Exponential functions,rates of change,and the multiplicative unit

Conventional treatments of functions start by building a rule of correspondence betweenx-values andy-values, typically by creating an equation of the formy=f(x). We call this acorrespondence approach to functions. However, in our work with students we have found that acovariational appraoch is often more powerful, where students working in a problem situation first fill down a table column withx-values, typically by adding 1, then fill down ay-column through an operation they construct within the problem context. Such an approach has the benefit of emphasizing rate-of-change. It also raises the question of what it is that we want to cal rate across different functional situations. We make two initial conjectures, first that a rate can be initially understood as aunit per unit comparison and second that a unit is theinvariant relationship between a successor and its predecessor. Based on these conjectures we describe a variety of multiplicative units, then propose three ways of understanding rate of change in relation to exponential functions. Finally we argue that rate is different than ratio and that an integrated understanding of rate is built from multiple concepts.