Approximating Area

04 Nov

I have been challenged this weekend to really think about goals and formative assessment, reflecting on how I change what I am doing mid-lesson (or mid-PD session) based on feedback from my students (or participants).

I have our M-STAR Teacher formative assessment tomorrow during my Calculus class. I have a plan, which I am going to share now. Then I will write a reflective blog post after the lesson to think about whether and how formative assessment during the lesson changed my plan.

The plan:

4-1 Approximating Area Lesson Plan – Jennifer Wilson, AP Calculus

Essential question: What is the area under the curve?

Objectives: 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.
• Appropriate integrals are used in a variety of applications to model physical, biological, or economic situations. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the emphasis is on using the method of setting up an approximating Riemann sum and representing its limit as a definite integral.

Note: Objectives are from the AP Calculus Course Description.

 Bell 4-1 (if needed, #6 will be for problem solving points) 25 minutes The Old Pump – problem for students to work alone. Will send QP to collect responses and then have some share their methods. 15 minutes Overview of answering our essential question “what is the area under the curve”, using flipchart as guide. 15 minutes Practice Riemann Sum Rectangles – Right, Left Midpoint Students have choice of paper or TNS document Use observation, Class Capture, & QP as needed for formative assessment 15 minutes How can we get a better approximation? 5 minutes Trapezoidal Rule 15 minutes Closure – Plane Crash Application 5 minutes

While I have thought about some “I can” statements for students, I am not sharing those with students before this lesson. While I want them to be able to say “I can approximate the area between two curves using left, right, and midpoint Riemann sums” and “I can approximate the area between two curves using the Trapezoidal Rule” and “I can use an infinite number of rectangles to get the exact area between two curves”, I don’t want to give that away before the lesson. I want them to figure some of those things out through our exploration. What I’m going to tell them is that we are beginning to explore the area between curves, and that we are going to focus on the mathematical practice “reason abstractly and quantitatively”.

I’ll report back after the lesson to see what happens as the journey continues ….