We have started our unit on the definite integral for a few years now with Lin McMullin’s The Old Pump.
I love watching students work without yet having developed Riemann Sums. Many use areas of rectangles to approximate the amount of water in the tank, but even then, they don’t all do it the same way.
That work leads us to developing the idea of estimating area between a curve and the x-axis using Riemann Sums and the Trapezoidal Rule. And then we are finally ready to determine the exact area between a curve and the x-axis using a Riemann Sum with an infinite number of rectangles.
We practice reason abstractly and quantitatively throughout these lessons.
Once we’ve thought about numerical approximations for area between a curve and the x-axis, we spend some time writing a Riemann Sum to represent area and evaluating its limit as the number of rectangles approaches ∞. I want them to be able to go backwards, too. So we start with a limit, and I ask them what definite integral will have the same value.
Which is apparently not as difficult as the groans suggested when I first gave it to them.
But we are always working on our Mathematical Flexibility, and while I was pleased that everyone can get a definite integral, I was disappointed that they all did it the same way. Jill Gough has provided us with a leveled learning progression for Mathematical Flexibility.
Can you write another definite integral for which the area can be calculated using the given limit?
It took a while. But students made progress. Some made use of the symmetry of the graph of y=x2 to write a second integral.
Some figured out that translating the parabola and the limits of integration one unit to the right would result in a region that has the same area.
Those were the types of answers I was expecting. But I also got answers I wasn’t expecting.
Some of my students were on the path to Level 4 of reason abstractly and quantitatively, beginning to generalize the idea of translating the parabola and the limits of integration c units to the right, resulting in a region that has the same area. They didn’t quite make it, as their limits were shifted to the right c but their parabola shifted to the left c. I was still impressed by their jump to Level 4, finding connections between pathways.
Our TI-Nspire CAS software let us check our results and helped us attend to precision.
And so the journey continues … learning more from my students and our technology every day about mathematical flexibility.