# From Two Dimensions to One

In my last post, I provided some dilation challenges and linked to a Dilation Function Family activity. In that activity students manipulate independent and dependent variables, observe their relative rate of change, restrict the domain, and use meaningful function notation.

This and similar activities involving “technologically embodied geometric functions” are designed to familiarize students with important function concepts in an active and concrete way corresponding to cognitive science findings. Still, students’ conceptual understanding in the geometric realm may not transfer well unless it’s explicitly linked to their work with families of symbolically defined algebraic functions.

Making that connection was the topic of the presentation that Daniel Scher and I gave at last April’s NCTM Annual Meeting. In that session we proposed a series of activities to make the link explicit: Students move their two-dimensional geometric functions into one dimension (by restricting the variables to a line), mark the line as a number line, and connect the geometric behavior of the variables to their numeric values.

Reduce the Dimension, shown below, is the first of those activities. The task for students in this activity is to take the initial step by turning a two-dimensional function into a one-dimensional function by putting both variables on the same line.

The activity is available in two forms:

• A Sketchpad activity in which students create the functions from scratch, restrict their domains, and so forth. The worksheet is here.
•   A prepared web sketch like the one above (but also incorporating the student worksheet in a scrollable window), available directly at http://geometricfunctions.org/wsp/tegf/reduce-the-dimension

I much prefer the build-from-scratch choice, because students’ understanding and retention are enhanced by constructing the mathematics themselves. (This choice requires Sketchpad 5; if you don’t already have Sketchpad, you can download the free preview.) But I decided to also create the web-based choice because of its convenience and accessibility.

This activity is new, and there are no teacher notes available yet. We’d love to have your comments on the activity, and your suggestions for improving it.

In future posts, we’ll number the line so that our variables have numeric values, we’ll turn the line into a dynagraph with separate parallel input and output axes, and we’ll conclude by turning the dynagraph axes perpendicular to each other and generating the Cartesian graph corresponding to our original geometric function.