In Germany, one usual example to start teaching about integrals is to look at a simple (piecewise constant or with constant slope) functions that make up a water flow vs. time diagram and ask about the total flow during a given time span. This leads to the idea that the area under the curve is of interest and represents exactly the desired quantity.

I was thinking about a discretisation of this example because my dog requires pain medication and I plotted a daily dose vs. day graph. This is basically the same as the water flow diagram, but already with discrete steps, probably leading to the "rectangular stripes" method.

Is there any research or experience about this approach?


Some of my students have a real a-ha moment when they first graph the integral of a constant function (using the area-so-far concept), seeing a new line of non-zero slope emerge. Then having them graph piecewise constant functions reinforces the height $\rightarrow$ slope connection they will depend on when they learn the short-cut rules for antiderivatives.

No research to speak of here, but I think beginning with something simple and concrete like this is an excellent idea.

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