical portion of a distance time graph.

Student one seems to recognize the difference between distance and displacement. I wonder what they would say if the graph was labeled differently (distrance from rather than distance?).

I agree with others. In this student’s physics or science class they may have stressed that distance, position, and displacement are different things, and s/he is trying to apply that here. If that’s the case then the s/he is correct, “distance” is what we use for cumulative ground covered and it monotonically increases. “Position” is how I would label this graph, but if that’s too jargony then distance from the wall etc would be fine.

The second student is concerned about what it means for there to be nothing on the graph. That seems fair to me! Either the object remains at that position or goes to zero or something, but it has to be somewhere. I think this could be a fruitful insight. So often I have seen students do the opposite — when the value they need to plot is 0 they just don’t plot anything instead (as though 0 were not a value on the y-axis).

ill-posed problem. does one mean “total distance travelled”.
the right answer here is “of course it’s possible… there it *is*!”

The real problem with the graph is not its completeness on time, but that you can’t have an immediate change in position. You can have an extremely fast change of position, but it’s not instantaneous. The distance “traveled” is a product of the position relative to time, so the graph is wrong because of that. Also, distance only adds up. Displacement is what can “go back”, relative to the starting position.

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