Exponential functions, rates of change, and the multiplicative unit
 Jere Confrey,
 Erick Smith
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Conventional treatments of functions start by building a rule of correspondence betweenxvalues andyvalues, 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 withxvalues, typically by adding 1, then fill down aycolumn through an operation they construct within the problem context. Such an approach has the benefit of emphasizing rateofchange. 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.
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 Title
 Exponential functions, rates of change, and the multiplicative unit
 Journal

Educational Studies in Mathematics
Volume 26, Issue 23 , pp 135164
 Cover Date
 19940301
 DOI
 10.1007/BF01273661
 Print ISSN
 00131954
 Online ISSN
 15730816
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Industry Sectors
 Authors

 Jere Confrey ^{(1)}
 Erick Smith ^{(1)}
 Author Affiliations

 1. Department of Education Mathematics Education 422 Kennedy Hall, Cornell University, 14853, Ithaca, NY, USA