Approximate area under a curve using geometric and numerical methods
I can use geometric shapes to find area under a curve
I can use left Riemann sums, right Riemann sums, and midpoint Riemann sums to approximate area under a curve with uniform partitions
I can determine if a Riemann Sum approximation is an overestimate or underestimate
Quick Lesson Plan
Three familiar scenarios lead students through increasingly complex investigations of the area between a rate of change curve and the bounding axis. Students initially work with linear graphs before developing a method for approximating the area beneath the graph of a quadratic equation. Each example requires students to assign meaning in context to their computations.
Our students attacked today’s activities with high interest, creativity, and energy. Groups were allowed to develop their own strategies for approximating areas and we were rewarded with amazing ideas: some groups “discovered” left Riemann sums, right Riemann sums, midpoint Riemann sums, trapezoidal methods and other ingenious (and legitimate) strategies. The debrief segment is both enlightening and important today: do not rush through this part of the lesson! Connecting the graphical representations with the analytic, numeric and verbal (context) results is vital.
The AP Test often presents information about the rate of change curve in tabular form. And, often, the values reflect non-uniform subintervals. Presenting multiple opportunities to work with subintervals of varying width will prepare your students to handle these questions on the AP Test.
One of the most common frustrations students experience with rectangular approximation methods is identifying the appropriate x-value to use when finding the height of a rectangle. Providing extra practice may be necessary. Day 2 of Topic 6.2 is designed for review: revisit this skill as you introduce trapezoidal techniques and strategies for working with data in tabular formats.