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Riemann Sum Investigation – Teacher’s Guide Lesson Objectives: 1. Investigate left, midpoint, and right Riemann sums using GeoGebra. 2. Investigate when left Riemann sums overestimate or underestimate. 3. Investigate when right Riemann sums overestimate or underestimate. 4. Draw conclusions about which of the three generally used types of Riemann sums (left, midpoint, and right) is most accurate. 5. Draw conclusions about how the number of rectangles used in a Riemann sum affects the accuracy of the estimate. Use this conclusion as an introduction to a definite integral (defined as the limit of the sum of an infinite number of rectangles). Standards: 1. The following objective is given in the AP Calculus Course Description, published by the College Board: Numerical approximations to definite integrals Use of Riemann sums (using left, right, and midpoint evaluation points) and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values Pacing Guide: Introduction (10 minutes) Lab activity – GeoGebra (25 minutes) Conclusions Materials/ Resources: Computer Lab GeoGebra software installed on lab computers GeoGebra demonstration module on Riemann Sums Student worksheets Prep Work: 1. You should do the student worksheet in advance in the lab to make sure everything works and to familiarize yourself with the content. 2. Make sure students have logon id’s, etc. to use the lab. Activity Launch: 1. Introduce the idea of “finding the area under a curve” by drawing a curve on the board and asking “how might we estimate the area under this curve?”. Start with a trivial example (like y = 3), then move to a line (y = x +2) and then to a curve (y = x2). 2. After a brief period of brainstorming, tell them about the method used by (and named after) Georg Riemann. The idea was to use rectangles to estimate the area under a curve. The area could be estimated by summing up rectangles and hence is called a Riemann Sum. 3. Today’s activity in the lab will be to investigate several aspects of using rectangles to approximate areas under curves. In the Lab: 1. Pass out the student worksheets. 2. Have students launch the GeoGebra applet that goes with this lesson. 3. Give them 20-25 minutes to complete the worksheet and draw their conclusions. 4. Walk around the lab and help if students are having difficulty. Concluding Activities: 1. As a concluding exercise, ask the students “What concepts have you learned today about using Riemann sums to estimate the area under a curve?” 2. Write summary statements on the board. 3. Briefly extend the concept to the definition of a definite integral, using the last question on the student worksheet as a springboard.
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