**Title: **Rolle’s Theorem and the Mean Value Theorem

**Discipline(s) or Field(s):** Mathematics, Statistics, Computer Science

**Authors:** Joy Becker, Petre Ghenciu, Matt Horak, Helen Schroeder, University of Wisconsin-Stout

**Submission Date: **April 1, 2008

**Executive Summary: **The topic of the lesson is Rolle`s Theorem and the Mean Value Theorem.

*Learning Goals.*

- Students will understand the meaning of Rolle`s Theorem and the Mean Value Theorem, including why each hypothesis is necessary.
- Students will complete problems and applications using Rolle`s Theorem and the Mean Value Theorem.
- Students will appreciate the discovery process of developing mathematics and have a better understanding of the construction and proof of mathematical theorems.

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Lesson Design.* The lesson was designed in order to emphasize the discovery process and the role of proof in mathematics. The first major piece of the lesson is an activity that asks students, in several steps, to draw graphs of functions satisfy various hypotheses. The last graph that students were asked to draw is impossible to draw, because any graph satisfying all of the required conditions would violate Rolle`s Theorem. Rolle`s Theorem is introduced in this way. A second activity involving graphs related to the Mean Value Theorem is used to introduce or study the Mean Value Theorem. These graphing exercises are intended to help students discover for themselves the two theorems and help them to appreciate the discovery process in mathematics.

The second major part of the lesson is to work problems involving the theorems to better understand how the theorems are used and apply in practice. The variety of problems is intended to emphasize different aspects of the theorems, including why the hypotheses are necessary and how to apply the theorems to modeling applications and more abstract settings.

The final part of the lesson is to prove the Mean Value Theorem assuming Rolle`s Theorem. This portion of the lesson is expected to be difficult for students, so ample time should be allotted for question and discussion.

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Major Findings*. During the first round of the lesson, we learned that students seem to catch on quickly that the second graphing exercise is almost identical to the first and that therefore the last graph is impossible to draw. This seemed to cause a significant reduction in their engagement with the lesson. However, when this activity was changed for the second round, the decrease in performance on certain quiz and homework problems suggests that the repetition may actually have served its purpose of emphasizing the hypotheses present in the two theorems.