Calculus project – real-world data

I recently assigned and graded a take-home project for my Calc I class. Here is the assignment I gave them:

This is a group assignment.  You must find a partner in the class to complete this assignment with.  If you cannot find a partner, let me know and I will help you find one.
1. Find a real data-set that represents some quantity measured over time.  This can be population data, the position of some object/particle, height, weight, temperature, etc.  Look at the exercises from section 2.1 for examples of the type of data you’ll need.  You should provide the original data and a citation for where you found it.
2. Sketch a plot of the data over time.  This can be done by hand or on a computer, but it should be neat and legible.
3. Using the data, pick one of the times.  Then find the average rate of change for six different time periods, where the first time you pick is one of the end points for each time period.
For example, if your data uses the times 1900, 1910, 1920, 1930, 1940, 1950, and 1960, you might choose 1920 as your fixed time.  Then you’d find the average rate of change for 1900-1920, 1910-1920, 1920-1930, 1920-1940, 1920-1950, and 1920-1960.  Show your calculations for at least one of these, and create a table displaying all of the results.
4. Estimate the instantaneous rate of change at your chosen time, and justify where this number came from.  You should write a few sentences here explaining what the instantaneous rate of change represents and how you tried to find it.
5. Discuss briefly the difficulties in using discrete data to estimate instantaneous rates of change, and suggest in what ways the data collection could be changed to make your job easier.

If you have any questions about this assignment, please ask me, sooner rather than later.

I think my instructions for part 3 could have been more clear, but the class seemed to understand, and I was happy with most of their submissions. Several groups happened to pick “population of Atlanta” as their data set, but some were more creative, and this helped me get to know their personalities better. For example, one group did “number of YouTube followers PewDiePie has per month, so now I know these two students are gamers.

I was also happy to see different methods for answering part 4. Some students took the left and right estimates and averaged them (e.g. averaging the 1910-1920 and 1920-1930 rates), while others found a middle rate (e.g. calculating the 1910-1930 rate). People who chose an endpoint for their fixed time struggled here, so in the future I might recommend they always choose a middle time to fix, but it was nice seeing how these students justified their answers.

The answers to part 5 were of varied quality, but I generally saw the answer I wanted: having more frequent data points would help get a more accurate instantaneous rate of change. I think that it’s important for students to interpret their answers in words instead of just providing a number, which might be meaningless to them.

I liked this project for several reasons, not least of which being that now every one of my students knows someone in class, has their contact info, and knows where/when is a convenient time to meet with them. Studies show that one of the best indicators of student success in math classes is studying with peers. As I assign more projects throughout the semester, I’ll probably require students to have new partners each time, so they can continue to meet more classmates and hopefully find one they work best with, with whom they can study in the future.