The whole concept behind the limit definition of the derivative can really elude kids. They get that slope is rate of change and they get the mechanics of the derivative rules, but they can be very unsure as to why we're taking a limit and what that "h" means anyway. I decided to tackle this by building up the concept, slowly and intentionally.
Phase I: Have students experiment using what they know about average rate of change and see how they relate it to instantaneous rate of change (on their own, with no intervention from me)
Phase II: Make sure they understand average rate of change and the structure behind the notation
Phase III: Move students slowly from their understanding of average rate of change to instantaneous, moving notationally from middle school to college level
I was really happy with how it went and truly impressed with the ingenuity of some of my students. While there are things I will definitely tweak for next year, I felt like my kids walked out with a firm grasp on what the hot mess of notation known as the limit definition of the derivative means. We did no evaluating, no calculating. We just worked on concept and structure.
Phase I: Engineering Design Challenge
There are profound benefits to having a work wife who is a physics teacher and that is being able to raid her lab room for supplies when you're feeling inspired in math class. I just wrapped up a graduate course about utilizing engineering and engineering practices in the STEM classroom and this phase was very much inspired by that. It also helped serve as my final exam project, so should out to my students for writing at least half my paper for me.
I told the students that their objective was to find the instantaneous velocity of a marble on a track using only items found in the classroom. No downloading a radar gun app on their phones. No running to the physics lab to grab a photogate.
Groups took all different approaches, some of which suprised me! I knew many would time the whole length of the track and soon realize that the smaller their interval, the more accurate their approximation should be. Some tried to control for the initial velocity by creating ramps, using the acceleration due to gravity to help them calculate. They really got creative with their reasoning, worked well together, and gave each other great feedback.
I used a BeeSpi to determine the actual velocity of the marble and we found a percent error to determine how accurate groups were in their measured value. And basically, the coolest thing happened. Since we used the engineering design process- which uses iteration of trial, redesign, feedback, etc- the students continually tweaked their process. And each and every group independently determined that you wanted an interval as tight to the actual BeeSpi as possible. We wanted the size of this interval to approach 0. Without that, we'll never have the most accurate prediction. Couldn't have lobbed it in for me any better the next day to build the idea of the limit definition.
If you're interested in the activity and actual documentation of my kid's work, check out the whole document here:
Phase II: Emphasize Average Rate of Change
Here's the document I used for notes and partner tasks:
I wanted kids to understand not only how to calculate slope and it's meaning (duh, they've been doing that forever), but I wanted them to really examine the structure. Here's what the filled in notes looked like:
I also wrote a matching activity here where students would match the structure to the interval and function to the solution, but I wrote it at 11 pm and when I opened it the next morning I realized it made no sense so we will be revising that for next year :)
Phase III: Moving from Average to Instantaneous
These notes and activities are found in the notes above.
We talked about how the slopes of these secant lines (average rates of change) approached the slope of tangent line (instantaneous rate of change) and the groups chatted about how this all related to what we did the day before. Smaller interval, more accurate prediction.
Partners then worked through this exploration (adapted from Calculus God Mr. Korpi ), showing the impact on the slope of the secant line as the size of the interval approaches 0. Students were able to articulate this easily and knew the slope was approaching 1. (They also had a good debate over what the word astute meant, so add that to the list of things they learned today). The very last question took some prompting and we never got to a formula, but the idea you see written here is a good summary of the conversation we had. They made sense of this themselves, I just wrote it down.
Then, we went to the big momma. I made her a few years ago and I love her. She builds from the middle school understanding of slope to the limit definition of the derivative.
The kids walked out today and could explain this in their own words, could write the limit definition for a given function, and could identify the function and x coordinate from a "disguised derivative" given to them. All in all, I'd call that a big win.