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Technologically Embodied
Geometric Functions

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:

Use the Materials, Support the Work

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.

Investigate Geometric Transformations as Functions

This unit provides an introduction to geometric transformations from a functions point of view.

Students construct geometric transformations that take a point in the plane as input and produce another point as output, they vary the input and observe the output, they analyze the relative movement of the two points, and they use the language of functions to describe the behavior of the point variables.

▿︎︎︎︎ Overview

Students begin this unit by dragging points to explore how dragging one point can affect another, and then solve several 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).

This introductory video shows short highlights from some of this unit’s activities.

▹︎︎ Status

All of these activities include websketches, most have classroom-tested worksheets, and many include performance-based assessment games. None of them yet incorporate teacher-support materials, and we continue to refine the websketches, worksheets, and assessment materials as we use them with students and observe students’ successes and difficulties.

▹︎︎ Activities

The unit includes seven activities, each requiring one or two class periods:

1. Identify Functions: Students experiment with points that are related to each other by geometric transformations, dragging them to discover which points depend on which other points. By dragging, they treat the points as variables, and they develop a working definition of function in the context of geometric transformations.
2. Identify Function Families: Students explore similarities and differences in the behavior of transformation families through a series of puzzles in which they’re asked to figure out which of four transformations is not like the other three.
3. Reflect Family: Students construct and explore reflections. They begin by reflecting a point across a mirror and describing the relative motion of the two points, follow up by restricting the point to a polygon in order to reflect an entire set of points (the polygon) across a mirror, and then undertake a series of challenges to develop a deeper understanding of the relationship between the pre-image, the mirror, and the image.
4. Rotate Family: Students construct and explore rotations. They begin by rotating a point about a center and describing the relative motion of the two points, follow up by restricting the point to a polygon in order to rotate an entire set of points (the polygon), and then undertake a series of challenges to develop a deeper understanding of the relationship between the pre-image, the center and angle of rotation, and the image.
5. Dilate Family: Students construct and explore dilations. They begin by dilating a point about a center and describing the relative motion of the two points, follow up by restricting the point to a polygon in order to dilate an entire set of points (the polygon), and undertake a series of challenges to develop a deeper understanding of the relationship between the pre-image, the center and scale factor of dilation, and the image.
6. Translate Family: Students construct and explore translations. They begin by translating a point by a vector and describing the relative motion of the two points, follow up by restricting the point to a polygon in order to translate an entire set of points (the polygon), and then undertake a series of challenges to develop a deeper understanding of the relationship between the pre-image, the translation vector, and the image.
7. Four Families: (Under construction!) Students construct, manipulate, and investigate example functions from each of the four families above. Students observe several members of each family to find similar behaviors, and compare the families with each other to find their differences. This activity is a more open-ended, independent group-inquiry version of the individual family activities. (The worksheet and other support are not yet available.)
8. Function Dances: Students experience the relative rate of change of transformations from each of the families. In each dance, an independent point moves around a shape (its restricted domain) and the student drags another point, trying to behave as the transformed image point. In the course of these dances, students develop a deeper intuitive understanding of the effect of each of the transformations and develop a sense of covariation (the way in which two variables move in concert).
9. Dancing Leopards: This page contains two sketches. The first is a demonstration sketch showing a computer animation based on the geometric transformations included in this unit. It can be shown to students at the beginning of the unit. The second sketch contains all the tools students need to create a similar animation of their own design.
10. Unit Summary: (Under construction!) It will have tools and pages on which to review the big ideas of this first unit on function concepts.

▹︎︎ Objectives

These activities are designed to accomplish a number of objectives. In doing them, students will:

• Create and investigate members of several families of geometric transformations (reflection, rotation, dilation, and translation).
• Experience these transformations as functions that take a point on the plane as input and produce another point as output.
• Drag independent variables of geometric functions while observing the motion of dependent variables.
• Attend to function behavior, and particularly to the relative rate of change of the independent and dependent variables.
• Use function notation with geometric transformations.
• Restrict the domains of functions and observe the effect on the range.
• Drag a point to play the role of the dependent variable as the independent variable follows a given path.
• Use several different functions in creating a computer animation of their own.

▹︎︎ The Fine Print

Requirements:

These activities require web access using a browser that supports HTML5 and JavaScript. (That means almost any current browser.) No purchase is required, and there’s no advertising anywhere.

Release Information

Update History:

09 Mar 2017: Added an introductory video.
05 Mar 2017: This page now hides & reveals content, and is reorganized to be clearer and more readable.
16 May 2016: Modified entire unit: label-edit dialog box now requires a longer press-and-hold before it appears.