In a post a few weeks ago, I told the story of Doctor Dimension, the two-dimensional scientist who spends his fictional life filling two-dimensional containers with two-dimensional liquids, and has been doing this since 1994. In the comments, two different Bay Area math teachers directed me to Water Line, a wonderful Dan Meyer / Desmos collaboration. Water Line is a highly interactive lesson, in which *the height of the liquid in different containers is graphed as a function of time*. The context (filling containers with liquid) is very much the same as the Doctor Dimension lessons, but there are some significant differences, because in Dr D we are graphing *amount of liquid (area) as a function of height*. Therefore:

– Water Line is more “real”, because in the real world, the question we ask ourselves as we fill a container with liquid is “how long will it take?”

– In one sense, Water Line makes is easier to see the relationship of the graph to the situation being modeled, since the height variable is represented on the y axis.

– In another sense, Dr D makes it easier to see the relationship between the situation and the graph, because *the rate of change can be seen* as a thin horizontal slice of the area. So a wider section of the container yields a steeper curve. In fact, Dr D offers a *preview of integration*, though it is not readily recognizable, because our independent variable (the height) changes in a vertical direction.

The math is different enough that Doctor Dimension and Water Line do not overlap, and students can profitably do both activities.

Of course, the interactivity is much fancier in Water Line, as students get to draw their guessed graphs, and can share the vessels they create with each other. In my 1994 Dr. Dimension lesson I did ask students to create their own containers, but in the absence of suitable technology, that was a bit too hard. No longer! Inspired by Water Line, I created a fifth Dr. Dimension applet, in which students are asked to modify a container in order to get these graphs.

This activity should make for fruitful discussions of rate of change, concavity, and differentiability. Check it out!

–Henri

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