Inquiry based Calculus Lesson

I try to make my lessons where the students have to participate and interact with the material. I can not always achieve that, but I was able to create a lesson where the students really got to explore and make predictions in my class.

So to start off the unit on applications of derivatives I use a directed reading and thinking activity(DRTA). I have students look at the title of the chapter, the subtitle, the example problem, and more. After each I have them write on their personal whiteboards about what they think they will learn and how it relates to what we have learned so far. I keep track of predictions up front on the whiteboard. I do this at the beginning of each new chapter or each new unit. Then I read the overview at the beginning of the chapter and I pause to recognize the student that had the correct prediction. So then I begin to define extreme values and the different terms for these. Then I have students find the extreme values of x^2 when the domain is all real numbers, from [0,2], from (0,2], [0,2), and (0,2). This way students can see the function may have one extreme value in most of those intervals, but only has two extreme values in the closed interval. However I don't simply tell them this. After the students have filled out a table with their answers I put the correct answers on the whiteboard up front. Then I have the students look at the table and make up their own theorem based on the table. They have look at the data and come to a conclusion on their own. So after letting them have time to think and write on their whiteboards. I call on several students. I usually start with someone who was able to make a really general statement about the data and move on to students who got more specific. What I am looking for the students to do is to come up with the Extreme Value Theorem on their own. After we have talked about the students conclusions I tell them that they just came up with this theorem or came close to getting the theorem. I give the formal definition and explain when it will be used. I often get really good discussions about the theorem even after I have revealed. Several students had to give examples of when it worked and counter examples to some students predictions about the theorem. With almost every class it has sparked really good discussions all done in English. I also tell them in this lesson that this is what real math is. It is more than just memorizing formulas and solution processes. It is looking at data and information, then drawing your own conclusions. Then later discussing them with your fellow mathematicians.

I then move on to discuss and define local extreme values or relative extreme values. Then I put up a graph and have them copy the graph onto their personal whiteboards. I have them label the points as local maximums and local minimums on their whiteboards. Then I have them hold them up so I can see the students responses. I call on the students to go over the right answers and reveal the answers on the power point. However once this is done I have them write about what those points all have in common and how they could find those points without looking at the graph. I give them time to think and write on their whiteboards. Some students respond with that is where the first derivative is zero or where the first derivative does not exist, which is what I am looking for. However I have also had students notice that the function switches from increasing to decreasing or decreasing to increasing at those points. So the students come up with the definition of critical points and the idea behind the first derivative test on their own. Then after we are discussing that and student response I put up the formal definition of critical points.

I wish I could make all of my lessons this inquiry based, but I have yet to be inspired with how to do so. However as I teach more I think I can come up with more lesson plans like this. I have had my supervisors observe this lesson several times because it is a good sample of my teaching and I am really proud of this lesson. I got really good feedback and impressed them with this lesson. I think this lesson sums up my view on mathematics education. I want to engage students and make them think. I want them to be involved and to challenge them. This incorporates a lot of things that I learned about in a lot of my classes for my Masters.

Lesson Plan and Materials