I’m excited to be making my first presentation of the 2013-14 school year next week in Baltimore. Daniel Scher and I are presenting Picturing Functions and Functions of Pictures. We’ll be discussing the connections between pictures and functions.
These connections are even richer than I realized when I first began to work with pictures in Sketchpad. In the digital age pictures are just collections of numbers: every photograph we take, every TV show we watch, and (soon) every movie we view exists as a collection of numbers that specify the color at every point—and each of those points corresponds to an (x, y) location on the image. Thus the geometric and numeric elements—the points, the coordinates, and the color values—are inseparably intertwined.
The first connection we’ll explore is that of picturing functions. We make a picture of a function every time we create a graph, which is really just a picture that expresses something important about the function. We work with such graphs all the time, and making and interpreting graphs becomes second nature to our students. Like other pictures, we interpret graphs visually to understand something about the underlying object that the picture represents. But the graph is a static picture that omits the dynamic process that created it. Though the graph arises from the coordinated motions of two variables, it depicts only the static results of that motion, and not the motion itself. In our session, we’ll address this disconnect and propose some enjoyable solutions.
The second connection is that of functions of pictures: the use of functions to manipulate pictures, to edit them, to turn them into special effects. When we transform a picture in any way, mathematical functions are at work. For instance, a picture is enlarged or reduced in size by applying a dilation function to every point in the picture. When special effects are created for a TV show or a movie, those “effects” are the result of mathematical transformations. Mostly, this looks like magic to our students, and we’ll demystify some of the magic in our presentation.
Finally, we’ll use functions to create pictures other than graphs. We’ll make pictures of families of functions, and we’ll see how iterated function systems can create fractal pictures like Barnsley’s Fern and Sierpinski’s Gasket.
As we explore the connections between functions and pictures, we’ll not neglect the Common Core; the activities we’ll look at emphasize not only the Standards for Mathematical Practice, but have content mainly drawn from the two conceptual categories for high school that are newly emphasized in the Standards for Mathematical Content: functions and transformations.
Finally, I’d like to draw your attention to another presentation Daniel and I are doing, at 11 am on Friday: Give Puzzles a Starring Role in Your Math Class. Daniel is the lead speaker, and we’ll be showing, and having participants work on, a number of puzzles for students and teachers in the elementary grades. It’s also at the Hilton, in Key Ballrooms 1 & 2. We hope to see you there.
This material and the activities I’ll make available through my presentation are based upon work supported by the National Science Foundation under KCP Technologies Award ID 0918733. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.