One of the courses I teach this year is a course called Honors Calculus. It is a non-AP course and we made a decision about 6 years ago to make this a course in Differential Calculus. While the AP AB course completes a college semester in a high school year, this course completes an AP AB semester in a high school year. This allows us to remedy some bad habits, fill in some gaps in understanding or mechanics and, most importantly, really slow down and think about what we are exploring. I have an activity that I do on the first day of class where we explore motion. This year we went in the hallway and rolled a whiffle ball, a softball, and a lacrosse ball down the hall a fixed distance. We tried to roll each with approximately the same force and had a good conversation about what the data told us. We made some physics based observations that I did not plan for and we talked about what we knew and what we did not know about the rate at which the balls were rolling. My goal was to arrive at the conclusion that we could talk about average rates of change but not so much about the rate at a particular instant. The conversations went reasonably well, but we got distracted a bit by conversations about bounciness of balls and air resistance. In any event, I think I planted some decent ideas to consider as we embark on a conversation about average rates of change of a function on an interval (our text calls this the AROC) and we are about to wrestle with the limitations of being able to know much about the instantaneous rate of change (the IROC)
This summer I rented a car whose dashboard gave information that I knew would work well for this class. A picture below will prompt your teacher brain as well I think.
I took six such photos during the course of my trip. Tomorrow, I intend to give each of my groups of students (I have them in groups of three) three of the pictures. I’ll scramble them up a bit so different groups should have different subsets of the data. I intend to ask them some pretty simple questions that should generate some good conversations. I want to ask them the following questions:
- What was my average speed between any of the two pictures? (So each group should have three answers for this)
- Can you determine my maximum velocity in that time interval?
- I want to raise the average to 42 MPH. How far would I have to travel at 60 MPH for this to happen?
These are not terribly deep questions, but they feel rooted in an example of real world data (I was inspired by Denis Sheeran’s wonderful book Instant Relevance for this data driven experiment) I also think that this will continue to scratch at the itch that will make the breakthrough of being able to find the IROC feel more meaningful.
I have all the photos together in a WORD doc on my dropbox. You can find that file here. I would love to hear any clever ideas about how to play with these images/this data.
2 thoughts on “Exploring Rates of Change”
Seems like there is an opportunity to ask some informal mvt based questions here, and push at these ideas:
what can you deduce/not deduce from discrete data?
The idea that you iroc is continuous. (What situation could produce a non continuous roc? Maybe students could chew on that for a bit)
Even if you don’t go all those places this time, you should be able to return to this context to flesh out understanding throughout the year.
Thanks for dropping by and sharing some ideas. We have not named MVT yet, but we have already touched upon it. Once I reassured the students that all pictures were taken while the car was at rest, they realized that I must have exceeded the intervals AROC at some instant. A follow up post will highlight some of the good questions that came up.