Ultimate Stem: The Book

This is a book that we collectively wrote over the summer that includes 49 examples of amazing STEM lessons.

Bio:
Tim Nuttle worked as a copywriter in an advertising agency before enthusiastically jumping to his current career as a high school math teacher. He is National Board Certified Teacher and teaches AP® Calculus and Precalculus to a group of wonderful students on the north side of Chicago, where he also lives with his wife and daughter.

STEM Lesson:
This lesson relates to the Mean Value Theorem, and it implemented over the course of one class period, in my AP Calculus class.

Students had already been instructed to record data from a trip in the car with someone. The data consisted of:

  • The starting value on the odometer
  • The ending value on the odometer
  • The speed limit(s) on the road(s) traveled
  • The odometer reading of the car every 30 seconds for 10 minutes (20 points of data)
  • The velocity of the car every 30 seconds (which we will not use here)
We had already graphed and investigated the average rate of change between two points many times. We also found a best fit curve (regression) for the data and had an approximate equation for the data. We also had learned how to find the derivative at a point and use our calculators to find derivatives at points. Finally, the students had just learned the mean value theorem, which states that the derivative of the continuous function at some point must equal the average rate of change of the function.

Students worked in groups of four and chose the best set of data they had. Then they exchanged data sets, graphs of the data and the best-fit function with another group. Once all of the this was exchanged, the goal of each group is to determine whether the car (based on the data provided) was speeding.

The students must use the Mean Value Theorem to figure out if the car was speeding or not. They need to find the average rate of change (the average speed) which must equal the derivative of the car's position (the velocity, at a certain time). If the average speed is over the speed limit, the car must have been speeding.

This may seem difficult to some students, since they are not explicitly told to use the Mean Value Theorem to find their answer, and therefore they must relate the problem at hand to the application of the theorem. This application of the theorem allows students to demonstrate their understanding in a real world context.