Introducing Integral Approximation Methods

So I wanted to share with you one of my favorite lessons that I came up with a year ago and have done several times now with several Calculus classes.

So I wanted a clever way to introduce the Right Riemann Approximation Method, Left Riemann Approximation Method, and Trapezoid Approximation Method.

So I went over an example of how to use each method. I ask students to make predictions about which method comes next and about whether the approximations are an overestimate or an underestimate. I make sure they understand why the approximation is an overestimate or an underestimate. This where I ask them to connect to the graphs behavior and whether the graph is increasing or decreasing (a major point on the AP test). So after introducing the method I have the students practice one of the methods. I split the class into either groups of three or groups of four. Each person in the group picks a different method to use and so in each group you have one expert on each method. However I make the groups according to ability. I do this because each group has to approximate the integral using a different number of subintervals. The smarter students get a greater number of subintervals. So the lowest students approximate the integral using three subintervals and then the subintervals increase by one as the students ability increases.

I have the groups put up their answers in a table on the white board. Once everyone is done we talk about what patterns they see. They talk about how the numbers are increasing and decreasing. They compare the four methods and talk about which one they think is more accurate. I put up the approximations for the integral using 25, 50, 100, and 1,000 subintervals which are listed in the textbook I use. We talk about how little the numbers change the more subintervals use. I finally put up the exact answer and see how close the groups got to that answer and which number of subintervals got the closest. Analyzing this table easily leads into how if we let the number of subintervals go to infinity we get the actual integral. So in the next class I talk about integrals and show how Riemann sums lead to integrals.

Last years 11th graders groaned and moaned about using that many subintervals. Then with this years 11th graders wanted to go above what I assigned them. I gave them a certain number of subintervals and they were like no we want to use even more than that. Then I did this same activity with this years 10th graders and had an interesting response. In one of the classes the entire class wanted to pick on the very top students and make them use 100 subintervals. I talked the class down to making them use 15 subintervals. I checked with the group if it was ok and they ended up not having a problem with it. It really isn't that difficult using more it just means smaller and decimals and more numbers to keep track of. When showing examples of the different methods in one of this years 10th grade classes a really good discussion started about which method was more accurate and whether the approximation was an underestimate or overestimate. They really analyzed the function, its behavior, and the methods. They were doing most of the discussion in English which was great to hear. So I am looking forward to teaching them more and having lots of in-class discussions with them.

Materials for lesson. Please look in the notes section of the powerpoint for more details.