Finding Areas in AP Calculus (without talking about calculus)

I’ve probably mentioned this elsewhere in my blog but one of my goals this year is to introduce each major topic using an exploration or by allowing students to “play” with the math. In that theme I considered different ways to introduce the topic of finding areas under curves in calculus.

I felt like each time I’ve either learned it or taught it, this idea is just dropped on the student. It’s actually a profound idea and technique that we use to find these areas. I wanted to solidify the idea that, by using areas of “normal” shapes, we can get decent estimations of areas of abnormal. In addition, I wanted students to see that the more shapes you use and the smaller the shapes the more accurate your measurement of the area. To do this, I gave students four shapes that had varying degrees of “squigglyness”. They had to use a ruler and formulas they already new to get measurements for the areas as accurate as possible. They also had to explain their method for finding the areas.

The activity went really well. I found a lot of value in not helping at all. Students asked “what’s the best way to find the area of this?” and I said “I don’t know.” I made sure to point out that there was no correct method for finding the area and many students used different methods. We finished by comparing all the areas in this google doc and discussing who had the most accurate method. What I loved about the activity is that students engaged in problems with no obvious answers that required them to think critically. It was then a natural segue into this activity, where we look at finding areas under curves.

Below are some samples of the students’ work. I definitely enjoyed the different methods and thought processes that students demonstrated. As usual, any feedback you can give would be much appreciated!


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