The cost of saying yes is an analysis that mathematicians are always exploring. What does this purchase actually cost me? What kind of fruit comes from this relationship? or work? class? How do we know when to say yes and when to pass? Learning to observe, predict, and calculate possible benefits and drawbacks can help us make wise decisions.
These ideas can be explored with functions. A simple input and output relationship can reveal some difficult truths for us. If I put in thirty minutes of time with my brother, what will be the result? Is this linear? quadratic? or the most powerful of all, exponential?
This week we explored the world of exponential functions. Did we practice computing powers of 2 and graphing? Of course. Did we spend a large chunk of class exploring the power of exponential growth and decay? Yes. Exponential growth is one of those relationships that we study in mathematics often without context. Context brings the importance of this particular abstract idea to my students. Why does an exponential growth function appear to cuddle up close with zero as the input approaches negative infinity? Why does the output shoot off towards infinity so quickly? We all have lives filled with the fruit of exponential growth or decay. What we do now impacts the next generation and few things in mathematics land that idea into the heads of our students more clearly than a good old fashioned exponential growth graph.