Dynagraphs
Dynagraphs—an idea invented by Paul Goldenberg, Philip Lewis, and James O'Keefe[1]—are dynamic representations of functions designed to introduce students to the ideas of variable and of functional dependency. In dynagraphs, unlike in symbolic expressions or Cartesian function plots, a variable actually varies. The motivation comes from the often-noted difficulty students have in seeing the graphs of functions not just as static pictures but as dynamic representations of functional relationships between two quantities. By "decoupling" the input and output axes and having an arrow connect points on either (as opposed to having the axes arranged perpendicularly with single points representing particular input-output dyads), students are better able to see the "input-output machine" view of functions expressed graphically. Being able to drag the input pointer gives students the further advantage of seeing functions as dynamic relationships between quantities.
Dynagraphs can be thought of as a bridge between the "input-output machine" model with which students are often introduced to functions and function graphs in the Cartesian plane. To use dynagraph activities effectively, it's important to first make sure students understand that dynagraphs are representations of functions with all that entails and, second, to connect dynagraphs to other representations of functions. Several of the activities in the Sketchpad curriculum modules Exploring Algebra 2 and Exploring Precalculus are based on dynagraph explorations.
[1] The term dynagraph was coined by Paul Goldenberg, Philip Lewis, and James O’Keefe in their study “Dynamic Representation and the Development of a Process Understanding of Functions” published by Education Development Center, Inc., and supported in part by a grant from the National Science Foundation.