Purpose: To build a definition of instantaneous rate of change.
Classroom Procedure: Students work on the first four questions in small groups. It is then appropriate to have a whole class discussion before moving on to Q4. Q4 definitely needs to be worked on in class - it is important to have time to consider how good the estimates the students come up with really are and what their flaws are. It is also important to spend time moving from the process to the limit definition - some students do this very quickly and others require a lot more time. Timing may make it appropriate to send students away to finish the Glucose in the Bloodstream question at home and come back to class ready to discuss that question and then finalize the definitions that conclude this activity.
Ideas this Activity Builds On: Students non mathematical understanding of speed and their work previously in the course on rates of change.
Introduction/Motivation of the Activity: The motivation for this activity is really included in the activity. But I do briefly point out that so far in the class we have studied functions and how they behave with regards to average rates of change. We are now interested in moving to an understanding of instantaneous rates of change - how that is defined, what it means and how different functions can be characterized using this.
Need to Establish by the End of Activity/Wrap-Up: By the end of this activity we need to have established a definition of instantaneous rate of change and the ability to numerically estimate this quantity.
Additional Notes: There are no synthesis questions for this activity. The Glucose question is carried through to the next activity where we move to a different representation and define the derivative.