Purpose: Discover the characteristics of exponential and linear growth.

Classroom Procedure: This activity should take about two days in class - one day for pages 1-3 and one day for the rest. This year I sent them home to do pages 4 and 5 in between these two days of class work. I leave students to read and work on Q1 in small groups and then have a class discussion around this question - gathering answers from the class for each part before moving on to Q2. Again students work on this in small groups and gather answers when they are done. They often need help on 2c - seeing patterns numerically as a lot will just say rate of change is increasing - it is important that they not think that this characterizes exponential growth. Also I emphasize in both Q1 and 2 what it means to interpret parameters in the context of the model - it is not enough to say that 26.57 is the slope for instance but instead what does that mean for this scenario. I finish the first day of work on this activity with a class discussion on Q3 and 4 after giving students about 5 mins to work on them in their small groups. They are usually pretty capable of doing Pages 4 and 5 then at home without a lot of questions the next day. We start on Page 6 and students work through f and g before we talk as a whole class. Some students will have trouble with figuring out the form of the function in g. After that I leave them to work on Pages 7 and 8 independently and collect the activity to grade what we did not discuss as a class.

Ideas this Activity Builds On: This activity builds on the work we did on linear functions and average rate of change with the bottle graph activities. Also on students high school work with linear and exponential functions.

Introduction/Motivation of the Activity: I introduce this activity saying that we have already learned about general functions now we want to study some particular classes of functions in more detail. In particular we want to study what characterizes how these functions grow/decay.

Need to Establish by the End of Activity/Wrap-Up: We need to have established that:

• Linear functions have a constant rate of change. This is the slope of the function. When the function is increasing the slope is positive, decreasing slope is negative. Connect this with how to move from data to a linear function that models that data.
  
• Exponential functions have constant ratio of successive y values. This is the growth factor of the function. It is greater than 1 when the function is increasing and less than one when the function is decreasing. Connect this with how to move from data to an exponential function that models that data.
  

Additional Notes: