With Technologically Embodied Geometric Functions, students develop conceptual metaphors that directly relate computer-based sensory motor experiences of abstract function concepts. This approach relies on four foundations:
We’re working hard to create the sketches, worksheets, and support materials for these Web Sketchpad activities. Ours is a volunteer effort on the part of everyone involved, both curriculum developers and field testers, and we make these activities freely available for you to download and use with your own students.
As we develop, revise and expand the activities, we need your help. Email your commentary (What worked well? What didn’t?) and your suggested improvements to the webmaster.
The Geometric Functions activities are designed to help students learn about functions and geometric transformations through direct experiences constructing and manipulating them. (See the four Foundations pages for the mathematical, cognitive science, technology, and pedagogical foundations of this approach.)
These activities are drafts. They all include student worksheets and websketches, and a number of them include performance-based assessment games. None of them yet incorporate teacher-support materials, and we expect the worksheets, websketches, and assessment materials to change substantially as we continue to use them with students and observe students' successes and difficulties.
Investigating Geometric Transformations as Functions
In this unit students explore geometric transformations as functions that take a point in the plane as input and produce another point as output. They begin the unit by dragging points to explore how dragging one point can affect another, and then solve some puzzles, figuring out which of four transformations belongs to a different family (reflection, rotation, dilation, etc.) from the other three. In the central activities of the unit, students construct transformations from each of the families, describing their behavior in detail and solving challenges such as finding a hidden mirror or a hidden center point. The unit concludes with a series of transformation "function dances," in which an independent point follows a defined domain (such as a polygon) and the student drags another point, trying to match the motion of the transformed image point.
In the course of the unit, students connect transformations and function concepts. They develop a sense of points as variables, label the points using meaningful function notation, observe and describe the relative rate of change of the points, and restrict the independent variable to a geometric domain (a polygon) in order to trace the geometric range (the transformed image of the polygon).
The unit includes seven activities, each requiring approximately one class period:
Connecting Functions in Geometry and Functions in Algebra
In this unit students transfer their understanding of geometric functions that take a point in the plane as input and produce another point as output—reflection, rotation, dilation, and translation in “Flatland”— to linear functions that take a real number as input and produce another real number as output. Students restrict the domains of the geometric functions to number lines (“Lineland”), measure the values of the variables, and discover that composing and restricting just two geometric functions enables them to create any linear function geometrically, and to represent it in either Dynagraph or Cartesian graph form.
Geometric Functions in the Complex Plane
Dynamic Number Activities
These activities from the Dynamic Number NSF project are designed to help students learn about functions and geometric transformations through direct experiences constructing and manipulating them. Each activity includes a student sketch, a student worksheet, and extensive teacher notes. These activities require The Geometer's Sketchpad. (Many of these activities are also available in Web Sketchpad form in the unit on Introducing Geometric Transformations.)
In these activities students drag a dependent variable to match the behavior of a particular function such as the sine function. In the process, students gain a tactile sense of how the rate of change of a function varies across its domain.