It’s been a while since I’ve posted but I had to make time for an activity I did last week in AP Calculus since it went so well. We have been calculating Riemann Sums (left, right, midpoint, and trapezoid) and students felt very comfortable doing these. We also used this Desmos file to show how increasing the number of rectangles makes our approximation closer to the actual area under the curve. (I took this Desmos file from someone else who blogged about it but I can’t find the original blog, only the Desmos file, so let me know if it was you!) At the end of the previous class we went through the typical way to bring in integrals. This is outlined at the bottom of ThinkThankThunk’s post here, however I did not yet connect it with the anti-derivative.
To connect it with the anti-derivative I used an activity from Mark Howell, a teacher from Washington, D.C. If you go to the College Board site you can find many of his publications there. I attended his AP Calculus training at an AP Calculus Summer Institute and learned a great deal from him. The students homework the previous night was to go for a 15 minute drive where they were the passenger. They first recorded the initial odometer reading. Then every 30 seconds they recorded the velocity of the car, making note of special conditions. For example, making a note if they were at a stop sign. They then recorded the final odometer reading.
They brought this data into class and graphed it by hand. They then used the midpoint rectangular approximation method, with a width of 1 minute, to find the area under the curve. Since they recorded the velocity every 30 seconds and used the midpoint method, the height of each rectangle was always the actual velocities they recorded. When finding the area of each rectangle they had to make sure they converted the width of each rectangle from 1 minute to 1/60th of an hour so the area under the curve would be in miles. Once students completed this, without prompting they immediately realized this was almost the same number as their final odometer reading minus their initial odometer reading.
Once everyone had completed this I brought them make together to make the connection to the anti-derivative. From last class, students were fine with saying the integral was just the area under the curve, the Riemann sum. They just saw that the area under the curve for a velocity vs time graph was their final odometer reading minus their initial odometer reading. I then asked them, how is velocity related to position? (Pause…) Then a bunch of light bulbs went off. We had spent a week earlier in the year during derivatives working position/velocity/acceleration problems. Position is the anti-derivative of velocity!
Now, there was no problem writing this and all students agreed.
Here is some of their graphs and write ups. Almost all students concluded we could make our approximation better by increasing the frequency at which we recorded our velocities. (i.e. increasing the number of rectangles) Many also noted having a digital speedometer would be best to get more accurate velocity readings.